My gratitude goes to Lynda Brasier for all her help and intuition, especially for

the 10/17. Heartfelt thanks also go out to Raphiem, to Gary Val Tenuta,

Norma Smith, and ultimately one’s thanks can go on forever! Therefore the

most thanks go to the Creator, whatever that is.

Contents

Introduction

1 Birth of triangles 1

2 The Vedic Square 20

3 9 over 8 dilemma 43

4 The invisible aspect of the triangle 47

5 Fibonacci numbers 51

6 The Lambdoma 56

7 Fibonacci mode boxes 63

8 Both sides number flows 77

9 The switch in binary 81

10 Divisions and the Vedic Square number pairs 84

11 Where else? 90

12 Note to number grids 97

13 The Dorian Connection 102

14 Shades of dark and light 110

15 A tonal fountain 120

16 Visible invisible Dorian 127

17 Opposing forces 129

18 Swings four ways 133

19 Dorians at the tritone 136

20 Chords through the triangle 138

21 Triangles, the circle of tones and the number nine 145

22 In and out of the mirror 150

23 Uninteruptus 167

24 144 major scale grid 171

25 Visible and invisible axis points within numbers 186

26 Pimirrorrorrimip 196

27 Caos/non-relating Vedic square 198

28 Central interval 200

29 Exponentials and the 369 positions 203

30 Differences triangles 212

31 The surface of the torus universe 215

32 I Ching hexagrams and the 4.5 219

33 The Tzolkin, Vedic Square and scales 225

34 Magic squares and the 45-degree angle 242

35 Alphabet and overtones 247

36 Working conclusion 251

Copyright 2002 – 2014

Permission is given to other researchers to use any of the charts and examples within this

book in their own work, with credit given to the original source.

Introduction - The Unbroken Thread

My own personal journey into studying the mirror side of music and number cycles started in 1989.

The ideas grew on me rather slowly, so I do not expect a sudden “aahaa!” proclamation from the

reader! Well, maybe one or two will see it instantly. After all, it took a fair few years of information

slowly gathered, and so the hope is that others can process a lot quicker, what was rather

torturous at times in terms of understanding what the data was suggesting.

The idea to reverse a formula came to me whilst I was practising scales on the guitar one evening.

It is easy to let the mind wander when doing one’s musical exercises! And during one of those

moments there was rather a loud thought that came flying into my mind, shouting out “reverse the

formula of the major scale”. The thoughts preceding that one had been allowed to recede back

into the sub conscious, but this particular thought had made an imprint, so that deeper still was

something that was urging me to actually try what the voice had suggested. I wouldn't let it go, in

other words.

Even though it seems obvious to many that a musical scale is a cycle of notes, very few have

wondered about a mirror side to these cycles.

I am self-taught when it comes to playing the guitar or even within the realms of music theory. This

has its drawbacks as well as its blessings I suppose. It may have been that I would have met a

guitar teacher that had told me about scales that had been mirrored. As it was, my musical

knowledge at that time extended as far as the building of a major or minor scale, chords and one

or two ideas about musical form (verses, choruses, bridges, intros, AABA form etc). This had been

gleaned from guitar magazines and the odd music book from charity shops.

I picked up some paper and simply wrote out the notes that comprise a C major scale. I also wrote

the formula for this scale above the notes. Each step of the formula was reversed. I began to like

the experiment when I could see that different notes were appearing on the mirror side.

Being a self-taught musician I regarded the idea of mirroring scales as something new. I had never

come across that idea in any of the music books I had acquired, and no musician I knew had ever

mentioned it. I spent some four years gathering up as much information on the subject as I could

imagine. This included the mirroring of well-known scales and songs, and discovering a way of

creating many new scales by merging mirror side with non-mirror side.

I didn’t even know the term “Mirror music”. I called the mirror side the “Shadow side” for the first

four years. At one point I heard of certain organisations that offered grants to people doing musical

i

research. I wrote to one of these organisations and told them what I was up to with said formulas.

They wrote back and told me they wished me the best of luck in my chasing shadows, but that

their grants were not available for such undertakings.

By the time a new colleague informed me that scales had been mirrored in the past, I had taken

the idea into a different direction, and was actually looking for structure on the mirror side, rather

than getting on with two handed mirror playing on the piano, or using it as some avant garde

musical approach.

The outline of the book of course does not really sequence the way in which the information was

gathered. This already opens up one problem, and that is the impact that the results shown within

the book will give the reader. Within a few weeks, I was in total awe of what a simple major scale

had just taught me! All one’s life as a musician you play it, nurture it and grow from it, and then you

see it mirroring to a minor scale, which you had also spent many hours exploring. It really did feel

like a mirror world had been penetrated. The fact that major and minor scales had been known as

masculine and feminine tonalities left me with the image of a man staring into a pool and a woman

staring back, not just a reflection of himself. This in some ways set the tone of the research, in that

it felt as though one was glimpsing into the workings of masculine and feminine forces within

nature, as expressed through the music scales, the seven modes/cycles of the major scale, for

example.

Through the process of using this mirroring technique, a structure that seems to exist at the heart

of all number and vibration can be seen to emerge. It is to bring this particular structure to the

attention of other researchers that is the main motivation for presenting the data.

The theory of simple numbers is more or less accepted universally. There is a general consensus

that one is one, two is two etc. We see one sun, one moon, two eyes, five fingers, and a £10 note.

Therefore, in presenting the idea that numbers have a “mirror” side, it is more a question of firstly

wishing to log this data, rather than to present any new theory as such, although one would not be

human if they did not contemplate a theory as to what the result of mirroring implies. It is clear that

the simple mirroring technique unearths an intrinsic structure within the nature of music scale and

number cycles.

Mirroring brings to light another type of structure at a layer beneath the number cycles and the

Modes within Music. This 'Mirror Structure' unites both sides of the duality that is inherent within

cycles, that flow both clockwise and anti-clockwise. This duality is united at one specific position,

that is common to every example given in the book, and is called the 4.5 axis. This position shows

ii

perfect symmetry in terms of the information surrounding it, be it musical notes or numbers that

are mirror pairs of each other. The 4.5 axis is a state of marriage, where the mirror pairs, either

musical notes or numbers, are in perfect symmetry. It is also another view of a zero point, as

relating to the indig number system - +1 +2 +3 +4 -4 -3 -2 -1.

The book traces the one consistent result that mirroring brings, through number and vibration, and

a host of other grids and systems, including the Fibonacci numbers, the Phi Ratio (also known as

the Golden Mean), the Pi ratio, overtones, fundamental number sequences, even such grids as

the Mayan calendar and the I-Ching hexagrams (when viewed as binary numbers), and of course

the music system as we use it this part of the world.

I hope that, regardless of the fact the book is not exactly presented in academic form, that other

researchers may benefit from these examples.

So what exactly is mirroring? Mirroring, or symmetrical reflection, simply means that the various

major and minor type scale formulas will be applied in a reverse fashion, instead of their usual

ascending direction. This technique is explained fully, for those who are unfamiliar with it, and it's

pretty easy to understand the process involved, although having the rudiments of basic music

theory under one's belt will help.

Even though music scale formulas are used, the focus of this book really isn't about making music.

It is true that scale formulas are often mirrored and tunes are created that way. Yet for the

purposes of this book music scales are taken out of their music making domain, and the focus

instead will be how the scales are representative of the law of Position, the flow of cycles,

clockwise flows and anti-clockwise flows, and ultimately on the structure that emerges that renders

both sides of the mirror into one whole unit..

Once a music scale has been cycled, and the mirror cycle has also been exposed, there is a focus

on the symmetrical relationships between the various positions. For example, what is the mirror of

the 3rd cycle, and what are the mirror note pairs either side of the mirror/axis point? The picture is

built up using mirror logic.

The law of position and the clockwise/anti-clockwise flows go on to show that there is a structurally

based mirror side, and that it is not an isolated system, but one whose flow of information is

interlocked with the information on the opposite side of the mirror to itself. Both sides of these

mirrored formulas will be seen to need each other in order to function as a whole unit.

iii

Are you for real?

All musicians are well aware of musical scales. Almost everyone is aware of the sound of a

musical scale, especially when one hums the doh-reh- meh-fah-sol-lah-teh-doh notes. A large

majority of the music we listen to in our western society is based on the major (and minor) scales.

This scale is a cycle of notes, that one can repeat over and over, into lower and lower sounding

notes or higher and higher ones. This is possible mainly because of the nature of certain intervals.

The most prolific interval in a musical cycle is the Octave.

The Major scale (the doh reh meh fah sol lah teh doh sound) is a cycle that contains additional

inner cycles, which musicians call Modes. The Major scale cycle itself is a Mode. Beginning the

scale from Reh, instead of Doh , will begin a new cycle that ends on the Reh one octave higher, or

lower. All these seven possible individual cycles have a mirror partner.

In the western music system there are twelve major scales, all evolved from what is known as the

circle of 5ths. These individual major scales are based on a system of sharps and flats that

distinguish them one from the other. The major scale can be written as a series of seven letters, to

replace the doh reh meh symbols. The very start of the circle of 5ths, and the twelve major scales

is on the letter C. Applying the major scale formula from here creates the C major scale, which , as

yet, contains no sharps (#) or flats (b) in its make-up.

C D E F G A B C

It would be beneficial for any non-musician to search various sources for information on the circle

of 5ths, and other basic music theory.

These seven letters (note pitches) can also be roots of their own seven note scales. Here another

scale is produced from the above information, merely by starting at the note D:

D E F G A B C D

This is the second cycle, or second Mode, known as the Dorian Mode.

E F G A B C D E

This is the third Mode, known as the Phrygian Mode, and so on.

Each of the cycles are built using strict ratios. It is the symmetrical reflection of these mathematical

ratios that open up a mirror side. And this mirror side is not a disjointed affair, but actually the

iv

missing half of the complete unit of information.

The mirror structure, that is the main focus of the book, also emerges within various types of

number cycles, for example, the Fibonacci numbers. Therefore, one is led to believe that there is

an intrinsic structure that binds the two sides of the mirror together, which is expressed through

different approaches and examples.

The rule of symmetry used within this book is that of equally mirrored moves around an axis point.

If a musical note moves forward/upward by one tone to another note, then that move is simply

mirrored by a reverse move of one tone from the same note.

mirror point

Bb – C - D

The same note either side of the mirror here is C, because it is the axis point. C to D is an

ascending movement of one tone. C to Bb is a reverse descending movement of one tone. And

now the note D has a mirror partner, Bb, in that they exist at a symmetrical point either side of the

Root C mirror point. This relationship will carry far more meaning than at first may seem the case

here. Regardless of the ratio one employs for finding this D note, that ratio is simply mirrored and

one will find the appropriately tuned Bb, according to which tuning system is used. The C to D

move may have been accomplished using a 9:8 ratio, or 200 cents (used in Equal temperament).

The mirror move will simply be an 8:9 ratio or 200 cents, from C down to Bb. Any further moves

are always mirrored in the same manner, by equal movement relative to an axis/mirror point.

Obviously, the most apparent thing here is that no musical note is ever alone. Well, how about the

C note, that is the same both sides of the mirror? The axis point too has a partner, an axis-point

partner, which will be shown in the first chapter. This other axis point is not a visible aspect of the

two scales being mirrored, but it still reflects the very same pairs that are reflected around the C

note. It was this fact that became the driving force to experimenting with the mirror side of

formulas much more, because they led to the uncanny result of only one unifying structure, even

though totally unrelated systems were being used. It will be seen that what can be termed the

visible information will swap over to the “invisible” mirror side, and back again continually. I call this

simply 'in and out of the mirror'. It is the real journey of the mirror structure.

It may be wise perhaps to look at a vital difference between two concepts of what a mirror image

of music actually is.

v

It is said that one would put a scale to the mirror, like this one:

C D E F G A B C

And in the mirror it would become:

C B A G F E D C

And joining the two together:

Mirror point

C D E F G A B C/C B A G F E D C

Here it is on the music stave:

C D E F G A B C / C B A G F E D C

This music stave is known as a ‘treble clef’, which ascends and descends in movements of the

first seven letters of the alphabet used in music, be it a line to a space to a line etc, or vice versa.

Composers who are looking for something different have mirrored music this way often. The art of

placing music manuscript paper in front of a mirror and then playing the tune “backwards” was

pretty rampant in Brahms’ day. It is certainly a very valid form of mirroring when it comes to

manuscript paper. Holding a set of ratios, based on a given formula to the mirror, is quite different.

Mirroring the actual ratios means that one applies equal proportions on either side of an axis point

(the mirror point). This would mean that any ratio must be reciprocal. 1:2 must become 2:1 in the

mirror. A move forward by one tone is simultaneously a reverse movement of one tone. Imagine

the next diagram is a depiction of a major scale formula being placed in front of a mirror, and it

should be noticed how the mirror notes are different, and not just a reverse flow of the notes being

placed in front of the mirror.

vi

The mirror point has the note C twice. Musically what this means is that the mirror note of the note

C remains as C. From there one of the scales evolves outside the mirror, and the other one travels

through the mirror in reversed equal steps. Following the arrows and working each dual step at a

time should help one see how the formula is symmetrical, but the notes are having to change in

order to accommodate the formula's requirements.

From this axis point at the note C there is a leap upwards in pitch to the note D. The letter T over

this note is part of the formula, or instruction as to how far to move upwards in pitch. The T simply

means TONE.

The mirror note is that of Bb. It is not the same as the note B. The note B is a white note on the

piano, whilst the note Bb is the next black note downward in pitch (travelling to the left of the

piano)

Let's continue with the next note of the C major scale:

C D E

vii

From D to E is yet another move of one tone. Therefore the mirror move must be one tone to the

left of the Bb.

T T T T Ab Bb C/C D E

At first one is entitled to think that surely this is not a true mirror image. Yet it is as far as the

reversal of the formula is concerned. And it will be seen that this is actually a first step to exposing

a unified mirror structure.

The first chapter will focus on this major scale and show how it produces the said mirror structure

from the formula reversal of all of its seven different cycles/modes, which create a 7*7 mode box.

There are various animations provided with this book. They are labelled according to the

corresponding chapter within the book. It is best not to watch all of them first, but rather read a

chapter and then watch the relevant animation.

There are also a few additional animations that cover material not in the book. Again these are

labelled accordingly.

viii

Chapter One

The birth of Triangles

In this chapter the major scale and its Modes will be shown to exist as part of a larger mirror world,

half the elements in a tapestry of mirror relationships. A musical tree of information evolves from

just the one main major scale. It will be seen that each of these individual formulas/cycles/modes

have their mirror partner. Once these mirror partners are exposed, the focus is on highlighting the

natural flow of musical information from one side of the mirror to the other. This flow can be made

up of triads or the flipping over of modal relationships constantly from one side of the mirror to the

other.

Observe in this next diagram how the same looking formula flows in both directions and creates

two different scales. The T (tone) signifies 2 steps on a piano (or two frets on a guitar), and the S (semitone) signifies a 1 step movement.

C Major having its formula reversed (Mirrored)

mirror point

S T T T S T T T T S T T T S 3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1 (C Phrygian) C Db Eb F G Ab Bb C D E F G A B C (C Ionian)

If you have viewed the first few animations that accompany this book, you will have seen how this

mirrored dual scale has been achieved.

Start at the center C, and follow the logic of the tones (T) and semitones (S) as they are mirrored

around that C axis note.

One can see the red formula is a mirror of the black formula. A few minutes on an instrument like a

piano and this should pose no problem in understanding. One can mentally imagine the center C,

or play it on piano, and simple utter that D is in symmetrical reflection to Bb (or play the two

notes ), E is in symmetrical reflection to Ab, and so on.

1

These are two scales/cycles that coexist on one formula. By applying the formula leftwards as well

as rightwards from a center axis, one is cutting through into a mirror world tonally speaking, and it

will be seen that there is structure within this other world of information, interlocked and swapping

its information with the non-mirror side.

Be aware of the names of these scales and how they will begin to interrelate. Be aware of the

mirror relationships between relative positions, as the C major scale mirror pair show here:

1/3 2\2 3/1 4/7 5/6 6/5 7/4 C/C D/Bb E/Ab F/G G/F A/Eb B/Db

Notice how the 1 2 3 4 5 6 7 ascending sequence has mirrored to an anti-clockwise descending

sequence, starting at number 3 – 3 2 1 7 6 5 4. This in itself already shows that the two sides of

the mirror are inter-related, and have structure of some kind, with contrary flows being evident.

The C Ionian scale formula, when a mirror axis is placed at the root note, builds a C Phrygian

scale in the mirror. The first partnership is then 1 IONIAN/3 PHRYGIAN. These Greek sounding

names are simply representing the other possible cycles within a parent major scale formula. A

quick explanation of these names, referred to in music as Modes, might prove handy.

Here is a list of the traditional seven modes, as they are constructed from the parent major scale

1. C D E F G A B C = Ionian Mode (parent major scale)

2. D E F G A B C D = Dorian Mode

3. E F G A B C D E = Phrygian Mode

4. F G A B C D E F = Lydian Mode

5. G A B C D E F G = Mixolydian Mode (Dominant scale)

6. A B C D E F G A = Aeolian Mode (Relative Minor AKA Natural Minor)

7. B C D E F G A B = Locrian Mode

The first scale is the parent major scale. It is also referred to as the Ionian mode. The next scale is

really the same notes as those of C major but the root is being shifted upwards by a tone and

beginning its cycle from the D note. This is not the scale of D major as a result, but a modal scale

referred to nowadays as the Dorian mode. This applies to all seven scales, they commence from a

different note of the parent major scale, as shown in this example:

2

C D E F G A B C = Ionian

D E F G A B C D = Dorian

E F G A B C D E = Phrygian

F G A B C D E F = Lydian

and so on..

All twelve major keys have the same cycle of modal relationships. Here they are in D major.

1. D E F# G A B C# D = Ionian Mode

2. E F# G A B C# D E = Dorian Mode

3. F# G A B C# D E F# = Phrygian Mode

4. G A B C# D E F# G = Lydian Mode

5. A B C# D E F# G A = Mixolydian Mode

6. B C# D E F# G A B = Aeolian Mode

7. C# D E F# G A B C# = Locrian Mode

Only the note names differ, but the flow of modes is the same, meaning the same tonal colour is

occurring..

The C Phrygian mode (mirror of C major/Ionian mode) would be the 3rd mode of the Ab Major

scale:

1. Ab Bb C Db Eb F G Ab = Ionian2. Bb C Db Eb F G Ab Bb = Dorian3. C Db Eb F G Ab Bb C = Phrygian4. Db Eb F G Ab Bb C Db = Lydian5. Eb F G Ab Bb C Db Eb = Mixolydian6. F G Ab Bb C Db Eb F = Aeolian7. G Ab Bb C Db Eb F G = Locrian

Step one shows that the C Ionian Mode (the Major Scale) becomes the C Phrygian Mode (a Minor

scale) when the flow of Tones and Semi-tones are reversed. Yet there is a seed being exposed

here that will allow one to continue with this mirroring. The question arises as to which Major scale

houses this mirror Phrygian Mode. The modes of any major scale always flow in the same order,

3

regardless of the major scale in question. The 3rd mode is always Phrygian, and it is always part of

a parent major scale that carries the number 1, which is always known as an Ionian mode.

As you can see the C Phrygian mode resides in the scale of Ab Major. It is the 3rd mode of that

scale (the Major scale carries the number 1).

From mirroring the formula that creates C major we have ventured into the scale of Ab Major, by

means of a Phrygian/Ionian modal mirror relationship. We can now set Ab as the new axis point,

and the major scale formula will be mirrored once more from that new axis.

S T T T S T T T T S T T T S 3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

Phrygian Ab Bbb Cb Db Eb Fb Gb Ab Bb C Db Eb F G Ab Ionian

The above would look extremely complicated for a non-musician. As with the scale of C major, all

that is occurring here is that the Ab major scale formula is being reflected across the mirror point,

reflecting all the information held within the scale on the right the opposite way according to the

spacings defined by the given formula.

Ab Phrygian is more commonly known as G# Phrygian, because hardly anyone thinks in the key

of Fb Major, which would house Ab Phyrygian. It is easier to think of Fb as the note E, its

enharmonic equivalent.

Replace these mirror notes on the left hand side with the other enharmonic equivalent of each:

Ab Bbb Cb Db Eb Fb Gb Ab

(G#) (A ) (B) (C#) (D#) (E) (F#) (G#) = G# Phrygian, from E Major

As you can see G# Phrygian is the third mode of the E Major scale (E- F- G#, again a

Phrygian/Ionian partnership).

Numerically this will look similar to the first example of C major and its mirror scale. That is the

beauty of a single formula (and the equal temperament system). You may not have seen the

double flat sign before (bb), but it is simply an instruction to lower the note in question by two semi-

4

tones (each “b” representing one semi-tone). In the above example, Bbb is an instruction to lower

the B note by two semitones, which gives the note A. This way of writing scales is to keep things

as tidy as possible, so as not to use the same letter name twice.

The mirror note partners around the Ab/G# axis (in red) are obtained in a similar fashion to those

found around the C axis previously. We will use the G# as the axis in order to highlight the note

partners:

1/3 2/2 3/1 4/7 5/6 6/5 7/4

G#/G# Bb/F# C/E Db/D# Eb/C# F/B G/A

If you study the scales above, you should have no trouble seeing that these note pairs are in

symmetry around the root note axis/mirror-point. Again notice there is a contrary flow between

either side of the mirror. The whole process has jumped up a minor 6th interval (or down a major

3rd), C d e f g Ab.

C Ionian/Phrygian and Ab/G# Ionian/Phrygian share what may be called a `mirror link'.

Symmetrical association links them.

The mirror link can now be extended by mirroring the formula of the new mirror scale that has

emerged, E major (carrying the number 1 on the mirror side, with Phrygian its usual number 3

position)).

The formula and numbers for mirroring E major will read the same as the previous two.

S T T T S T T T T S T T T S

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

Phr. E F G A B C D E F# G# A B C# D# E Ion (from C major)

As you can see the mode of E Phrygian is the result. This is the third mode of the C Major scale,

and we are back to the beginning. We have established a thread where three major keys relate to

5

each other through the mirror via the Phrygian modes, up a minor 6th or down a Major 3rd. As the

movement is anti-clockwise we can assume that it is the latter movement that is occurring.

Three Mirrors

C

Maj 3rd Maj 3rd

Ab/G# E Maj 3rd

Here’s another way of looking at it:

Three in One

6

C major mirrors to Ab major. Ab Major mirrors to E major, and E major mirrors back to C major.

The link is through the third mode of each major scale, the Phyrygian, a minor sounding scale. A

contrary flow-like environment has been seen to emerge, rather than no systematic flow at all.

This is only one of quite a few ways of achieving this triangular effect that is evident when musical

cycles are mirrored. This is still not the whole mirror structure however, as will be shown in the

next section on Mode Boxes.

7

MODE BOXES

Each Mode of the Major Scale is based on its own unique formula and it is the formula which is

reversed as if put through a mirror. One simple major scale will produce all the information below,

which I call a Mode Box. Use this mode box as a reference. You will need to check back to it often

so it would be best to draw out your own. It is amazing how it will start to make sense if you do this

mirroring for yourself! Instead of using the T and the S, I have opted to use numbers throughout

this box. Mainly because it will be easier for the eye to relate to the flow of numbers rather than a

series of T and S letters. The 2 refers to the Tone, and the 1 refers to the semitone. This can be

equated to the movement of one or two steps on a piano, or one or two frets on a guitar, for

example.

The right hand side of the above box consists of the seven Modes of the Major scale. Most

musicians who have studied a little music theory will recognize these seven different scales, all

generated from the parent major scale. For non-musicians, it should be relatively clear that the

8

right hand side consists of seven different orders of the notes C D E F G A B C. The important

thing is to understand that these seven cycles are being individually mirrored, using the axis

positions down the center of the grid.

The left hand side, being the mirror side, instantly looks different, made up of different notes to

their counterparts on the other side of the mirror.

To confirm the results within the above Mode Box, it is best to take each formula in turn and mirror

it. Using a piano to work out how the mirroring is done, will of course allow the reader to fully

understand how the mirror modes are emerging.

Take each individual line separately and notice how the same formula (the 2s and 1s) is flowing

both leftwards and rightwards from the center of the grid. Then take note of the modal names and

how they pair up across the mirror, the flow of numbers vertically and the individual note partners

from each line that partner each other.

One always begins in the center of this mode box. From this viewpoint it should be easy to spot

that each formula is reflected from here as if a mirror were put to it. Yet the mirror is not reflecting

the pitches themselves, because these change in the mirror, as you can see. The note D, which is

on the C Ionian line, becomes Bb by an equal movement of a tone away from the red C/C axis

point. Therefore it is the proportions or ratios that are being mirrored.

If one study the numbers within the mode box occurring vertically it will be seen that the two sides

of the mirror move contrary to each other, as shown by the arrows either side. There is a clockwise

flow on the right, being the modes 1 through to 7, and an anti-clockwise flow of Modes on the left.

The anti-clockwise flow begins at the 3rd Mode, the Phrygian, and moves the opposite way to the

right hand side’s flow. After 3 will come the 4, which is actually at the bottom left of the box, and

then the 5, 6, 7, 1, 2 and finally the 3.

This next example represents the default mirror pairs of the top line. This list represents the whole

mode box, holds true for all twelve Major keys, and within those keys one may begin the mirroring

process from any Mode, and it will evolve similarly. The names of the modes will be abbreviated,

as is general procedure:

9

1 3 2 2 3 1 4 7 5 6 6 5 7 4

Ion/Phr Dor/Dor Phr/Ion Lyd/Loc Mix/Aeo Aeo/Mix Loc/Lyd

It's the same dual flow as seen when only C major was mirrored. In the Mode box, one is dealing

with the major scale and its seven modes on both sides of the mirror, with one vital difference; the

non-mirror side is a major scale with its seven modes, and the mirror side is seven modes

belonging to seven different major scales. This means that definite structure has appeared. We

are looking at a similar Major scale type structure on the left but all its logic is opposite and it

commences from the 3rd position. All the mirror modes are similar types of modes as on the right

hand side.

The two flows of numbers above, the red on the right and the black on the left, are like two wheels

in motion:

Flow of Mode box

1 3

7 2 2 4

6 3 1 5

5 4 7 6

The left hand circle starts at the number 3, signifying the Phrygian Mode, and it flows anti-

clockwise through the modes. The right hand circle starts at the number 1, signifying the Ionian

mode, and it flows clockwise. Also of note is the fact that the Ionian is a major sounding scale and

the Phrygian is a Minor sounding scale. All the relationships share this major/minor duality

together. The 4/7, for example, signifies a Lydian Mode (major sounding) and a Locrian Mode

(minor sounding). Again there is the 5/6, which signifies a Mixolydian Mode (major) and an Aeolian

Mode (minor) pair. When one can see this relationship, they may be pleasantly astounded that

such a major/minor duality is so consistent whilst moving through the seven dual positions.

Let’s concentrate only on the left hand side. These seven mirror modes do not belong to only one

major scale as the seven modes on the right hand side do. There are in fact six major scales

10

within this mirror side of the Mode Box (if we come to count Gb and F# Major as the same scale).

We have seen the first mirror major scale on the top line was that of Ab Major, home of C Phrygian

above. We also see that D Dorian, the next mirror scale down, is from the same C major scale

opposite. The Dorian, therefore, is also a symmetrical mode.

Moving down the list to E we see that the mirror scale is that of E Ionian, which is the actual E

Major scale itself. Here are all the mirror major scales that have emerged on the mirror side:

Ab C E Gb Bb/A# D F#

Here is another depiction of the mode box, showing these same notes and parent scales also

replicate along the 45-degree angle:

Left hand side of Mode Box All different Major keys Seven Modes of C Major

Ab Major

C Major

E Major

Gb Major

Bb Major

D Major

F# Major

The 45-degree angle has replicated the parent keys on the mirror side, and some of the keys have

changed from a flat (b) key, to as sharp key (#). For example, G# has swapped for Ab, A# has

11

3

C 4

Db

5

Eb

6

F

7

G

1

Ab

2

Bb

3/1

C2

D

3

E

4

F

5

G

6

A

7

B

1

C2

D

3

E4

F

5

G

6

A

7

B

1

C

2/2

D3

E

4

F

5

G

6

A

7

B

1

C

2

D1

E

2

F#

3

G#4

A

5

B

6

C#

7

D#

1/3

E4

F

5

G

6

A

7

B

1

C

2

D

3

E7

F

1

Gb

2

Ab

3

Bb4

Cb

5

Db

6

Eb

7/4

F5

G

6

A

7

B

1

C

2

D

3

E

4

F6

G

7

A

1

Bb

2

C

3

D4

Eb

5

F

6/5

G6

A

7

B

1

C

2

D

3

E

4

F

5

G5

A

6

B

7

C#

1

D

2

E

3

F#4

G

5/6

A7

B

1

C

2

D

3

E

4

F

5

G

6

A4

B

5

C#

6

D#

7

E#

1

F#

2

G#

3

A#4/7

B1

C

2

D

3

E

4

F

5

G

6

A

7

B

swapped for one of the Bb notes, and F# has swapped for Gb. The reasons for this swap are

intriguing, but will be left for much further on.

We have seen that by simply mirroring and re-mirroring the Major scale, three times in all, we find

a triangular type relationship that cycles through many octaves. The keys that emerged were C E

Ab major, which create the cyclic triangular flow. What is interesting about the arrangement at the

45-degree angle of the left hand side of the mode box is that two of these triangles have appeared.

The second augmented triangle is Bb D F# (the note A# is the same as the Bb). Therefore, there

are two augmented triangles of keys (1 3 #5) on the vertical line, and also on the 45-degree angle, that also turn flat keys into sharp keys.

Be aware of all these things as we move deeper through the mirror. We have already found the

notes C E G# (Ab) in the previous experiment so this 45-degree angle must be of significance

when trying to understand the mirror environment.

As seen, these seven mirror modes belong to seven different major scales. Vertically the major

scales are Ab C E – Gb – Bb D F# (the Major scale always carries the number 1 above it).

These triangles can also be written down as a scale – C D E F# Ab Bb.

There are no semitone movements within this scale, each movement being that of one tone. There

is a scale that musicians already play, called the Whole Tone Scale, so to distinguish this overall

mirror structure that is emerging from the major scale I have called it the Circle of Tones Structure. This mirror structure can be viewed in various guises, as a circle of tones, or two

triangles that flow through a mode box, and if one were to plot these triangles onto a circle, what

emerges is a star of David symbol.

12

It is because the mirror cycle is started off-base commencing from the third position of the parent

scale that the modes are sent on some kind of a tonality journey through a series of seven mirror

major scales. The ensuing loop that is created sees the emergence of this triangular-based

structure. Every major scale comes with this structure.

There are only four possible triangles within the twelve Key Major and Minor scale system. Each

Major scale can mirror 3 times in an unbroken link and produces a triangle relationship (like C Ab

E).

C Db D Eb

Ab E A F Bb F# B G

It is these same four triangles that we will consistently find through various examples that are

music theory and number sequences based.

Here's another example of the Mode Box of C Major, with the numbers signifying the note's

frequency (cycles per second).

Another view of Mode Box

13

One can see here how the frequencies fit in within a major scale formula, and also the formulas for

each of its modes. Ratios are used in order to ascertain each step of the scale. As there are many

web sites that convey this information on how to build scales and the frequencies involved, the

reader is urged to further explore this for themselves. One could view the very center of the major

scale as 374 (the mean between the F and the G on the first line) on one side, and 187 on the

mirror side.

My thanks go to Dale Pond for compiling this.

This next example is the mode box with the note names removed, showing only the numerical

positions. Notice how 1/3 and 3/1, for example, are always symmetrically linked at every point of

the mode box, and trace a 45-degree angle. This is true for the other pairs 4/7 and 7/4, 5/6 and 6/5

, and 2/2. Each number signifies the position the note occupies within the parent major scale. One

can also see how all the numbers on either side of the mode box are moving in contrary flow, and

all is very systematic.

The number positions of the Dual Modal partnerships

14

This 45-degree angle is related in some way to the Tri-tone position of a scale. The tri-tone is an

interval that cuts the octave exactly in half. In the case of C Major the tri-tone position will be at the

note F#/Gb. Other names for this interval are a sharp 4th (#4) or flat 5th (b5), which one can

simply call a 4.5 position, and 4.5 is half of 9, as 45 is half of 90.

1 2 3 4 4.5 5 6 7 8 C D E F (Gb/F# ) G A B C

There is the oddity that Gb(F#) is the only note missing from the C Ionian and C Phrygian mirror

pair:

C Db Eb F G Ab Bb C/C D E F G A B CPhr Ion.

no F#/Gb

Eleven of the chromatic notes are evident between the two scales, but as you see the F#/Gb is not

present in the mirror when the scale of C Major is symmetrically reflected. The F#/Gb are at the

center of both scales at a tri-tone interval and not part of the tonality of either scale, and yet they

do end up appearing on the mirror side of the mode box, as the major scale that houses a mirror

mode (at F Locrian and B Lydian).

The F Locrian is the 7th mode from the Gb Major scale, and the B Lydian is the 4th Mode of the F#

Major scale. This symmetrical link is very interesting, because the Locrian is regarded as the

darkest mode of the major scale, whilst the Lydian is regarded as the brightest. This relationship

will be studied in depth a little later on, in order to show how all the shades of tonal light and dark

are distributed and linked across the mirror this way.

The relationship that C and F#/Gb share can be viewed as an invisible tri-tone relationship ( C to

F# are three whole tones from each other). The reason that this tri-tone position is of interest is

because it is another axis point, and it means that whatever information C reflects around its axis

is also reflected at this center point of the scale. It will also be shown that information is being

15

passed on from the visible axis and the invisible axis, back to the visible and so on. What this

eventually comes to mean was that information is travelling in and out of the mirror.

There also exists a visible tri-tone relationship amongst the notes of a major scale. In C major it

is the notes B and F that form the visible tri-tone (again three whole tones apart from each other).

Invisible Tri-tone axis

C D E F (F#) G A B C

Visible Tri-tone

Each major scale has an invisible axis tri-tone relationship and a visible axis tri-tone relationship.

Above, the C to F# points to the invisible axis, and the F to B points to the visible axis. Whatever

musical information C reflects around its axis, the F# will reflect. Likewise, whatever F reflects, the

note B will also reflect. Examples of this will be given further on in the book.

The first example of the visible and invisible being related is the fact that the F#/Gb on the left

hand side of the mode box were represented at the F and B lines, F Locrian/Lydian and B

Lydian/Locrian. Here they met, exchanged tonal information at this merging point, and it will be

seen that they swapped over the opposite side of the mirror axis too.

For simplicity only the note F# will be used whenever possible as in the Equal Tempered system

both F# and Gb serve for each other. When dealing with other tuning systems it will be shown that

the F# and Gb are independent and part of a wider picture of interlocking Major scale mirror

relationships that extend a fair way past the12 Major keys that are generally used.

16

Triangles of note pairs

C

B Db

Bb D

A Eb

Ab E G F# F

Going back to the very first mirror pair, the scale of C major and its mirror we find, for example,

that the note B mirrors to Db, as indeed it does in the above diagram. D mirrors to Bb and A to Eb,

E to Ab, G to F. Only C and F# are axis points and therefore have no mirror partner. That is

because they are both axis point partners. The C is the visible axis, and the F# is the invisible axis

at the 4.5 position of the C major scale.

Here is a straightforward colour grid found in most word processors, for example.

The colour spectrum

17

What we see here is the C and the F# axis points being a tri-tone apart, on opposite ends of the

spectrum. The frequencies for each note do more or less correspond with the frequencies of the

colour spectrum. It is a question of raising the frequencies by many octaves, starting at around

2^26.

A Pythagorean tuning for the major scale would look like this:

9/8 9/8 256/243 9/8 9/8 9/8 256/243

To this tuning would be attributed a series of pitches, shown as alphabetical symbols, for example:

C D E F G A B C = C major

256/243 9/8 9/8 9/8 256/243 9/8 9/8......9/8 9/8 256/243 9/8 9/8 9/8 256/243

C Db Eb F G Ab Bb C D E F G A B C

Here we see the logic that, regardless of the tuning used, a mode box will be presented using the

same note symbols. However, it would also be a mistake to think any tuning delivers the same

kind of result on one's mind. Obviously, when experimenting on the properties of the mode box,

different tunings will produce different perceptual results.

One rather psychedelic looking example of the mirroring of the major scale formula is this “leaf

effect” caused when all seven cycles are continually mirrored. The color-coding should help one

trace these modal formulas. The Dorian, for example begins on the central yellow, with the

Phrygian commencing on the red etc.

18

In this section, hopefully this mirror structure has stuck in the mind, and also that you agree the

results of mirroring scales do produce these triangular key relationships. The real key to

understanding how both sides of the mirror create a whole unit of information is to focus on the tri-

tone.

There is something uncanny about the 45-degree angle. By applying this mirroring technique to

the major scale, and then also to the modal system contained therein, it's been shown that a

hidden structure is at play, which can be seen to merge both sides of the duality/mirror together

and represent it as a whole unit.

After a few years the obvious question occurred to me; what would happen if one created a living

mode box within a laboratory? Would it trigger off the deeper mirror structure made up of these

triangles of keys, which are a way of viewing the mode box in terms of balance, a whole unit. This

invariably led to the question as to what effect this may have on consciousness itself.

19

Chapter two

The Vedic Square

The Vedic Square is a part of Vedic mathematics, popular in countries like India, but is becoming

increasingly popular in the west, mainly because of the speed one can learn to mentally attain

results to fairly complicated multiplications, divisions, subtraction, addition, and even algebraic

equations. In effect, a Vedic square is no more than a 9 by 9 multiplication table, with the totals

reduced to single digit, creating nine number sequences that apply throughout the number line into

infinity.

Back in 1993 I had no clue that such a grid existed. I discovered these nine possible number

sequences whilst laying on the bed playing around with numbers. Again, playing with numbers

was not a habit of mine, but I was hoping to find a way of justifying the results that mirroring the

major scale and its modes brought. It was quickly discovered that there were mirror flows within

these nine number sequences, and that a similar 4.5 axis existed that performed the same role

within these number sequences as it did in the major scale mode box. I eventually found out, once

on the internet, that these sequences of numbers were known as a Vedic square, and this led to

my acquaintance with it and other Vedic mathematics.

What will be shown in this chapter is a series of clockwise and anti-clockwise sequences of

numbers. They work for the general number line, and obviously other bases will throw up other

sequences. The reason for focusing on the nine number sequences that emerge with this

particular base system is that it throws up the same triangles of frequencies as in the mirror music

theory example of the major scale and its seven modes.

All numbers can be broken back down to a number between one and nine. Here is an example:

37 = 3 + 7 = 10 = 1 + 0 = 1

Therefore the number 37 can also be broken down (reduced) to and represent the number 1. The

number 1 will be seen as the beginning of a cycle of numbers based on 37. What we do now is

cycle the number 37 and reduce it down again to a number 1 to 9. This is like multiplication tables.

37 + 37 = 74 = 7 + 4 = 11= 1 + 1 = 2

20

If we keep adding on the original number to the total we will uncover a "number sequence" which

will be related to the number 37 and its natural cycle, that is, doubled, tripled, quadrupled etc. The

sequence will cycle over and over like this:

37 = 3 + 7 = 10 = 1 + 0 = 1 74 = 7 + 4 = 11 = 1 + 1 = 2

111 = 1 + 1 + 1 = 3 148 = 1 + 4 + 8 = 13 = 1 + 3 = 4 185 = 1 + 8 + 5 = 14 = 1 + 4 = 5 222 = 2 + 2 + 2 = 6 259 = 2 + 5 + 9 = 16 = 1 + 6 = 7 296 = 2 + 9 + 6 = 17 = 1 + 7 = 8 333 = 3 + 3 + 3 = 9

As you can see the number sequence that has emerged here is - 1 2 3 4 5 6 7 8 9, which is

obtained by cycling the number 37. Other numbers that share this same ‘cyclic’ sequence are 10,

19, 28,46,55 etc. Not all sequences are as straightforward as this and you will see that some will

be mirrors of others.

In fact there are only nine possible number sequences and these sequences represent every

number into infinity. It is the first nine numbers that house all the possible sequences (the zero is

teamed with the number 9 as will become apparent). The number 10 would obviously represent

the number sequence 1; the number 11 would represent the sequence 2 and so on. Here is the

number two cycled.

2 = 2 4 = 4 6 = 6 8 = 8 10 = 1 + 0 = 1 sequence = 2 4 6 8 1 3 5 7 9

12 = 1 + 2 = 3 14 = 1 + 4 = 5 16 = 1 + 6 = 7 18 = 1 + 8 = 9

21

Other numbers that share this same sequence will be – 11 20 29 38 etc. If we follow this

procedure with the number three we uncover another sequence:

3 = 3 6 = 6 9 = 9 sequence = 3 6 9

12 = 1 + 2 = 3 15 = 1 + 5 = 6

18 = 1 + 8 = 9 etc.

When this is done with the first nine numbers we will uncover all the sequences. Instead of listing

each number in the above manner I will only give the sequence that the first nine numbers

uncover, the first nine numbers being all that is needed:

1 2 3 4 5 6 7 8 9 2 4 6 8 1 3 5 7 9 3 6 9 3 6 9 3 6 9 4 8 3 7 2 6 1 5 9 5 1 6 2 7 3 8 4 9 6 3 9 6 3 9 6 3 9 7 5 3 1 8 6 4 2 9 8 7 6 5 4 3 2 1 9 9 9 9 9 9 9 9 9 9

One can spend half an hour or so satisfying themselves that any number cycled in a similar way

will only produce these nine number sequences.

Producing a 9 by 9 multiplication table would yield the same grid. For example, 1*1 = 1, 1*2 = 2,

1*3 = 3 etc, forming the first number sequence of the Vedic Square. 1*10 = 10, but would be

broken down to the number 1 when cross adding the digits, and so the sequence would repeat all

over again. Then 2*1= 2, 2*2 = 4, 2*3 = 6 etc, forming the second number sequence of the Vedic

square, and so on.

The number sequences emerge horizontally as well as vertically. Four of the number sequences

mirror the flow of another four; the first flows contrary to the eighth, the second flows contrary to

the seventh, the third flows contrary to the sixth, and the fourth flows contrary to the fifth - 1/8 2/7

22

3/6 4/5 . At this point there is a swap over at the 4.5, as the pairs work their way back to the

beginning, 5/4, 6/3, 7/2, 8/1.

1/8

2/7

3/6

4/5

-----------4.5 axis

5/4

6/3

7/2

8/1

9/0

One flow is constant and you can see that it is that of the 9. The zero needs to accompany the 9,

and is its own constant, yet cannot stand on its own after its initial beginning. After the 9 will come

the 10, which is the return of the 1, and therefore the zero has occurred “within” the 9.

Buckminster-Fuller showed how the 9 is also a zero by using indig numbers, which will be shown

shortly.

Observe how the first number sequence mirrors the eighth sequence (flow in opposite directions

to each other).

1 2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1 9

One sequence flows left to right (clockwise) whilst the other sequence is in contrary flow. As you

can see the numbers are mirrors of each other both vertically and horizontally. The numbers 1 and

8, for example, are mirrors of each other up/down and left/right. Vertically the result is always a 9

when the two numbers are added together. This is true when all number sequence partners are

displayed as above.

23

2 4 6 8 1 3 5 7 9

7 5 3 1 8 6 4 2 9

Using such a simple set of number sequences allows access to seemingly unrelated numbers due

to their clockwise/anti-clockwise relationship. Numbers like 776567 can be related to a number like

47 just because of their hidden number sequence cycles. In this case they both belong to the 2 4

6 8 1 3 5 7 9 sequence. Therefore both these numbers have a clockwise spin inherent within

their doubling and tripling etc. An anti-clockwise partner to these numbers would need to break

down to a 7, and be part of the 7 5 3 1 8 6 4 2 9 sequence. Something like a 99999799999

perhaps?! Cycling this number will produce the 7th number sequence, so there is a relationship

there.

What struck me first is that when you add up a number from both sides of the mirror the total is

always a 9. 1/8, for example, or 2/7, 3/6, they all add up to 9 together. This is precisely the same

numerical relationship that is found within musical inversions. Here is an example of how to invert

an interval in music:

C to G is a 5th interval (C d e f G)

G to C is a 4th interval (G a b C)

These too are adding up to 9. The relationship is known as an Inversion. Other possibilities include

a 2nd and a 7th, or a 3rd and a 6th.

C to D = 2nd

D to C = 7th

These number sequences of the Vedic Square also correspond to the indig numbers +/-1 +/-2 +/-3

+/-4, as devised by Buckminster-Fuller. It is reckoned that every number can be made up of these

indig numbers. It will be seen that between numbers 4 and 5, at the 4.5 position, there is a

swapping over of partnerships, or clockwise and anti-clockwise flows.

This is what Buckminster-Fuller has to say about these four possible plus/minus integers:

24

‘one produces a plus oneness

two produces a plus twoness

three produces a plus threeness

Four produces a plus fourness

Where after we reverse:

Five produces a minus fourness

Six produces a minus threeness

Seven produces a minus twoness

Eight produces a minus oneness

1 2 3 4 5 6 7 8

+1 +2 +3 +4 -4 -3 -2 -1

We can see that a 2, for example, produces a plus twoness when the number is cycled – 2 4 6 8 1

3 5 7 9. One can see straight away that the cross over point was at the 4.5, between the fourth

and fifth number. Yet it was from a clockwise positive state to an anti-clockwise negative state, so

to speak. This 4.5 is a 9 halved after all. Also the plus/minus signs refer to the dual flows,

clockwise and anti-clockwise, as seen in the number sequences.

The oneness is plus at the number 1 and minus at the number 8. The 1 and 8 are number

sequence partners of the Vedic square and flow in an opposite direction to each other. This is true

for the 2/7 (twoness is at 2 and 7) 3/6 and 4/5. The number 9 does indeed behave in a similar

fashion to the number zero.

1 = +1

2 = +2

3 = +3

4 = +4

5 = -4

6 = -3

7 = -2

8 = -1

36 = 9 = 0

25

All the number partners can be seen to emerge from the center, as well as the number 9 itself,

acting like some kind of axis point. Here is the second number sequence of the Vedic Square to

highlight this:

2 4 6 8 -1 3 5 7 9 2 4 6 8 -1 3 5 7 9

In symmetry around the 9 axis are the correct + and - numbers according to Buckminster-Fuller. In

between the 8 and 1 is the “invisible” or uninvolved axis, also producing the right number partners

in symmetry, the same as around the number 9. Around the uninvolved axis one will find the 1/8,

2/7, 3/6, 4/5 pairs, which are the same relationships as around the number 9. This is really the first

correlation that the number sequences have with the mirroring of the musical Modal system, in that

there is a visible axis and an invisible/in-between axis.

The 2 and the 7 number sequence, for example, are what Buckminster-Fuller recognizes as +2

and -2. He states that the 2 has a plus twoness about it, and you can see that it does, in the light

of the sequence of numbers it creates. The 7 has a minus 2 about it, and you can see from the

above number 7 sequence that this is true. For it to remain true there has to be a hidden 9 in

between the 1 and 8 in the above sequence. This is where the visible and invisible axis really

make sense. The mean number between the 1 and 8 is going to be 4.5. This is true for the 2/7

sequence, the mean between those two numbers being 4.5. Here is a diagram to show the

relationship between number sequences of the Vedic Square, the 4.5 and the Indig numbers:

26

Another reoccurring theme within the mirroring of natural cycles will be seen to be the swap-over

factor.

4.5 4.5

1/8 – 2/7 – 3/6 – 4/5 - 5/4 – 6/3 – 7/2 – 8/1 - 9 - 1/8 – 2/7 – 3/6 – 4/5 – 5/4 - 6/3 – 7/2 – 8/1

Sequencing numbers alone will obviously only show them increasing or decreasing. But running

them as symmetrically related pairs, shows them crossing over mirror points. They delve back into

zero point at the 4.5 axis, get swapped over the mirror axis and continue their flow. It should be

noted that around the 4.5 axis and the 9 axis everything is symmetrical. This is no different to the

previous set of numbers around the 9 and 4.5 axis, but in the last example it seems that there is a

four-way relationship functioning when the mirror flows are brought together. The 1/8 for example

occurs twice around the 9 axis. Does the 1 to the right of the 9 mirror to the 8 on the left, with the

remaining 1/8 being the remaining pair? The musical examples that follow also suggest this four

way relationship.

The whole pattern is established within the first 9 numbers. After this the number 10 begins the

contrary cycles all over again. The 4.5 is shifted and is represented as the number 13.5. This 13.5

is now perfect symmetry for the contrary cycles existing between the numbers 10 and 17. This

next diagram aims to show this simple expansion/contraction process through all number.

1 2 3 4 (4.5) 5 6 7 8 9

10 11 12 13 (13.5) 14 15 16 17 18

18 20 21 22 (22.5) 23 24 25 26 27

28 29 30 31 (31.5) 32 33 34 35 36

4.5, 13.5, 22.5, 31.5 are all 4.5s. 13.5 is 1+3 = 4, plus the .5, for example. The magnitudes are

forever changing, but the mirror number pairs are forever rushing out of the 4.5 toward the 9 and

from the 9 toward the 4.5. In the first example we see the 1/8 on the outskirts, begin to compress

inwards toward the 4.5, traveling through 2/7, 3/6 and 4/5. As the 1/8 begin to compress toward

27

the 4.5, the 4/5 begins to expand outward from the 4.5, traveling through the 6/3, 7/2 and 8/1. This

flow meets the axis at 9, is switched over, and begins to compress to the 13.5, whilst from the 13.5

there is an expansion through 13/14 to the outskirts at 10/17.

The next 4.5 axis is then at 22.5, being perfect symmetry between the numbers 19 and 26,

beginning another expansion/contraction. And so on throughout number, into infinity.

This is how clockwise and anti-clockwise cycles keep themselves in perfect symmetry around the

4.5 and 0/9 axis points.

So far it has been seen that the Vedic Square and the Mode box have the 4.5 in common.

One instant parallel between the Vedic Square and the mode boxes comes from this tendency in

the number 9 to replicate itself and so remain unchanged, as indeed is the role of the Dorian mode

within a mode box. Dorian is Dorian either side of the mirror, and one would suspect it is a kind of

axis point of its own.

9 = 9, 18 = 9, 27 = 9, 36 = 9, 45 = 9 etc.

Perfect symmetry in the C major Mode Box happens at the D Dorian Mode position and by some

coincidence the number 9 falls on the Dorian position within a Major scale:

1 2 3 4 5 6 7 8 9 C D E F G A B C D

The note D, within a C major scale, is always the Dorian mode position, regardless of which

octave one plays the mode in. This 9th position is the same as the 2nd position. Both positions

occupy the Dorian Mode position within the parent major scale. The Dorian is the balanced mode,

as the number 9 is the “balanced” number, and one sits above the other this way. The Dorian

carries the mode numbers 2/2 within the Mode box, but it could also carry 9/9 were it to be a mode

box that evolved past the octave either side. Please take a moment to satisfy yourself of this fact

by studying the Mode Box of C major again, because the relationship the Dorian shares with the

number 9 is intrinsic.

28

The 4.5 is a swap-over point. As an example, the second sequence of the Vedic square is plotted

together with their indig number equivalent. Here we will see how positive and negative always

swap-over around the 4.5. Here is pyramid number 1's additions:

2 = +2

4 = +4

6 = -3

8 = -1

(4.5 axis)1 = +1

3 = +3

5 = -4

7 = -2

9 axis

9 acts as a zero, as does 4.5.

Here's the number sequence running at the 45-degree angle of the Vedic square:

1

4

9

7

(4.5) 7

9

4

1

9

The mirror point happens between the 4th and 5th number sequence. All of this inner space is

4.5, as there are the 1/8 2/7 3/6 4/5 surrounding it, and the mean number between all these pairs

is 4.5. This in effect is a single digit sequence representing the first 9 square numbers:

29

1 * 1 = 1

2 * 2 = 4

3 * 3 = 9

4 * 4 = 16 = 1 + 6 = 7

5 * 5 = 25 = 2 + 5 = 7

6 * 6 = 36 = 3 + 6 = 9

7 * 7 = 49 = 4 + 9 = 13 = 1 + 3 = 4

8 * 8 = 64 = 6 + 4 = 10 = 1 + 0 = 1

9 * 9 = 81 = 8 + 1 = 9

It has been seen in the Vedic Square that the 1/8 are number sequence partners. It is interesting

to note that at 1*1 and 8*8, in the above examples, the result is the number 1. This is also true for

the other Vedic square number sequence partners, 2/7 , 3/6, 4/5. Here is a list:

1 * 1 = 1 2 * 2 = 4 3 * 3 = 9 4 * 4 = 7

8 * 8 = 1 7 * 7 = 4 6 * 6 = 9 5 * 5 = 7

One could superimpose the number sequence partners on top, and it would show that at the mirror

point is the 4.5

m. p. 1 2 3 4 5 6 7 8 9

1 4 9 7 7 9 4 1 9

Also implied in this array are the multiples of 11. For example, 4/5 falls on the 77, 3/6 falls on 99

etc. These are all multiples of 11. When studying the overtone grid known as the Lamdoma, the

11:1 and 1:11 overtones will be seen to function as a 4.5 like axis.

30

Gnomons

The same 1 4 9 7 7 4 1 9 sequence will also exist in what was known as Gnomons.

Gnomon means carpenter's square in Greek. This is the name given to the upright stick on a

sundial. For the Pythagoreans, the gnomons were the odd integers, which also represented the

masculine numbers. Starting with the monad, a square number is obtained by adding an L-shaped

border, called a gnomon.

Thus, the sum of the monad and any consecutive number of gnomons is a square number.

1 = 1

1 + 3 = 4 = 4

1 + 3 + 5 = 9 = 9

1 + 3 + 5 + 7 = 16 = 7

1 + 3 + 5 + 7 + 9 = 25 = 7

1 + 3 + 5 + 7 + 9 + 11 = 36 = 9

1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 = 4

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 1

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 = 9

Again, this also implies the 2nd sequence of the Vedic Square, even though it begins 1 3 5 7 9 2 4

6 8. It occurs when each new number is added to the sequence above, 1, 3, 5, 7, 9, 11 (2), 13 (4),

etc.

mirror point

1 4 9 7 7 9 4 1 9 , 1 4 9 7 7 9 4 1 9

(3) (6) (9) (3) (6) (9)

31

The number sequence 3 6 9 is acting like a junction point for the whole sequence. See how the

sequence begins 1, 4 and then a 9, then 7, 7 and then another 9, and lastly 4, 1 to yet another 9.

This junction points will be explored further in the coming chapters because of their ability to swap

information over either side of the mirror.

32

Placing an axis down the center of the Vedic square (the 4.5 point) highlights the symmetrical

qualities of the number 9 yet again:

Vedic square dissected at center

1 2 3 4 5 6 7 8

2 4 6 8 1 3 5 7

3 6 9 3 6 9 3 6

4 8 3 7 2 6 1 5

5 1 6 2 7 3 8 4

6 3 9 6 3 9 6 3

7 5 3 1 8 6 4 2

8 7 6 5 4 3 2 1

At the 4.5 point all numbers either side add up to 9 , and as they ripple outwards still add up to 9

when symmetrically paired.

Here is more of the type of symmetry that the 4.5 axis creates:

1 2 3 4 5 6 7 8

2 4 6 8 1 3 5 7 3 6 9 3 6 9 3 6 4 8 3 7 2 6 1 5

* 5 1 6 2 7 3 8 4

6 3 9 6 3 9 6 3 7 5 3 1 8 6 4 2 8 7 6 5 4 3 2 1

The mirrors are now four digit columns moving out from the center. The numbers that are not

shown in bold are also performing similar symmetrical duties. Some of the mirroring was

highlighted to show how it all manifests from the 4.5 invisible axis point. It literally is a numerical

and tonal fountain. This next diagram shows how the above number pairs are related due to this

4.5 axis. Trace the numbers back onto the grid below and the pairs will always equal 9:

33

Within the Mode box of a Major scale, like the skeleton view of the C Major Mode Box below, the

45-degree angle holds the ‘circle of tones structure:

34

If we separate the 369 from the Vedic Square we will be left with the numbers that emerge when

certain numbers are either doubled or halved, which is the 124875, or variations of it. Binary

numbers also hide this particular sequence:

1 2 4 5 7 8

2 4 8 1 5 7

4 8 7 2 1 5

5 1 2 7 8 4

7 5 1 8 4 2

8 7 5 4 2 1

In the middle of the diamond shape is the 72 or 27, equaling 18, 144, 54 or 99, depending on the

available angles for adding the numbers together.

One finds the numbers 1 2 4 8 5 7, that are left within the Vedic square after the 3 6 9 is taken out,

when the number 7 is the divisor.

1/7 = .142857 2/7 = .285714 3/7 = .428571 4/7 = 571428

5/7 = 714285 6/7 = 857142 7/7 = 1

It should be noted that all these fractions are recurring totals that will produce the same six

individual digits infinitely. Also worth noting is that each fraction is based on a consistent ‘this side’

flow followed by the mirror side flow. Take the 1/7 as an example:

1/7 = .142857

This six number relationship can be shown as two triplets that are mirror number partners of the

Vedic Square:

142

857

35

The 1 partners the 8, the 4 partners the 5, and the 2 partners the 7. If one repeats the fraction

there will also be the usual evidence of swapping-over an axis point:

.142857142857

The journey through the mirror number pairs is like a sine wave flipping either side of the mirror

continually. The arc begins from the 1, and curves its way to the 8, flips over and the 8 forms an

arc and curves the opposite way to the 1, flips over, and so on. The other two numbers do the

same thing, 4 to 5, then flip, 5 to 4 etc.

There were two “invisible” 4.5 axis points between the moves, firstly between 1/8 and then

swapped to 8/1. And this is true for the other two number pairs, as indeed it would be for any of

the number pairs.

Notice also the 142 and 857 flows appear in reverse dividing the number 6 by the number 7.

6/7 = 857142

One will find that there are three of these reverse relationships, 1/7 + 6/7, 2/7 + 5/7, 3/7 + 4/7.

These pairs always balance to 7.

As far as hidden sequences go, what the Vedic square exposes is the Position quality of a

number, and its standing amongst other numbers. But it also shows this in terms of clockwise and

anti-clockwise cycles, which are linked in some way to the musical mode boxes. These links

extend further out and embrace things like the Fibonacci numbers, the Phi ratio, and various other

grids. And it is the Circle of Tones structure that will be seen to unite all these various things

together.

Its association with the Vedic square has been established. Here it is in another guise, flowing

through the shifting 124875 sequence.

36

.1 2 4 8 7 5 = C

1 . 2 4 8 7 5 = Eb 1 2 . 4 8 7 5 = G 1 2 4 . 8 7 5 = B 1 2 4 8 . 7 5 = Eb

1 2 4 8 7 . 5 = G

1 2 4 8 7 5 . = B

The decimal point is rather interesting. If the 124875 equation were made to look like a square, the

decimal point would be flowing across it at a 45-degree angle. In many respects this is no different

to the flow of the Triangles of Keys across a Mode Box.

Here we see one of the triangles, Eb G B from the circle of tones structure appear. This particular

triangle of frequencies was seen to apply to the body's frequency responses.

Number octaves

The Vedic Square defines an “octave” of information in the same way a major scale also defines

its own octave. So this would be the Vedic Square in its next octave:

10 11 12 13 14 15 16 17 18 11 13 15 17 10 12 14 16 18 12 15 18 12 15 18 12 15 18 13 17 12 16 11 15 10 14 18 14 10 15 11 16 12 17 13 18 15 12 18 15 12 18 15 12 18 16 14 12 10 17 15 13 11 18 17 16 15 14 13 12 11 10 18 18 18 18 18 18 18 18 18 18

Stripped of the number 18, and also showing the 4.5 axis.

10 11 12 13 14 15 16 17 11 13 15 17 10 12 14 16 12 15 18 12 15 18 12 15 13 17 12 16 11 15 10 14 * 14 10 15 11 16 12 17 13 15 12 18 15 12 18 15 12 16 14 12 10 17 15 13 11 17 16 15 14 13 12 11 10

37

All the relationships in the previous Vedic Square are intact, so looking at a basic Vedic Square is

looking at the whole of number.

Horizontally and vertically the mirror axis is 13.5. But at the 45 degree angles, 11:11 and 16:16 it

is zero.

38

Vedic Square Crosses

Imagine an army of grains of sand. They are lined up in a straight regimented line. The first line

moves forward one pace at a time, or one measure of Planck's quantum nanometer! The next line

moves two grains/paces at a time; the third line moves three grains at a time. This is what one will

end up with if one only deals with a reduced matrix, like the Vedic square:

4.5 *

No manipulation would be required at this stage, as this is the natural state of cycles expressed

progressively from a zero point. Both 9 and 4.5 can be equated with zero.

The same would hold true if one did the same experiment with wavelength ratios, 1:1 2:1 3:1 etc,

then 2:1 4:1 6:1 etc.

So the Vedic square is no more than the fundamental movement of all the possible cycles within

number, and would be expressed if everything in nature moved accordingly. But of course it

doesn't. All the Vedic square does is show the perfect symmetry where all movement would be

39

generated from. Nature tends toward breaking the symmetry and so manifests life, which invariably

is made up of vibratory movements to and from the symmetrical point of rest. To say there is a

mirror world would mean to take any perceived movement on this side of the mirror and give it the

anti-move. Again the Vedic square shows clockwise and anti-clockwise cycles, that are in

symmetry together. For all things to sum back to zero there is an insight within the Vedic square,

showing that things generate from a starting point of symmetry.

If a wavenumber were 3454, it's mirror point along the cycle would be 3449, but it would be a sine

wave traveling in a contrary cycle in a "mirror world". It isn't related to entanglement, for example.

Photons that affect each other at great distances will mimic the state of the other. A mirror photon

may have its entangled partner in a mirror world, and perhaps all four relationships are in

symmetry together.

40

The Nine number sequences and the Triangles

1 2 3 4 5 6 7 8 9 = Bb

1 = C12 = G123 = B1234 = Eb12345 = G123456 = B1234567 = D12345678 = F#123456789 = Bb1234567891 = D12345678912 = F#

3 6 9 3 6 9 3 6 9 = F

3 = G36 = D369 = F#3693 = Bb36936 = D369369 = F#3693693 = A36936936 = C#369369369 = F3693693693 = A36936936936 = C#

5 1 6 2 7 3 8 4 9 = B

5 = E51 = Ab516 = C5162 = E51627 = G516273 = B5162738 = Eb51627384 = G516273849 = B

2 4 6 8 1 3 5 7 9 = Bb

2 = C 24 = G246 = B2468 = Eb24681 = G246813 = B2468135 = D 24681357 = F#246813579 = Bb2468135792 = D24681357924 = F#

4 8 3 7 2 6 1 5 9 = Bb

4 = C48 = G483 = B4837 = D48372 = F#483726 = Bb4837261 = D48372615 = F#483726159 = Bb4837261594 = D48372615948 = F#483726159483 = A4837261594837 = C#48372615948372 = F

6 3 9 6 3 9 6 3 9 = Eb

6 = G63 = B639 = Eb6396 = G63963 = B639639 = Eb6396396 = G63963963 = B639639639 = Eb6396396396 = G63963963963 = Bb639639639639 = D6396396396396 = F#

41

7 5 3 1 8 6 4 2 9 = F

7 = Bb75 = D753 = F#7531 = Bb75318 = D753186 = F#7531864 = Bb75318642 = D753186429 = F7531864297 = A75318642975 = C#

9 9 9 9 9 9 9 9 9 = Bb

9 = D99 = G999 = B9999 = Eb99999 = G999999 = B9999999 = Eb99999999 = G999999999 = Bb9999999999 = D99999999999 = F#

8 7 6 5 4 3 2 1 9 = Ab

8 = C87 = F876 = A8765 = C#87654 = F876543 = A8765432 = C87654321 = E876543219 = Ab

42

Chapter three

9 over 8 will go

The Circle of Tones structure, which emerges through mirroring of the major scale and its modes,

can also be explained this way; the scales consist of note numbers 1 to 8, which completes an

octave, whereas within the context of the number sequence cycles of the Vedic Square there are

single digits of 1 to 9. This “dilemma” between the 9 and the 8 is yet another way to arrive at the

said twin triangle structure of the circle of tones.

If we superimpose the numbers 1 to 9 over the Major Scale Formula the result is an over spill of

one number every time each individual cycle is complete.

9 numbers

1 2 3 4 5 6 7 8 9 T T S T T T S T C D E F G A B C D

one octave

The scale is that of C major, extended to the 9th. We will assume there is a movement of a Tone

between 8 and 9, similar to that between 1 and 2. We can continue this experiment by repeating

the above scale formula from the next note, E:

1 2 3 4 5 6 7 8 9 T T S T T T S T E F# G# A B C# D# E F#

43

The scale becomes that of E Major, again extended to the 9th. You can see, then, that if we

repeat this experiment again we will unveil G#(Ab) Major as the next key and this will complete

one triangle of Keys - C E G#(Ab) .

1 2 3 4 5 6 7 8 9

T T S T T T S T

Ab Bb C Db Eb F G Ab Bb

The next example would start on the note C again, so we have discovered the Triangle of Keys by

simple use of single digits 1 to 9 superimposed over the Major Scale. This is only one of the

augmented triangles of the Circle of Tones.

There are 27 moves in all:

C major Ab Major

C D E F G A B C D E F# G# A B C# D# E F# Ab Bb C Db Eb F G Ab Bb 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

Uninvolved key of E major

C

D Bb

E Ab

F G

C Major G F Ab Major

A Eb

B Db

C C

D Bb

E F# G# A B C# D# E F# G#

E Major

44

When number sequences and scale formula are viewed in tandem, the number 18 plays an

inevitable role. These 18 account for the two sets of 9 flowing either side of an axis point. These

two sets are needed to take into account both scale formula (8) and number sequences (9). If one

entertain the fact that 1 to 9 is the basic cycle throughout number, because 10 is the return to the

1, then this is how the 9 affect the scales, starting from the central C:

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

Bb C Db Eb F G Ab Bb C D E F G A B C D

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

D E F G A B C D E F# G# A B C# D# E F#

9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9

F# G# A B C# D# E F# G#/Ab Bb C Db Eb F G Ab Bb

Notes either side of the red axis points are symmetrically reflecting. These scales and numbers

run as a three-in-one set. As already shown, one begins at C and applies the nine numbers over

its major scale formula. The next round begins at E and the third round at G# (the next move after

this would be to C, which is a return home). This will have defined one of the Triangles of Keys.

Vertically it is numbers like 111, 222, 333, 444 etc, that produce all the four triangles, C E Ab, Bb

D F# etc.

The scales appear on both sides of the mirror in different orders. This too implies a swapping-

over, because what was on the left hand side of the mode box, can appear on the right hand side

of the above three sets of 9 (as is the case with the E major scale above, which normally is found

on the left hand side of the mode box). Another implication here is that it is not so much the major

or minor tonality that matters as much as Position. Position can change the quality of a previously

major tonality to minor, and vice versa. The “ground” on which a relationship exists is where the

quality, major or minor, is born.

45

C and Ab are two of the major scales that form part of the triangle of keys. They are also the two

major scales that are involved in the C Ionian/C Phrygian mirror pair. E major is not only the

uninvolved amongst this mirroring, it also house the F# Dorian position, and this will be seen to

have its axis at the 4.5 positions of both C Ionian and C Phrygian.

46

Chapter four

The Invisible aspect of the Triangle of Keys

We have seen that for every Major type scale there is a mirror partner that is a Minor type scale:

Lydian (major) mirrors to Locrian (minor)

Ionian (major) mirrors to Phrygian (minor)

Mixolydian (major) mirrors to Aeolian (minor)

Dorian mirrors to Dorian – the odd one out!

The Dorian is not really the odd one out here. It is actually the most important of all positions,

where the major/minor tonalities are able to swap-over and receive their opposite quality. It

happens to carry the modal mirror relationship of 2/2, Dorian mirrors to Dorian, which implies that

for F# at the 4.5 of the C major Mode box, to remain F# in the mirror, it too may be connected to

the Dorian position in some way. Is this F# a 2/2 from another major key, for example? After all,

we have witnessed a Mode Box made up of seven major key tonalities either side of the mirror.

The answer is obvious when we remember that we are dealing with a series of triangles of keys

that are interrelated through mirroring. Mirroring a major scale in effect produces another major

scale (albeit commencing from its 3rd modal position) . The keys used, for example, are C major

mirroring to Ab major, leaving only E major as some uninvolved aspect of the C Ab E triangle of

keys. In the key of E major the F# position is certainly Dorian.

Ion Dor Phr Lyd Mix Aeo Loc

E F# G# A B C# D# E – E major

It will be seen that the E major aspect of the triangle can be seen to join up with C and Ab through

the 4.5 tri-tone position. Here is line one of the C major Mode Box, with the E major scale

dissecting at the tri-tone position either side of the mirror:

47

F# Dorian

E mirror axis D# C# B A G# C Db Eb F F# G Ab Bb C D E F F# G A B C E C Phrygian E major D# C major (Ab major) C# B A G#

F# (Dorian) E (major)

What we have here is the usual C major scale, and its mirror, with the E major scale dissecting

both at the 4.5/tri-tone positions either side of the mirror. The Dorian is perfectly symmetrical,

therefore it must be an F# Dorian on the other side too.

This would also make it logical that on one side the E major scale is ascending through the tri-tone

area, and on the other side it is descending through the tri-tone area. Therefore at F#/F# the

relationship has to be Dorian/Dorian.

The Dorian is the point of access to the other side because it is the only dual modal partnership in

a mode box that carries the same number either side, 2/2, establishing itself as an axis/access

point. The number 9 is perfectly symmetrical, and it comes to rest on the Dorian within a mode

box:

1 2 3 4 5 6 7 8 9

C D E F G A B C D

The number 9 seems to be a catalyst to its perfect symmetry, and the two 4.5s are related to the

9. The Dorian Mode seems to possess a special place amongst the Positions.

By studying the two E major scales in red it will be seen that their opposite partners still adhere to

the correct dual modal partnerships. If the axis is placed horizontally at the F# positions, the E

from the right hand side scale mirrors the G# from the left hand side scale, for example, and this

48

will equal the usual Ionian/Phrygian mirror pair as seen through the mode box of C major. Then

the D# from the right hand side (Locrian) will mirror to the A on the left hand side (Lydian) etc.

E (Ion) D# (Loc) C Ionian C# (Aeo) B (Mix) A (Lyd) G# (Phr) C Db Eb F F# (Dor) G Ab Bb C D E F F# (Dor) G A B C E (Ion) C Phrygian E Ionian (Major) D# (Loc) (Ab major) C# (Aeo) B (Mix) A (Lyd) G# (Phr)

Establishing this link at F# with the Dorian mode will explain the link it has with the number 9 at the

tri-tone position. The 4.5 aspect is like a halving of the 9.

At arriving at this point we have treated the mirror as though it held an individual life of its own. Yet

this individuality unites with its opposite partner on the other side of the mirror through the tri-tone

position. Here the churning of clockwise and anti-clockwise cycling superimposed over each step

of the Major scale formula takes the Major scale Modes on a `tonality journey' through seven other

Modes/Major Keys. And having gone through this perfectly symmetrical ride through the tri-tone, it

appears on the opposite side, once more unbalanced in terms of numerical modal partnership

(1/3, 7/4, 6/5 etc), until it eventually arrives at a 2/2, finds unity, and carries on in this vein

continually, swapping over once more in an infinite dance between expansion/contraction,

Light/Dark, Positive/Negative, clockwise and anti-clockwise. Every example given in this book

always lead to this same result.

At the 4.5/tritone positions of each major scale will live this invisible aspect of each triangle. The

F# will be at the 4.5 of the C major scale. The Bb will be at the 4.5 of the E major scale and the D

will be at the 4.5 of the Ab major scale. Together this equals – C E Ab, with F# Bb D. These are

the two triangles that will create the circle of tones mirror structure, and this is the structure that

unites the mirror and manifests its whole unit of information. A whole unit must include the mirror

49

side. Seeing the major scale as seven modes is not the whole unit, but observing it as a 7 by 7

mirror mode unit is. Doing this also allows access to the mirror sides.

All these keys which form both triangles are 4.5 partners of each other, so it is possible to see the

circle of tones structure as a series of six Dorian modes. These are the hidden symmetrical flows

that unite two sides of a system into one whole, according to the nature of the major scale and its

modes, and also to the nine number cycles of the Vedic square.

50

Chapter five

Fibonacci Numbers and the Swap

The Fibonacci numbers are intrinsic to nature in many ways. They are a flow of numbers that can

represent such things as the breeding habits of rabbits, cause the spiral effect we see as galaxies,

conifer seed arrangement, or in the design of nautilus shells. The Fibonacci numbers show up in

the arrangement of flower petals or leaves, or in the arrangement of our finger joints and body

according to its ratios. Here are the first few Fibonacci numbers:

1 1 2 3 5 8 13 21 34 55 89 144

The preceding two numbers always become the next new number:

1+1 = 2 1+2 = 3 2 + 3 = 5 , 3 + 5 = 8 , 5 + 8 = 13 and so on.

Nature seems to obey this growth principle and it is also seen that the Fibonacci numbers are

closely associated with the PHI ratio, an infinitely recurring fraction whose first few digits are

1.618… This number is closely approximated within the Fibonacci series, for example:

21/13 = 1.615, 34/21 = 1.619 , 55/34 = 1.617, 89/55 = 1.618

As one moves on toward the higher numbers from the series, this Phi ratio becomes more

consistent. Phi is also referred to as the Golden Mean or Golden Ratio.

The nine number sequence partners of the Vedic square will be teamed up with the flow of the

Fibonacci numbers (FN). The FN series will be broken down to single digit, and this will equate

them with the nine number sequences of the Vedic square. This is done in order to highlight a

swapping over of the number cycles to the other side of the mirror.

In doing this it will give the FN flows a clockwise or anti-clockwise characteristic. When a single

digit value for a FN is ascertained, its mirror number partner will be placed underneath, in order to

highlight how the overall flows are swapping sides of the mirror at the number 144.

51

When we get to the number 13 (8+5)) in the series, for example, we cross add the digits and

reduce this number down to the number 4 (1+3). This means that a number between 1 + 9 always

represents the series. NS = Number Sequence, and MNS = Mirror Number Sequence.

FN - 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657

NS - 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 MNS- 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 1 8

one cycle

The NS – 1 1 2 3 5 8 4 3 7 1 8 9 becomes the MNS, and the swap-over is at the twelfth position, at

the number 144. As you can see, the number 13 is represented as a clockwise cycling number 4.

To say the 4 is clockwise cycling is referring to the number sequence that the number 4 is a root

of, 483726159, the 4th number sequence of the Vedic square. It has also been seen how scales

have their own clockwise and anti-clockwise flows, with major tonality always mirroring to minor

tonality and vice versa. And above we see the number 9 allowing a number flow from this side of

the mirror to switch to the other side, and the switch is performed at the 4.5, the mean number

running through the mirror number pairs.

The mirror number sequence above, 8 8 7 6 4 etc, still adheres to the FN. Here they are as if they were a FN series:

FN - 8 8 7(16) 6 (24) 13 (40) 19 32 51 83 134 217 351 568 919

MNS - 8 8 7 6 4 1 5 6 2 8 1 9 1 1

Here the 16+8= 24 (but broken down to a 6 in the MNS flow), the 16+24 = 40, etc.

It is known that the FN produce a repeating 24 single digit number sequence, whereas this mirror

number method is speaking of a 12 single digit number sequence along with a mirror 12 digit

sequence that swaps over across an axis point. There is no real conflict between this dual 12-

number sequence and the 24-number sequence. The only difference is that there is some case

here to suggest that actually the transmission of these numbers is being performed by a

symmetrical and contrary flowing affinity that one number has with another.

52

To help show the above swap-over in the Fibonacci numbers clearer still, here it is again

according to the Indig number system as devised by Buckminster-Fuller. “IN” means Indig

Number, and “MIN” means Mirror Indig Number:

IN - +1 +1 +2 +3 -4 -1 +4 +3 -2 +1 -1 0 -1 -1 -2 -3 +4 +1 -4

FN - 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 etc

MIN- -1 -1 -2 -3 +4 +1 -4 -3 +2 -1 +1 0 +1 +1 +2 +3 -4 -1 +4

It is worth remembering that the Phi ratio is associated with the Fibonacci numbers, as well as the

fact that there is swapping over the two sides of the mirror. This swapping over of sequences is

ongoing, and this equates to a journey of information flowing in and out of the mirror.

The 4.5 dominates the inner space between the number pairs. The mean number between all the

number pairs is always 4.5. It is also the zero in between the plus/minus indig numbers, which is

seen to accompany the number 144. This number is actually a very important one in the study of

Gematria.

Dividing by the number seven, for example, also shows the same result of dual cycles swapping

over a mirror point:

1/7 = 142857142857....

1 4 2 8 5 7 1 4 2 8 5 7

8 5 7 1 4 2 8 5 7 1 4 2

The 2/7 mirror number p-air seem to be catalysts for this swap -over, convening on to the 4.5 that

sits at the centre between them. The 142 travels in and out of the mirror point as the sequence

unfolds, with its mirror partner, 857 also cycling in and out of the mirror, tied to its partner by the

symmetry of the number 9, with its swap-over axis at the 4.5/0.

53

Here is the 3 6 9 number sequence, as if it were the beginning of a Fibonacci number sequence.

The preceding two numbers, 3 and 6, have equaled the next number, 9, so the sequence can

evolved showing the swapping over effect once more:

FN - 3 6 9 15 24 39 63 102 165 267 432 699 1131 1830 2961 4791

NS- 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3

The overall sequence is – 3 6 9 6 6 3 9 3. When added together this sequence equals 45:

3 + 6 + 9 + 6 + 6 + 3 + 9 + 3 = 45

The 3 6 9 sequence is now shown with the MNS and this will exhibit the same tendency of

swapping each number cycle across the mirror point, and it has to be at a 4.5 axis because this is

always the mean number between any mirror number pair.

FN - 3 6 9 15 24 39 63 102 165 267 432 699

NS - 3 6 9 6 6 3 9 3 3 6 9 6 etc.

MNS - 6 3 9 3 3 6 9 6 6 3 9 3

Between the number 24 the 3 6 9 6 sequence begins to appear on the mirror side, whilst the 6 3

9 3 also manifests on its opposite side of the mirror. And so the journey continues infinitely through

this Fibonacci type flow.

Remember that these number partners (1/8, 2/7,3/6,4/5) are cycling in opposite directions to each

other, clockwise and anti-clockwise. Also that the Phi ratio is still maintained between the

Fibonacci-like flow of the 369 sequence ( for example, 267/165 = 1.618618...). A later section will

show many more of these mirror flows existing within divisions.

54

Mirror flow of the Fibonacci numbers

Here is another way to mirror the FN series. The very same result as previously ensues, but it may

help to visualize it this way, and also put the result within a different context. We can set an axis at

the 0 and use the clockwise and anti-clockwise number partners of the Vedic Square to highlight

the evolution of the Fibonacci numbers on the mirror side.

Mirror point

(5) (4)135 82 62 29 24 14 1 4 6 7 8 8 0 1 1 2 3 5 8 13 21 34 55 89 144

One would be forgiven in thinking that the mirror side bares no relationship with the Fibonacci

numbers on the right. What should have followed the 8 is the number 16 (8+8=16). The 16 breaks

down to the number 7 (1+6=7). The next number after the 7, on the mirror side, is a 6. This should

have been 16+8=24. Obviously the 6 is the single digit value one obtains when cross adding the

24. It would also work as 7+8 = 15 (1+5=6). The mirror side is expressing a form of Fibonacci

number growth but expressing the result as mirror number values according to the clockwise and

anti=clockwise flowing number pairs of the Vedic Square. It’s a different kind of flow, but it does

seem to relate consistently.

For the 1 to become a 14 is a gain of 13, and this is the number in symmetry to it. It creates a 5/4

mirror number pair. The 14 goes to a 24, an increase of 10. On the other side the 13 rises to a 21,

an increase of 8. This is actually establishing the 1/8 mirror number pair, the 10 becoming the

single digit value of 1, and the 8. Following the same logic shows that there is a Vedic Square

correlation in the dual FN line.

Amongst all these number flows also lurks the circle of tones mirror structure, the two triangles of

keys.

55

Chapter six

The Lambdoma

Below is an overtone grid known as the Lambdoma, which is said to have first been constructed

by Pythagoras. Each ratio is taken from the perspective of the C axis/fundamental. But with

careful inspection it will be seen how F# can also be considered an axis point.

Notice how at 11:1 and 1:11 the overtone is F#. The overtone F# is F# in the mirror, and again

behaves like an axis point.

56

To be noted is that the F# and the Gb are being taken for the same note, but they are also

partners. F# Major is a key that contains six sharps, whilst Gb Major is a key that contains six flats.

They are known as the poles, because they hold the maximum sharps and flats within the circle of

5ths.

It has been seen that at the 4.5/Tri-tone position of the major scale there is a second mirror axis

point. Could the same be true for this 11:1 and1:11 within the Lambdoma? There is a similar

arrangement of axis points here, the C and the F#.

I am sure this kind of grid may not make sense to some, so here is, hopefully, a simple

explanation of how this grid describes sound within material objects.

When one bows and plays a violin string, or clangs on a big bell, one will hear a musical pitch,

which is called the fundamental tone. Yet this tone that we all hear is accompanied

by many other tones as well, called Overtones. They are fainter than the fundamental tone, and

can be written as whole number ratios. The Lambdoma shows how C is the fundamental note.

This is shown as a 1:1 ratio. Note also that the note's frequency here, or Hertz, is seen as One

Cycle Per Second (CPS). The next ratio is that of 2:1, or cps. This first overtone derived from the

fundamental produces the same note, except that the second note is one octave higher in pitch,

and not as loud as the fundamental. Most musicians can hear this octave of the fundamental when

they pluck a string. In fact here is a diagram of, say, a violin string, showing how octaves evolve

when that string is progressively cut in half:

The whole string signifies the Fundamental pitch, 1:1. The 2:1, being the string cut midway is the

ratio for an Octave above the fundamental. The 4:1 is the half string being cut in half again (or 4 of

these slices would be required to complete the whole string length), and signifies two octaves

above the fundamental. Again the 8:1 signifies a C three octaves above the fundamental. All

these, within the Lambdoma will be shown as the note C. Where will this halving of the string end?

57

In between the main octave ratios are other ratios, such as 3:1. This ratio will produce the interval

of an octave plus a perfect 5th above the fundamental, which in this case will be the note G. Here

are the first few overtones plotted on the music stave:

The bottom note here is a low C. The next note is the C one octave above, then the G above that,

another C at 4:1, then the E above that is on the treble staff, and so on.

Colour and other electro-magnetic forces are all related to frequency. These ratios are all to do

with wavelengths, typically sine waves. Should a sine wave be cut and cut and cut, it will enter the

domain of colour frequencies, which have cycles per second in the trillions. Using the law of

octaves one may cut the fundamental C1 a total of 2^26, and it would fall into the wavelength

required to create the colour red, at 450 Terahertz. The question remains whether sound waves

and electromagnetic waves are the same. In simple terms as wavelength relationship they would

both adhere to similar type ratios, by which waves sum up or divide.

By using the law of octaves, scientists are able to determine a musical note that is being emitted

by the sun. This note is really too high for us to hear at its original frequency (somewhere in the

region of 50 to 100 Mhz) but the law of octaves can be applied in order to equate the frequency

with a musical note. All one has to do is halve or double the number of the frequency continually,

until our ears pick up a note.

Not all objects, however, produce all the overtones all the time. Brass instruments only produce

the odd numbered overtones, for example.

So far we are dealing with multiplications, 2:1, 3:1, 4:1 etc. What about the divisions? Here is the

fundamental note C set as an axis, with the multiplication ratios on the right of it, and the division

ratios on the left:

58

Again we see that the 2:1 ratio to the right produces an octave, and this complies with the law of

doubling a number to produce the same note an octave higher. In symmetry to this is the 1:2 ratio

to the left. This too complies with the law of halving a number in order to produce the same note

one octave lower in pitch. If one played the 1:1 C note on the violin, the wavelength would divide

itself along that string, and one would hear a faint G note, at a 3:1 ratio, the fundamental string

divided into three parts/separate wavelengths. In symmetry to this, the 1:3 ratio would produce the

overtone F (it's being played by a violinist on the mirror side universe so we can't hear it!).

Studying the Lambdoma above again , hopefully it will begin to make more sense.

The F/G pair are the same mirror note partners as in the C major scale and its mirror scale. This is

what we see amongst the multiplication ratios and the divisions, the same mirror note pairs as

around the C axis of the Major scale, C/C, D/Bb, E/Ab F/G G/F etc. D and Bb, for example, are 9:1

and 1:9 respectively.

What is also confirmed in the Lambdoma is that at one point after the C there is another axis point.

That point occurs at the 11:1 and 1:11. As mentioned, this is the same note, that of F#. Obviously

the other notes do not qualify as axis points because they share different note partners across the

mirror.

Remember that this 11:1 and 1:11 axis is from the point of view of the C axis at 1:1. There are

other ratios for F# within the grid, but these are from the point of view of other root notes. One axis

59

point is the visible fundamental (the C axis), and the other is the “invisible” or in between axis (the

F# axis), and is also the F#/Gb poles of the circle of 5ths.

Here is the scale of C major and its mirror again:

C Db Eb F G Ab Bb C D E F G A B C

The note F# sits in the very center of both sides of the mirror, in between the F and the

G, four and a half moves to the left and right. This is the 4.5 axis, whereas in the Lambdoma it is

at the F# 11:1 and F# 1:11. As seen in the Mode box (chapter one) the C on the mirror side starts

its cycle on the number 3, and on the number 1 on the right hand side. The F#, therefore can also

be seen as positioned at the number 6.5, as well as its 4.5 function. This of course equals an 11.

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

C Db Eb F G Ab Bb C D E F G A B C

Around the C note axis appear mirror note pairs, C/C D/Bb E/Ab G/F etc. F# = F# through the

mirror (it symmetrically reflects itself). The note C is an axis point because C = C in the mirror.

If F# reflects to F#, and this functions as an 11 within the Lambdoma, then is the major scale also

endowed with an 11 connection? The 4.5 + 6.5 of the F# axis points did equal an 11.

The musical note names will be omitted, and only the number position pairs will be shown:

3/1, 2/2, 1/3, 7/4, 6/5, 5/6, 4/7,

The 7/4 and 6/5 already show they add to 11. The others do as well, if we shift some of the

relationships up by an octave. Here are two octaves of the C major scale:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

C D E F G A B C D E F G A B C

60

The 1 can also be an 8. Therefore the 3/1 mirror pair can also be shown as 3/8. Then the 1/3 can

be shown as 8/3. Together this equals an 11:11.

The 2/2 can be designated as 9/2, again an 11.

Here is the Major scale formula written as ratios relating to the Lambdoma. These ratios apply to

any major key, not just C Major:

1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:1 9:1 5:1 2:3 3:2 5:3 15:1 1:1

One can see that the top line of a mode box shown as ratios reflects perfectly around the 1:1 axis.

If the above were the key of C Major then one would find, for example, that the 9:1 would

represent the note D and the 1:9 would represent the note Bb along the top line. One can check

this against the Lambdoma, and then against the C major scale and its mirror.

After the second line of a Mode Box there will be an asymmetrical picture. This is because one

Lambdoma is related only to one Fundamental, as in the Lambdoma shown in this chapter, which

is all related to the note C. In a mode box of course, one exposes seven major scales, that is,

seven root fundamentals, two of which are enharmonic. Therefore one would need to draw out at

least six Lambdoma charts to prove that the mode box does reflect in a natural way, and therefore

the resulting triangles of frequencies that emerge are part of the overall picture within a Mode Box,

and they also penetrate the Lambdoma in some sense..

Here is what would seem a rather complicated looking set of ratios, but the only result needed to

focus on is that along the 45-degree angle. This ratio grid represent two left hand sides of a Mode

Box. It begins with a Bb 1:9 on the far right, then traces two octaves of each mirror mode.

61

1:1 15:1 5:3 3:2 2:3 5:1 9:1 - 1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:9 1:5 3:2 2:3 3:5 1:15 1:1 - 1:9 1:5 3:2 2:3 3:5 1:15 1:1 1:5 1:11 5:1 3:5 1:15 15:1 5:3 - 1:5 1:11 5:1 3:5 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:15 15:1 5:3 - 3:2 1:11 5:1 9:1 1:15 15:1 5:3 2:3 3:5 9:1 1:1 1:9 1:5 3:2 - 2:3 3:5 9:1 1:1 1:9 1:5 3:2 3:5 1:15 15:1 1:9 1:5 1:11 2:3 - 3:5 1:15 15:1 1:9 1:5 1:11 2:3 1:15 15:1 5:3 3:2 1:11 5:1 9:1 - 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:1 15:1 5:3 3:2 2:3 5:1 9:1 - 1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:9 1:5 3:2 2:3 3:5 1:15 1:1 - 1:9 1:5 3:2 2:3 3:5 1:15 1:1 1:5 1:11 5:1 3:5 1:15 15:1 5:3 - 1:5 1:11 5:1 3:5 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:15 15:1 5:3 - 3:2 1:11 5:1 9:1 1:15 15:1 5:3 2:3 3:5 9:1 1:1 1:9 1:5 3:2 - 2:3 3:5 9:1 1:1 1:9 1:5 3:2 3:5 1:15 15:1 1:9 1:5 1:11 2:3 - 3:5 1:15 15:1 1:9 1:5 1:11 2:3 1:15 15:1 5:3 3:2 1:11 5:1 9:1 - 1:15 15:1 5:3 3:2 1:11 5:1 9:1

Here is the 45-degree angle from bottom left to top right:

1:15 1:15 9:1 9:1 1:15 1:15 9:1 1:15 1:15 9:1 9:1 1:15 1:15 9:1

All patterns flowing from bottom left to top right along the 45-degree angles have repeating

sequences of ratios about them. As do all the patterns flowing from top left to bottom right. Along

this angle the triangles of keys/circle of tones mirror structure is being created. Here are the notes

and ratios together along this 45-degree angle:

1:1 C, 1:5 Ab, 5:1 E, 9:1 D, 1:9 Bb, 1:11 F#, 9:1 D, 1:1 C

At this point the pattern repeats. We have also found yet again the two triangles of the C major

Mode box, known as the Circle of Tones, C Ab E and D Bb F#. In terms of ratios, the reversing of

the C major formula complies fully with the symmetry within the Lambdoma, which is a grid that

represents natural sound and its wavelength relationships expressed as whole number ratios.

62

Chapter seven

Here the Fibonacci numbers will be cycled as if they were part of a Mode Box, and each number

will relate to an overtone, or cycles per second. In doing this one unearths the same triangle flows

as within the mode box, and again across a 45-degree angle, as the notes in bold show. The

mirroring is from the centre 1:1 at the note C, stretching outwards both left and right.

Fibonacci Mode Boxes

55

Eb34

B

21

G

13

E

8

C

5

Ab

3

F

2

C

1:1

C2

C

3

G

5

E

8

C

13

Ab

21

F

34

C#

55

A1

C

55

Eb34

B

21

G

13

E

8

C

5

Ab

3

F

2:1

C3

G

5

E

8

C

13

Ab

21

F

34

C#

55

A

1

C2

D

1

D

55

F34

C#

21

A

13

F#

8

D

5

Bb

3:1

G5

E

8

C

13

Ab

21

F

34

C#

55

A

1

C

2

C3

C#

2

G#

1

G#

55

B34

G

21

D#

13

C

8

G#

5:1

E8

C

13

Ab

21

F

34

C#

55

A

1

C

2

C

3

G5

Ab

3

F

2

C

1

C

55

Eb34

B

21

G

13

E

8:1

C13

Ab

21

F

34

C#

55

A

1

C

2

C

3

G

5

E8

E

5

C

3

A

2

E

1

E

55

G34

Eb

21

B

13:1

Ab21

F

34

C#

55

A

1

C

2

C

3

G

5

E

8

C13

D

8

Bb

5

Gb

3

Eb

2

Bb

1

Bb

55

C#34

A

21:1

F34

C#

55

A

1

C

2

C

3

G

5

E

8

C

13

Ab21

A

13

F#

8

D

5

Bb

3

G

2

D

1

D

55

F34:1

C#55

A

1

C

2

C

3

G

5

E

8

C

13

Ab

21

F34

F

21

C#

13

Bb

8

F#

5

D

3

B

2

F#

1

F#

55:1

A1

C

2

C

3

G

5

E

8

C

13

Ab

21

F

34

C#

All the ratios from the central 1:1 column travelling to the right are based on that fundamental tone,

rising as 2:1 3:1 etc. To the left from this central column, each interval created on the right has

been mirrored, giving rise to 1:2 1:3 etc. Analysing this Fibonacci mode box is rather tricky. What

is worth focusing on for now is that the same triangle of frequencies are appearing on the mirror

side along the 45 degree angle.

It will require quite a lot of focus in order to fully take in the many patterns within the box above.

One will need to see that, in order to maintain the true mirror notes either side of the central

column, it is a question of bringing in other overtone series built on different fundamental notes,

because the remaining notes down the central column do not begin from the C fundamental. It will

63

hopefully become apparent, with enough study, how wonderful the tapestry is that Nature plots

out, through its symmetries.

Starting at the 55 on the top left side, after the initial Eb and F, the two triangles that have

appeared are:

B Eb G

C# F A

These notes are in bold along the 45 degree angle on the left hand side. These particular two

triangles will appear in six of the twelve possible mode boxes, based on the major or natural minor

scales. Plotting these frequencies on a circle will produce one of the dual triangle symbols shown

in the first chapter, which look like a star of David.

For the same mirror structure to occur within the Fibonacci series does strengthen the idea that

this circle of tones structure is intrinsic, signifying a point of unity between the two sides of the

mirror. After all, there are two sides exposed within the above grid, and Nature is known to use the

sequence information on the right hand side in order to regulate certain growth patterns, like

spiralling galaxies, conifer seeds, the nautilus shell, the distribution of leaves and branches on

trees, and many other things.

The very first thing that needs to be focused on is the different tonalities either side of the mirror.

As in the very first experiment, where we found C major mirroring to the C Phrygian mode, we find

that the overtones built on the C fundamental (first two lines) adhere to the same principle. The

line commencing as 2:1 on the central red note C is the same as the 1:1 line above it, because the

second line is merely an octave higher than the first. The mirror side does not create the same

overtone series that belongs to the non-mirror side. It is the mirror side's undertones that reveal

the overall twin triangle structure.

One cannot call the mirror side on the first line, the C Phrygian Overtones. Overtones do not obey

the Modal structure this way. On the one hand, each move on one side of the mirror is

reciprocated on the other side, multiplication of ratios becoming divisions. In turn this creates

tonality that really belongs to other overtone series, built on different fundamental tones. In

determining these overtone series one is led into a mirror world that fits consistently with this sides

logic, yet also with its own mirror logic.

64

Any chromatic note can become the fundamental, and therefore each note can build a Lambdoma

grid of its own. In fact, building twelve Lambdoma charts would ultimately help one understand

what is going on.

The first nine movements of the Fibonacci numbers have been taken and cycled in the same way

the Mode Box of C Major was cycled. The cycles have been mirrored according to the musical

notes and how they are positioned around the axis point of each new line. The axis is always the

red notes in the centre. As always the diagonals on the left hand side display the Circle of Tones.

This can be highlighted with one or two examples. At the top left is the note Eb. The diagonal here

runs from the Eb all the way to the A, at the bottom centre:

Eb Eb F, B Eb G, C# F A,

After the initial three notes there are the two triangles of frequencies, B Eb G and C# F A, and

when combined they will produce the Circle of Tones – B C# Eb F G A. It doesn’t matter which

note one begins with; the result will be a circle of tones. Below the top left Eb is the note C. Along

this diagonal the other Circle of Tones is produced:

C D, G# C E , Bb D F#

Firstly they are grouped up as two triangle of frequencies, and then when put in sequential order

produce the Circle of tones – C D E F# G# Bb. Also the top left Eb and the centre A is a tri-

tone interval. This is also true for the C and F#.

If we now continue the Fibonacci numbers for another nine movements, it is uncanny that the

thread will be picked up by the note F# on the number 89. The first Fibonacci mode box begins

with C, then after nine it begins with F#, establishing a strong link with the results in the C major

mode box, where C and F# are the invisible tri-tone relationship. The 8:9 ratio would be that used

for the movement of one tone, and six of these are what produce the circle of tones. Although the

Pythagorean comma would come into play, and in actual fact it would be seen that the structure

has slipped into another cycle separated by this 23 cent interval of a comma. And the Fibonacci

mode box will continue to evolve this way, never ending:

89 144 233 377 610 987 1597 2584 4181 C Ab E C# A F D Bb F# D Bb G Eb B G# E C

Bb E C G# F C# A F# D Bb G Eb B G# E C F#

65

The triangles of frequencies have not only appeared along the 45 degree angle. With careful

inspection one will notice them attempting to appear along the horizontal line as well. No where

more so than along the F# line. On the far left here one sees the first triangle -C Ab E. It then

gives way to the next triangle from the other circle of tones , C# A F. After this, the first circle of

tones appears again with the second triangle from it – Bb F# D. And lastly, there is swapping once

more to the second circle of tones with the appearance of the G Eb B triangle of frequencies. This

in-and-out-of-the-mirror phenomena is a basic trait of the structure.

The numbers on the left hand side are rather difficult to determine. The first move from F# to D is

that of a minor 6th. Therefore the reciprocal move to the left is also a minor 6th. This resulting Bb

may be determined by simply lowering the 233 Bb note by octaves.

233/2 = 116.5, 116.5/2 = 58.25

The resulting 58.25 would fit because it is lower than the F# 89. What then of the D next to it?

Halving 144 will give 72, and halving again will give 36. Even though 36 will fit one can see that

the left hand side is not producing a Fibonacci sequence. Well not quite anyway. If we add the 36

and 58.25 together we get 94.25. This is quite close the F# 89, and given that we are dealing with

approximate minor/major 6th intervals it may be close enough to go along with. But then comes the

question as to whether 58.25 is a Bb pitch. The notes have been determined by their cycles per

second, so that 89 cps is the note F#, for example. Will this be true for the numbers on the left?

Certainly so! Because doubling or halving a number will produce an octave of the same pitch.

When we remember that a note can be spread across a few numbers within its approximate range

(Bb commences at 58, and is a Bb until 61 cycles per second. An octave higher and Bb

commences at 116 cps until 122 cps.

The note F on the top left will probably be the F 21. This is because we can add it to the D 36 and

then add both numbers together and get close to the 58.25. Moving on to the A we can assume it

is an A at 13.75 (55 halved and halved again). This 13.75 then adds to the 21 to make 34.75. As

you can see this is nearly the total for the note C# at 34. We can start deducing from all this that

the left hand side is really producing some in-between pitches to the ones written. This is not

surprising because the Phi ratio at 1.618 is only an approximation of a minor/major 6th, one

choosing either in order to keep the flow of notes going. Therefore it is more a question of playing

to the numbers, because as mentioned, a note can cover more than the number that it is being

represented with, just like the colour red has many shades. In other words, the right hand notes

are already approximations, so there is little chance that things will add up perfectly when these

notes are mirrored to the other side. Then the Fibonacci numbers are equated with the Phi ratio

66

even though that too is an approximation, and also given he fact that the Phi ratio is not really a

finite number as far as we know.

Yet because the arrow is pointing right to left there is more of a subtraction system to the left hand

side of the mode boxes above. It is more 89 less 58.25 equals 30.75, further confusing the issue,

yet a workable structure nevertheless, depending on how one may wish to use it.

Should the second Fibonacci mode box be completed, starting at the F#, one would find the same

Circle of Tones structure along the diagonals on the left hand side.

This is by no means the end of the relationship that the Fibonacci numbers have with the Circle of

Tones. Next are discussed what are called Summation tones, within the overtone series. This

simply means that two pitches/wavelengths can come together and between them they add up

and create a third pitch/wave from that union. The same is true for Difference tones, where two

pitches/waves subtract and create a third pitch/wave from that.

67

Summation Tones and the Fibonacci numbers

Summation tones equate closely to the triangles of frequencies that were found to exist flowing

through a major scale mode box.

A summation tone is created when we add any of the overtone numbers together. An example

would be G3 (three cycles per second) and E5 (5 cps) add together to produce the C8, and so the

overtones’ frequencies speed up it seems. The summation tone4s table below shows that the

Fibonacci number series is being produced down all three columns (the summation tones

produced are those on the far right). The first example, C1 and C2 produce G3. Then, just like the

Fibonacci number series, the previous two tones produce a third tone, in the shape of C2+G3

producing the E5:

C 1 + C 2 = G 3

C 2 + G 3 = E 5

G 3 + E 5 = C 8

E 5 + C 8 = A 13 (nearly A)

C 8 + A 13 = F 21 (nearly F)

A 13 + F 21 = Db 34

F 21 + Db 34 = A 55

Db 34 + A 55 = F# 89

By 55 + F# 89 = D 144

F#89 + D 144 = Bb 233

D 144 + Bb 233 = G 377

B 233 + G 377 = Eb 610

G 377 + Eb 610 = B 987

Eb 610 + B 987 = Ab 1597

C 987 + Ab 1597 = E 2584

Ab 1597 + E 2584 = C 4181

E 2584 + C 4181 = A 6765

This way of combining the overtone series and the Fibonacci numbers leads straight to the

triangles of frequencies and keys evident within the mirror side of a mode box. After G E C, which

forms the grand triad, the proceeding triplets are those same triangles that have been found on

68

the mirror side of the Major scale, that go on to define two Circles of Tones, each of which is two

triangles combined - A F Db, F# D Bb, G Eb B, Ab E C..

There is also an “in and out of the mirror” flow occurring here. We can say that the triangles are

swapping from one COT to the other. A F Db, which belongs to one COT gives way to F# D Bb,

which belongs to the other COT. This then gives way to G Eb B, which re-establishes the first

COT again, and finally this yields to the last triangle, Ab E C, before commencing the whole

journey again.

Summation tones also lead to the same mirror structure. Simply by summing up adjacent notes of

one circle of tones the opposite circle of tones will be created.

Here is the chromatic scale using the whole number overtone ratios:

C 128, D 144, E 160, F 168, G 192, A 216, B 240, C 256

Db 136, Eb 152, F# 176, Ab 208 , Bb 224

Here are the harmonics summed together:

C 128 and D 144 come together and create Db 272

D 144 and E 160 come together and create Eb 304

E 160 and F# 176 come together and create F 336

F# 176 and Ab 208 create G 384

Ab 208 and Bb 224 create A 432

Bb 224 and C 256 create B 480

One Circle of Tones has created the other. C D E F# Ab Bb (one circle of tones) creates Db Eb F G A B (the other circle of tones).

69

Fibonacci as overtones

Every move of an octave and a 5th produces a Fibonacci like sequence.

1 2 3 = 1, 1:2 and 2:3

Are they the first three overtones or are they the first three Fibonacci numbers? As far as the first

three overtones are concerned it is a movement of an Octave (1:2 ) and a 5th (2:3). As far as the

Fibonacci numbers are concerned it is a male and female rabbit having a baby rabbit! Ok so the

number 1 should repeat, but that would only be a replication of the fundamental as far as the

overtones are concerned.

The idea is that the numbers 1 2 3 set off a period of growth that is relative both to overtones and

to numbers like the Fibonacci. In fact the 1 + 2 coming together is the very same as the

phenomena of Summation Tones. A scale can be created on any number and then evolved

according to a Fibonacci like sequence, where the preceding two numbers equal the new number.

Here is the number 21 evolved that way:

21, 42, 63, 105, 168, 273, 441, 714, 1155, 1869, 3024, 4893 7917 etc

These will be the notes:

21 cps = E +33

42 = E +33

63 = B +35

105 = G# +19

168 = E +33

273 = C# -26

441 = A +4

714 = F +38

1155 = D -29

1869 = Bb +4

3024 = F# +5

4893 = Eb -30

7917 = B +3

12810 = G + 36

Once the root triad has been established (the triad of E major, comprising the individual notes E

G# B), the Fibonacci sequence settle back to producing the triangles belonging to the circle of

tones, which once more shows that it is an intrinsic structure.

70

The approximation of the Phi ratio is still very much evident as the distance from one number to

the other:

273/168 = 1.625, 441/273 = 1.615384615384.., 714/441 = 1.619047619047...

The E +33, for example, means that the note E is 33 cents sharp. +/-50 cents equals a semi-tone.

There are thousands of scales opened up that we have never heard by using this approach. They

will undoubtedly sound pretty harsh for musical purposes, but vibrations are not only for

musicians to compose with. Scientists too are experimenting with different frequencies.

This circle of tones structure exists within any number that evolves according to a Fibonacci

number sequence.

We will pursue the swapping-over of information across mirror points in the following examples

2 4 6 = Octave and 5th – 2:4 and 4:6

It is also a beginning point for a new Fibonacci style number sequence to emerge.

The following shows the Fibonacci sequence on the top line, the single digit number equivalent,

and the mirror single digit number. The number partners, as seen, are 1/8 2/7 3/6 4/5 5/4 6/3 7/2

8/1 9/9:

2 4 6 10 16 26 42 68 110 178 288 466 752 1218 1970 3188 5158(19)

2 4 6 1 7 8 6 5 2 7 9 7 5 3 8 2 1

7 5 3 8 2 1 3 4 7 2 9 2 4 6 1 7 8

The total at the number 9 is 288. It is the place that the number sequences swap over the mirror

point. Again the Phi ratio will be approximated as the space between each adjacent number. And

also the triangles from the circle of tones will evolve in a similar manner as shown before. This

sequence merely doubles the original FN sequence.

This happens with the number 3 as well. The number three is doubled to become an octave and

then the two totals are added together in order to create a 5th interval. This in turn creates the

beginning to another Fibonacci type sequence;

3 6 9 15 24 39 63 102 165 267 432 699

3 6 9 6 6 3 9 3 3 6 9 6

6 3 9 3 3 6 9 6 6 3 9 3

71

Here the flows swap over at number 24 then at number 165 etc. Next the number 4:

4 8 12 20 32 52 84 136 220 356 576 932 1508 2440 3948

4 8 3 2 5 7 3 1 4 5 9 5 5 1 6

5 1 6 7 4 2 6 8 5 4 9 4 4 8 3

This will only double the flow of the 2 4 6 sequence. The swapping over will also occur in like

fashion. Perhaps like everything else there will only be nine different strands of Fibonacci like

sequences? The number sequence and mirror number sequence swap over after the total 932.

5 10 15 25 40 65 105 170 275 445 720 1165 1885 3050 4935

5 1 6 7 4 2 6 8 5 4 9 4 4 8 3

4 8 3 2 5 7 3 1 4 5 9 5 5 1 6

The mirror number sequence pairs from the previous example have appeared again, only

swapped over. The 516742685494 sequence and the 483257314595 sequence are on opposite

sides of the mirror now. So why is this happening? The 4/5 are number partners from the Vedic

square, and perhaps this is why the produce the same sequences only in reverse. If this were

true then the next number, which is the Fibonacci sequence beginning at 6, will echo flows of the

369 above in opposite manner:

6 12 18 30 48 78 126 204 330 534 864 1398

6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3 3 6 9 3

This is indeed what has happened. The 6396 of the last example has appeared on the opposite

side of the mirror, and establishes a link with the 3/6 partners of the Vedic Square.

There are some special ways of combining numbers with their mirror number counterparts. Take

any number, say, 33, and run that number according to the Fibonacci number ratios, so that the

numbers are representations of both overtones and the Fibonacci numbers:

33 66 99 165 264 429 693 1122 1815 2937 4752

6 3 9 3 3 6 9 6 6 3 9

3 6 9 6 6 3 9 3 3 6 9

As you can see the number sequences at the bottom are the same as that of the previous

example. The 3/6 are as much partners as the 30/33, and will produce the exact same single digit

72

number flows. This in effect shows how numbers move in units of nine, steadily progressing into

infinity, and never loosing the thread that is maintained with the original nine number relationships.

Each unit replicates the original sequence ascertained within the first nine numbers, and because

of this a link is actually provided between very small numbers and very large numbers, for

example. As we can see above; even though the two number sequences below the main numbers

are identical in both examples, nevertheless the musical scale created by each example is very

different. Here is the mirror number of 33, the number 30, which will be treated in exactly the same

way:

30 60 90 150 240 390 630 1020 1650 2670 4320

3 6 9 6 6 3 9 3 3 6 9

6 3 9 3 3 6 9 6 6 3 9

This series produces a harmonic of 432.

One can see how the number sequences underneath each main number of the 30 and 33 series

have swapped places. The top sequence under the 33 series is 6 3 9 3 etc, and the bottom mirror

sequence is 3 6 9 6. Under the number 30’s series the sequences have swapped over. This is all

logical blending of nine original numbers, through their attraction for their opposing partners. A

clockwise sequence always finds its anti-clockwise sequence partner.

Adding any two numbers together, like the 30 and 33, will always give a distilled value of 9. This

again is quite logical because partner numbers from either side of the mirror will always add to a 9.

63 126 189 315 504 819 1323 2142 3465 5607 9072

9 9 9 9 9 9 9 9 9 9 9

Fibonacci/Overtone scales contd.

These are Fibonacci/Overtone scales. The technique for unearthing them opens up frequencies

that are not in use.

Below is a scale beginning on the number 4. Just like 1 is doubled to 2, so 4 is doubled to 8. Then

1 and 2 are added together to produce the next number in the FN, which is 3. The 3 is also the

Summation tone being the 1st and 2nd overtones coming together. Likewise the 4 and 8 are

joined together to produce 12. From here on the scale builds up organically just like a Fibonacci

series. Each number is a note. These notes will not be exact pitches as we are used to hearing

73

them, in fact some will sound more like quarter tones. So this scale evolves organically according

to the number 4.

4 8 12 20 32 52 84 136 230 366 596 962

Here are the notes:

C

C

G

D#

C

G#

E

C#

A#

F#

D

B

You won't find the notes on a piano pertaining to the scale built from the number 4 that has grown

from applying the Fibonacci proportions to it. The vibrations fall within the 'main' notes of standard

scales. Synthesizers nowadays can accommodate these scales, because most synthesizers allow

for offsets of pitches up or down 50 cents.

Here is another scale built on the number 76. Even though 76 breaks down to a 4

(7+6=13=1+3=4) the scales unearthed using this approach are very different.

76 152 228 380 608 988 1596 2584 4180 6764 etc

D#

D#

A#

F#

D# B G

E C G#

74

This resulting scale is likely to sound quite discordant. The first three notes are typically an Octave

76:152, and a 5th 152:228. From then on the Fibonacci numbers take the vibrations on a course

of growth that is natural for them. Notice also that, as always, the triangles of the circle of tones

will begin to emerge.

Here is an idea for an overtone/Fibonacci scale based on the Phi ratio. The first few notes based

on the first three numbers of Phi are:

1.618

3.236

4.854

8.090

12.944

This would fulfil three criteria, the Fibonacci expansion of numbers coupled with the

overtone sequence following the phi ratio. These notes would fall in between

the well known notes in use.

A circle of 5ths and a scale for all

Having seen that the Fibonacci like sequences create the circle of tones, it is quite possible to

define many circle of 5ths based on unique tonalities. Any number that exceeds the hearing range

of humans can simply be halved so as to produce an octave of the same note.

The first Fibonacci number/circle of tones/circle of 5ths can be this one:

A 13

F 21

Db 34

F# 89

D 144

Bb 233

G 377

Eb 610

B 987

Ab 1597

E 2584

C 4181

75

Twelve notes have emerged, in their triangle formation. These can then be made into twelve

chromatics. Obviously the A13 is too low for the human ear to distinguish. This is where doubling

or halving numbers can be used in order to create twelve audible chromatics. For example, we

can begin at A 208

A 208, Bb 233, B 246.75, C 261.3125, Db 272, D 288, Eb 305,

E 323, F 336, F# 356, G 377, Ab 399.25, A 416

The major scale can be extracted – C D E F G A B C , for example. This can be harmonized

according to the usual rules of music. The next scale of G major can also be extracted, and so on,

completing a circle of 5ths based on Fibonacci/Circle of tones tonality.

It must be mentioned that these frequencies are not reckoned by the A=440 rule, but rather

A=432. This aligns the frequencies with the Lambdoma overtone grid. Even so, regardless of 440

or 432 or any other number, the relationships between the notes will be maintained.

Therefore, any three numbers that can be seen as an octave and a 5th apart, can either be the

beginning of an overtone/harmonic series, or the beginning of a Fibonacci number series. Here

are some more examples – 7 14 21 - 11 22 33 - 27 54 81 etc. As always, the Phi ratio will be

evident in between two adjacent numbers, and as the numbers increase the Phi ratio will

approximate more closely.

76

Chapter eight

Both sides number flows

One will be entitled to wonder why the number 13, say, should mirror to the number 14. Hopefully,

it will be shown that this relationship is quite natural in the evolution of ‘both sides’ number flows.

But it is only the first way a number can be given a mirror number partner. It all begins with the

number 1 having the number 8 as a contrary cycling number partner.

Just because the 1 is partnered with the 8 does not mean it is unbalanced. The 1 is a +1 flow and

the 8 is a -1 flow, according to the indig numbers. In their positions the1/8 are contrary flowing and

they will cycle through asymmetry because they are sympathetic toward each other. What is

holding the balance is the 4.5 invisible axis running in between.

The case for mirror numbers can be seen as valid when one puts the number pairs together and

builds them up progressively.

1

What is the next total that breaks down to a 1?

10 = 1+ 0 = 1

The opposite cycling number partner of the 1 is the number 8. What is the next total that breaks

down to an 8?

17 = 1+7 = 8

Therefore 10 and 17 are mirror numbers. After the number 9 the whole square is born again from

10 to 18 in effect:

The 9's being like junction points give rise to a double sine wave that can be traced through each

set of sequences. There must be something that completes cycles, and that something is the 9,

and a cycle consists of a "both sides" journey.

77

0=9 1=8 2=7 3=6 4=5 5=4 6=3 7=2 8=1

9=0

10=1711=1612=1513=14 14=1315=1216=1117=10

The idea of a mirror number table is really quite logical, and the following one can be called the

default one, because it is the steady parallel progress of the mirror number pairs as they weave

through the fabric of all number.

This table of number pairs can be seen as having a dual flow, clockwise and anti-clockwise. The

numbers coloured red can be seen as the negative numbers beginning their infinite flow in the

opposite direction. They will relate to the positive numbers in the same way as always. For

example, one would need to establish a plus/minus relationship, as in the indig numbers.

And again the four-way mirror relationship is hinted at, because these plus/minus pairs are on

both sides of the mirror. There is the contrary flow this side, and there is the mirror contrary flow

on the other side.

78

Please do not attempt to memorize this list! There is an easier way to know any mirror number.

The rule is, if the first number is less than 4.5, then the mirror number is the next greater number

according to the mirror number pairs of the Vedic Square.

to infinity 3=6 4=5 5=4 6=3 7=2 8=1

axis 0=9 1=8 28=35 55=62 82 = 89 2=7 29=34 56=61 83 = 88 3=6 30=33 57=60 84 = 87 4=5 31=32 58=59 85 = 86 5=4 32=31 59=58 86 = 85 6=3 33=30 60=57 87 = 84 7=2 34=29 61=56 88 = 83 8=1 35=28 62=55 89 = 82 9=0 36=27 63=54 90 = 8110=17 37=44 64=71 91 = 9811=16 38=43 65=70 92 = 9712=15 39=42 66=69 93 = 9613=14 40=41 67=68 94 = 95 14=13 41=40 68=67 95 = 9415=12 42=39 69=66 96 = 9316=11 43=38 70=65 97 = 9217=10 44=37 71=64 98 = 9118=9 45=36 72=63 99 = 9019=26 46=53 73=80 100 =10720=25 47=52 74=79 101 =10621=24 48=51 75=78 102 =10522=23 49=50 76=77 103 =10423=22 50=49 77=76 104 =10324=21 51=48 78=75 105 =10225=20 52=47 79=74 106 =10126=19 53=46 80=73 107 =10027=18 54=45 81=72 108 = 99

109=116 112=113 115=110 110=115 113=112 116=109 etc 111=114 114=111 117=108

79

Here in the numbers one can also see a continual switch over across a mirror point at the “4.5”

positions. It begins in the first strip of numbers in between the 4 and 5. The 4.5 reappears in

between the 13 (4) and 14 (5). The relationship 13/14, is swivelled round and becomes 14/13 at

this point.

The picture will not complete without the mirror side and its structural components. That is what

gives one the focus of the twin cycles and their relationship one with the other. Because all is

number, and all is wavelength, then all evolves out of primary contrary cycling relationships.

Focusing only on the one side of the mirror isn't exposing the full unit and how it expresses itself.

This is shown through musical cycles, and through the basic number cycles that exist. The truth

about these things are held in a simple seed.

One can ascertain the mirror coordinate of anything with a cyclic nature. For example, something

representing 138 in some way, its mirror partner can be ascertained to be at the 141 position and

cycling in a contrary fashion. Groups of numbers also have their mirror coordinates. For example

the points 6 14 4 28 39 7 will become 3 13 5 35 42 2. One could create a mirror universe,

theoretically speaking.

In mimicking these natural flows and building devices that are focused on the dual mirror process,

it may be possible to bring about the natural balance point that unites both sides of the mirror,

which is brought about by the triangles of keys and the circle of tones mirror structure.

All these apparent dualities of cycles exist at some point as merged/united/married. This merging

may bring about a third force, or it may neutralize the forces inherent in the cycle.

It isn't quite enough to ask where along a wavelength is this switching over to the mirror

performed. One has to imagine that there are two mirror wavelengths that are contrary to each

other.

80

Chapter nine

Dorian Binary

This next example shows the perfect symmetry of the Dorian again, but this time using binary

numbers:

The frequency number for each note on the first line of the C Major Mode Box will be represented

as a binary number.

C 132 = 1 0 0 0 0 1 0 0 Db 141 = 1 0 0 0 1 1 0 1 Eb 158 = 1 0 0 1 1 1 1 0 F 176 = 1 0 1 1 0 0 0 0 G 198 = 1 1 0 0 0 1 1 0 Ab 211 = 1 1 0 1 0 0 1 1 Bb 235 = 1 1 1 0 1 0 1 1 C 264 = 1 0 0 0 0 1 0 0 0 D 297 = 1 0 0 1 0 1 0 0 1 E 330 = 1 0 1 0 0 1 0 1 0 F 352 = 1 0 1 1 0 0 0 0 0 G 396 = 1 1 0 0 0 1 1 0 0 A 440 = 1 1 0 1 1 1 0 0 0 A 432 = 1 1 0 1 1 0 0 0 0 B 495 = 1 1 1 1 1 0 0 1 1 C 528 = 1 0 0 0 0 1 0 0 0 0

The C at 264 is the axis. If you look at the binary number for the note D you will notice it is

perfectly symmetrical, like the Dorian mode within the Mode Box. The note D represents the

Dorian in the C major mode box, the 2/2. The note D at 297 cps is the only symmetrical binary

pattern to emerge in the above list. What is equally fascinating is to view the central zero as an

axis point

1 0 0 1 – 0 – 1 0 0 1

1001 is the binary for the number 9. It couldn't describe the Dorian better, as the 0 is also a 4.5.

81

The switch in binary numbers

64 = 1000000 0 = 0 32 = 100000 1 = 1 16 = 10000 10 = 2

8 = 1000 100 = 4 4 = 100 1000 = 8 2 = 10 10000 = 16 1 = 1 100000 = 32 0 = 0 1000000 = 64

The switch is between the 4th and 5th position, another 4.5.

Binary numbers can be associated with the Vedic square. In the Vedic Square the number 1 is in

contrary flow to the number 8. Likewise, this matches with the indig numbers, where the +1 is in

contrary flow to the –1. The table below reflects this logic:

Dual binary pairs

0 = 1001 = 0/9 pair 1 = 1000 = 1/8 = +1 / -1 10 = 111 = 2/7 = +2 / -2 11 = 110 = 3/6 = +3 / -3 100 = 101 = 4/5 = +4 / -4 101 = 100 = 5/4 = -4 /+4 110 = 11 = 6/3 = -3 /+3 111 = 10 = 7/2 = -2 /+2 1000 = 1 = 8/1 = -1 /+1

1001 = 0

1010 = 10001 = 1/8 1011 = 10000 = 2/7 1100 = 1111 = 3/6 1101 = 1110 = 4/5 1110 = 1101 = 5/4 1111 = 1100 = 6/3 10000 = 1011 = 7/2 10001 = 1010 = 8/1 10010 = 1001 = 9/0)

and so on

The 10010 is the binary for the number 18, which breaks down to a 9. The 9 and the 0 reflect

similarly when set as an axis, still maintaining the rule that all number partners either side of the

mirror add up to a 9.

82

Bases and their number sequences:

Taking each possible base, and the positions each number in any particular base occupies, a

simple single digit grid appears.

In base 2 the repeating single digit sequence is 1 2 4 8 7 5

In base 3 it is (2) 3 9 9 9 9 9

In base 4 it is (3) 4 7 1 4 7 1......

In base 5 it is (4) 5 7 8 4 2 1......

In base 6 it is (5) 6 9 9 9 9........

In base 7 it is (6) 7 4 1 7 4 1........

In base 8 it is -(7) 8 1 8 1 8.........

In base 9 it is - (8) 9 9 9..........

In base 10 it is - (9) 1 1 1 1 1.........

in base 11 it is - (1) 2 4 8 7 5 1...........

In base 12 it is - (2) 3 9 9 9 9 9

In base 13 it is - (3) 4 7 1 4 7 1

In base 14 it is - (4) 5 7 8 4 2 1

The grid:

1 2 4 8 7 5 1

(2) 3 9 9 9 9 9

(3) 4 7 1 4 7 1

(4) 5 7 8 4 2 1

(5) 6 9 9 9 9 9

(6) 7 4 1 7 4 1

(7) 8 1 8 1 8 1

(8) 9 9 9 9 9 9

(9)1 1 1 1 1 1

83

Chapter ten

Divisions and the Vedic Square number pairs

It was seen in the Fibonacci number examples how a sequence will swap from one side of the

mirror to the other. This tendency is also prevalent within the division of numbers. There are many

constants that actually hide the flow of these number sequence partners belonging to the Vedic

Square. Here are some examples:

11/29 = 0.37931034482758620689655172413793

This recurring 28-figure constant is dominated by the number 9. Yet after the first fourteen

numbers there is a switch-over across the mirror point, and the fourteen mirror numbers of the

original fourteen emerge. Dissecting the equation in the centre where the swap-over occurs, and

then placing the next fourteen numbers under the first fourteen, can show this. All vertical number

pairs, as always, will be seen to add to 9, and be the correct number pairs from the Vedic Square:

37931034482758

62068965517241

To show the cross-over even more clearly:

37931034482758 62068965517241 62068965517241 37931034482758

And the 4.5 is always the mean number between each vertical number pair. For the cross-over to

occur there has to be a 4.5 somewhere in the space, so it doesn't seem to be the 1 and the 6 that

are responsible for the flip over, but the 4.5 mean number between the adjacent columns.

37931034482758

4.5 ------------------------------------

62068965517241

Why such a consistent mirror imprint of the original fourteen numbers is a mystery. But it is a very

common theme too.

11/19 = 0.57894736842105263157894736842105

84

If we take the first three numbers, 578, we will need to find a 421 in order to suspect that this

constant too is hiding a structure based on the Vedic Square number sequence flows. There is

indeed a 421 in the equation. The pairs as are follows:

578947368

421052631

11/38 = 0.28947368421052631578947368421053

Ignore the first number and then the pairs emerge: 894736842

105263157

The number 19 used as a divisor previously is a prime number. Doubling this 19 doesn’t produce

a prime number, but it still produces mirror flow number pairs of the Vedic Square, after the initial

number 2. Many prime numbers possess these flows.

11/28 = 0.39285714285714285714285714285714

After the initial 39 this sequence is based on numbers that are divisible by 7. Therefore we know

that the 285714 is a movement where the 2 and the 7 are partners, the 8 and the 1, and the 5 and

the 4. These pair together every third number.

285

714

11/14 = 0.78571428571428571428571428571429

Again this is a replication of numbers divisible by 7.

11/35 = 0.31428571428571428571428571428571

This one too shows the same set of numbers, after the initial 3. Again this is because seven

divides into thirty five.

11/13 = 0.846153846153846153846153846153

85

This one shows the mirror flows in the shape of 846 partnering 153 (8 with 1, 4 with 5, and 6 with

3):

846

153

11/26 = 0.42307692307692307692307692307692

After the initial 4 the sequence 230769 is another one hiding a number sequence pair, 2 with 7, 3

with 6 and 0 with 9.

230

769

11/23 = 0.47826086956521739130434782608695

Again the first three numbers are 478, so we look for a 521, which sure enough is in the total, so

the signs are that this constant too contains a mirror number sequence pair. The two sets will be:

47826086956

52173913043

11/17 = 0.64705882352941176470588235294117

The first three numbers are 647, so we look for a 352, which indeed is in the total:

64705882

35294117

11/34 = 0.32352941176470588235294117647058

Here we have to ignore the first two numbers and then the pairs emerge:

35294117

64705882

11/47 = 0.23404255319148936170212765957446

86

Yet again there are mirror flows evident within this constant:

23404255319148936170212

76595744680851063829787

11/49 = 0.22448979591836734693877551020408

Again this will go on to produce number pairs relating to the Vedic square (or indig number system)

224489795918367346938

775510204081632653061

11/52 = 0.21153846153846153846153846153846

Ignore the 21 then:

153

846

There is no grand scientific claim to all this. It is merely interesting that certain fractions show the

swap-over effect that the nine cycles of the Vedic square highlight. Of course one can also find the

same nine single digit contrary flowing cycles in a 9*9 multiplication table. This is in effect what a

Vedic Square is. There is nothing mystical about it!

The indig numbers again can be applied to these constants in order to represent them with a

clearer picture of the opposing flows prevalent within the number pairs. The first division will be

highlighted again in order to show this:

11/29 = 0.37931034482758620689655172413793

37931034482758

62068965517241

This can now be shown as follows:

+3 –2 0 +3 +1 0 +3 +4 +4 -1 +2 -2 -4 -1

-3 +2 0 -3 -1 0 -3 -4 -4 +1 -2 +2 +4 +1

Or as a graph:

87

Dividing by the prime number 17 also gives a constant, which adds up to 72:

1/17= .0588235294117647

2/17= .1176470588235294

3/17= .1764705882352941

Yet, as seen, these constants hide the opposing flows of the Vedic Square mirror cycling number pairs:

05882352

94117647

11764705

88235294

17647058

82352941

These are only the first three examples. The number 19 also behaves similarly:

1/19= .052631578947368421

2/19= .105263157894736842

3/19= .157894736842105263 etc

Again this shows one overall cycle of numbers is emerging, beginning at different points. This

recurring number cycle adds up to 81.

052631578

947368421

88

105263157

894736842

157894736

842105263

The number 23 is another number with a recurring number cycle, which adds up to 99, and a

hidden mirror number sequence flow:

1/23= .0434782608695652173913

2/23= .0869565217391304347826

3/23= .1304347826086956521739

04347826086

95652173913

08695652173

91304347826

13043478260

86956521739

There are endless other examples of this process but these examples should be enough to show

that hidden mirror flow swapping is evident within the division of numbers. This is the process of

things flowing in and out of the mirror at the 4.5 swap-over point, as seen in music scales,

Fibonacci number flows and now divisions.

The line between numerology and natural structure can be mighty fine. The fact that the common

number line has this inbuilt structure may be an open invitation by nature to ponder its mirror side,

including a pathway to the mirror universe! Well I wouldn't want to go.

89

Chapter eleven

Where else?

In this chapter there will be shown small examples of how to achieve the triangle of frequencies.

These triangles of frequency relationships are engrained in fundamental ways , flowing through

various number grids, music scales and overtone structures.

We begin by what can be called number pyramids. How they are built is self explanatory. The

thing to focus on is the totals and the resulting frequency relationships.

1

1 * 1 = 1 = 1 11 * 11 = 121 = 4 111 * 111 = 12321 = 9 1111 * 1111 = 1234321 = 7 11111 * 11111 = 123454321 = 7 111111 * 111111 = 12345654321 = 9 1111111 * 1111111 = 1234567654321 = 4

11111111 * 11111111 = 123456787654321 = 1 111111111* 111111111 = 12345678987654321 = 9

1 + 1 = 2 = 2 11 + 11 = 22 = 4 111 + 111 = 222 = 6 1111 + 1111 = 2222 = 8 11111 + 11111 = 22222 = 1 111111 + 111111 = 222222 = 3 1111111 + 1111111 = 2222222 = 5

11111111 + 11111111 = 22222222 = 7 111111111 + 111111111 = 222222222 = 9

Treating each total as a frequency number we have these notes appear; 1 * 1 = 1, which is the

note C, for example. This assumes the number one as one cycle per second. Then 11*11 = 121 =

the frequency for the note B, and so on.

Multiples pyramid = C, B G Dqt, Bb F# D, A F Db, etc

Additions pyramid = C, F A Db, F A Db, E G# C etc

90

The pattern soon settles down and notes, grouped in triplets, are seen to represent the Triangles

of frequencies that emerged from the Mode Box. After the initial C the multiples are moving by the

interval of a minor 6th, or approximately, and after every three minor 6th movements there is a

descent by a quarter tone. If isolated in threes, these notes all add up to one of the triangles of

frequencies of the mode boxes. The additions are moving by the inversion of the minor 6th, which

is the major third, F to A, A to Db(C#), but after every three moves there the interval ascends by a

quarter tone.

One can build many different number pyramids, and the triangles of frequencies will emerge

throughout. Here is another number pyramid to highlight this:

4 4 * 4 = 16 = 7

44 * 44 = 1936 = 1 444 * 444 = 197136 = 9 4444 * 4444 = 19749136 = 4 44444 * 44444 = 1975269136 = 4 444444 * 444444 = 197530469136 = 9 4444444 * 4444444 = 19753082469136 = 1

44444444 * 44444444 = 1975308602469136 = 7 444444444 * 444444444 = 197530863802469136 = 9

4 + 4 = 8 = 8 44 + 44 = 88 = 7 444 + 444 = 888 = 6 4444 + 4444 = 8888 = 5 44444 + 44444 = 88888 = 4 444444 + 444444 = 888888 = 3 4444444 + 4444444 = 8888888 = 2

44444444 + 44444444 = 88888888 = 1 444444444 + 444444444 = 888888888 = 9

Multiples of 4 = C, B G Dqt , Bb F# D, etc

Additions of 4 = C, F A Db, F A Db, E G# C etc

These pyramids can be extended further and all four triangles of frequencies will emerge. These

four triangles can be seen as two circle of tones structures, or two star of David like symbols.

91

Any set of numbers will produce both the circle of tones structures. The following set of numbers

equal the note's frequency:

1 = C11 = F#111 = A1111 = Db11111 = F 111111 = A 1111111 = Db11111111 = E111111111 = G#1111111111 = C

11111111111 = E 111111111111 = G# 1111111111111 = C 11111111111111 = E 111111111111111 = G# 1111111111111111 = B 11111111111111111 = Eb 111111111111111111 = G

Another example:43.2 = F

432 = A 4320 = Db

43200 = E

432000 = G# 4320000 = C 43200000 = E

etcAnd another: 100 = G

1000 = B10000 = Eb100000 = G

1000000 =B

10000000 = Eb

100000000 = G

1000000000 = Bb10000000000 = D100000000000 = F#

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Within fractions:

Related to divisions by the number 7:

.1 2 4 8 7 5 = C

1 . 2 4 8 7 5 = Eb 1 2 . 4 8 7 5 = G 1 2 4 . 8 7 5 = B 1 2 4 8 . 7 5 = Eb

1 2 4 8 7 . 5 = G

1 2 4 8 7 5 . = B

1 2 4 8 7 5 1 . = Eb

1 2 4 8 7 5 1 2 . = G

1 2 4 8 7 5 1 2 4 . = Bb

1 2 4 8 7 5 1 2 4 8 . = D

1 2 4 8 7 5 1 2 4 8 7 . = F#

1 2 4 8 7 5 1 2 4 8 7 5 . = Bb

Continuing in this fashion will see all four triangles of frequencies emerge.

1 2 4 8 7 5 can also be seen as the recurring sequence through binary, or the sequence that

emerges when numbers are doubled or halved (and the totals reduced to single digit again)

10/17 = 0.5882352941176470

After the number 9 the sequences flow flips over and begins mirroring its previous journey:

58823529

41176470

Inversions within music follow this rule, in that a 5th will invert to a 4th, for example, or the 2nd

inverts the a 7th. And of course these are the mirror numbers as evident within a Vedic Square.

The number 9, as always, is the sum of each mirror pair.

93

The triangle of frequencies emerge if we continually move the decimal point to the right, like this:

0.5882352941176470 = D5.882352941176470 = F#58.82352941176470 = Bb588.2352941176470 = D

5882.352941176470 = F#

58823.52941176470 = Bb

588235.2941176470 = D

5882352.941176470 = F#

58823529.41176470 = Bb

588235294.1176470 = C#5882352941.176470 = F58823529411.76470 = A588235294117.6470 = C#

5882352941176.470 = F

58823529411764.70 = A

588235294117647.0 = C5882352941176470 = E58823529411764705 = G#

94

Phi:

1 . 6 1 8 0 3 3 9 8 8 7 4 9 8 9 4 = G#1 6 . 1 8 0 3 3 9 8 8 7 4 9 8 9 4 = C1 6 1 . 8 0 3 3 9 8 8 7 4 9 8 9 4 = E1 6 1 8 . 0 3 3 9 8 8 7 4 9 8 9 4 = G#

1 6 1 8 0 . 3 3 9 8 8 7 4 9 8 9 4 = B1 6 1 8 0 3 . 3 9 8 8 7 4 9 8 9 4 = Eb1 6 1 8 0 3 3 . 9 8 8 7 4 9 8 9 4 = G

1 6 1 8 0 3 3 9 . 8 8 7 4 9 8 9 4 = B

1 6 1 8 0 3 3 9 8 . 8 7 4 9 8 9 4 = Eb

1 6 1 8 0 3 3 9 8 8 . 7 4 9 8 9 4 = G

1 6 1 8 0 3 3 9 8 8 7 . 4 9 8 9 4 = B

1 6 1 8 0 3 3 9 8 8 7 4 . 9 8 9 4 = D1 6 1 8 0 3 3 9 8 8 7 4 9 . 8 9 4 = F#1 6 1 8 0 3 3 9 8 8 7 4 9 8 . 9 4 = Bb1 6 1 8 0 3 3 9 8 8 7 4 9 8 9 . 4 = D

Here is the grid presented in the chapter “Vedic Square” that shows the sequences created by

every number base system:

1 2 4 8 7 5 1

(2) 3 9 9 9 9 9

(3) 4 7 1 4 7 1

(4) 5 7 8 4 2 1

(5) 6 9 9 9 9 9

(6) 7 4 1 7 4 1

(7) 8 1 8 1 8 1

(8) 9 9 9 9 9 9

(9) 1 1 1 1 1 1

95

Having seen just how these triangles emerge through the whole of number, with the involvement

of the 45 degree angle as well, and the in/out mirror effect, it should be no surprise that the above

list of sequences will also expose them. These triangles signify balance of a dual process. The

sequences shown signify how much can come about from just a small seed like arrangement. If

experiments are the food of science, we can see that a method for creating balance amongst

frequency relationships could be applicable.

96

Chapter twelve

Note to Number Grids

The nine number sequences of the Vedic square can be further tied in with the Major scale by

using number/note grids. The table below consists of musical notes of the C Major scale together

with the nine number sequences of the Vedic square.

NOTE TO NUMBER GRID 1

Here we see how the note D and the number 9 dominate the outer edge of the grid. The D of

course occupies the Dorian position within the major scale. It is also possible to show the grid

without the number 9, as this number is already implied when the symmetrically related number

pairs are added together.

A very simple way to begin merging the number sequences of the Vedic Square in with the C

major scale and its mirror, the C Phrygian mode, is by finding one relationship between the two

that is definite. The number 1 and 8 are numbers that represent octaves, for example, from the

1 C

2 D

3 E

4 F

5 G

6 A

7 B

8 C

9 D

2 D

4 F

6 A

8 C

1 C

3 E

5 G

7 B

9 D

3 E

6 A

9 D

3E

6A

9D

3E

6A

9D

4 F

8 C

3 E

7 B

2 D

6 A

1 C

5 G

9 D

5 G

1 C

6 A

2 D

7 B

3 E

8 C

4 F

9 D

6 A

3 E

9 D

6A

3E

9D

6A

3E

9D

7 B

5 G

3 E

1 C

8 C

6 A

4 F

2 D

9 D

8 C

7 B

6 A

5 G

4 F

3 E

2 D

1 C

9 D

9 D

9 D

9 D

9 D

9 D

9 D

9 D

9 D

9 D

97

note C to the next note C one octave higher. This 1/8 partnership occurs within a major scale and

also between the numbers of the Vedic Square, as you can see in the above grid. This is true for

all number pairs and note pairs. If this logic is applied to the mirror of C major, there is no conflict,

and the two, music and numbers, flow in synch.

The number/note grid number 2, coming next, is a change from grid number 1 in that it now also

reflects the mirror side of C Major, the notes belonging to the C Phrygian mode. You will see the

numbers and notes exist in perfect accord. Also the “flip” is at the 4.5, in between the 4th and 5th

number sequence. Here again is the scale of C major and its mirror:

(C Phrygian) C Db Eb F G Ab Bb C D E F G A B C (C Ionian)

NUMBER/NOTE GRID 2

1 C

2 D

3 E

4 F

5 G

6 A

7 B

8 C

9 D

2 D

4 F

6 A

8 C

1 C

3 E

5 G

7 B

9 D

3 E

6 A

9 D

3E

6A

9D

3E

6A

9D

4 F

8 C

3 E

7 B

2 D

6 A

1 C

5 G

9 D

5 G

1 C

6 Ab

2 Db

7 Bb

3 Eb

8 C

4 F

9 Bb

6 Ab

3 Eb

9 Bb

3Ab

6Eb

9Bb

3Ab

6Eb

9Bb

7 Bb

5 G

3 Eb

1 C

8 C

6 Ab

4 F

2 Db

9 Bb

8 C

7 Bb

6 Ab

5 G

4 F

3 Eb

2 Db

1 C

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

9 Bb

98

It would be easiest to relate the number pairs vertically, as that will show, for example, how the D

(2) mirrors to the Bb (7). As seen the D/Bb are mirror pairs around the axis of the C major scale

and its mirror. This also applies to the F/G, A/Eb, B/Db and Ab/E. Basically, what has happened is

that between the 4th and 5th number sequence there has been a swap-over to the mirror scale

notes belonging to the scale of C Phrygian.

There are two mirroring processes in one here. There is the number sequences which mirror:

1 +8 2 + 7 3 + 6 4 + 5 5 + 4 6 + 3 7 + 2 8 + 1 9 + 9

And there are the symmetrical note pairs:

C (1) + C(8), D (2) + Bb (7), E (3) + Ab (6), F (4) + G (5),

G (5) + F (4), A (6) + Eb (3), B (7) + Db (2), D (9) + Bb (9).

As you can see the number 9 sequence also mirrors between D and Bb, the two Dorian Mode

positions. Even though the number nine plays no initial role within the Major scale formula it does

play an important hidden or uninvolved role. All symmetrical possibilities, as always, add up to 9.

Yet if there were seven separate mode boxes drawn for the one major scale, each mode box

beginning on any one of the seven modes, one would find that the quality of each aspect of the

circle of tones structure to be continually shifting between major and minor identity. For example,

begin a mode box with the Dorian/Dorian 2/2 as the first position. Then build another mode box

with the E Phrygian/Ionian 3/1 as the first pair. The 45 degree angle, which expresses the two

triangles of keys/circle of tones will show major tonality and minor tonality, depending on the

beginning position of the Mode box.

If one embrace the idea of mirroring formulas in order to observe the unified aspect of that specific

duality they will no doubt find many examples leading to what was once a CEG# triangle of keys in

the position of major, to being in the totally opposite position of minor, as each particular example

is cycled, through the seven modes. Also, one should bear in mind that both triangles are major

keys at one point, and are all Dorian at another point. They are all major when one sees them

existing within the unit of a Mode Box. They are all Minor when it is realized that they are all Dorian

positions of each other:

99

C major has D Dorian

D major has E Dorian

E major has F# Dorian

F# major has Ab Dorian

Ab major has Bb Dorian

Bb major has C Dorian

These flows are two circles of tones.

To give a better feel for the positive and negative number positions according to the notes of the C

major and its mirror, here is the number note grid represented by Buckminster-Fuller’s indig

number system:

NUMBER/NOTE GRID 2a

Notice in the first number sequence that the F#/Gb poles would be tucked in between the +4 and

the –4 (this is between the F and the G). This grid can be associated with any type of mode box,

and gives an alternative view of where the symmetry is broken. But is symmetry only some linear

aspect here? If we thought of each individual number pair as two blinking lights then one would

+1 C

+2 D

+3 E

+4 F

-4 G

-3 A

-2 B

-1 C

0 D

+2 D

+4 F

-3 A

-1 C

+1 C

+3 E

-4 G

-2 B

0 D

+3 E

-3 A

0 D

+4 F

-1 C

+3 E

-2 B

+2 D

-3 A

+1 C

-4 G

0 D

-4 G

+1 C

-3 Ab

+2 Db

-2 Bb

+3 Eb

-1 C

+4 F

0 Bb

-3 Ab

+3 Eb

0 Bb

-2 Bb

-4 G

+3 Eb

+1 C

-1 C

-3 Ab

+4 F

+2 Db

0 Bb

-1 C

-2 Bb

-3 Ab

-4 G

+4 F

+3 Eb

+2 Db

+1 C

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

0 Bb

100

find that, as the +1 blinked then so would the -1, regardless of its vicinity. This would be a kind of

quantum entanglement. Does the mirror partner even blink within “this side’s” matter/anti-matter

make-up? Where exactly is this “mirror” world?

101

Chapter thirteen

The Dorian connection

In this chapter we will come to see the perfect symmetry held around the Dorian Mode axis, and

how it is a secret catalyst to the creation of the major and minor scale system, as used in the

West.

It will be shown how the Dorian mode, the second mode of the major scale, becomes ‘pregnant’

with the sharps and flats, and gives birth to the key signatures symmetrically around its axis. I

realize that sounds rather “out there”, but it is justified in terms of what is to follow.

It is true that the circle of 5ths, as we generally use it today, establishes keys whose contents of

sharps and flats increase by one at a time, rising to a maximum of six sharps and six flats. Below

is a circle of 5ths, showing how many sharps or flats are in any particular Key Signature.

Major keys in a circle of 5ths

C

(b) F G (#)

(bb) Bb D (##)

(bbb) Eb A (###)

(bbbb) Ab E (####)

(bbbbb) Db B (#####)

Gb/F# (bbbbbb/######)

Sharps and Flats

0 1 2 3 4 5 6

C - G D A E B F# - sharps

C- F Bb Eb Ab Db Gb - flats

102

Interestingly, one can run the F#/Gb notes along the center in-between the two music staves. This

shows how C and F# reflect the same mirror note positions. More importantly, if you follow the

arrows, you will see that in reaching the F#/Gb axis the process happens by contrary motion. One

would intuit that it is a changing from clockwise to anti-clockwise flow.

Notice something rather uncanny here, as we plot the scale of C major and its mirror again:

C Db Eb F G Ab Bb C/C D E F G A B C

The note pairs around the C axis are the same as the major keys that have the same number of

sharps and flats within their signature. D on one side, for example, is in reflection with Bb on the

other side. The D major key has two sharps and the Bb major key has two flats. Again E/Ab are

mirror note partners. The E major key has four sharps, and the Ab major key has four flats. And so

on through the mirror pairs. This consistency is rather uncanny, and it will be seen that the idea of

a “Dorian tonal pregnancy” is not as far fetched as first may seem, and is actually empirically

based.

We have already seen how the Dorian dissects the 4.5, at the hidden F# position either side of the

mirror. Does the D Dorian mode hold any symmetrical secrets of its own? You will see that each

new flat and sharp needed to build a new major key is born in symmetry around the D Dorian axis.

When building major scales, each new major scale in turn has only one note that is different to the

proceeding major scale. The only difference in notes between C major and G major, for example,

is the note F#. C major is the right hand scale as shown above. Here is G major:

G A B C D E F# G = G Major

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F#/Gb---------------------------------------------------------------------------------

Think of this major scale as any doh-reh-meh-fah-sol-lah-the-doh sounding scale. The only note

that is different to the C major scale's notes is the F# in place of the natural F.

The Dorian Mode takes into account both sides of the Duality and deals with both types of cycles,

clockwise and anti-clockwise. It is at the Dorian where the journey begins into more and more

subtle structures and inner mirror patterns.

All twelve major keys have a special link with the D Dorian Mode.

The F# and Gb major keys are at full expansion and full contraction in the circle of 5ths. The F#

major scale contains six sharps, and the Gb major scale contains six flats. These two places are

known as the poles within music, and they will be at 180% to C major, the key with no sharps or

flats. As each new key progresses and gains more sharps and flats, this distribution is reflected

around the Dorian axis point, as will be seen. This type of symmetry happens nowhere else but

within the D Dorian Mode.

The Dorian Mode which belongs to the scale of C Major possess no sharps or flats, in other

words, it is all the white notes of the piano beginning at D and ending on D an octave up or down

from that point.

T S T T T S T D E F G A B C D = D Dorian

The intervals that make up this Mode (the Tones and Semi-tones) are then 'mirrored'. The note D

is set as an Axis point and the notes are written as mirror partners around this axis.

T S T T T S T T S T T T S T 2 - D E F G A B C D E F G A B C D - 2

The red D is the axis point. See the formula on the right of D reverse itself to the left of D. The

same mode emerges either side of the axis point, but the note partners will be different. In effect,

this is the second line of the C Major Mode box.

Mirror note partners are D/D, E/C, F/B, G/A, A/G, B/F, C/E, which are visible mirror pairs, and

D#/Db, Gb/A#, G#/Ab, A#/Gb, C#/Eb, which are the 'in between' mirror note pairs. Some of the

in-between note partners are shown in parenthesis next, so you will see how they are in symmetry

around the D axis:

104

center center

D E F G (Ab) A (A#) B C (Db) D (D#) E F (Gb) G (G#) A B C D

Now we focus on each increase of sharp or flat in Key signature that is connected to one of the

two poles, F# and Gb.

G Major contains one sharp and F Major has one flat:

G A B C D E F# G

F G A Bb C D E F

The sharp and the flat appear at the F and the B notes respectively. These two notes are mirror

partners around the Dorian Mode. You can add the sharp and the flat and see that they mirror

perfectly around the D axis point. Here is the D Dorian Mode and its mirror partner again:

b* #* D E F G A B C D E F G A B C D

One note is a sharp one side of the mirror and the other note is a flat the other side of the mirror.

The F has ascended by a semitone and the Bb is the result of the B note descending by a

semitone.

The Key of D major contains two sharps and the Key of Bb major contains two flats:

D E F# G A B C# D

Bb C D Eb F G A Bb

The sharp and flat notes are F and C, and B and E respectively. We know that F and B are mirror

partners; so are C and E.

b* # b #* D E F G A B C D E F G A B C D

This is the pattern as we add more sharps and more flats. The altered notes (the flats and sharps)

are always mirror partners in the perfectly symmetrical Dorian Mode. There is also an in and out of

the mirror process above, as the roles between flat and sharp reverse.

The Major scale of A has three sharps and the scale of Eb has three flats.

105

A B C# D E F# G# A

Eb F G Ab Bb C D Eb

The new altered notes are Ab and G#. Again these are reflected around the axis note D. This is

also halfway point, and the very center of the D Dorian Mode, which is like an axis point too.

G#/Ab is the same when symmetrically reflected around the D axis.

b* # b #* D E F G (Ab) A B C D E F G (G#) A B C D

The role of the sharp and flat have reversed again.

E Major has four sharps and Ab major has four flats:

E F# G# A B C# D# E

Ab Bb C Db Eb F G Ab

The new altered notes are Db and D#. This is again the result within the D Dorian mode too where

D# mirrors to Db.

b* # b #* D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

Two more sharps and flats and we will arrive at the poles of F# and Gb.

B Major has five sharps and Db major five flats:

B C# D# E F# G# A# B

Db Eb F Gb Ab Bb C Db

The new altered notes are the Gb and the A#. These two notes are also mirror pairs around the D

Dorian Mode.

# b* # b #* b D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

106

Lastly F# Major has six sharps and Gb Major has six flats:

F# G# A# B C# D# E# F#

Gb Ab Bb Cb Db Eb F Gb

The new altered notes are E# and Cb. This last result too is similar to that within the D Dorian

Mode and its mirror partner.

# # b* # b #* b b D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

These are the way the sharps and flats progress. The note C is the beginning of the journey but

has no sharps or flats. Every addition of a sharp and a flat within the key signatures, which

develop toward the musical poles, is seen to reflect perfectly around the D Dorian mode axis.

To verify the growth of sharps and flats there is then a second witness, the Major scale of C itself,

as seen earlier. This is the home key which gives rise to the D Dorian Mode.

Look at the C Major scale, its mirror, and the note pairs either side of the axis, together with the

amount of sharps and flats required to build the major scale from each root note:

0 5 3 1 1 4 2 0 2 4 1 1 3 5 0 C Db Eb F G Ab Bb C D E F G A B C

Gb (6) F# (6)

Mirror partners - C/C D/Bb E/Ab F/G A/Eb B/Db. The center of both scales being F#/Gb means

that this is another axis point. It is interesting that the two poles live at the center of this natural of

Major scales. The results are akin to a seed set in symmetry in order for the duality to contain the

potential of self unity, or a kind of neutral state, that will swap over at the poles and then continue

infinitely to higher or lower circles of twelve keys? If we were to create the next twelve cycles of

keys, this indeed would be the evident structure.

This next diagram shows all the above information in a graphical way and may be easier to

imagine when it comes to the wonderful symmetry that Nature displays around the Dorian Mode.

107

The Dorian Distribution of Sharps and Flats

An example is the C# note pointing rightwards with its arrow to the D note. This means that C# is

the new sharp required to construct the D major scale, and it is born in symmetry as the Eb note

points itself toward the left, in order to construct the Bb Major scale. The number 2 that joins them

up signifies the amounts of flats or sharps within their respective key signatures. The major scales

are reckoned vertically at these points, with Db and B major being at opposite ends. The notes

colored blue are not usually within either the C major (on the right), or C Phrygian (on the left)

scales.

There was a time when major tonality was called masculine and minor tonality was called

feminine. If C were in the masculine role and D the feminine, it would also explain why D is the

note 'pregnant' with the six sharps and six flats. The C major scale itself displays the roots of all

twelve major scales. The story being told when the key of C is mirrored is that of symmetrically

linked major scale Key signatures either side of its axis point.

In the D Dorian mode is the story of how the sharps and flats are born and distributed amongst the

two poles (of six flats and six sharps) at opposite ends, and in doing so the journey creates

naturally occurring Keys around a circle of 5ths. This creation is a working system that complies

108

with the law of symmetry, and it is done within the Home key of 'home' Keys, C Major. The

distribution within the symmetrical Mode of D Dorian is musical expansion and contraction

reaching to the two poles, where they swap over.

The Dorian also allows the sharps and flats to journey in and out of the mirror. Here are the sharps

and flats as they evolved through the keys:

Bb F#

C# Eb

Ab G#

D# Db

Gb A#

E# Cb

The second flat (Eb) is born on the other side of the mirror, as is C#, and this continues

throughout.

Another way of seeing the Dorian's involvement in the construction of the triangle of keys is like

this:

Dorian’s pregnancy

b5

C D E F G A B C D E F G A B C D

b5 E F# G# A B C# D# E F# G# A B

b5 Ab Bb C Db Eb F G Ab Bb

C D E F G A B C D

The first line above is the scale of C Major, the second line that of E Major, and the third line that

of Ab major. The red broken arrows point to three Dorian Modes, F# , Bb and D. They are also the

central points of all three scales of the triangle of keys (the tri-tone/flat 5th position).

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Chapter fourteen

Shades of Dark and Light

Every music scale is different. For every scale there is a blending of colours, each contributing to

one overall flavor that contributes to the overall sound of the scale. Some scales take a trained

ear in order to distinguish them apart but other scales are easily recognized by the masses. In the

western culture the major scale and the relative minor scale are heard within almost all the music

we listen to. One can usually tell the difference between a bright sounding scale and a dark

sounding scale. Some scales sound very “country” and with a happy “major” sounding feel whilst

other scales sound mysterious and mellow, which is the trait of the minor scale. So what gives a

scale this quality, where each shade of colour held within the scale induces an appropriate

response within the listener, and produces such powerful emotions as to make an audience weep

unashamedly, or a person alone in their room reflect and relive memories that the music is stirring

within, as if it somehow contained the power to make this happen? The answer lies within the

sometimes over-feared field of music theory and is based on the scale’s make-up, that is, its

formula. The formula, or set of ratios based on a chosen temperament, provides the “colouring”

aspect of tonality.

Imagine a bow striking a low note on a violin. Then the player moves the first finger of his left hand

down the violin string, and the note is heard rising higher and higher in pitch (until it sounds like a

cat squeal!). The player can simply slide his finger up and down the string and will always produce

a note. This is true for bicycle pump playing as well, and highlights the fact that Nature’s Sound is

a continuous tone that is there to be sliced up to our liking. There are no gaps between pitches

within natural sound. This slicing up technique is what is meant by a scale’s formula. Start off with

a low tone and then, instead of sliding with the finger, leave definite gap between each step that is

played. These steps then are like musical slices of the continuous tone. Each step is given a letter

between A and G. There is a numbering system too that is used to describe each step of the

overall formula.

Continuous tone rising and falling

Above is continuous pitch traveling higher and higher, or lower and lower. Below, the sound is

sliced up into a scale.

C D E F G A B C

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A chosen set of ratios will define each slice, and at that point the pitch is then determined. The

brightness and darkness of a scale depend on the arrangement of these steps. Almost everyone

can hum the doh reh meh fah sol lah teh doh scale. Each of these seven steps that exist within

this scale is predetermined and not accidental. Each step is based on a leap upwards or

downwards in pitch by a certain ratio.

Slicing the continuous tone creates steps and in turn steps create formulas, that is, collections of

steps. As soon as one understands this they will no doubt imagine how many different sounding

scales there are. Musical scales are not made up of exactly the same arrangements of steps and

so will sound different to each other, and also give rise to the different effects felt by the listener.

It is the formula that reigns within the building of scales and not necessarily the actual pitches of

the notes themselves. The formula is the provider of colour, and therefore the scale’s place on the

musical light/dark spectrum.

The modal system within the major scale can be known by its individual light or dark tonal

qualities. Here are the different shades relative to the Modes of any major scale, beginning with

the brightest sounding mode:

Lydian brightest

Ionian

Mixolydian

Dorian axis

Aeolian

Phrygian

Locrian darkest

It should be no surprise that the perfectly symmetrical mode, the Dorian, contains the axis point

between light and dark tonality. In this example we will start with what is regarded as the darkest

mode of a Major scale, the Locrian. This mode actually mirrors to the Lydian mode, which is

regarded as the brightest mode of the Major scale. The Lydian mode is the brightest because of

the formula it is built on. There is what musicians call a sharp 4th within its make-up so it is even

brighter than the Major scale (Ionian Mode) itself. This is because the Major scale contains what is

called a Perfect 4th, which is one semi-tone lower than a sharp 4th, and it is that extra raising of one

semitone that has made the Lydian Mode even brighter than the Major scale (you will see this in

the next diagram where the “#” shows expansion/light, and the “b” shows contraction/dark). What

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makes the Modes darker in quality within this major scale is the addition of flatted notes (shown as

a “b” sign). The last Mode in the diagram below contains a b2 b3 4 b5 b6 b7 within its formula and

one can see why it is the darkest Mode. You can see this shift from light to dark if we show the

formulas required to make up each mode.

To establish such a table one must use the same root note for every one of the seven formulas,

for that is how the shades of colours can be seen to be flowing through the seven modes. This

next list uses the note C as the root of all seven modes:

1 2 3 #4 5 6 7 8 (C Lydian mode formula) - Brightest

1 2 3 4 5 6 7 8 (C Ionian mode formula)

1 2 3 4 5 6 b7 8 (C Mixolydian)

1 2 b3 4 5 6 b7 8 (C Dorian) Axis

1 2 b3 4 5 b6 b7 8 (C Aeolian)

1 b2 b3 4 5 b6 b7 8 (C Phrygian)

1 b2 b3 4 b5 b6 b7 8 (C Locrian) Darkest

The Ionian mode is the very root of the parent major scale. The scale above it, the Lydian has its

raised 4th and is the brightest because every scale within the structure of the seven modes is

darker in quality than the Lydian mode. The Mixolydian contains one flat, and the Dorian contains

two etc, getting progressively darker sounding tonally speaking.

To really explain this one would again need to go back to music basics. I do realize all this may

sound rather confusing for non-musicians, and study will be required in order to understand why

these particular formulas mean what they do. When the above list is represented on the music

stave, as shown shortly, it should become clearer. Basically , these C based scales in the above

list are getting darker and darker sounding, started from the Lydian mode, which is the brightest.

The bottom mode in the above diagram, the Locrian, contains five flats. Therefore it resides within

a major scale that contains these five flats as part of its Key signature. We know that Db major is

that major scale. The C Locrian will be the seventh mode of that major scale, because the Locrian

mode is always the seventh mode. Here is the Db major scale, with its modal positions written

above the notes:

Ion Dor Phr Lyd Mix Aeo Loc Ion

Db Eb F Gb Ab Bb C Db = Db Major

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Therefore the previous list based on modal colour is still related to the circle of 5ths, as the major

scales evolve, and the shades of light and dark that it creates are a natural progression, with the

Dorian mode position being its central axis point.

Here is another way of seeing the above information. All the modes are generated from the C root

position (C Ionian, C Dorian, C Phrygian, C Lydian etc):

It is no surprise now that the Dorian Mode is the Constant (symmetrically reflects as Dorian both

sides of the mirror) as this is where light and dark meet, and it will be directly over the F# point

within the C Dorian Mode. It affirms the Dorian as having an axis quality of its own as seen

previously. The tri-tone position demands it and it demands it at the dissection of 9 into two equal

halves either side of the mirror, at the 4.5 positions. The Dorian can gain access to the mirror side,

through its perfect symmetry.

Also you will find that the above Modes, all built on the root note C, belong to the following Major

scales:

Db Ab Eb Bb F C G

The first mode of C Locrian, for example, resides within the Db Major scale, the C Phrygian

resides within the Ab Major scale, the C Aeolian resides within the Eb major scale, and so on. If

we put these Major scales in sequence starting from the note C we have this list of scales:

C Db Eb F G Ab Bb

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And these are the notes required to build the C Phrygian mode, which is the mirror scale of C

Major, as shown in the mode box. This C Phrygian Mode brings us back to the very first step of

the mirror process, a cycle of light to dark and dark to light complete, perhaps only spiraling faster

and faster and slower and slower.

This next diagram takes the C major modes through a circle of 5ths beginning with the brightest

mode, F Lydian. These shades are then mirrored, and in effect, this is a mode box arranged in the

right sequence of light and dark, that is, through the cycle of 5ths. If you have made it this far into

the book you should probably know how to mirror scales by now.

Mode Box in shades of light to dark through cycle of 5ths

7 LOC F Gb Ab Bb Cb Db Eb F G A B C D E F LYD 4

3 PHR C Db Eb F G Ab Bb C D E F G A B C ION 1

6 AEO G A Bb C D Eb F G A B C D E F G MIX 5

2 DOR D E F G . A B C D E F G A B C D DOR 2

5 MIX A B C# D E F# G A B C D E F G A AEO 6

1 ION E F# G# A B C# D# E F G A B C D E PHR 3

4 LYD B C# D# E# F# G# A# B C D E F G A B LOC 7

7 LOC F Gb Ab Bb Cb Db Eb F G A B C D E F LYD 4

The right hand side of this mode box shows the progressive shades of colour, with F Lydian (4)

being the brightest sounding scale, then the C Ionian being the next brightest, all the way down to

the B Locrian (7), which is the darkest sounding scale. This then correlates to an opposite journey

on the mirror side. The thing to look for here is that the top left scale starts with maximum

contraction, in the form of six flats, at F Locrian (7). The next line has only four flats, and the line

after that has two flats, until the Dorian, which is where light and dark meet, balances out and

switches over the contraction for expansion. The journey continues with two sharps, four sharps

and six sharps. At this point, if the cycle is repeated, we see a point where maximum contraction

becomes maximum expansion, between B and F. This B to F is the visible tri-tone of the C major

scale.

This diagram says a lot more than at first seems to be the case. Here we have tonality distributing

expansion and contraction as the circle of 5ths evolves. There is no faltering, or creation of

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gibberish on the mirror side, but actually a well laid out structure, showing how the Dorian is

perfect point of symmetry, and swap-over axis. As it has been shown that the 4.5 is also the swap-

over axis, then it should be quite clear that there is a Dorian aspect to the 4.5, as seen in the

chapter “the invisible aspect of the triangle of keys””. The Dorian and the 4.5 is where light and

dark have their axis, and the swapping over of qualities is performed. Swapping from major type to

minor type, from expansion to contraction, from tonal light to tonal dark. Positive to negative, as in

Buckminster-Fuller's indig number (where -4, for example, switches to +4, as in full contraction

becoming full expansion), clockwise to anti-clockwise. The Dorian/4.5 is where nature has a unity,

between both sides of the mirror. It is perfectly symmetrical at this point, and one can imagine it

having the ability to witness the both sides of duality.

Here is a simple chart showing the evolution of light/dark within the above left hand side of the

mode box in true shades:

bbbbbb

bbbb

bb

DORIAN

##

####

######

bbbbbb

bbbb

bb

DORIAN

##

####

######

This compares to the way the light/dark quality of the indig numbers expand and contract across

the 4.5 axis point, shown in the “Vedic square” chapter, where it is seen how maximum expansion

immediately becomes maximum contraction at the swap-over point, and the 4.5 is in effect the

zero point. Here it is shown in music and number fashion, as nature would inevitably leave its

imprint on such structures. It is a map describing its own inner process.

It just so happens that F Lydian , the brightest mode, mirrors to F Locrian, the darkest. They exist

at the 7th and 4th position, or the 4th and 7th position of their respective major scales. The

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light/dark relationship between the two F's is echoed in a similar light/dark relationship through the

two B's, also a 7/4.

4 F Lydian/Locrian 7

7 B Locrian/Lydian 4

dark light

F Locrian - F Gb Ab Bb Cb Db Eb F G A B C D E F - F Lydian

light dark

B Lydian - B C# D# E# F# G# A# B C D E F G A B - B Locrian

One journeys in through the F and comes out through the B. Both B/F and F/B are the visible tri-

tone positions of the C Major scale and its mirror. Notice that F Locrian and B Lydian are in effect

the same scale.

Gb = F#

Ab = G#

Bb = A#

Cb = B

Db = C#

Eb = D#

F = E#

In one manifestation the scale is full expansion, and in the other manifestation it is full contraction.

These two scales are modes and therefore they each belong to a parent major scale. The scales

that these two modes belong to are F# and Gb Major, the two poles. This is what ties in the visible

tri-tone aspect with the invisible. C to F# is the invisible tri-tone axis relationship, and F to B is the

visible tritone axis relationship. The F# is not visible in the C major scale, nor the mirror of the C

major scale. The F and B are visible in that they are two notes that belong to C Major and are a tri-

tone apart (F - G -A - B = three tones). When thinking of a pole shift, we can see how it is possible

musically.

Shades from the Center

Again, there is a hidden structure behind the distribution of these shades of tonal colors, and it is

emerging from the center of each formula. Observe this diagram:

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Darkest mode

Phr. Lyd 4

7 Loc.

S T T S T T T

Brightest mode

The formula from left to right is that of the Locrian Mode (STTSTTT). Reversing the flow of the

formula gives us the Lydian mode (TTTSTTS). So the brightest sounding mirrors to the darkest.

Here is a Lydian scale starting on the note C

T T T S T T S

C D E F# G A B C

This is the brightest mode of the major scale family. Now if we use this formula in reverse, the

result is the C Locrian mode, the darkest.

S T T S T T T

C Db Eb F Gb Ab Bb C

The flows meet in the middle. Commencing the next scale from the central semi-tone in the

formula above would create the Phrygian mode (S T T T + S T T). If we continue our experiment

using the Phrygian mode as our new starting point we will see that it is the next darkest mode after

the Locrian that emerges, and it is paired with the next less brightest mode. And in the middle

waiting to start the next formula is the Aeolian mode, which is the correct shade that develops

next.

Aeo 3 Phr. Ion. 1

S T T T S T T

The Phrygian mode is one shade lighter than the Locrian mode, and the Ionian mode is one shade

darker than the Lydian mode. As you can see the two respective modes flow left to right and right

to left, creating the respective shade; S T T T S T T creates the Phrygian mode, and in reverse the

T T S T T T S creates the Ionian mode. Therefore this is simply the Ionian/Phrygian relationship

we have uncovered before. The formulas meet in the middle, and if we make this Aeolian point

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(built using TSTT+STT) the new starting point of the next scale we get even brighter on one side

whilst getting darker on the other. The Aeolian will mirror to the Mixolydian mode.

Dor

6 Aeo Mix 5

T S T T S T T

As well as the dual looking formula, which creates shades of tonal colour, the next mode is waiting

in the center in order to evolve the next pair of shades between light and dark tonality. Let’s

continue in this vein with the other modal partners. Dorian is the next starting point in the dual

journeys:

2 Dor Mix Dor 2

T S T T T S T

5 Mix Ion Aeo 6

T T S T T S T

1 Ion Lyd Phr 3

T T S T T T S

4 Lyd Loc Loc 7

T T T S T T S

So, the Modal structure of the Major scale is seen as evolving in sequential shades of brightness

or darkness from a central point within each formula, which can be read either left to right, or right

to left. The above information is related to the Mode Boxes in a subtle way. The modal partners

are consistent with those found in the C Major Mode Box, like Ionian/Phrygian, Dorian/Dorian etc,

and if anything we are seeing how the inner qualities given to the different shades of the Modes

springs from a hidden axis at the central tri-tone position. This center, 4.5 position, is in effect a

Tonal Fountain.

The last example in the above list throws up a rather interesting oddity. The Locrian mode is

mentioned twice. It’s as if the journey is about to commence again through the mirror at this point,

with the Locrian mode waiting in the middle to start the process all over again. The Locrian Mode

on the mirror side of the C Major Mode Box is that of F Locrian which resides within the Key of

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Gb Major. This interplay between the invisible axis and visible axis, swapped over at the tri-tone

position, will be seen in the chapter “In and Out of the Mirror”. It is the place where things are

made manifest on the opposite side of the mirror. The way in which these relationships flow is not

necessarily confined to musical scales. One must surely ask in what other fields of study viewing

information as swapping over from mirror/non-manifest to non-mirror/manifest may be of benefit.

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Chapter fifteen

A Tonal Fountain

Having seen the power of the Dorian Mode, with its ability to symmetrical birth the sharps and flats

of the circle of fifths, it brings us now to another hidden Dorian aspect. The F#, 4.5 connection, is

indeed a Dorian Mode connection. As shown in the chapter “The Invisible aspect of the Triangle”, it

is the F# Dorian mode that dissects the C major and C Phyrygian scales at the 4.5 positions either

side of the mirror point.

The invisible axis is at the 4.5 position, and it replicates the relationships around the visible axis,

which is C/C. As it occupies a position that is not part of the two cycles/scales involved, it gives it a

quality of being the Tonal Fountain, where the contrary cycles first emerge in mirror pairs and

journey toward the visible tritone (F# to C), swap-over and journey toward the next invisible tritone

and so on.

At the F# 4.5 axis point the dual mirror note and mode partners are in perfect symmetry, as will be

shown:

4.5 4.5

C Db Eb F (f#) G Ab Bb C/C D E F (f#) G A B C

The F, descending by a semitone from the f# on the left, mirrors to the G ascending by a semitone

from the f# on the right. This G/F mirror pair also happens around the C axis (F/G is bolded).

After the F/G, the next note on the left from f# is the descending movement of one tone to the Eb,

which is mirrored by an ascending movement of one tone to the A on the right. Again A/Eb are

mirror pairs around the C axis. There is a visible/invisible axis partnership occurring here, between

the C axis and the F# axis, and it is the 4.5 tri-tone position that creates it.

Db will mirror to B, around the two f# positions, and one returns to C/C again. The rest of the

symmetrical relationships ascend from the f# on the left, and descend from the f# on the right.

Here is the sub-section in order to show this:

4.5 4.5

(f#) G Ab Bb C/C D E F (f#)

One can begin with the F to the left of the f# on the right hand side. This symmetrically pairs up

with the G, to the right of the f# on the left hand side. The rest of the dual mirror relationships are

E/Ab, D/Bb and C/C again. Again, these are exactly the same pairs as around the C/C axis.120

As has been seen, f# mirrors to f# on the other side, remembering also that the note Gb is also

included in this as the other pole. The importance of the F#/Gb poles, as reference to full tonal

expansion and contraction will be dealt with further on.

The symmetrically related partnerships emerge from this f# central point. In order for F# to remain

F# across the mirror point it can only be identified as the modal relationship of Dorian/Dorian. This

Dor/Dor, as mentioned, is not to be confused with the usual Dor/Dor along the horizontal line (that

will be the D/Bb Dorian mode partners). Together with the D/Bb Dorian mode partners, this F#

Dorian completes the second triangle of the circle of tones structure, in the form of D Bb F#.

The next series of examples is the same information again, and will hopefully make things as clear

as possible. The 'm' and the 'f' stand for masculine and feminine respectively. It’s not so popular

now to equate minor and major tonality with either a feminine quality or a masculine one, but

nevertheless these qualities have been included in order to show how feminine always mirrors to

its masculine mirror partner and vice versa:

The above is a picture of a flash animation. It represents the two F# positions either side of the C

major scale and its mirror. These are the two 4.5 positions.

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As the note F emerges on the left and descending, the G emerges from the right ascending. The

modal relationships are the Aeolian/Mixolyfdian (6/5), and their quality is feminine(f)/masculine(m).

Next, the Eb emerges from the left, as the A emerges from the right. This represents the other

Mixolydian/Aeolian partnership, and also the feminine/masculine quality has been swapped over.

Here are the others, which show that F# can be considered the tonal fountain, in relation to the

symmetry it holds around which are all the mirror note and mode pairs in their root positions.

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123

The Dorian is the marriage point of the masculine/feminine.

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One can see, for example, the Ion/Phr pair (red arrows). The Ionian will be on the left hand side of

the mirror, at Ab, and the Phrygian will be on the right hand side, at E. Both will be in symmetry

around the F# axis at the 4.5 positions. Then when the modal pair swaps over to a Phr/Ion there

will be the C Phrygian one side of the mirror mirroring to the C Ionian on the other side. This

accounts for the Ab/E and C/C positions.

The Lyd/Loc pair is another example. This will equate to the notes and modal positions of B/Db

(Lyd/Loc) and G/F (Loc/Lyd)

If we mirror the scale of F# Major another intriguing result emerges:

S T T T S T T T T S T T T S

Phr - F# G A B C# D E F# G# A# B C# D# E# F# - Ion

As always, the major scale (Ionian), mirrors to a Phrygian mode. Here you have similar note and

modal pairs, as is the case when C is the axis, except that enharmonic pitches are in use. When C

major is mirrored the note partners either side are:

C/C D/Bb E/Ab F/G (F#/F#) G/F A/Eb B/Db C/C

When F# major is mirrored the results are similar:

F#/F# G#/E A#/D B/C# (C/C) C#/B D#/A E#/G F#/F# (Ab) (Bb) (Db) (Eb) (F)

D/Bb and A#/D are in effect the same combination, with only the difference of a sharp replacing a

flat. Perhaps it signifies musical expansion turning to contraction? The relationships have swapped

over the mirror point.

Again F/G and E#/G are the same combination within Equal Temperament, the E# also being the

note F. In this combination we see the role that Key signatures play within this system of things. F

has one flat within its key signature, whereas E# has eleven sharps within its key signature. This is

really a story of a pole shift and it will be shown in another section when we deal with Major scales

possessing a greater number sharps or flats than the six held at the F#/Gb poles (see the chapter

144 major scale grid).

What is clear here is that flats are swapping over for sharps but that the original bonding of modal

pairs still relates. They are now in Inversion and in doing so have changed propensity. In swapping

Bb with A#, for example, the relationship has also been inverted/swapped-over and is now A#/D,

instead of D/Bb. Musically this really is a form of expansion (toward a bright scale) to contraction

(toward a dark scale)125

In a tonal sense, this F# (and Gb) position is akin to a neutral centre, and from there it holds the

expansion/contraction , light/dark qualities. It is the place of the poles, and as such also has a

swap-over quality.

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Chapter sixteen

Visible/Invisible Dorian

As has been seen the Dorian mode and the number 9 share some symmetrical link together, with

the Dorian being perfectly symmetrical as a mode and can be seen as the 9th of the major scale,

so too the number 9 is a perfectly symmetrical number sequence, that flows through an infinite

amount of number cycles. Observe how in this next diagram it is at the Dorian position where a

strange symmetry occurs between the Modal positions:

Axis Axis Axis

1 3 2 2 3 1 4 7 5 6 6 5 7 4 1 3 2 2 C D E F G A B C DIon/Phr Dor/Dor Phr/Ion Lyd/Loc Mix/Aeo Aeo/Mix Loc/Lyd Ion/Phr Dor/Dor

#4/b5 D G#

(These two notes, D and G#, correspond to the two Dorian positions from C and F# Major

These relationships apply to all twelve major scale keys. The symmetry exists at the visible 2/2

relationship, and in between the two 6s (follow the red arrows as well as the black arrows). The

second symmetrical axis is in between the modal tonality here, in between the notes G and A,

hence the G#. The b5 (tri-tone) interval creates this particular pattern. Perhaps it is this dance

between formula and Position that creates Nature's capacity for beholding two sides of a formula

at once, the tangible and the abstract.

Here is another view of the point of balance at the Dorian 2/2 position:

4/7 5/6 6/5 7/4 1/3 2/2 3/1 4/7 5/6 6/5 7/4

And then looking carefully one will also notice an invisible like axis in between the two sets of the

6/6 pair. The symmetry around the 2/2 axis is 23147566574132, a loop effect that takes in both

sides of the mirror. The symmetry in between the 6 axis would generate 65741322314756, which

is the same cycle taken up at the half way mark..

With the 7/4 and 4/7 being the Lydian/Locrian modal pair, and brightest/darkest tonality pair, one

can see once again the point of that unity between tonal light/dark is at the Dorian position.

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This is all a glimpse into how natural cycles flow in a mirror environment and become united at

the Dorian axis in order to switch sides of the mirror. The only candidates for a swift passage

through to the “other” side are the Dorian modes and the number 9. The Dorian carries the

influence of the 9 , as does the 4.5 position, which is the position the tri-tone resides at. For a tri-

tone position to be named an axis, when its partner is the other side of the mirror, can only mean

that the tri-tone position is a hidden Dorian.

Here is another way of getting an invisible axis point and a visible axis. It is done by multiplying the

number nine until ninety-nine:

Invisible visible Invisible

09 18 27 36 45 54 63 72 81 90 99 09 18 27 36 45 54 63 72 81 90 99 49.5 49.5

49.5 to 54 = 4.549.5 to 45 = -4.5 49.5 to 63 = 13.5 49.5 to 36 = -13.549.5 to 72 = 22.549.5 to 27 = -22.549.5 to 81 = 31.549.5 to 18 = -31.549.5 to 90 = 40.549.5 to 09 = -40.549.5 to 99 = 49.549.5 to 00 = 49.5

All these totals imply a 4.5. One can see the symmetry from the 99 as it mirrors all the numbers

either side. If 90 were made the axis then both the axis points become invisible.

09 18 27 36 45 54 63 72 81 90 09 18 27 36 45 54 63 72 81 90

In effect it is the number 99/0 axis that has become invisible.

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Chapter seventeen

Opposing forces

To show how opposing Triangles of Keys affect each other one would need to draw out all twelve

mode boxes, one for each major scale. What follows is a more concise procedure for

understanding the way the mirror relates each triangle with the other three triangles. This list

shows all possible four triangles contained within the mirroring of each major scale and its seven

modes. The circle of 5ths is used to highlight this. Therefore all these triangle pairs form the two

circle of tones structure, each comprised of its two triangle relationships. The (1) and (2) beside

each pair of triangles designates which circle of tones they belong to:

Major scale Related triangles C C E Ab – D F# Bb (1)

G G B Eb – A Db F (2)

D D F# Bb – E G# C (1)

A A Db F – B Eb G (2)

E E Ab C – F# Bb D (1)

B B Eb G – Db F A (2)

F#/Gb F# Bb D – Ab C E (1)

Db Db F A – Eb G B (2)

Ab Ab C E – Bb D F# (1)

Eb Eb G B – F A Db (2)

Bb Bb D F# - Ab E C (1)

F F A Db - G B Eb (2)

It must be remembered that a triangle is primarily three major keys, chords or notes all threaded

together through symmetry. Each major key can be mirrored three times and will produce one of

the triangles. The mode box then shows that, on the mirror side, the triangles are the result of

interrelated modal positions.

Each triangle of keys, say C E Ab, is progressively mirrored through the formulas of the triangles it

is not related to. Let’s begin by mirroring the key of F major. This is part of the F A Db triangle:

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S T T T S T T T T S T T T S F Gb Ab Bb C Db Eb F G A Bb C D E F

F Phrygian F Ionian

Here is how the triangles show opposing force with each other (the notes of one triangle are

mirrored across the F axis position):

CC = Bb

E = Gb Bb D

Ab = D (tri-tone)

Ab E

Gb

Here we can see that one triangle equals the other through the mirror. The tri-tone relationship

exists between Ab and D.

The F major scale is not associated with these two triangles, as seen in the table above. It will

become clear that two triangles from each major key are able to swap to the other two unrelated

triangles of the other circle of tones structure. The next key to be symmetrically reflected will be A

major (the second aspect of the F A Db triangle).

A Bb C D E F G A B C# D E F# G# A

A Phrygian A Ionian

C = F# (tri-tone)

E = D

Ab/G# = Bb

This has yielded a similar result, except that the tri-tone relationship is now between C and F#.

Here is the third key from the triangle, C# major and its mirror:

130

C# D E F# G# A B C#/Db Eb F Gb Ab Bb C Db

C# Phrygian Db Ionian

C = D

E = Bb (tri-tone)

Ab = F#

Now for the second triangle associated with F A Db, which is G B Eb. Together, they make up a

circle of tones:

G Ab Bb C D Eb F G A B C D E F# G

G Phrygian G Ionian

C = D

E = Bb (tritone)

Ab = F#

There is another interesting result here. The key of G above produces the same results as the key

of C#/Db. This is therefore another tri-tone relationship – G to C# is a tri-tone.

B C D E F# G A B C# D# E F# G# A# B

B Phrygian B Ionian

C = A#(Bb)

E = F#

Ab(G#) = D (tritone)

Again the result is consistent with the opposing triangles. Also the result for B Major is the same

as for F major, which is yet another tri-tone relationship – B to F is a tri-tone.

Eb Fb Gb Ab Bb Cb Db Eb F G A Bb C D Eb

Eb Phrygian Eb Ionian

C = Gb (tritone)

E = D

Ab = Bb

131

Lastly one sees that the result in Eb major is the same as in A major – Eb to A is a tri-tone.

One would expect that the triangles of C E Ab and D F# Bb will show a similar opposing force with

the triangles just used to expose them. Let us only mirror C major to show this is the case, as we

know that the other five major keys show similar results:

C Db Eb F G Ab Bb C D E F G A B C

C Phrygian C Ionian

F = G

A = Eb

Db = B

As you can see this produces the other two triangles that do not belong within the C major key and

its inbuilt circle of tones structure. Therefore:

C Db

Bb D B Eb

= Ab E A F

F# G

Pick any root note of any major key, and, in symmetry, it will create triangles, which then oppose

each other through the mirror. The triangles are mirrors of each other, and this is an insight into

how vibration can be transformed from one side of the mirror to the other. The gateway is through

the triangles, at the tri-tone, and related to the access point through the Dorian Modes.

132

Chapter eighteen

Swings four ways

This next example hopefully adds further witness that the two sides of the mirror are well

structured together.

After the single notes of a scale, the next step in music theory is usually to create the seven

primary triads from that scale. The root triad will represent the home chord of the Key.

A major scale is also known to possess a Relative Minor. The Relative Minor is a scale closely

associated with the major scale, through its 6th position. C Major and A minor are two such scales

that are said to be relative to each other.

The major triad and the relative minor triad from both sides of the mirror swap over with each other

across the axis point. Here is the major scale again, and its mirror:

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

C Db Eb F G Ab Bb C D E F G A B C

This was also the very first major scale mirrored. The left hand scale has already been seen to be

a Phrygian Mode. But here we are concerned with how the seven primary triads relate to each

other through the mirror. Here is a list of the seven primary triads:

I II III IV V VI VII

C major, D minor, E minor, F major, G major, A minor, B-diminished

Here they are as they appear on a music stave:

Chords are normally numbered according to the Roman numeral system. Individual notes are

numbered normally, as seen in the mirror scales above. The C major triad is composed of the

notes CEG. If a non-musician study the C major scale, it should be quite clear that there is a

sequence within the choice of the notes used to make the C major triad; the root note, the 3rd and

the 5th. The pattern established here is to set one note as root, miss another, choose the next,

133

miss the next, and choose the next. And this pattern is extended greatly within the construction of

chords with more than three notes in them. One simply misses the next note after the 5th, and

adds the 7th, then the 9th (same as the 2nd) then the 11th (same as the 4th ) etc. Here is an example

of a C major 9th chord

C E G B D = 1 3 5 7 9

This is the basic way of harmonizing scales. The mirror triads exist and are built by reversing the

flow of the intervals that make up each individual triad.

The mirror triad of the C major triad would be calculated this way:

C E G = C Ab F

The note C is the axis, so is the same both sides of the mirror. The note E reflects around the C

axis and becomes the note Ab in the mirror (both a major 3rd away from the axis). Likewise the

note G reflects around the C axis and becomes the note F in the mirror (both a perfect 5th away

from the axis). This triad is rather well known of course; it is that of F Minor. Therefore, what has

happened is that a major triad has mirrored to a minor triad. This is consistent with the fact that a

major scale mirrors to a minor scale. More importantly, it is the root triad mirroring to the relative

minor triad on the mirror side. The C triad normally has the A minor triad as its relative minor on

the same side of the mirror. The F minor triad is relative to the Ab major triad. F minor is at the 6th

position of the Ab major scale, making it the relative minor of that scale.

The 6th position of the C major scale is the position for the A minor triad, which is its Relative

Minor. The mirror triad of the A minor triad is calculated:

A C E = Eb C Ab

We are still using the key of C Major, and that is the visible axis that holds the symmetry around

itself. Therefore the A note reflects to the Eb note, the C note is still C, and the E note mirrors to

the Ab note. This is the triad of Ab major, the 135 of its own key. And therefore, relative minor

swaps for mirror relative major, or A minor triad swaps for Ab major triad.

The resulting mirror triads of Ab major and F minor, in the mirror key, function as the root chord

(Ab major) and its relative Minor chord (F minor). This is what creates the four-way relationship. It

is found that the root chord from one side of the mirror reflects to the relative minor chord from the

other side of the mirror. And the root chord from the other side of the mirror reflects to the relative

134

minor chord from this side. That is some coincidence, given the fact that the modes move in a

reverse flow on the mirror side, and even begin that flow disjointed, with the 3rd mode reflecting to

the 1st mode on the other side of the mirror. And here amongst that disjointedness we find a

perfect relationship:

Root Relative Minor

C major triad A minor

Ab major triad F minor

Things crossover at specific axis points in order to gain access to the other side.

Earlier in history the major and minor tonalities were associated with feminine and masculine.

Here we would see the masculine having its counterpart on the other side of the mirror, and vice

versa. Alternatively, it should be obvious that both sides of the mirror would require their own

duality.

135

Chapter nineteen

Dorians at the tri-tone

As seen earlier the notes F and B represent the visible tri-tone relationship within the key of C

major. F occupies the Lydian mode position, which is the brightest sounding mode of the major

scale. The note B occupies the Locrian mode position, the darkest sounding mode of the major

scale. Running B Loc/Lyd in parallel will open up an interesting result. In the following example,

the two scales are separated by an octave. The B Locrian descends through the rest of the modal

relationships within C major, whilst the B Lydian ascends through its modal relationships of F#

major, forming a unison note at the end.

7 6 5 4 3 2 1 7 B Loc A Aeo G Mix F Lyd E Phr D Dor C Ion B Loc

B Lyd C# Mix D# Aeo E# Loc F# Ion G# Dor A# Phr B Lyd 4 5 6 7 1 2 3 4

Each note’s position within both scales also represents a modal position. One finds by studying

the above scale pair that the original mode pairs are still maintained. There is still an

Ionian/Phrygian relationship, for example, at symmetrical points between the two scales. When

the Dor/Dor modal pair is analysed it is seen to be a tri-tone interval that defines the relationship,

between the notes G# and D. This is one more clue as to why the F# 4.5 position is a Dorian

mode, and why it is perfectly symmetrical within itself, and possible why it is the catalyst for

distributing sharps and flats to the proper key signatures in turn (as seen in the chapter “Dorian

Symmetry”). The above G# and D are the Dorian positions of F# and C major, which is yet another

tri-tone relationship.

The same process can now be applied to the F Lydian/Locrian pair, from the same C Major mode

box:

4 3 2 1 7 6 5 4 F Lyd E Phr D Dor C Ion B Loc A Aeo G Mix F Lyd

F Loc Gb Ion Ab Dor Bb Phr Cb Lyd Db Mix Eb Aeo F Loc 7 1 2 3 4 5 6 7

The Dor/Dor pair display the same tri-tone interval separating them, albeit with an octave in

between. The Ab and G# are the same note within the equal temperament system, so again the

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notes in question are Ab and D. Of course the G# swapping for Ab would make sense when one

sees the result as a swap between expansion and contraction, which is the duty that the tri-tone

positions perform within the world of symmetry.

Also worth noting is that the F and B notes from both scales begin an octave apart and are

therefore axis points too. What is clear is that the Dorian, with the number 9 resting over it, is

creating an access point at the Tri-tone, which is also a 9 in the form of two 4.5’s, one either side

of the mirror.

137

Chapter twenty

Chords through the Triangle

In these examples triads will be symmetrically reflected through the Keys that comprise each

individual triangle. This will further highlight the transference of information from one side of the

mirror to the other. More importantly, it will show how the two individual circles of tones, each

comprised of their two triangles of keys, interact either side of the mirror, and at what point they

swap over.

The first example starts with the root triad of C Major, which is comprised of the notes C E G, and

it will be cycled through the triangles of keys, with which it shares a mirror link – C Ab E, D Bb F#.

Firstly we will mirror the C triad through the key of C Major and its mirror scale:

* * * * * * C Db Eb F G Ab Bb C D E F G A B C Phr Ion

The chord C equals Fm (F Ab C).

Now we mirror the chord C through the key of Ab Major, the major scale that houses C Phrygian

above:

* * * * * * G# A B C# D# E F# G#/Ab Bb C Db Eb F G Ab

Phr (C) (E) Ion

Here the chord C equals Am (made up of the notes ACE). The E note on the right hand side

does not appear within the scale as such, but is in between the Eb and F. Likewise, it mirrors to

the C note on the left hand side, in between the B and C# (as the asterisks show). Think of it as an

equal distance either side of the mirror point, falling on a chromatic note not part of either scale.

138

We then take the chord C through the key of E Major which is the key that houses G# Phrygian

above:

* * ** * *

E F G A B C D E F# G# A B C# D# E Phr (G#) (C#) (G) (C) Ion

The chord C equals C#m (made up of the notes E C# and G#). Again the G and C notes are in

between the pitches that make up E major. And on the mirror side the C# and G# are not notes

contained within the E Phrygian mode, but in between the C/D and the G/A.

The above triangular relationship consisting of the keys C E and G#/Ab is only one of the

possible triangles here, the other triangle that makes up the Circle of Tones being D F# Bb. By

mirroring the chord C through the Keys of the other Triangle an interesting result emerges. We

start by mirroring D Major and trace the C triad over it and the mirror triad through the mirror scale:

* * * * * * D Eb F G A Bb C D E F# G A B C# D Phr Ion

The chord C equals Am. This result is similar to that when the chord C is mirrored through the key

of G#/Ab Major. Interestingly, D and Ab are a Tri-Tone apart. Again we see information replicated

in two keys that are a tri-tone apart.

The keys of F# and Bb Major also yield similar replication of the first triangle’s results. Through F#

major the chord C equals Fm, establishing a tri-tone link with C Major (C/F# are a tri-tone interval

apart), and through Bb Major the chord C equals C#m, establishing a tri-tone link with E Major

(and E/Bb are a tritone apart).

Triangles - C/F# Ab/D E/Bb

Chord C equals - Fm Am C#m

139

Notice how the minor chords are three relative minors to the above major scales of C Ab and E,

one of the triangles. The major/relative minors are C/Am, Ab/Fm and E/C#m. Notice also the tri-

tone relationship each Key pair shares.

The F A and C# is also another triangle of the other possible Circle of Tones.

We can also try this experiment on the triad Ab ( the notes for this triad are Ab C Eb), and E

(E G# B), as they too are root triads of one of the triangles.

Triangle of Keys in Ab/D in E/Bb in C/F#

Ab triad mirrors to C#m Fm Am

in E/Bb in C/F# in Ab/D

E triad = Am C#m Fm

The root triads belonging to this triangle all mirror to the same three minor chords in different

orders. We also know that if we put the three root chords through the second possible triangle of

that particular circle of tones the three minor chords will still emerge as Fm Am and C#m. We

must also bear in mind that one triangle from one circle of tones combines with the other two

triangles from the other possible circle of tones. So:

Triangles C/F# Ab/D E/Bb (both triangles of one circle of tones)

triad C = Fm Am C#m

triad Ab = Am C#m Fm one triangle of other circle of tones

triad E = C#m Fm Am

At this stage, we will take the seven possible primary triads through each of the triangle of keys.

This experiment will be to show how the two individual circle of tones run into each other and

swap-over across the mirror.

Let’s now turn our attention to the second triad of the Major scale, D minor, and mirror it in a

similar fashion through the two triangles of keys, C E G# and D Bb F#: One table should suffice,

as the process is similar to above.

Triangles C/F# Ab/D E/Bb

Dm triad = Eb G B

140

As you can see we have discovered the missing triangle, Eb G B. Together with the F A and C#

these two triangles make up the other circle of tones.

F A C#, Eb G B = second possible circle of tones

Let’s proceed by applying this process to the third triad of the Major scale, E minor, through the

triangles of C E Ab(G#) and D F# Bb:

C/F# Ab /D E/Bb

Em = C# F A

Here we have returned to the first triangle from the other possible circle of tones, but the chords

are major instead of minor (because a major triad always mirrors to a minor triad). Now for the

fourth triad, the F:

C/F# Ab /D E/Bb

F triad = Cm Em G#m

This is the first time the resulting triangle has not ventured into the other possible circle of tones.

This is a very subtle clue that another swap-over of some kind has occurred. The circle of tones,

does not possess a perfect 4th in the shape of F. Its next destination note is F#, according to its

own formula. As it is, the Major scale employs a semi-tone movement at this point, from E to F so

the flow of one circle of tones is broken. But it isn't lost.

We find that from the triads F, G and Am, that the same mirror triads emerge, being Cm, Em and

G#m (this reverts to a C E G# major triad when the A-minor triad is mirrored through the circle of

tones). When we put the seventh triad of the major scale through the triangles, that of B

diminished, the result is G dim. Eb dim and B dim. This has re-established the link with a Triangle

from the first circle of tones again, because again there is only a semi-tone between the B and C

notes of the major scale here. Here the second circle of tones that had occurred through the F G

and A, has its next destination, after B, to the note C#, so again the swap-over has occurred.

141

COT 1 COT 2 COT 1

C D E F G A B C

Gap Gap

The first circle of tones will move to F#, and the second circle of tones will move to C#. The cycle

eventually becomes that of a movement of three places within one circle of tones followed by a

movement of four places within the other. The above can also be seen to be occurring on the 4/7

relationship of the F Lydian/B Locrian, which is the ‘visible’ tri-tone relationship within a major

scale.

Here is another view of how the circle of tones movements flow and swap-over at the semi-tone

points of the major scale.

All movements of a tone

COT 1

F# G# A#

C major - C D E F G A B C D etc

B Db Eb Db(C#)

COT 2 All movements of a tone

Imagine that there is an invisible in-between journey in the flows above. The circle of tones flowing

through the notes C D E is in visible relationship with this major scale. When that particular circle

of tones dives out of sight at the note F, it flows through the F# major key instead, which is a tri-

142

tone away from C major. Hence the notes F# G# A# are the next moves of its journey. The circle

of tones has switched at the F#, due to the disturbance of the semi-tone movement of the Major

scale formula. This is one of its functions with the invisible tri-tone relationship. It will reappear

when the next semi-tone movement of the major scale happens, between the B and C (its other

function with the visible tri-tone relationship).

The second circle of tones was flowing on the other side of the mirror, but the switch over at the tri-

tone was access point to this side. Again a hidden Dorian aspect has caused this to happen,

because the F# is a hidden Dorian modal position.

To recap all this, what has been seen is that in mirroring the seven triads of C major through the

triangles of keys led to the triads of F A C#, until they reached the F note of the major scale. At this

point there was a swap into the second circle of tones, which had been flowing through the mirror

of C major until that point. The F A C# was being replicated on the F# position, so it disappeared

from C Major, and became visible again at the next semi-tone area, between B and C. Whilst this

was happening to one circle of tones flow, the other circle of tones made its appearance in C

major, from the triads built on the root notes of F G A and B. The interaction between these two

circle of tones structures is by flowing in and out of each other's mirror side, the visible tritone

being the catalyst.

The next images also show how the F C# A triangle can be seen to emerge from the arrangement

of the other two triangles:

E A#Bb

F

C C# D A

G#/Ab F#

143

E A#/Bb

C F# G# D

G#/Ab F#

The second example shows how the tritone of C and D can emerge from the vertical note pairs,

between E and G# vertically, and F# and A# vertically.

144

Chapter twenty one

Triangles, the circle of tones and the number nine - ( 369 )

The Mode Box of C major is shown once more, including the names of the Modes that occupy

each individual position.

1. ION = IONIAN MODE 2. DOR = DORIAN 3. PHR = PHRYGIAN 4. LYD = LYDIAN 5. MIX = MIXOLYDIAN 6. AEO = AEOLIAN 7. LOC = LOCRIAN

3

Phr

C

4

Lyd

Db

5

Mix

Eb

6

Aeo

F

7

Loc

G

1

Ion

Ab

2

Dor

Bb

3/1Phr/Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C 2

Dor

D

3

Phr E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2/2Dor/Dr

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D 1

Ion

E

2

Dor

F#

3

Phr G#

4

Lyd

A

5

Mix

B

6

Aeo

C#

7

Loc

D#

1/3Ion/Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E 7

Loc

F

1

Ion

Gb

2

Dor

Ab

3

Phr Bb

4

Lyd

Cb

5

Mix

Db

6

Aeo

Eb

7/4Loc/Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F 6

Aeo

G

7

Loc

A

1

Ion

Bb

2

Dor

C

3

Phr D

4

Lyd

Eb

5

Mix

F

6/5Aeo/Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G5

Mix

A

6

Aeo

B

7

Loc

C#

1

Ion

D

2

Dor

E

3

Phr

F#

4

Lyd

G

5/6Mix/Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A4

Lyd

B

5

Mix

C#

6

Aeo

D#

7

Loc

E#

1

Ion

F#

2

Dor

G#

3

Phr

A#

4/7Lyd/Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

145

By studying the names of the modes above each note you will see how the original dual modal

relationships are kept intact throughout this Mode Box. Everything is so systematically laid out

that it is worth focusing on some of the results. For example, the Ion\Phr relationship is maintained

throughout. Every time Ionian is represented on one side of the mode box there is a Phrygian

mode in symmetrical reflection to it (a 3/1 or a 1/3 relationship spreading out at two 45-degree

angles). Here are the Phrygian Modes found on the left hand side (mirror side) of the whole Mode

box. For example, E Phr is found on the Dorian mode line (second line down in the mode box). G#

Phr is found within E Ionian (third line down). Bb Phr is found on the F Locrian line etc. As

mentioned they also flow at a 45-degree angle across the mirror side of the mode box.

C Phr. E Triangle of Keys G# (the first triangle found from the mirroring of C Major)BbDF# Triangle of KeysA#

The result here is similar to the Circle of Tones structure. Also exposed are the same two Triangle

of Keys when the major scale modes were mirrored.

The next table shows the Modes as they occur throughout the left hand side of the Mode box

along the 45-degree angles. Follow the individual numbers across this angle, and it should be

easy to see them:

Ion. Dor. Phr. Lyd. Mix. Aeo. Loc.

1 Ab 2 Bb 3 C 4 Db 5 Eb 6 F 7 G 1 C 2 D 3 E 4 F 5 G 6 A 7 B 1 E 2 F# 3 G# 4 A 5 B 6 C# 7 D# 1 Gb 2 Ab 3 Bb 4 Cb(B) 5 Db 6 Eb 7 F 1 Bb 2 C 3 D 4 Eb 5 F 6 G 7 A

1 D 2 E 3 F# 4 G 5 A 6 B 7 C# 1 F# 2 G# 3 A# 4 B 5 C# 6 D# 7 E#(F)

These notes are, as always, all representations of the triangle of keys that find their home on the

mirror side of a mode box. The individual triangle relationships, such as Ab C E, flow either side of

the central red notes , until at the phrygian mode there is a switch to the other two triangles. It is at

this point that the semitone move occurs, between the notes E and F, for example. At the F note

146

position of the Major scale (the perfect 4th) the directional flow changes to meet that of the other

two triangles/circle of tones (as seen in the chapter “Chords through the Triangle). In other words,

the semitone movement within the major scale flips the two COT flows over. From the Lydian until

the Locrian position the other two triangles are part of the mode box. Within each column there are

two triangles of keys rotating around a central axis point.

Notice how except for one movement of a tone between E and F# how all the other movements

within one cycle are that of two tones. It may be this shifting up of a tone (9:8) at a time through

the cycle of modes of the Major scale Mode Box (mirror side) that satisfies both formula, that is 1

to 8, and number sequences of the Vedic Square, 1 to 9. Here is the flow of the Ionian modes

again (the number 1s on the mirror side of mode box)

Ab Maj 3rd

C Maj 3rd

E Notice the movement of just one Tone at these points

Axis point F#

Bb Maj 3rd

D Maj 3rd

F#

Ab

C

E

F#

Bb

D

F# etc.

Here we are looking at the first Mode box again with the relative Major scales representing the

modal positions (Ab major represents C Phrygian, for example). It is noticed that F# acts as a

central axis point for two Triangles of Keys. By suggesting that at the point where there is only a

147

Tone movement between two adjacent notes a new central axis point emerges we begin to

uncover the familiar mirror structure found within the previous examples.

The new central axis point in the above diagram would be the note Ab, because that is where the

movement of only a Tone exists. It was this movement (from E to F# ) which allowed F# to act as

a central axis point for the two triangles, so it is only logical to make the Ab the next central axis

point.

The next list is a representation of this gradual movement of this central axis point within the COT

like structure. At the new central point ( Ab for example ) we repeat the formula either side (T =

tones) :

2T 2T 1T (Ab) 2T 2T 2T

We then make the next point where there is only a tone movement the new central axis point for

the triangles of keys. A journey in and out of the mirror commences, looking rather like a tonal

double helix!

i) Ab ii) Bb iii) C iv) D v) E vi) F# 2

C D E F# Ab Bb 2 E F# Ab Bb C D 1 * F# G# A# C D E 2 ii) Bb iii) C iv) D v) E vi) F# i) Ab 2 D E F# Ab Bb C 2

F# Ab Bb C D E *

This same procedure can also be applied on the remaining modes, 2 to 7. Each strip is a left hand

side of a Mode Box relating to one of the mirror Major keys.

148

In all, the central notes are F# Ab Bb C D E, a Circle of Tones. The Triangle of Keys always

swap over either side of the mirror point and weave themselves back to the beginning. As you

have seen this is also a characteristic shared within numbers, including the Fibonacci numbers.

149

Chapter twenty two

In and Out of the Mirror

Visualizing how information swaps to the other side of the mirror may or may not prove useful.

Even if it were obvious, that would be more beneficial and easier to understand. If bringing a mode

box to life, with an array of resonators or wiring, were to cause such a swap of information at the

tri-tones/4.5, there would be many ways to take advantage of this phenomena.

If the non-musician can at least understand the principle behind this next series of diagrams, they

will get an insight into how one may view information as existing firstly on one side of the mirror

and then on the other side, in a continual sequence, which is held together by the function of the

Circle of Tones structure, that is, the two triangles of frequencies. There have been other musical

examples that have shown this process, and there are of course also numerical ways of seeing

this information (as seen in the flow of the Fibonacci numbers, for example). There is a continual

swapping between what I have termed the invisible axis to the visible axis. One moment the

information is part of the visible scale or sequence, and then next it is emitting from the in-between

axis, 4.5, from the other side of the mirror.

To help weed through the jungle of music theory, the non-musician may like to bear in mind that

they merely need to be satisfied that the 45-degree angle of these music scale boxes, and the

tritone positions show the flow of this circle of tones structure, and how it is able to access the

mirror side. It isn’t only the 45-degree angle that carries this structure. It in fact occurs vertically

along the left hand side of the Mode box, as the major keys that house the seven modes, and then

transfers itself to the 45-degree angle. Cyclic events discussed so far always have this structure

flowing through them.

The other two Mode Boxes of a triangle of keys

The whole mode box of C Major can be transposed using the other keys that made up one

Triangle of keys (C Ab E). You will see all the modal relationships that occurred in C major’s

mode box remain intact in the next two mode boxes, even though other notes are being used; the

formulas that build the modes being exactly the same. Therefore any musical relationships

established in the C major mode box will remain true for the other two Mode boxes. Plus we are

also able to observe how the triangles of keys continue their flow both vertically along the mirror

side of the mode box, and at the 45-degree angles. All this will come to highlight the points at

which the information swaps over to the other side of the mirror.

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The same two triangles are flowing at this angle within the left hand side of the Ab Major Mode

Box, yet the arrangement of notes and the axis pitch have changed. In the C major mode box, the

axis pitch that held the two triangles around it was the note Bb. Even though there has been a

change, the axis point at Gb still maintains two triangles of keys either side of it. This circle of

tones structure is alive at the 45-degree angle.

This looks rather complicated but it is the same type of information as in the first mode box. The

only difference is that other notes are being used, but all the formulas and relationships are

exactly the same as the C Major mode box. There is an Ionian/Phyrgian, Dorian/Dorian,

Phrygian/Ionian, Locrian/Lydian, etc. They will be paired numerically exactly the same as in the C

Mode box.

The circle of tones is now grouped up along the 45-degree angle as – Ab C E – Gb - Bb D F# -

with the Gb acting as an axis between them. This was actually the flow of notes along the vertical

line in the C major mode box, and the implication is that the 45-degree angle has become the

swap-over point, as the journey through the triangle of keys continues. Vertically at the C Mode

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Box, 45-degree angle at Ab Mode Box, but what would happen through the third key of the

triangle? The triangles that appeared vertically along the C major mode box, have now reached

the 45-degree angle and will swap over to the other side of the mirror..

The central axis position between the two triangles is always made up of the enharmonic pitch

held at the Lydian position on the mirror side (Gb in this case is an enharmonic of F#), the last

relationship shown as F# in the above mode box. There is a swap between expansion and

contraction, musically speaking. The axis appears at the Locrian mode position, the Gb sitting

there in the center of the two triangles above.

Remember that the flow of these triangles is cyclic. If one drew out a 32 by 32 mode box, one

would see the Gb and F# continually swap-over and it will be a continual cycle of contraction and

expansion.

This 4-Lydian/7-Locrian relationship occupies the interval that has been shown to being the visible

tri-tone within a major scale. Earlier this was established as the notes F and B. In the above major

scale it is between the notes Db(4) and G(7). The distance between these two notes, as always, is

that of a tri-tone (Db – Eb – F – G, is three whole tones). This is now the visible tri-tone of the Ab

major scale, always at the 4/7 positions: Visible tri-tone

1 2 3 4 4.5 5 6 7

Ab Bb C Db (D) Eb F G Ab = Ab Major

Invisible tri-tone

It is at the visible tri-tone position of a scale that the invisible tri-tone meets. When understood,

one will find this overall set of relationships quite uncanny. The invisible hands over to the visible,

in a continual swapping-over effect, which occurs at a special point within the cycles. You will see

this occurring within all Major scales when mirrored. The invisible tri-tone in the scale above would

be between the root note, Ab, and the note D (in between the Db and Eb, at the 4.5 position). This

handing over from visible to invisible will be witnessed again when the mode box for the scale of E

is drawn. But it occurs because the two circles of tones have swapped-over at that point.

The Lydian is known as the brightest mode of the major scale, whilst the Locrian is known as the

darkest mode. So here we also have a combination of Light and Dark, wherein a hidden

symmetrical link with the tri-tone axis positions is seen to swap from contraction to expansion.

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We have plotted two of the mode boxes that relate to one triangle of Keys. The E Major Mode box

is the third such mode box that exists in an unbroken thread within the C E Ab triangle of keys.

This unbroken thread could also be viewed as a kind of seed, or three things in one.

Here the same circle of tones exists at the 45-degree angle, albeit mostly as enharmonic

equivalent notes. The Cx is really a D note (the x means double sharp), and the B# is the

enharmonic of C. So here we still have the E, Ab/G#, E, and the D, F#, Bb/A# triangles.

F# Dorian is the mode that comes to light as the invisible aspect of the triangle, as shown in

chapter four. True to its 4.5 quality, it seems to have swapped most notes over from flats to

sharps. Notice how A Locrian is the only mode containing flats. This is another clue as to how

information swaps over. The Locrian/Lydian is the visible tri-tone, 7/4.

The central axis at the 45-degree angle has shifted again. This time it is D. The vertical flips to the

45-degree angle after every cycle of seven modes, using three major third moves that defines one

triangular relationship. It does this for as long as vibration and number can be counted and plotted.

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As mentioned, the note D, which was the invisible tri-tone note of the previous scale, becomes the

visible note that is axis to the two triangles along the 45-degree angle. More examples should

make this quite clear. Here is the scale of E major with the visible and invisible tri-tone

relationships:

Visible tri-tone

E F# G# A (Bb) B C# D# E

Invisible tri-tone

The invisible tri-tone 4.5 axis points of all three modes boxes have been F#, D and Bb, which itself

is one of the two triangles that make up the circle of tones. The three roots are of course the C Ab

E triangle.

Below is a visual of the three mode boxes and their 45-degree angle activity. In effect they are a

series of Ionian/Phrygian relationships traveling along the diagonal of the mode box.

Helter skelter mode boxes

C Major Ab Major E Major

C C Ab Ab E E

E C C Ab G# E

G# C E Ab B# E

Bb C Gb/F# Ab D E D C Bb Ab F# E

F# C D Ab A# E

A#/C F#/Ab D/E

The first triangle starts as C Phrygian, then across the 45-degree angle becomes E Phrygian, and

G# Phrygian, with the Bb Phrygian acting as axis point to the other triangle, then D Phrygian, F#

Phrygian, and A# Phrygian. This modal relationship covers both triangles, so this 45-degree angle

is describing a set of seven Phrygian Modes. Two of the Phrygian modes are enharmonic of each

other, for example, A# and Bb. And, as mentioned, these enharmonic equivalent pitches are

necessary for the transference of information from visible axis to invisible axis, and likewise,

contraction to expansion. The cycle would actually continue past these three mode boxes. The B#

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of the E major mode box is signifying the beginning of a new circle of 5ths, where B# becomes the

start of a new triangle of keys.

We know that these triangles of keys that keep emerging are connected to clockwise and anti-

clockwise cycles. They represent a system where a major/clockwise cycle is always mirrored to

the opposite minor/anti-clockwise cycle. To be ongoing this procedure would require a traveling

from one side of the mirror to the other, back again and so forth. And it may be that this is what the

zigzagging motions of these three mode boxes laid out next to each other are showing, one 45-

degree angle becoming the swap-over point into the other 45-degree angle. This would cover all

three mirrors. It could be seen as three generations of matter, according to their frequencies, but

each generation swapping from one side of the mirror to the other.

Further evidence of this two-way mirror relationship that continually swaps over can be seen

musically in this next example, where the three mode boxes that comprised one triangle of keys

relationship will be looked at more closely.

Let's start with the Ionian mode. We move down the C Major mode box and find the Ionian on the

mirror side (left hand side of mode box). It is at the note E. We then move to the Ab Major mode

box and find the Ionian residing on the mirror side there too, and it falls on the note C. Lastly we

repeat this procedure with the E Major mode box. The Ionian modes appeared on these notes:

E C G#

As you can see this equates to an Augmented Triangle of Keys. No surprise there really. Yet if we

follow this procedure with all the Modes we will see how this triangle remains constant and how it

combines with the other triangle found on the right hand side of the mode boxes to form the Circle

of Tones. In fact both possible circles of tones are involved.

To ascertain the other triangular modal relationships we follow a similar procedure to the first

example by focusing on which notes within the three mode boxes each particular mode is found.

This is also the first clue that the augmented triangles flow from one side of the mirror to the other,

and the main reason for focusing on the results. The same numerical relationships are also

involved, and may explain the natural double reflection principles between the relationships.

Imagine the first line as a 1/3, then the second line as a 2/2, just like in the mode boxes. Here is

the full list:

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Modes occurring on the mirror side of the three mode boxes non-mirror side

Ionian (on the notes) E C G# C Ab(G#) E

Dorian D Bb F# D Bb F#

Phrygian C G# E E C G#

Mode's Major keys - (F#) (D) (Bb) (C) (Ab) (E)

Lydian B G Eb F Db A

Mixolydian A F C# G Eb B

Aeolian G Eb B A F C#

Locrian F Db A B G D#(Eb)

Take the Phrygian line as another example. Here we find that the Phrygian modes appear on the

mirror side of the three mode boxes at the notes C G# and E. On the non-mirror side they appear

at the notes E C and G#. Both sets have appeared on either side of the mirror.

The other triangles of keys are also replicating. All except at the Dorian, which shows that it has

direct access to the other side, by virtue of its perfectly symmetrical quality.

In the diagram above, from the Lydian mode onwards, the notes when put in sequence form a

Circle of Tones – Db Eb F G A B. This is the other possible circle of tones (which is also the other

two possible triangles of keys). The red notes in brackets above are the Major scales that house

the particular modes creating this circle of six Lydian keys. By doing this the other circle of tones

emerges. The B Lydian resides as a mode within F# Major, for example, and the G Lydian mode

resides within D major, and so on.

What is quickly apparent here is the Triangle of Keys relationship existing between all the modes.

Each set of modes is one of the four triangles that make up the circle of tones. Each note is

separated by a minor 6th interval (the inversion of a Major 3rd), which eventually defines each

series of notes as an Augmented type Chord/Triangle. The triangle of notes, chords and Keys are

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always defined by the intervals 1 3 #5 , in reverse or some inversion of them which is still the

formula for an Augmented chord.

This process of swapping over can also be called Double Reflection, meaning, for example, that if

we mirror the Ionian mode we get the Phrygian mode and if we mirror the Phrygian mode we get

the Ionian mode, or 1/3 becomes 3/1.

T T S T T T S S T T T S T T

1 ION - C D E F G A B C Db Eb F G Ab Bb C - PHR 3

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Swapping the invisible tritone for the visible tritone

The next set of diagrams further highlights the visible and invisible axis relationships that exist

through the symmetrical reflections of a plain C major scale and its modal structure. Here is a

stripped down version of the mode box of C major, showing only the relationships along the 45-

degree angles:

Mirror point

C - PHR C C C C - ION

D - DOR E D C D - DOR

E - ION G# E C E - PHR

F - LOC Bb F C F - LYD

G - AEO D G C G - MIX

A - MIX F# A C A - AEO

B - LYD A# B C B - LOC

C

The note C running down the 45-degree angle is in symmetrical reflection to a series of notes

throughout the mirror side of the mode box. In analyzing these relationships one gains an insight

into the cyclic events that occur either side of the mirror, and how axis points interact within the

whole picture. Here is a list of these relationships:

C = C in C Phr/Ion

C = E in D Dor/Dor

C = G# in E Ion/Phr

C = Bb in F Loc/Lyd

C = D in G Aeo/Mix

C = F# in A Mix/Aeo

C = A# in B Lyd/Loc

One can see that every enharmonic pair (the Bb and A# in the above example) actually relate to

what can be called the visible tri-tone interval, F to B in this case, or F Lyd/Loc – B Loc/Lyd. What

is interesting here is that the Bb and A# are positions occurring within the Gb and F# major keys,

which is the invisible tri-tone interval from C. The Bb resides at the F Lydian (which belongs in the

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key of C major), and F Locrian (which belongs in the key of Gb major). The A# resides at the B

Locrian (which is C Major), and the B Lydian (which is F# major). Here is the scale of C once more

to show the two Tri-tone points:

Invisible Tri-tone axis

C D E F (F#/Gb) G A B C

Visible Tri-tone

1. F Lydian - C Major

F Locrian - Gb Major

2. B Locrian – C major

B Lydian – F# major

In the mode box diagram above it is seen that on one side of the mirror is the root note along the

45-degree angle, that is, the note C. In reflection to the root is the Circle of Tones, flowing as two

triangles of Major thirds either side of a central position. Bb acts as centre to C E G# one side and

D F# A# on the other. In fact every note on the left hand side creates triangles along the 45-

degree angles. If one were to focus on the note B occurring along the 45-degree angle on the right

hand side of the mode box, there too in symmetrical reflection to it would be a circle of tones.

Db B

F B

A B

Cb(B) B

Eb B

G B

B B

Along this particular 45-degree angle we see clearer evidence of the two tri-tone axis points

playing the visible/invisible role. The Cb(B) position on the mirror side is occurring at the F Lyd/Loc

position, and in reflection is another B on the non-mirror side. This is one tri-tone axis (the Cb/B is

at the tri-tone position in either F scale both sides of the mirror). The B/B also occurs at the

bottom, at the Loc/Lyd position. It was already seen how this tri-tone is a hidden Dorian aspect

too, and it is this ability to have direct access to the other side of the mirror that allows the Dorian

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to swap the full Light/Expansion for the full Dark/Contraction in a tonal sense. This means,

musically, that the brightest Mode (the Lydian) swaps for the darkest mode (the Locrian). It is also

a two-way swapping, Lydian for Locrian, and Locrian for Lydian.

What we see is the same information around the root position axis being replicated at the tri-tone

axis. For example, if one look at the original Mode box for C major, they will notice that at the B

Locrian/Lydian position the notes C and A# are mirror partners. This is also true for the F

Lydian/Locrian position in the same mode box, where C and Bb are mirror partners. Sharps and

flats have swapped, which signifies a swap from light to dark, tonally speaking.

C = A# in B Loc/Lyd

C = Bb in F Lyd/Loc

This means that a Mode Box is a complete unit unto itself displaying all twelve notes of the

chromatic scale. In fact if one count notes like Db and C# as two separate notes there are

displayed in the Mode Box in all twenty different notes, including a Cb and E#.

Each unit is replicated twelve times and is representative of all twelve Major and Minor keys that

evolves through the circle of 5ths. The next diagrams again shows how every note of C Major

along its 45-degree angle creates a Circle of Tones on the mirror side, made up of two triangles

separated by the major 3rd interval.

F G

A G

C# G

Eb G

G

Following each notes individual 45-degree angle exposes how triangles are always formed on the

mirror side at the corresponding 45-degree angle. F A C# is one triangle of keys, which belongs to

the second circle of tones. The Eb is probably a central axis here for the second possible

augmented triangle along this line. If the flow were uninterrupted by the mirror point we would see

the notes G B D# go on to produce the second triangle of keys. In fact those notes appear at the

Aeolian modal position on the mirror side (along the 45-degree angle). Then the Eb and D# would

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be the enharmonic pair involved, where the swapping between contraction and expansion would

occur, due to the replication of information at the visible and invisible tri-tone positions.

The Major 3rd intervals will meet in the center of the Mode Box at the 45-degree angle, and this will

define the most perfectly symmetrical point between the two triangles:

C

E

G#

A

Bb

D

F#

A#

If the axis point around the triangles were to be the note A then the other notes would be seen to

mirror around this axis point. Here is the key of A Major mirrored in order to highlight this:

A Bb C D E F G A B C# D E F# G# A

A Phrygian A Ionian

It is in this key/axis point that the note C equals F#. If this procedure were carried out on the other

two Mode Boxes of the triangle of keys the three axis points will be A C# F. If you observe the

Major scale above again you will see that these Augmented triangles or triad relationships

naturally occur by moving the first Major 3rd interval away from the root axis, A to C# left to right

and A to F right to left.

F A C#

The fact that this relationship uses both sides of the mirror at once, in order to form provides

further evidence of relationships existing in and out of the mirror, turning both halves of the mirror

into one whole unit.

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The Swap-over point

It has been seen that the Lydian/Locrian positions are the visible tri-tone, whilst the invisible tri-

tone axis is at the 4.5 position. There is a swap-over point between the two sides of the mirror that

is rather subtle, but the logic is there, and it will require some study to fully understand how this

occurs.

The tri-tone position is a place of replication. What this means here is that the symmetry between

note partners around the note F, for example, is exactly the same as around the note B. Likewise,

the symmetry around C is the same as the symmetry around the F#/Gb poles. And in a sense it

will mean that F or B can play either role, either Lydian or Locrian. This seems to be the way

information can flow either side of the mirror and for the unit to remain whole.

To show how the Lydian/Locrian play each other’s roles, here are other examples. If we take a

closer look at the two Circles of Tones we will find another intriguing result:

F

C D E F# G# A#

F Lydian and B Locrian display symmetrical characteristics that other note pairs/modal

partnerships do not display. One of the reasons for this is their Tri-tone interval relationship. The

Lydian/Locrian position is really a partnership of perfect light/dark, the Lydian being the brightest

Mode of a Major scale whilst the Locrian is the darkest sounding. In C Major the Lydian mode is

generated from the note F whilst the Locrian is generated from the note B. These two notes

perform an interesting function within the above Circle of Tones, at the point where the note F

would normally be (even though it is not used within the scale). The notes of the Circle of Tones

are in symmetry around the note F.

C D E F F# G# A# min 2nd min 2nd

min 3rd min 3rd

4th 4th

Replacing the above F with the note B also yields the same symmetry.

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C D E B F# G# A# 5th 5th

Maj 6th Maj 6th

Maj 7th Maj 7th

The intervals reflected are the inversions of the original intervals (although one will need to start at

the bottom to tie together the min 2nd and maj 7th, and then work upwards for the rest).

This experiment can be carried out on the other Circle of Tones.

F#/Gb

Db Eb F G A B

This will produce the same set of intervals as in the first example. The note C can be used

between F and G instead. The C takes the place of the axis F#/Gb, its Tri-tone partners, and it will

then be seen that the notes either side of C are in symmetrical reflection, as in the previous

examples:

Db Eb F C G A B 5th 5th

Maj 6th Maj 6th

Maj 7th Maj 7th

It is the tri-tone that really brings this ability to swap-over about. Any musical information

established at one root, can be replicated a tri-tone interval away. It won't work with any other

interval.

We will now see how the swap-over to the mirror side is achieved because of this inherent

symmetrical ability for the Lydian/Locrian to replicate each other’s information around their axis

positions. One can become the other, just like poles can switch from positive to negative. They

contain within them both the visible and invisible tri-tone relationships, because F and B Lyd/Loc

belong in the keys of C and F#/Gb.

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Lyd Loc

C D E F (F#/Gb) G A B C - C major

Also be aware of the arrangement on the mirror side of the C Major Mode box. At the note F on

that side is the Locrian Mode position. This is the F Locrian mode, and it belongs to the Gb major

scale. The note F on the right hand side is on the Lydian mode position, as shown above. Yet it is

also partnered with the B Locrian on the same side, and can take on its qualities in terms of

reflection, as seen. It does this, and the swap-over to the mirror side occurs, by the F switching to

Locrian, which then identifies it with Gb at the invisible tritone position. From here, the mirror side

is entered, and this Gb becomes the axis that supports the two triangles of keys around it.

Fragment of C Major Mode box:

Lyd/ Loc/

Loc Lyd

C D E F (Gb) G A B C

Loc/ Lyd/

Lyd Loc

F Gb Ab Bb Cb Db Eb F = (Gb major or F Locrian)

The Cb is an enharmonic of the note B

The F Loc is from the Gb major scale. Here we see the roles of Lyd/Loc swap-over, and access to

Gb from one side of the mirror to the other accomplished. And it is accomplished by the visible F

Lyd/Loc entering the invisible tri-tone area at F#/Gb, the poles) on the right hand side, and re-

establishing itself at the F Loc/Lyd on the mirror side.

Deeper than this is the fact that Cb will really be taking the whole process through a tonal spiral,

and will do this because of the slight difference between itself and the note B. For our purposes, it

is just a question of spotting that certain positions take on each other's qualities and so are able to

swap these qualities to the mirror side, through the tri-tone gateways.

This whole scenario is then repeated in the Ab major mode box.

Fragment of Ab Major mode box:

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Lyd/ Loc/

Loc Lyd

Ab Bb C Db (D) Eb F G Ab

Loc/ Lyd/

Lyd Loc

C# D E F# G A B C#(Db) (D major or C# Locrian)

Again the Db Lyd/Loc enters the mirror side by the invisible tri-tone route of D, and again the

enharmonic is used.

This access to the mirror side isn’t only between the C and Gb, and the Lydian/Locrian along

those scales. The note F# is also involved, as it is the other pole within the major scale system:

Lyd/ Loc/

Loc Lyd

C D E F (F#) G A B C

Lyd/ Loc/

Loc Lyd

B C# D# E# F# G# A# B

(F)

Here the F Lyd/Loc of the right hand enters the mirror side by the invisible tri-tone route of F#. The

mirror side is the F# major scale, beginning from the B Lydian position at the seventh line of the

mode box. And this because identities can be swapped, at a cost of also swapping expansion for

contraction (through enharmonic equivalents) and vice versa.

The next image attempts to portray the whole pod of information, the two triangles, the function

Lyd/Loc pair, and the number 9, being more a dual 4.5 junction point in the centre.

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Triangle Wheel

The two points, C and F# can be imagined to rotate in a contrary fashion to each other, through

the modal positions, the two meeting at the tri-tone 4.5 and swapping over all information for the

opposite quality, major to minor, expansion to contraction etc. The flows meet at D#, and after

swapping over at the tri-tone they meet again at the A position. These two positions are known as

the Secondary Dominant positions, and are also a tri-tone apart.

Important meeting points always seem to be at the 4.5/tritone position, the very center of each

scale. All these pairs are at that relationship with each other:

C and F# , D and G# , F and B , A and D# , E and A#

The fact that musical data is seen to appear in and out of the mirror at certain strategic points is

obviously what one would consider a musical exercise. Yet, with the added insights regarding the

Fibonacci numbers, and the Phi ratio, as well as many other number based examples, one can

begin to entertain the idea that there is a system in place that is more intrinsic than the musical

framework it is also found in. There is at least some justification for experimenting along those

lines.

Whatever the implications, one cannot dismiss the fact that the circle of tones is playing a vital

cyclic role in connecting both sides of the mirror, in music based examples as well as number

sequence based examples.

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Chapter twenty three

Un-Interruptus – the journey of the Comma

In this example the Major Keys flow in a Circle of 5ths, uninterrupted. Normally the circle is kept to

12 by ignoring the Pythagorean Comma, which is so small an interval that the now tempered circle

of twelve works wonderfully, as it should. The sharp keys and the flat keys would evolve

individually. These lists will help one understand the coming chapter that deals with finding a 144

major scale grid:

0 1 2 3 4 5 6 7 8 9 10 11 C G D A E B F# C# G# D# A# E#

12 13 14 15 16 17 18 19 20 21 22 23 B# Fx Cx Gx Dx Ax Ex Bx Fx# Cx# Gx# Dx#

24 25 26 27 28 29 30 31 32 33 34 35 Ax# Ex# Bx# Fxx Cxx Gxx Dxx Axx Exx Bxx Fxx# Cxx# (5#)

36 37 38 39 40 41 42 43 44 45 46 47 G(5#) D(5#) A(5#) E(5#) B(5#) F(6#) C(6#) G(6#) D(6#) A(6#) E(6#) B(6#)

48 49 50 51 52 53 54 55 56 57 58 59 F(7#) C(7#) G(7#) D(7#) A(7#) E(7#) B(7#) F(8#) C(8#) G(8#) D(8#) A(8#) 60 61 62 63 64 65 66 67 68 69 70 71 E(8#) B(8#) F(9#) C(9#) G(9#) D(9#) A(9#) E(9#) B(9#) F(10) C(10) G(10)

72 73 74 75 76 77 78 79 80 81 82 83 D(10) A(10) E(10) B(10) F(11) C(11) G(11) D(11) A(11) E(11) B(11) F(12)

CThe numbers above the notes represent the Sharps each contain within its Major Key signature. It

is actually quite easy to work out this table. Accented notes move in groups of seven. There will be

seven root key notes shown with one sharp, then seven with two sharps etc. Fx (F double sharp),

for example, is an enharmonic of the note G, and it contains thirteen sharps in its key signature. It

belongs to another circle of twelve 5ths.

In working out the keys that contain flats it is best to imagine that they are being created at the

same time in symmetry to the sharps. The note C is obviously the axis point containing neither a

flat nor a sharp. This is what the first few sharp and flat Major Scale Keys would look like as they

evolve symmetrically:

3 2 1 0 1 2 3 Eb Bb F C G D A etc.

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This next table then is occurring in 4ths as the 1st table is occurring in 5ths, in other words,

clockwise and anti clockwise simultaneously.

Table 2 occurring anti-clockwise with table 1

0 1 2 3 4 5 6 7 8 9 10 11 C F Bb Eb Ab Db Gb Cb Fb Bbb Ebb Abb

12 13 14 15 16 17 18 19 20 21 22 23 Dbb Gbb Cbb Fbb Bbbb Ebbb Abbb Dbbb Gbbb Cbbb Fbbb Bbbbb(4b)

24 25 26 27 28 29 30 31 32 33 34 35 E(4b) A(4b) D(4b) G(4b) C(4b) F(4b) B(5b) E(5b) A(5b) D(5b) G(5b) C(5b)

36 37 38 39 40 41 42 43 44 45 46 47 F(5b) B(6b) E(6b) A(6b) D(6b) G(6b) C(6b) F(6b) B(7b) E(7b) A(7b) D(7b)

48 49 50 51 52 53 54 55 56 57 58 59 G(7b) C(7b) F(7b) B(8b) E(8b) A(8b) D(8b) G(8b) C(8b) F(8b) B(9b) E(9b)

60 61 62 63 64 65 66 67 68 69 70 71 A(9b) D(9b) G(9b) C(9b) F(9b) B(10) E(10) A(10) D(10) G(10) C(10) F(10)

72 73 74 75 76 77 78 79 80 81 82 83 B(11) E(11) A(11) D(11) G(11) C(11) F(11) B(12) E(12) A(12) D(12) G(12)

C

There are 165 Major Keys (counting C only once) in all before each note stops producing

enharmonic equivalents.

One could say, for example, that when C = C there are no sharps or flats. Yet when

Dbb = C there are 12 flats, and when Ebbbb = C there are 24 flats and so on.

Also this is not the end of the journey. We have come to a stand still because the note C has been

reached again. Yet there are more cycles, as there has not been an octave of comma intervals

used so far. More letters of the alphabet will need to be employed to complete the journey of the

Comma. 624 major keys versus 26 letters of the English alphabet!

A Comma Interval is some 23 cents. In an octave there are 1200 cents. There is approximately 52

commas in an octave. Fifty two times twelve is 624 major keys in total. A to G deals with 165 of

the major keys. H to N, and O to U, and V to Z account for approximately all 624 major keys. Do

not worry, the focus will only be on the first 144.

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Cycles of twelve fifths with C and F# as gateways through each

11:11 Tri-tone axis A Bbb Gx Cbbb D E Fb Ebb Cx Dx Gbbb Fbbb G B Cb Abb Fx Ax Dbbb B4b

C F# Gb Dbb B# Ex Abbb E4b

F Db C# E# Gbb Ebbb Bx Dx#

Bb Ab G# A# Cbb Bbbb Fx# Gx# Eb D# Fbb Cx#

Cycle - 1 2 3 4

There is the difference of a Comma in between each new circle. To reach an Octave of Commas, one needs approximately 52 circles of twelve

fifths. This equates to around 624 possible major scales. What these circles show is that it is the Tri-tone gap that swaps one circle of 12 with

another. You will see that it is all about the axis points C and F#, and their enharmonic equivalent, which build up each successive circle of

twelve 5ths.

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Continual triangles

T T S T T T S T T T S T T T S T T T S T T T S T T T S T T T S T1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1C D E F G A B C D - E F# G# A B C# D# E F# - G# A# B# C# D# E# Fx G# A# - B# Cx Dx E# Fx Gx Ax B# Cx Dx etc

The keys continue this way for about 624 applications of the Major scale formula and numbers 1-9. The roots of the major scales (the number

1’s) for the first few triangles are as follows:

C E G# B# Dx Fx# Ax# Cxx Exx Gxx# Bxx# Dxxx - Fxxx# Axxx# Cxxxx - Exxxx Gxxxx# Bxxxx#

One Comma One Comma One Comma etc.

X = two sharps, or one whole tone

This procedure shows that the original triangle, C E G# is being maintained throughout, yet reappearing as enharmonic notes, such as B#

instead of C, Dx instead of E and Fx# instead of G# and so on into the next triangle. Follow the same procedure for the number two and the

result will be the D F# Bb triangle. The number 3’s produce the first triangle again, and the numbers 4, 5, 6, and 7, produce the other two

remaining triangles. Numbers 8 and 9 revert back to the original two triangles. All this is to do with the semitone gaps in a major scale, as

discussed previously.

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Chapter twenty four

144 Major scale grids

The process that unveils the 144 major scale grid is on the assumption that every twelve keys

throws up the Pythagorean Comma, a small gap', which helps commence a further circle of twelve

keys, and so on, until an octave of Commas is reached.

We begin with the first, and generally accepted circle of twelve keys that make up our music

system in this part of the world. In the following diagram, the twelve chromatic notes evolve around

the circle and in brackets are the amount of sharps or flats in their respective major key signature.

The key signatures of the first cycle of twelve 5ths

0 C

(5) B Db (5)

(2) Bb D (2)

(3) A Eb (3)

(4) (Ab E (4)

(1) G F (1)

(6) Gb/F# (6)

One begins at C, that contains no sharps or flats in its key signature. The first movement of a 5th is

to the G, which then contains one sharp. From there another movement of a 5th is to D, which will

contain two sharps, and so on. One finds that flats and sharps are mirroring each other when an

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axis is drawn from C through F#/Gb, the poles. Also, the same note partners of C major are

evident, D/Bb, A/Eb etc

The Poles Shift

F#/Gb are the two poles within the first circle of twelve, each of the two keys containing six sharps

and six flats respectively. These two notes were found at the 4.5, which is the tri-tone position

within both C Ionian and C Phrygian. We will see now that these two points are a gateway into

more and more cycles of twelve, all the way up to a 144 Major scale grid and far beyond, creating

more and more of these major scale grids.

6 1 3 5 7 2 4 6 8 10 5 7 9 11 6 F# G A B C# D E F# G# A# B C# D# E# F#

F# Phrygian F# Ionian

At the F# Ionian/Phrygian pair the numbers above the notes represent the amount of sharps or

flats required were each note to be the root note of a Major scale Key signature (so G#, for

example, signifies the root of the G# Major scale, which contains 8 sharps within its key signature).

What we have here is a representation of what occurred in C Major only there are enharmonic

equivalent note names being used. For example, either side of the F# is E and G#. The G# is an

enharmonic of the note Ab which contains four flats within its key signature. E Major has four

sharps (check with the diagram above of the first circle of twelve).

One would think this doesn’t tally. Yet within this second cycle things must add up to twelve flats

and sharps and not six, in order to reach the next poles. The two new poles will both be keys

containing twelve flats and twelve sharps, and the rest of the note pairs of the second cycle will,

together, add up to twelve sharps or flats, as already seen with E and G# adding up to twelve

sharps. Therefore we have around the F# axis the same note pair, E and Ab/G#, as we did

around the C axis, showing that the root axis and tri-tone axis replicate, with the extra insight of

seeing tonality swapping between flats and sharps.

As you can see every note above, around the F# axis, has a Major key Signature made up of

sharps. At every symmetrically reflected point around the root axis the tally is twelve, 8+4, 10 + 2,

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etc. The enharmonic equivalent notes have shifted the cycle into its second rotation. Even

between G#, say, and its enharmonic Ab (four flats) there is a tally of 12.

So where are the flat keys of the second cycle? Well we must remember that F# and Gb are both

poles so we find that it is Gb Major and its mirror scale which produces them:

6 11 9 7 5 10 8 6 4 2 7 5 3 1 6 Gb Abb Bbb Cb Db Ebb Fb Gb Ab Bb Cb Db Eb F Gb

Gb Phrygian Gb Ionian

Again the mirror note pairs around this axis are the same as the note pairs around C Major, except

that the pitch names are enharmonic. It is only at the F# Major and Gb Major points that this

symmetry appears. Doesn’t this seem like one has walked through a musical doorway into another

layer? One must remember that this second cycle is not quite like the first cycle. There is a

difference of one Comma, which is an 80:81 ratio, separating both cycles. The second cycle exists

as an entity in its own right, as will the following cycles.

To complete our second cycle we will need to build Major scales from the new accidentals that

appeared in the last two examples, C# D# E# Fb G# A# B# Cb Abb Bbb and Ebb, which are

the eleven other keys that accompany F#/Gb within this second cycle of twelve. The use of double

sharps and double flats will be required in order to define the different keys here. The sign for the

double sharp is ‘x’ placed after a note. The sign for the double flat is ‘bb’ placed after a note. A

double sharp will raise the pitch of a note by one tone, and the double flat will lower a pitch by one

tone. When these new keys are found it will be seen that they can be added to the first circle of

twelve above and they will fit perfectly in between the existing notes. We will now apply the familiar

major scale formula to these new notes. Before we do so, here is a list showing enharmonic

equivalent notes.

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C = Dbb or B#

C# = Db or Bx (add two semitones to the note B), or Ebbb (take three semitones

away from the note E – Eb - D - Db)

D = Ebb or Cx

D# = Eb or Cx# (add a tone and a semitone to the note C)

E = Fb or Dx

F = E# or Gbb

F# = Gb or Ex

G = Fx or Abb

G# = Ab or Fx#

A = Gx or Bbb

A# = Bb or Gx#

B = Cb or Ax or Dbbb

There will be a lot more enharmonic notes as the cycles progress. To calculate the correct note

names it is a simple matter of learning how to add sharps or take away flats from the note names.

Equal Temperament is an appropriate tuning system to use here. It doesn’t matter what ratios one

may want to use when building scales. The fact is that every set of ratios will have its mirror ratio,

whilst the letters used to name the scale remain consistent.

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The Major scales of the second Cycle

T T S T T T SCb Db Eb Fb Gb Ab Bb Cb - therefore Cb Major has 7 flats within key signature

C# D# E# F# G# A# B# C# - therefore C# Major has 7 sharps

Fb Gb Ab Bbb Cb Db Eb Fb - Fb Major has 8 flats

G# A# B# C# D# E# Fx G# - G# Major has 8 sharps

Bbb Cb Db Ebb Fb Gb Ab Bbb - Bbb Major has 9 flats (B) (C#) (D) (E) (F#) (G#) (A) - related to A major which has 3 sharps (9+3=12)

D# E# Fx G# A# B# Cx D# - therefore D# Major has 9 sharps (or Eb Major which has 3 flats – 9 +3=12)

Ebb Fb Gb Abb Bbb Cb Db Ebb - Ebb Major has 10 flats (D Major has 2 sharps – 10+2=12)

A# B# Cx D# E# Fx Gx A# - A# Major has 10sharps (Bb Major has 2 flats – 10+2=12)

Abb Bbb Cb Dbb Ebb Fb Gb Abb - Abb Major has 11 flats (G Major has 1 sharp)

E# Fx Gx A# B# Cx Dx E# - E# Major has 11 sharps (F Major has 1 flat)

Dbb Ebb Fb Gbb Abb Bbb Cb Dbb - Dbb Major has 12 flats (enharmonic of C Major which has

no flats or sharps)

B# Cx Dx E# Fx Gx Ax B# - B# Major has 12 sharps

It will be easier for you to see this info if we now plot these new keys onto the first circle of twelve

presented earlier. The original keys of the first circle are shown in bold. Take note of the mirroring

on either side of the C and F#/Gb axis.

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Cycle 2:

There is shuffling amongst all the pitches. The new poles are now the B# and Dbb, because they

contain the maximum sharps and flats of the second cycle. They are also enharmonic equivalents

of the note C. This is an example of how the poles swap musically speaking, as the 3:2 ratio

continues to develop these different key signatures.

To become more proficient at calculating a note’s new enharmonic pitch when more and more flats

and sharps are added it may be helpful to draw out C to C with no alterations:

C D E F G A B C

Using only the C as an example, it will evolve like this when more sharps are added:

C D E F G A B C

C C# Cx Cx# Cxx Cxx# Cxxx Cxxx# Cxxxx Cxxxx# Cxxxxx Cxxxxx# Cxxxxxx

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And here with more and more flats added (note the direction of the arrow)

C D E F G A B C

C12b C11b C10b C9b C8b C7b C6b C5b Cbbbb Cbbb Cbb Cb C

In the second circle of twelve everything commences from the key of C, containing no sharps or

flats within its key signature. C is the originator. Next is a movement to the right where Dbb is seen

to contain 12 flats. Mirroring this is a movement to the left where B# is housing 12 Sharps, and

together the Dbb/B# are the two new musical poles that contain maximum contraction and

expansion of the second cycle, taking over from the F#/Gb of the first cycle, north pole swapping

with south pole, negative polarity to positive. Next is a dual movement to the symmetrical keys of

C# and Cb respectively, one containing 7 sharps and the other 7 flats. Here is this complete Dual

movement within the second circle of twelve keys.

C Clockwise Anti-clockwise

Dbb 12 B# 12 19

C# 7 Cb 7 12

Db 5 B 5 7

D 2 Bb 2 12

Ebb 10 A# 10 19

D# 9 Bbb 9 12

Eb 3 A 3 7

E 4 Ab 4 12

Fb 8 G# 8 19

E# 11 Abb 11 12

F 1 G 1 7

F# 6 Gb 6

The 19 moving down to 7 is 12 descending major scale moves, for example. Then the 7 up to 19

is 12 ascending major scale moves. The above patterns are very consistent and are dominated by

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twelve-ness. The evidence shows so far that the thread commenced at C is unbroken. What is

began by C major and the D Dorian “pregnancy” is maintained throughout the cycles of twelve by

the enharmonic equivalent names for C and D. We will come to these enharmonic equivalent

notes very soon after the uncovering of the overall grid that emerges from all these slight shifts in

tonality. The symmetry exposed along the way is always consistent to one overall mirror structure,

the 'circle of tones'. The 144 major scale grid that emerges is really the reason for venturing on

into these nether regions of major scales. By the end of this chapter we will see how the nine

number sequences of the Vedic Square and the major scale formula merge together perfectly. It

takes 144 major scales to achieve this marriage of music and numbers, and this is achievable due

to the Pythagorean Comma interval, 80:81, which is the slight difference between each circle of

twelve major keys.

The next two signs will be the triple sharps and the triple flat. A triple sharp may look like this – Ex#

or Gx#. A triple flat will look less fanciful and be represented as –bbb (eventually signs like 5b will

be needed to denote five flats, eq F5b = Fbbbbb which is an enharmonic of the note C, like this- F

less 1 flat= E, less 1 flat = Eb, less I flat = D, less 1 flat = Db, less 1 flat = C).

The question now is how many flats and sharps are there in the major keys Abbb and Ex? These

notes are the enharmonic equivalent of Gb and F# respectively and would represent the new

poles of the third cycle of twelve. These two new poles would also switch from north pole, around

the C axis, to the south pole again, around the original F#/Gb axis.

When a Key signature contains 13 flats it will be one octave of key signatures higher. This means

that this Key signature will be an enharmonic of F Major, which has 1 flat. This key will be known

as Gbb Major. As notes have their octaves so do Keys. This following list is a representation of

the major scale keys as they evolve, first the flat keys:

Flats - 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Keys - C F Bb Eb Ab Db Gb Cb Fb Bbb Ebb Abb Dbb Gbb

The Gbb is an enharmonic of F, so they will be separated by twelve flats. Similarly the key

signature with 13 sharps will be an octave higher in key signature terms. So this new key must be

an enharmonic of G major. Fx (double sharp) Major will be this new key signature that has 13

sharps.

Sharps - 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Keys - C G D A E B F# C# G# D# A# E# B# Fx

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Here is a list of the new Major scales needed for the 3rd cycle. The enharmonic equivalent is

written under each note. After this cycle we should know enough to understand the 144 major

scale grid that emerges after twelve cycles of twelve are complete.

The first key in the list below is pronounced G double flat Major, and from there it is made up of

the notes Abb (A double flat) Bbb Cbb Dbb Ebb Fb Gbb. This is the major scale sound. F double

sharp Major is the next Key/major scale, and so on.

T T S T T T SGbb Abb Bbb Cbb Dbb Ebb Fb Gbb = 13 flats (key of F Major in its second cycle of twelve) (F) (G) (A) (Bb) (C) (D) (E) = F major, which contains 1 flat

Fx Gx Ax B# Cx Dx Ex Fx = 13 sharps (G Major in its second cycle)G A B C D E F# G = I sharp

Cbb Dbb Ebb Fbb Gbb Abb Bbb Cbb = 14 flats (Bb Major in its second cycle)Bb C D Eb F G A Bb = 2 flats

Cx Dx Ex Fx Gx Ax Bx Cx = 14 sharps (D Major in its second cycle)D E F# G A B C# D = 2 sharps Fbb Gbb Abb Bbbb Cbb Dbb Ebb Fbb = 15 flats (Eb Major in its second cycle)Eb F G Ab Bb C D Eb = 3 flats

Gx Ax Bx Cx Dx Ex Fx# Gx = 15 sharps (A Major in its second cycle)A B C# D E F# G# A = 3 sharps

Bbbb Cbb Dbb Ebbb Fbb Gbb Abb Bbbb = 16 flats (Ab major in its second cycle)Ab Bb C Db Eb F G Ab = 4 flats

Dx Ex Fx# Gx Ax Bx Cx# Dx = 16 sharps (E major in its second cycle)E F# G# A B C# D# E = 4 sharps

Ebbb Fbb Gbb Abbb Bbbb Cbb Dbb Ebbb = 17 flats (Db Major in its second cycle)Db Eb F Gb Ab Bb C Db = 5 flats

Ax Bx Cx# Dx Ex Fx# Gx# Ax = 17 sharps (B Major in its second cycle)B C# D# E F# G# A# B = 5 sharps

Abbb Bbbb Cbb Dbbb Ebbb Fbb Gbbb Abbb = 18 flats (Gb Major in its second cycle)Gb Ab Bb Cb Db Eb F Gb = 6 flats

Ex Fx# Gx# Ax Bx Cx# Dx# E#x = 18 sharps (F# Major in its second cycle)F# G# A# B C# D# E# F# = 6 sharps

The third cycle of twelve keys can also be integrated into the first circle, amongst the other twenty-

four keys. As that would look rather crammed packed and messy, here is the circle showing only

the twelve new keys:

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3rd Cycle:

The new poles are now back at the F#/Gb position, which are represented by enharmonic

equivalents in the form of Ex and Abbb.

An interesting table emerges from this information. If it is true that a key signature returns as an

enharmonic, as we have just discovered with Gbb consisting of 13 flats and being an enharmonic

of F Major consisting of 1 flat, then a series of numbers can be plotted down for F major and its

enharmonic Keys in succession. Remember, all these notes are enharmonic of the note F:

F =1, Gbb=13, Abbbb =25, B5b = 37, D7b =49, E9b = 61

There are six movements of 12 flats before the 73rd flat would arrive, but that would actually be the

return of the note F once more so in effect this would be the same as F=1 flat, unless more letters

are used. All of these six movements of 12 are evolving from the origin of F Major. Within the third

cycle of twelve keys, for example, will be a key known as Abbbb, and this will be an enharmonic of

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F and will contain 25 flats!! This complication doesn’t matter that much. The important thing is that

you get the idea and look at the resultant table that emerges.

C

72 60 48 36 24 12 Dbb/B# 12 24 36 48 60 72 67 55 43 31 19 7 C#/Cb 7 19 31 43 55 67

65 53 41 29 17 5 Db/B 5 17 29 41 53 65 62 50 38 26 14 2 D/Bb 2 14 26 38 50 62 70 58 46 34 22 10 Ebb/A# 10 22 34 46 58 70

69 57 45 33 21 9 D#/Bbb 9 21 33 45 57 69 63 51 39 27 15 3 Eb/A 3 15 27 39 51 63 64 52 40 28 16 4 E/Ab 4 16 28 40 52 64 68 56 44 32 20 8 Fb/G# 8 20 32 44 56 68 71 59 47 35 23 11 E#Abb 11 23 35 47 59 71

61 49 37 25 13 1 F/G 1 13 25 37 49 61 66 54 42 30 18 6 F#/Gb 6 18 30 42 54 66

So, for example, the first line travelling from top center to right would read like this:

(C is zero) B# = 12 sharps, Ax# = 24 sharps Gxx# (5 sharps) = 36 sharps F(7#) = 48 sharps E(8#) = 60 sharps D(10#) = 72 C=0(73)

All the red central notes will perform this enharmonic journey, as keys, in flat/sharp pairs (they are

also different manifestations of the usual note pairs that mirror around C The Cb with 7 flats, for

example, will become Dbbb major (D less on flat – Db less on flat – C less one flat - Cb) with 19

flats when it is seen to exist within its own particular circle of twelve keys. The C# with 7 sharps will

become Bx (B add one sharp - C add one sharp - C#), with 19 sharps in its key signature.

I have stopped at 12 rotations to show how the two lots of numbers either side of the grid (it can

also be represented as a circle) can hide number sequences. For example, let’s take the B# and

its series of sharps. Each number can be broken down to a single digit. Doing so would reveal the

369 number sequence.

12 = 1+2 = 3, 24 = 2+4 = 6, 36 = 3+ 6 =9, etc.

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Therefore the above 144 major scale table can be represented as single digits:

9 6 3 9 6 3 3 6 9 3 6 9 4 1 7 4 1 7 7 1 4 7 1 4 2 8 5 2 8 5 5 8 2 5 8 2 8 5 2 8 5 2 2 5 8 2 5 8 7 4 1 7 4 1 1 4 7 1 4 7 6 3 9 6 3 9 9 3 6 9 3 6

9 6 3 9 6 3 3 6 9 3 6 9 1 7 4 1 7 4 4 7 1 4 7 1 5 2 8 5 2 8 8 2 5 8 2 5 8 5 2 8 5 2 2 5 8 2 5 8 7 4 1 7 4 1 1 4 7 1 4 7 3 9 6 3 9 6 6 9 3 6 9 3

We can observe the mirror numbers in case they can shed light to more structure. Take the

number 45 from the first grid, for example. This applies to an enharmonic of Bbb, which is already

an enharmonic of A. Also on the mirror side of the previous table D# also contains this number. Its

mirror number, 54 is an enharmonic of F# and Gb. These points are in red above.

If we now write down the 144 major scale table in movement of the circle of 5ths we will discover

the 9 number sequences (the Vedic Square) flowing at 45-degree angles across the Keys. We will

find that clockwise the motion is 5ths but anti-clockwise it is a circle of 4ths motion. This is in

accord with the fact that a 5th ‘mirrored around an axis point is a 4th:

F 4th C 5th G

The sharps and flats are broken down to a number between 1 + 9 as in the last grid.

C

D(10b) C(8b) B(6b) A(4b) Gbb Fx Ex# D(5#) C(7#) B(8#) 7 4 1 7 4 1 F/G 1 4 7 1 4 7 8 5 2 8 5 2 Bb/D 2 5 8 2 5 8 9 6 3 9 6 3 Eb/A 3 6 9 3 6 9 1 7 4 1 7 4 Ab/E 4 7 1 4 7 1 2 8 5 2 8 5 Db/B 5 8 2 5 8 2 3 9 6 3 9 6 Gb/F# 6 9 3 6 9 3 4 1 7 4 1 7 Cb/C# 7 1 4 7 1 4 5 2 8 5 2 8 Fb/G# 8 2 5 8 2 5 6 3 9 6 3 9 Bbb/D# 9 3 6 9 3 6 7 4 1 7 4 1 Ebb/A# 1 4 7 1 4 7 8 5 2 8 5 2 Abb/E# 2 5 8 2 5 8 9 6 3 9 6 3 Dbb/B# 3 6 9 3 6 9

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Across the Gb/F# point is where one focuses in order to see the swapping over from

contraction/flats to expansion/sharps. Just below this point is the pair Cb/C#. The Cb, for example,

switches over across the Gb/F# axis, and becomes the note B on the opposite side. Whereas the

C# swaps over and become the note(or rather, Key), of Db. On swapping over the key pairs head

for the top, which is to merge with C.

Highlighted in bold is the 7 5 3 1 8 6 4 2 9 number sequence of the Vedic Square. The whole of

the left hand side grid is dominated by this sequence across the 45-degree angle. Vertically one

can see the 1 2 3 4 5 6 7 8 9 number sequence. Horizontally, amongst others, is the 3 6 9 number

sequence. Across the opposite 45-degree angle on the same left hand side of the grid is the 5 1 6

2 7 3 8 4 9 number sequence. The Vedic Square has fitted in to this 144 major scale grid.

On the other side of the grid the number sequences on the left are being mirrored. Along one 45

degree angle is the 4 8 3 7 2 6 1 5 9 number sequence. Across the other 45-degree angle is the

2 4 6 8 1 3 5 7 9 number sequence. Horizontally is the 3 6 9, and vertically is the 8 7 6 5 4 3 2 1 9

sequence.

If this grid was not meant to be, that is, exist in some manner, then it is such a waste of perfectly

symmetrical relationships that give one some understanding of the musical and numerical duality

of things. Also, a heap of information is represented in one simple grid, covering so many major

scales, each 12 differing by the Pythagorean Comma. For sci-fi fun a grid like this could be

programmed into a computer for in and out of the mirror journeys. One disappears at a certain

point in one locality and can predict where to appear on the other side, like a dimensional map!

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Dorian through and through

The Dorian Mode, as seen in the chapter “Dorian Pregnancy”, reflected the dual growth of flats

and sharps representing the first twelve keys in a circle of 5ths. Is this second cycle, then, also

influenced in a similar manner? Will the Dorian around the F#/Gb axis, which is the new starting

position for the next cycle, also be seen to create the sharps and flats required to build the twelve

key signatures of the second cycle?

The Dorian that resides within the F# major scale is that of G# Dorian:

T S T T T S T T S T T T S T (DOR) G# A# B C# D# E# F# G# A# B C# D# E# F# G# (DOR)

As always the Dorian is a mode generated from its parent major scale, and it is always the same

mode on the other side of the mirror.

What we are attempting to find out is do the major keys, as they develop past six sharps or six

flats, still obtain their new flat or sharp from the notes around the Dorian Mode axis? To confirm

this here is the Gb/F# note pair combination again, which is also the musical poles, and beginning

of a new cycle.

6 11 9 7 5 10 8 6/6 8 10 5 7 9 11 6 Gb Abb Bbb Cb Db Ebb Fb Gb/F# G# A# B C# D# E# F#

After the key signature of Gb Major, which contains six flats, comes Cb Major, which contains

seven flats within its key signature. You will find that Cb Major will have every flat that is contained

in the Gb Major scale plus a new one:

Cb Db Eb Fb Gb Ab Bb Cb = Cb Major (new flat is Fb)

Here it is on a musical stave:

Gb Major

Cb major

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Around Gb/F# axis, the C#, which is across the mirror from Cb, contains seven sharps, which will

be all the sharps contained within the F# Major scale key signature plus one new sharp:

C# D# E# F# G# A# B# C# = C# Major – new sharp is B#

F# major

C# major

What should happen next is that we marvel at the consistency of the structure with its inbuilt

symmetry. Fb and B# are indeed mirror partners within G# Dorian, but they are also still mirror

partners of D Dorian (as E and C, the enharmonic equivalents), establishing an unbroken link here

with C Major. It may6 not be easy to see the Fb and the B# mirror, but basically one has to raise

the note B on one side, whislt lowering the note E# on the other. The E# is really an enharmonic of

the note F.

D Dorian and G# Dorian are also a tri-tone interval apart.

Things have literally swapped over at the b5 (tri-tone) point. Enharmonic equivalent notes have

stepped in to continue the thread so that it may continue unbroken, as in the case of Fb being an

enharmonic of E, and B# an enharmonic of C. If you persist with studying both the G# and D

Dorian dual scales and the mirror note positions you will find that they are reflecting the same

symmetry only swapped over. And the only difference in pitch is the 80:81 ratio of the Pythagorean

Comma.

Therefore the Dorian pregnancy is akin to the creation of symmetrical relationships at every octave

of the spectrum. It teams up with the number 9 and 4.5, as visible and invisible axis, and keeps it

all united. But united by a continual in and out of the mirror journey. The 144 major scale grid

eventually marries together the Vedic square and the Mode box.

185

Chapter twenty five

Visible and invisible axis points within numbers

A number like 101 has its axis point visible, the middle 0 being the mirror axis, with the number 1

either side of it. Then will come 111 with the axis again being visible at the central number 1, and

this time the number 1 merging from both sides to its centre. There are nine three-digit numbers,

with a 1 either side, with this potential, as will be seen.

A number like 1001 has its axis invisible, in between the 00, with a 01 either side of this axis.

Again there are nine four-digit numbers with this potential. In fact, regardless of the amount of

digits there are always nine such sets of numbers with such axis points.

When these types of numbers belonging to their respective groups are added together one will

consistently see the emergence of the second number sequence of the Vedic Square, 2 4 6 8 1 3

5 7 9.

Firstly, here are the three digit numbers with the visible axis. 121, for example, has the number 2

as the mirror axis, with a 1 either side. The totals in red are further broken down to single digit, in

order to expose the second number sequence of the Vedic Square.

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Visible axis

101 202 303 404 505 606 111 212 313 414 515 616 121 222 323 424 525 626 131 232 333 434 535 636 141 242 343 444 545 646 151 252 353 454 555 656 161 262 363 464 565 666 171 272 373 474 575 676 181 282 383 484 585 686 191 292 393 494 595 696 1460 2470 3480 4490 5500 6510 2 4 6 8 1 3

707 808 909 717 818 919 727 828 929 737 838 939 747 848 949 757 858 959 767 868 969 777 878 979 787 888 989 797 898 999

7520 8530 9540 5 7 9

Invisible axis

1001 2002 3003 4004 5005 6006 1111 2112 3113 4114 5115 6116 1221 2222 3223 4224 5225 6226 1331 2332 3333 4334 5335 6336 1441 2442 3443 4444 5445 6446 1551 2552 3553 4554 5555 6556 1661 2662 3663 4664 5665 6666 1771 2772 3773 4774 5775 6776 1881 2882 3883 4884 5885 6886 1991 2992 3993 4994 5995 6996 14960 24970 34980 44990 55000 65010 2 4 6 8 1 3

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7007 8008 9009 7117 8118 9119 7227 8228 9229 7337 8338 9339 7447 8448 9449 7557 8558 9559 7667 8668 9669 7777 8778 9779 7887 8888 9889 7997 8998 9999

75020 85030 95040 5 7 9

In the next example the totals increase by a value of 10010, and have a visible axis at the centre.

Visible axis

10001 20002 30003 40004 50005 11111 21112 31113 41114 51115 12221 22222 32223 42224 52225 13331 23332 33333 43334 53335 14441 24442 34443 44444 54445 15551 25552 35553 45554 55555 16661 26662 36663 46664 56665 17771 27772 37773 47774 57775 18881 28882 38883 48884 58885 19991 29992 39993 49994 59995 149960 249970 349980 449990 550000 2 4 6 8 1

60006 70007 80008 90009 61116 71117 81118 91119 62226 72227 82228 92229 63336 73337 83338 93339 64446 74447 84448 94449 65556 75557 85558 95559 66666 76667 86668 96669 67776 77777 87778 97779 68886 78887 88888 98889 69996 79997 89998 99999

650010 750020 850030 950040 3 5 7 9

These next sets have the axis in between the central numbers:

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Invisible axis

100001 200002 300003 400004 500005 111111 211112 311113 411114 511115 122221 222222 322223 422224 522225 133331 233332 333333 433334 533335 144441 244442 344443 444444 544445 155551 255552 355553 455554 555555 166661 266662 366663 466664 566665 177771 277772 377773 477774 577775 188881 288882 388883 488884 588885 199991 299992 399993 499994 599995

1499960 2499970 3499980 4499990 5500000 2 4 6 8 1

600006 700007 800008 900009 611116 711117 811118 911119 622226 722227 822228 922229 633336 733337 833338 933339 644446 744447 844448 944449 655556 755557 855558 955559 666666 766667 866668 966669 677776 777777 877778 977779 688886 788887 888888 988889

` 699996 799997 899998 999999 6500010 7500020 8500030 9500040 3 5 7 9

The last two examples produce individual totals increasing by a value of 100010 and 1000010

respectively. One can see from these few examples that the pattern is set. The second number

sequence of the Vedic square, 2 4 6 8 1 3 5 7 9, will consistently appear. The last example is only

one of the possibilities within the set of six numbers evolving from either a visible or invisible axis.

The number 101101 is also a number with an invisible axis, and other lists are possible. The next

number in this particular list would be 102201 and so on. Here is a list covering these numbers:

101101 102201 103301 104401 105501 106601 107701 108801 109901

1049510 2

The number 2 is still the single digit number that this sequence distils to. Knowing that from now

on the same number sequence of the Vedic Square will emerge we can leave this particular list

and move on to a different mirror number combination.

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110011 – with its in-between axis point this too will produce the same number sequence of the

Vedic square.

We leave the mirror number lists at this point to concentrate on any possible musical applications

for such numbers. Because these numbers are related in some respects to the visible/invisible

aspects of the mode box, and to the number sequence of the Vedic Square, it is worth viewing

these numbers as cycles-per-second and listing the vibrations that would emerge. We begin with

the first mirror number total 1460, which was the very first total of the first example, and then move

on to the next total 2470 etc.

1460 cps = Gb 2470 cps = Eb 3480 cps = A 4490 cps = Db 5500 cps = F 6510 cps = Ab 7520 cps =Bb 8530 cps = C 9540 cps = D

Gb Ab A Bb C Db D Eb F

One can see the tri-tone relationship here between Gb and C. The overall scale also has a very

Phrygian type sound about it. One could view it as F Phrygian with the A and D as added notes.

Doubling or halving a number produces octaves so it would be possible to play this scale all within

one octave.

Notes can also be shown as belonging to each individual list:

101 = G-qt 111 = A 121 = B-qt 131 = C 141 = Db

151 = D-qt 161 = Eb-qt 171 = E-qt 181 = F-qt 191 = Gb-qt

This hexagonal cluster resonator consists of the frequencies for the numbers

101 1001 10001 100001 1000001 100001 10001 1001 101:

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G G G

B B B B Eb Eb Eb Eb Eb G G G G G G

B B B B B B B

G G G G G G

Eb Eb Eb Eb Eb

B B B B G G G

The cluster consisting of three G notes are all tuned to 101 cycles per second. The next cluster

consisting of B notes are all tuned to 1001, and so on. Observe that we have uncovered one of the

Augmented triangles of keys, discovered in the Mode Box – G B Eb, yet remember that the notes

are all a quarter tone flat. It would be a question of tuning a mode box down by this interval (not to

mention the myriads of other tunings, as not all the notes of any mode box would be strictly a

quarter tone). Therefore we can join this particular cluster resonator with the other triangle that

goes on to make the Circle of Tones. So where is this other triangle, namely F A C#? It will be

found within the numbers 707 7007 70007 700007 7000007 700007 70007 7007 707. To get the

image of this simply replace the above hexagram with the appropriate notes, which will be tuned

to the new frequencies. Any note can roots of both clusters of resonators.

The hexagonal cluster resonators are placed at 45% angles from their respective partners. Here

are the four prevalent relationships:

191

G B Eb C E G#

D F# Bb F A C#

The other two possible triangles that go on to make the other circle of tones structure, C E G# and

D F# Bb, will be found within the numbers 131 1331 13331 133331 1333331 etc for C E G#, and

181 1881 18881 188881 etc for D F# Bb. It must be held in mind some of the notes are close

approximations.

Here is a rough sketch of hexagonal resonator formations that are the triangular frequencies of the

circle of tones, following in/out of mirror movements. The format would be designed to be looped

continuously.

G# C E Bb D F# C E G# D F# Bb E G# C F# Bb D

F# G# Bb C D E

Bb D F# C E G# D F# Bb E G# C F# D Bb G# C E

The notes in red are the axis points that exist between the triangles of frequencies. Together these

axis points form yet another circle of tones, showing how the triangular frequencies shift through

every point of the scale. The size of this ‘machine’ would depend on the individual resonators that

are packed to form each hexagram cluster.

192

In the present form the largest resonator would be vibrating at 131 cps (for the C). The smallest

resonator would be resonating at 1888881 cps (for the D). This may be a hybrid Nano-resonator!

Here perhaps one needs to use the doubling and halving rule in music, as this produces octaves.

One can experiment with relative sizes of the resonators until something that has the potential of

being built emerges.

These visible axis and invisible axis number lists have exposed once again the Circle of tones.

One cannot call this structure a phenomena of no importance, as it is the only structure that

signifies unity of the mirror sides. These cycles exposing the Circle of Tones are based on number

sequences that are intrinsic to all of Number itself. There was correlation in the Fibonacci number

flows and every single example given so far.

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A link between a triangle of frequencies and the body's response

It had already occurred to me that there may be a link between the information in these mirrored

grids and the human body, and even perhaps consciousness itself. By uniting natural cycles,

exposing the mirror side and realizing that the system becomes one Whole, rather than two

disjointed sides, does lead one to wonder if the brain or mind itself has a relationship with this kind

of phenomena existing in these grids and scales. What is the relationship between the firing of

neurons at specific frequencies, and this overall Whole system that is exposed through mirroring?

Is there a unity aspect to mind or consciousness, in the same way there is this unifying mirror

structure within music and number?

The next picture is taken from a book by David Gibson, called “The art of mixing”

It is a guide to sound engineers who produce music for us to listen to. This picture below is a

sound engineer's guide to the various parts of the body that respond to specific frequency ranges.

I was rather surprised by the result.

The numbers given here relate directly to one of the triangle of Keys, namely, the G Eb B.

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40Hz = Eb

800 Hz = G

1000 Hz = B

5000 Hz = Eb

10000 Hz = Eb

Therefore, there is an aspect of the human response toward certain frequencies, and it does relate

to one of the triangles of this circle of tones, the G Eb B triangle..

Obviously some of the pitches involved are slightly sharp or slightly flat, but they do fall within this

G E Bb range. The mind and body seem responsive to this particular triangle of frequencies.

195

Chapter twenty six

Pi mirrorrorrim iP

A mathematician called Jerry Iuliano found that the first 144 Pi numbers added to 666.

The first 144 movements of the Pi number will have 144 mirror movements. Here are the first 144

Pi numbers

1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8

2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4

8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9

Here would be the mirror numbers. The first few would look like this. The 3 is in the axis position

together with the 5. The 3.1415 etc and 5.8584 etc are Pi mirror Pi.

4 8 5 8 .(5/3). 1 4 1 5:

Here are the 144 mirror numbers of Pi:

8 5 8 4 9 7 3 4 6 4 1 9 2 9 6 7 6 1 5 3 7 3 5 6 6 1 6 7 2 9 4 0 7 1 1 5 8 9 2 8 3 9 6 9 9 6 2 4 8 0 4 1

7 0 9 2 5 9 5 5 4 9 7 6 0 2 1 8 3 5 0 3 7 1 3 7 0 1 9 9 1 3 7 1 0 6 5 1 7 4 6 5 7 8 8 2 0 3 2 9 1 7 8 5

1 0 1 3 4 8 6 7 1 7 6 0 3 3 5 2 0 9 6 1 5 5 3 0 9 4 4 0 4 1 7 7 6 8 2 7 4 6 4 9

These 144 mirror numbers of Pi also add up to 666. The chances of this happening do not seem

that great when one analyse each step and mirror step, building up the dual totals.

(54) 6 4 3 7 9 4 8 5 8/1 4 1 5 9 2 6 5 3 (36)

The first nine numbers either side are out of synch by 18. Here are the next nine numbers either

side:

(45+54=99) 1 6 7 6 9 2 9 1 4/5 8 9 7 9 3 2 3 8 (54+36=90)

The right hand side has caught up by nine places and one can see that the number 9 itself is

already playing a central role in the flow of the Pi numbers either side. Every block of 9 so far has

produced a number that will break down to a single digit sum of 9.

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(42) 6 1 6 6 5 3 7 3 5/4 6 2 6 4 3 3 8 3 (39)

No 9 on the third cycle but we do have a 3 and a 6 being the 42 and 39 distilled to single digit.

(36) 5 1 1 7 0 4 9 2 7 /2 7 9 5 0 2 8 8 4 (45)

The two sides are still out of synch by 3. The dual nine digit totals again break down to a 9.

( 63) 9 9 6 9 3 8 2 9 8 /1 9 7 1 6 9 3 9 9 (54)

The two totals break down to a 9, and they are now out of synch by 12.

( 32) 0 7 1 4 0 8 4 2 6 /3 7 5 1 0 5 8 2 0 (31)

Obviously at some point the two sides will begin merging so that at the 144th number they are both

totalling exactly 666. One can see now just how uncanny it all is.

In the context of the examples in this book, the 666 signifies a point of balance between the two

sides of the mirror.

197

Chapter twenty seven

Chaos/ Non-relating Vedic Square

If the nine number sequences of a Vedic Square were like a natural seed from which many other

systems come into reality, then this order would possess a chaos counterpart. This chapter

speculates the possible nature of that chaos, as would be seen according to the Vedic Square

itself.

Here are some numbers consisting only of number sequence partners of the Vedic Square, plus

the 9 as an axis:

9 1 8 9 1 8 9 1 8 9 1 8Swap partner:

9 8 1 9 8 1 9 8 1 9 8 1

Here are other numbers that can be seen in a similar way:

297 - 792 - 396 – 693 - 495 - 594

We can see that what would give the Vedic square order is the mirror number pairs coexisting

around the 9 axis (and the 4.5 axis too).

Therefore to build the Vedic square of chaos (a shadow Vedic square), one would include non

mirror numbers around the nine axis.

192 193 194 195 196 197

Here the number 1 is joined around the 9 axis with every other number except its own number and

the number 8, which is the number 1’s correct mirror partner.

Here is the whole square that emerges by using this logic:

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Chaos Vedic square

192 193 194 195 196 197

291 293 294 295 296 298

391 392 394 395 397 398

491 492 493 496 497 498

591 592 593 596 597 598

691 692 694 695 697 698

791 793 794 795 796 798

892 893 894 895 896 897

If the Vedic Square proper is viewed as a “relating” square, then this shadow square can be

viewed as “non-relating”.

There is a profound result that is displayed within this square. The peculiarity exists at the 45-

degree angle. Here is the relevant section of the above diagram showing the flow at the 45-degree

angle, and the swap-over effect at the 4.5 position:

192

291 293

392 394

493/

596

695 697

796 798

897

Notice how the 596 and the 695 are opposites of each other, with the 5/4 and 6/3 sequences

finding themselves once more through this non-relating environment. The 394 and 596 are next to

join their true symmetrical flows. After this, the 293 and the 796 will be mirrors of each other, then

the 392 mirroring the 697, and so on. All of them have the 4.5 as the mean number, and it can be

seen as a swap-over point at the centre of this square.

Structure of a unique swapping-over kind is always displayed at the 45-degree angle, whether it

be with the Major scale and its modes, or the Vedic Square, or the Fibonacci numbers. Above, the

45-degree angle is the point that makes the “Chaos” square suddenly relate.

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Chapter twenty eight

Central interval

Some basic understanding of intervals will be needed in order to follow what is being said here.

All intervals and their inversions within music are born from the visible Root position and at the F#

invisible axis point, at the tri-tone/4.5 position. This can be shown clearly using the ratio

associated with the interval in question.

The tri-tone position can be seen to flip the intervals over across the two number 45s, even

perhaps at the usual 45-degree angle.

1:1 16:15 Diatonic semi-tone 9:8 Major 2nd 6:5 Minor 3rd

5:4 Major 3rd

4:3 Perfect 4th

64:45 Diminished 5th (also known as a tri-tone) 45:32 Tri-tone (also known as diminished 5th) 3:2 Perfect 5th

8:5 Minor 6th

5:3 Major 6th

16:9 Minor 7th

15:8 Major 7th 2:1

Let’s take the Major 2nd interval as an example. The ratio for this is 9:8. The ratio 16:9 is that of the

Minor 7th interval, which is the Inversion of the Major 2nd’, so it's across the mirror from the F# axis.

We can see that if we flip over the 9:8 ratio, and then double the number 8, we have the 16:9 ratio.

Doubling the 8 creates an octave of the same note. This method works for all intervals and their

inversions. There is even an in and out of mirror flow here, and one should see itif they focus on

the tritone position, and the two 45s. The number three above and below a 45 moves to a number

4 and 8 below. This shows there has been a flipping over of the ratio where the number on one

side was doubled and switched sides. These are the flows to focus on in order to see how the

intervals sit in their default positions being reflected around two axis points, the C and the tritone.

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The Centre of the Octave

Another way of viewing the previous information is to use Cents. An octave covers 1200 cents. At

this 600 cent mark the symmetrical reflection that occurs around the 1/1 will also occur here, and

this is linked to the square root of 2.

128:125 Diminished Second = 41.059 cents

256:243 Pythagorean Minor Second = 90.225 cents

81:64 Pythagorean Third = 407.820 cents

1024:729 Low Pythagorean tri-tone = 588.270 cents

Square Root of 2 = ET Tri-tone = 600 cents = Centre of Octave

729:512 High Pythagorean tri-tone = 611.730 cents

128:81 Pythagorean Sixth = 792.180 cents

243:128 Pythagorean Major Seventh = 1109.775 cents

125:64 Augmented Seventh = 1158.941 cents

The Many Tritones

These are all well known tritone interval ratios:

1.415882656 649539 / 458752 602.0420445

1.415738192 127575 / 90112 601.865397

1.414954501 281600 / 199017 600.9067958

1.414486748 16384 / 11583 600.3343924

1.414285714 99 / 70 600.0883238 600 sq root 21.414141414 140 / 99 599.9116762

1.41394043 11583 / 8192 599.6656076

1.413473011 199017 / 140800 599.0932042

1.412690574 180224 / 127575 598.134603

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These tritones exist within various tunings, such as meantone or just intonation. The very centre

would be 600 cents. If one concentrate on the invisible centre, it will be noticed that, within the two

facing equations, the .0883238 and the .9116762, are the mirror number pairs from the Vedic

Square (except for where they have been rounded off at the end).

0883238

9116762

In between each number pair is the 4.5 axis position. This means that, not only is the 600 another

4.5 position, but it is also equal to the number zero, when the mirror number pairs of the Vedic

Square are seen as Indig numbers instead.

This relationship with the mirror number pairs of the Vedic Square persists through the chart of

different Tritone ratios. The chart expands from the 600, 4.5, 0 mark, both above and below.

The .6656076 is complemented by the 3343924

6656076

3343924

Again all vertical pairs add up to 9, and there will be a 4.5 axis position between each pair.

The full Tritone ratio list is much larger than the one presented above. Yet the link with the mirror

number pairs is never broken. And the reason for this is quite simple. It is to do with the ratios

themselves. 99/70 above the central mark, and 140/99 below it. The number 70 from the first ratio

doubles to 140, and switches sides, from a denominator to a numerator. The 99 has switched

sides too, but has remained the same number. This is basically the whole technique required to

read the whole tritone ratio list, as it was for the chromatic scale interval list.

202

Chapter twenty nine

Exponentials and the 3 6 9 positions

Here are the first nine numbers multiplied together in a sequential manner:

1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 = 362880

Here it is again with each separate total along the way highlighted:

1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 1 2 6 24 120 720 5040 40320 362880

The totals broken down to single digit value:

1, 2, 6, 6, 3, 9, 9, 9 = 45

The numbers above, 1, 2, 6, 24, 120, 720 etc, will now be discovered in an unusual way. The

exponentials will be built up systematically, and then the totals filtered until only one number is

shown at the bottom of the grid (excluding the number zero). It will become obvious what this

means as we go along.

As usual the examples here are not brain-achingly difficult, but more an exploration of the simple

root structures that most things are a part of, and then to observe these things in their symmetries

together, looking for anything that may unite the two sides of the duality up into one cohesive

whole. With that in mind, here are a few observations on the simple behaviour of powers. We

start with the first nine numbers squared:

1^2 = 12^2 = 43^2 = 94^2 = 16 = (1+6) =75^2 = 25 = 76^2 = 36 = 97^2 = 49 = 48^2 = 64 = 19^2 = 81 = 9

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The totals are now laid horizontally and slowly filtered like this:

1 4 9 16 25 36 49 64 81

3 5 7 9 11 13 15 17

2 2 2 2 2 2 2

The difference between 1 and 4 is 3, so this becomes the in-between total on the second line, and

so on. The second line is really a snippet of the Vedic square’s second number sequence – 2 4 6

8 1 3 5 7 9, and as you can see, the number 2 is the final in-between total.

Also of interest is the appearance of the same number sequence running at the 45-degree angle

of the Vedic Square. 1 4 9 7 7 9 4 1 9. This sequence can be seen as containing a mirror point in

between the two 7’s, or at either end travelling toward the centre, which is the same behaviour

shown by the number sequences of the Vedic square and musical intervals.

The next totals that are focused on are the first nine numbers cubed:

1^3 = 1

2^3 = 8

3^3 = 27 = 94^3 = 64 = 15^3 = 125 = 86^3 = 216 =97^3 = 343 = 18^3 = 512 = 89^3 = 729 = 9

The single digit sequence that has appeared is quite obviously a 189. Again the 9 is seen to

appear at the 3rd 6th and 9th positions. Here are the totals laid out horizontally and filtered:

1 8 27 64 125 216 343 512 729

7 19 37 61 91 127 169 217

12 18 24 30 36 42 48

6 6 6 6 6 6

The number 6 is the grand in-between total, and it is also a total from one of the nine numbers that

were sequentially multiplied at the beginning of this chapter. There are more sequences appearing

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too. The second line horizontally is made up of the 7 1 1 sequence, and will remain a 711 along

this second line infinitely (the 19 = 1, the 37 = 1, the 61 = 7 and so on). The third line is made up of

a 3 9 6.

Next are numbers to the power of 4:

1^4 = 1 = 12^4 = 16 = 73^4 = 81 = 94^4 = 256 = 45^4 = 625 = 46^4 = 1296 = 97^4 = 2401 = 78^4 = 4096 = 19^4 = 6561 = 9

Again the 9 appears at the 3rd 6th and 9th positions. This sequence is really a rearrangement of

the sequence that emerged under the squares totals. One saw that the 45-degree angles of the

Mode Boxes also had a shifting sequence, made up of the Circle of Tones. The sequence there is

seen to contain another axis at the 4.5. Even in these number experiments we are seeing the 4.5

is once more catalyst for another layer of symmetry. If one places an asterisk at the 4.5 position in

the numbers above (in between the two 4’s), they will see the symmetry, especially if more than

the first nine totals were shown. The final 9 will be the next axis for the same symmetrical flow,

and the next 4.5 in that sequence will also be another in-between axis reflecting the same flow

combinations as the number 9.

Here are the totals laid out horizontally and filtered:

1 16 81 256 625 1296 2401 4096 6561

15 65 175 369 671 1105 1695 2465

50 110 194 302 434 590 770

60 84 108 132 156 180

24 24 24 24 24

The end total here is 24, another number from the progressive multiplication of the first nine

numbers. The third line hides its own number sequence, a 5 2 5. In fact every line really carries a

hidden sequence (the fourth line hides a 6 3 9 sequence), These sequences can be used either

205

as compositional aids and idea generators, or perhaps as a focus for scientific purposes. Another

coincidence with this 4.5 position occurs in the next example too.

1 ^ 5 = 1 = 1

2 ^ 5 = 32 = 5 3 ^ 5 = 243 = 9 4^ 5 = 1024 = 7 5 ^ 5 = 3125 = 2 6 ^ 5 = 7776 = 9 7 ^ 5 = 16807 = 4 8 ^ 5 = 32768 = 8 9 ^ 5 = 59049 = 9

The 9’s are at the 3 6 9 points, but the overall number sequence can only be seen as some form

of symmetry if we view the 1 and 8 being partners on the outer edge, and the 5/4 and then 2/7

partnering each other.

1 32 243 1024 3125 7776 16807 32768 59049

31 211 781 2101 4651 9031 15961 26281

180 570 1320 2550 4380 6930 10320

390 750 1230 1830 2550 3390

360 480 600 720 840

120 120 120 120

The number 120 is the final total here, and so far there has been a 2, 6, 24 and 120, as the end

totals for the first four powers.

Notice how the last two examples contain totals that are in close relationship to each other. For

example, at the same second position the 32 is twice the 16 of the previous example. Then the

243 is three times the 81, 1024 is four times the 256 etc.

Numbers to the power of one were omitted for obvious reasons, but here they would be in any

case:

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1 1

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So now we see that 2 is twice as much as 1, and that 6 is three times as much as 2, and that 24 is

four times as much as 6, and 120 is five times as much as 24. We can even predict the rest of the

numbers emerging here without needing to plot the totals. Yet in not doing that we would not know

the other hidden sequence at the far right, so it is a matter of plotting these anyway in order to

have them if only as reference and fun. Numbers to the sixth power must contain a final total,

which has been obtained by the in-between numbers, as shown in the other examples above, and

it must be six times 120, which 720.

1 ^ 6 = 1 2 ^ 6 = 64 = 1 3 ^ 6 = 729 = 9 4 ^ 6 = 4096 = 1 5 ^ 6 = 15625 = 1 6 ^ 6 = 46656 = 9 7 ^ 6 = 117649 = 1 8 ^ 6 = 262144 = 1 9 ^ 6 = 531441 = 9

The axis points are fairly self evident here.

1 64 729 4096 15625 46656 117649 262144 531441

63 665 3367 11529 31031 70993 144495 269297

602 2702 8162 19502 39962 73502 124802

2100 5460 11340 20460 33540 51300

3360 5880 9120 13080 17760

2520 3240 3960 4680

720 720 720

Here is the 720. For there to be consistent totals that move according to strict rules means that

there must be symmetry occurring from the very centre of each sequence, the 4.5.

The main reason for carrying out these simple experiments is to highlight the similarities between

the mode boxes and the Vedic square number sequences, both creating structure from the 4.5

axis, and to show how the 3 6 9 sequence consistently acts as junction points to the number 9 (or

4.5).

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1 ^ 7 = 1 2 ^ 7 = 128 = 2 3 ^ 7 = 2187 = 9 4 ^ 7 = 16384 = 4 5 ^ 7 = 78125 = 5 6 ^ 7 = 279936 = 9 7 ^ 7 = 823543 = 7 8 ^ 7 = 2097152 = 8 9 ^ 7 = 4782969 = 9

1 128 2187 16384 78125 279936 823543 2097152 4782969

127 2059 14197 61741 201811 543607 1273609 2685817

1932 12138 47544 140070 341796 730002 1412208

10206 35406 92526 201726 388206 682206

25200 57120 109200 186480 294000

31920 52080 77280 107520

20160 25200 30240

5040 5040

The 5040 is seven times the 720. It may well be that the eight times 5040 = 40320 will not appear

next, because there will not be enough lines to complete the differences, resulting in one last total

which will not be the 40320. But that remains to be seen!

1 ^ 8 = 1 2 ^ 8 = 256 = 4 3 ^ 8 = 6561 = 9 4 ^ 8 = 65536 = 7 5 ^ 8 = 390625 = 7 6 ^ 8 = 1679616 = 9 7 ^ 8 = 5764801 = 4 8 ^ 8 = 16777216 = 1 9 ^ 8 = 43046721 = 9

This particular number sequence appeared in the very first example when working with squares.

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1 256 6561 65536 390625 1679616 5764801 16777216 43046721

255 6305 58975 325089 1288991 4085185 11012415 26269505

6050 52670 266114 963902 2796194 6927230 15257090

46620 213444 697788 1832292 4131036 8329860

166824 484344 1134504 2298744 4198824

317520 650160 1164240 1900080

332640 514080 735840

181440 221760

40320

Well, there is the total after all. I decided not to correct the earlier suggestion that the final number

would not appear, mainly because I type at the computer and am actually working it out as I go

along! Nothing like a ‘here and now’ moment with a reader!

There is only numbers to the power of nine to go. Surely this example will run out of lines before

the final total appears? The total we would need is nine times the 40320 = 362880. You may

remember this total from the first example:

1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 1 2 6 24 120 720 5040 40320 362880

In fact you can see that all the totals that we have uncovered are mentioned here. This is perhaps

a little obvious to those who are proficient at mathematics, but for those looking for the simple

structures that hold nature up, these examples show how things originate before they are made

more and more complicated. Well perhaps not as obvious as some may suggest. To get the totals

above to match has been a journey of gathering in the in-between totals of the first nine powers,

using the numbers 1 to 9. Here are the totals for numbers to the ninth power.

1 ^ 9 = 1 2 ^ 9 = 512 = 8 3 ^ 9 = 19683 = 9 4 ^ 9 = 262144 = 1 5 ^ 9 = 1953125 = 8 6 ^ 9 = 10077696 = 9 7 ^ 9 = 40353607 = 1 8 ^ 9 = 134217728 = 8 9 ^ 9 = 387420489 = 9

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1 512 19683 262144 1953125 10077696 40353607 134217728 387420489

511 19171 242461 1690981 8124571 30275911 93864121 253202761

18660 223290 1448520 6433590 22151340 63588210 159338640

204630 1225230 4985070 15717750 41436870 95750430

1020600 3759840 10732680 25719120 54313560

2739240 6972840 14986440 28594440

4233600 8013600 13608000

3780000 5594400

1814400

Well it was coming, the fact that one would run out of lines here. The 362880 has not appeared. In

its place, however, is 362880*5 = 1814400. This would be enough to convince some structure

exists using this method calculating in-between totals and number sequences in general.

Surely this is all evidence that balance exists, a perfect symmetry behind all things, even though

seemingly asymmetrical at times, nevertheless at a deeper layer the symmetry is really unbroken.

A Universe may come “crashing” down, but the perfect symmetry is not affected at the level where

position can adjust and re-adjust around its dance with the number 9.

The logic behind the number sequences of the Vedic square is that one sequence is built in

contrary flow to another sequence, and both sequences must always equal 9. In music there are

what are called inversions, where an interval like a 4th becomes a 5th when inverted, and both

numbers also equal a 9. With this kind of logic we can combine powers in a similar way:

1 * 1 = 1 paired with 8 * 8 = 64, with a difference of 63

2 * 2 = 4 paired with 7 * 7 = 49, with a difference of 45

3 * 3 = 9 paired with 6 * 6 = 36, with a difference of 27

4 * 4 = 16 paired with 5 * 5 = 25, with a difference of 9

The number 18 separates all the differences.

63 + 45 + 27 + 9 = 144

If we add the two individual totals together instead we get:

65 + 53 + 45 + 41 = 204

The differences between these totals is 4 , 8 and 12.

210

Another way of pairing together exponentials and the number sequence partners of the Vedic

square is to use something like 1 ^ 2 and 8 ^ 7. This way there is a 1/8 pair with a 2/7 pair. Again

there should appear some structural significance centred around the number 9:

1 ^ 2 = 1 paired to 8 ^ 7 = 2097152, difference of 2097151 = 25 = 7 2 ^ 2 = 4 paired to 7 ^ 7 = 823543, difference of 823539 = 30 = 3 3 ^ 2 = 9 6 ^ 7 = 279936 279927 = 36 = 9 4 ^ 2 = 16 5 ^ 7 = 78125 78109 = 25 = 7 5 ^ 2 = 25 4^ 7 = 16384 16359 = 24 = 6 6 ^ 2 = 36 3 ^ 7 = 2187 2151 = 9 7 ^ 2 = 49 2 ^ 7 = 128 79 = 7 8 ^ 2 = 64 1 ^ 7 = 1 63 = 9 9 ^ 2 = 81 9 ^ 7 = 4782969 4782888 = 45 = 9

As always the 9 appears at the 3rd 6th and 9th positions.

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Chapter thirty

Differences Triangles

In chapter four the invisible aspect of the triangles of keys was seen to exist between notes 10 and

17. The approximation of this number is seen to occur here too, where the triangles touch. The

chart is read as 90/1, 90/2, 90/3 etc:

9th 8th 7th 6th 5th 4th 3rd half whole

10 11.25 12.85 15 18 22.5 30 45 90

1.25 1.6 2.15 3 4.5 7.5 15 45

0.35 0.55 0.85 1.5 3 7.5 30

0.2 0.3 0.65 1.5 4.5 22.5

0.1 0.35 0.85 3 18

0.25 0.5 2.15 15

0.25 1.6 12.85

1.35 11.25 9.9 13.5 (4.5)

17.1

15.75 7.65

14.15 7.4 8.75

12 6.85 8.5 8.75

27 6 8.15 8.65 8.9

22.5 4.5 7.5 8.35 8.7 8.8

33 1.5 6 7.5 8.15 8.45 8.65

54 12 1.5 4.5 6 6.85 7.4 7.75

99 54 33 22.5 27 12 14.15 15.75 17

It is the black triangle, and the number 90 that is further halved, then divided by three, then four

etc. The red triangle is the relative mirror number of each number from the black triangle.

Therefore the red triangle is the mirror reflection according to the clockwise and anti-clockwise rule

of the number pairs from the Vedic Square.

212

We see two triangles meeting at the 13.5, with what should be a 10 and 17 either side. The small

discrepancies in the fractions have made it 17.1 and 9.9, but nevertheless it is a 13.5 where the

meeting point between both sides of the mirror is, and this is a 4.5 axis.

The black triangle is distilled further and further until one overall number represents it at the

bottom, and it meets its mirror counterpart there. The 11.25, for example, crosses over and is

mirrored as15.75, whilst the 1.35 crosses over and mirrors the 7.65, and so on further up the dual

lists.

The red and black arrows show the two 45-degree angles where the results echo those of the top

and bottom horizontal lines of the two triangles respectively. Again we have replication or

assimilation at the 45-degree angle, as in the Mode boxes.

The top horizontal line of the black triangle adds up to 254.6. This number breaks down to 17

when the individual digits are added together (2+5+4+6=17). The bottom horizontal line of the red

triangle equals 294.4, which breaks down to 19 (2+9+4+4=19), which further breaks down to 10.

Eventually it breaks down all the way to its original Vedic Square number sequence pair, the 1/8.

Yet these totals are not absolutely exact. There will be minute differences because of some of the

fractions used. Nevertheless the mirror triangle is based purely on the non-mirror one and one can

see that they do agree with each other in some respects already.

We can also start with the top right hand side of the black triangle and add the totals from the

bottom left hand side, like this:

90 + 99 = 189 = 1 + 8 + 9 = 18 = 1 + 8 = 9

45 + 54 = 99 = 9 +9 = 18 = 1 + 8 = 9

33 + 30 = 63 = 6 + 3 = 9

22.5 + 22.5 = 45 = 4 + 5 = 9

18 + 27 = 45 = 4 + 5 = 9

15 + 12 = 27 = 2 + 7 = 9

12.85 + 14.15 = 27 = 2 + 7 = 9

11.25 + 15.75 = 27 = 2 + 7 = 9

10 + 17 = 27 = 2 + 7 = 9

Tracing the shape of the letter Z would show the effect of the 45-degree angles reproducing the

totals at the horizontal lines perfectly. In fact check carefully enough and you will see that

everything really adds up to the number 9. Even the touching points of the triangles on their own

213

add up to 27, which breaks down to 9. This may be a fair indication that the procedure for finding

the ‘mirror’ numbers is correct.

With a little patient study one should be able to spot axis points galore within the sub-totals

running along the 45-degree angles.

214

Chapter thirty one

The Surface of the Torus Universe

Below is an image created by Marco Rodin. Some of these sequences plotted exist because of

binary , and doubling or halving number. His work is highly recommended and can be found on

various web sites. The number 7 has been associated with many phenomena, none more

enticing than the claim that it exposes a number sequence that is identical to a numerical

sequence that flows across the surface of a torus. The universe is said to resemble a torus

shape.

215

Firstly, it is easy to see how the 9, in the centre of the above 3*3 square, is axis point to the

number sequence pairs of the Vedic Square. Around the 9 axis in the centre is the 1/8 and 4/5 at

45-degree angles. Then there is the 2/7 horizontally and the 3/6 vertically. Somewhere in all this

is the 4.5 invisible axis. Well, I would say it is invisibly sitting over the number 9 itself. The mean

number between the 1/8 pair, for example, is 4.5, and it would fall directly over the number 9 in

the centre. The same goes for the 4/5, the 2/7, or 3/6, the mean number is always 4.5.

This 3*3 square can actually be mirrored, and one can gain a glimpse into one aspect of its

“shadow self” as it were. All one does here is replace the numbers that make up the magic

square with their mirror equivalent. By mirror equivalent is meant its counter-cycling partner. The

first number sequence of the Vedic square is simply 1 2 3 4 5 6 7 8 9, for example. This can be

seen to be cycling clockwise as a sequence. Its mirror will be cycling counter-clockwise to this

first number sequence, and in the same proportions. The 8 7 6 5 4 3 2 1 9 sequence of the Vedic

square shows this. Therefore by building a mirror 3*3 square one is simply reversing the

clockwise and anti-clockwise relationships. Instead of the number 1, for example, the mirror

square will contain the number 8 in its place, the number 6 will be replaced by the number 3 , and

so on:

8 3 4

7 0/9 2

5 6 1

The two 3*3 squares can be joined together, and a rather interesting result ensues:

1 6 52 0/9 7

4 3 8

8 3 47 0/9 2

5 6 1

Do you notice the 124875 sequence running top to bottom on the left hand side, and bottom to

top on the right hand side? This sequence is part of the surface of the torus, as Rodin is

explaining. In the centre is the 396 number sequence ( a variation of the 369). Along some of the

45-degree angles is the numbers 432 or 198. In fact one does get the sense that the mirror magic

216

square is meant to be there. You will also find that the other number sequences that are being

mentioned in the diagram of the torus above are related to the Vedic Square. This torus, in other

words, has a mirror counterpart. The two 124875 sequences running in opposite directions is

another clue that there is a mirror torus. In fact a mirror universe can be plotted using mirror

coordinates. Perhaps it can be observed through computer simulation.

Where the number sequence is 4 8 3 7 2 6 1 5 9, for example, there will be a mirror sequence of

5 1 6 2 7 3 8 4 9, and so on with the other sequences, until the mirror torus is built.

For those that are not aware of the fact, the 124875 sequence is also associated with the number

7 (as seen in a previous chapter), and also with binary. One will also find the same sequence

when infinite numbers are either divided or halved, each total being broken down to single digit.

Another symbol that is known for its 124875 number sequence, as well as the 369 triangle is the

Enneagram:

The Enneagram symbol is said to be thousands of years old. It is composed of three parts, the

circle, the inner triangle, and the "periodic figure." According to esoteric spiritual tradition, the

circle symbolizes unity, the inner triangle symbolizes the "law of three," and the hexagonal

periodic figure represents the "law of seven." These three elements constitute the Enneagram.

The Enneagram is built in various ways, including the division of the number 7.

The 4.5 invisible axis is in between the 1/8 at the centre of each sequence. From that axis will be

seen the same mirror number partners as those around the 9 axis.

Back to Marco Rodin's Universal Meta-Physics Mechanics diagram, the last two sequences throw

up some interesting interplay.

217

3 4 7 9 2 5 6 1 1 6 5 2 9 7 4 3 8 8

3 4 7 9 2 5 6 1 1

6 5 2 9 7 4 3 8 8

These mirror cycles usually appear when division is used, so it would be interesting to know which it is.

The second sequence mirrors the above:

8 8 3 4 7 9 2 5 6 1 1 6 5 2 9 7 4 3

8 8 3 4 7 9 2 5 6

1 1 6 5 2 9 7 4 3

The 11/88 suck the mirror number pairs into the central 9.

1 1 6 5 2 9 7 4 3 8 8

The 4.5 shows the characteristics of something that will lead one through the looking glass into a

mirror world, if such a thing existed. But, as information is shown to use both sides of the mirror

on its theoretical travels, who is to say we could not also travel through space utilizing the mirror

side as a partner?

218

Chapter thirty two

I-Ching Hexagrams and the 4.5

The I-Ching hexagrams have long been associated with the binary number system. Here we focus

on the default layout of the I-Ching Hexagrams, which are then broken down to binary number

representation. First the default layout:

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

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If we treat each hexagram as a binary number we will eventually discover the mirror number pairs

of the Vedic Square, and also the 4.5 crossover point. The top left hexagram, for example, has a

value of 63. Putting the I-Ching to binary is very common, and one will find many links on the Web

that will confirm these numbers. Here are the first three examples:

= 1 1 1 1 1 1 1 as binary, which equals 63

0 1 1 1 1 1 as binary, which equals 31

32 16 8 4 2 1

1 1 1 1 1 1 = 63 in binary (top left)

0 1 1 1 1 1 = 31 in binary (one below top left)

1 0 1 1 1 1 = 47 in binary (two below top left)

Therefore, the 1 1 1 1 1 1 1 binary number is made up of 1*32, 1*16, 1*8, 1*4, 1*2, 1*1.

The following table shows all the hexagrams represented as totals that are comprised of the

relative binary number. The central 4.5 emerges where the central red and black arrows join along

the 45-degree angle, in between the numbers 49 (4), 14 (5), 9 and 54:

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Axis

1 2 3 4 5 6 7 8

63 = 9 59 = 5 61 = 7 57= 3 62 = 8 58 = 4 60 = 6 56 = 2

31 = 4 27 = 9 29 = 2 25 = 7 30 = 3 26 = 8 28 = 1 24 = 6

47 = 2 43 = 7 45 = 9 41 = 5 46 = 1 42 = 6 44 = 8 40 = 4

15 = 6 11 = 2 13 = 4 9 = 9 14 = 5 10 = 1 12 = 3 8 = 8

55 = 1 51 = 6 53 = 8 49 = 4 54 = 9 50 = 5 52 = 7 48 = 3

23 = 5 19 = 1 21 = 3 17 = 8 22 = 4 18 = 9 20 = 2 16 = 7

39 – 3 35 = 8 37 = 1 33 = 6 38 = 2 34 = 7 36 = 9 32 = 5

7 = 7 3 = 3 5 = 5 1 = 1 6 = 6 2 = 2 4 = 4 0 = 0

Start at the very center, the empty space, the 4.5 position, in between the numbers 9,14,49 and

54. From here there is a movement to the 49 and 14 simultaneously, which also add up to 63

together. There will be a 4 and a 5 for the single digit sum of the two numbers. This is the center

point where the other pairs are evolving, through symmetry.

Every symmetrical pair of totals adds to 63, as well as maintaining strict number partners from the

Vedic Square. The fourth and fifth columns should be easy to read, by following the relative

arrows. At the center of the third and sixth column one finds that the 4 (13) and 5 (50) have

swapped over. Following the arrows from this point, blue with blue and green with green, will again

produce the right number pairs of the Vedic Square. The second and seventh columns have a 2

and a 7 either side of the center place, and again from there the right number partners will emerge

by following the arrows. Lastly the first and eighth columns has a 6 and a 3 either side of the

center, and yet again following the arrows flow will produce the right number partners. All these

number partners, as always, adhere to the number partners of the Vedic Square. The whole mirror

process flows out from the central 4.5, and back to the centre.

The single digit sum of the entire first column adds to 37 (single digit of 1). The second column

adds to 41 (5), the third to 39 (3), the fourth to 43 (7), the fifth to 38 (2), the sixth to 42 (6), the

seventh to 40 (4) and the eighth to 35 (8).

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It should be expected that the 4th and 5th column are number partners. The 43 breaks down to a 7,

and the 38 breaks down to a 2, so indeed this is the case. The same is true for the third and sixth

columns, breaking down to 3 and 6 respectively. The second and seventh column break down to 5

and 4, and the first and eighth columns break down to a 1 and 8. Therefore one can only consider

this table to being perfectly symmetrical at the layer beneath the main numbers that are being

displayed.

Because each pair of hexagrams adds up to 63 it is a question of multiplying 32 by 63 to get the

grand total for the whole I-Ching. The result is 2016.

The I-Ching literally explodes into life from the center. Whatever the reason for the orderly placing

of the hexagrams and their associated number the I-Ching is in perfect symmetry beneath its

apparently asymmetric layer.

A mirror layout of the I-Ching

Perhaps it is even blasphemous to suggest that an I-Ching arrangement can be represented in

mirror form. Nevertheless, a way does exist to mirror the above arrangements of hexagrams. If

one views the mirror number of, say, 21 as 24, a 3/6 mirror pair, then one can replace the relevant

hexagram, and commence building a mirror I-Ching table that way. Because 63 can be broken

down to 9, and because 9 and the zero are mirror partners, then the 63 could be represented as

zero. This places the hexagram made up of lines and the hexagram made up of broken lines as

partners.

One can work in reverse here, and firstly create a number table based on the previous table,

which was for the default hexagram arrangement viewed as binary numbers. You will see that the

single digit equivalents are being replaced by the counter-clockwise partners of the Vedic Square.

So for example, the 59=5, is replaced by the 58=4, where the 5 and the 4 or counter-spinning

number pairs of the Vedic Square.

222

Axis

1 2 3 4 5 6 7 8

0 = 0 58 = 4 56 = 2 60= 6 55 = 1 59 = 5 57 = 3 61 = 7

32 = 5 27 = 9 34 = 7 20 = 2 33 = 6 19 = 1 35 = 8 21 = 3

52 = 7 38 = 2 45 = 9 40 = 5 53 = 8 39 = 3 37 = 1 41 = 5

12 = 3 16 = 7 14 = 5 9 = 9 13 = 4 17 = 8 15 = 6 1 = 1

62 = 8 48 = 3 46 = 1 50 = 5 54 = 9 49 = 4 47 = 2 51 = 6

22 = 4 26 = 8 24 = 6 10 = 1 23 = 5 18 = 9 25 = 7 11 = 2

42 – 6 28 = 1 44 = 8 30 = 3 43 = 7 29 = 2 36 = 9 31 = 4

2 = 2 6 = 6 4 = 4 8 = 8 3 = 3 7 = 7 5 = 5 63 = 9

And again the symmetry of the mirror number pairs will all be commencing from the 4.5 in the

centre, spreading outwardly and inwardly simultaneously. Of course, one is not attempting to

draw too many conclusions with this representation of the I-Ching. In some ways it is more an

experiment with a binary grid.

However, as there are implication of mirror side marriages within the data being exposed through

various grids, the I-Ching too throws up an intriguing pointer to the idea of marriage of the duality.

Surrounding the central part of the first I- Ching, at the invisible 4.5 axis, are hexagrams that

equal 9, 14, 49 and 54. These stand for KEN, HSIEN, SUN, and TUI.

KEN = Keeping Still, Mountain, Self-Restraining, Discerning

HSEIN = Influence (wooing), Restrained, joining

SUN = Reduce, Decreasing or Concentrating

TUI = The Joyous, Lake, Self-Reflecting, Stimulating, Intenseness

All four of these words can be represented by an alphanumeric number:

KEN+HSEIN+SUN+TUI = 189

223

As a binary this is 10111101. This can also be seen as 1011+1101, as it is symmetrical in the

centre. This would equal 11/13

11/13 = 0.846153846153…..

Here we have a mirror flow equation:

846

153

The 2 and the 7 is missing, and the 9 is implied in all the pairs, as an axis. This can imply another

number, the 297, which is seen (in binary form) to act exactly as the Dorian mode. It was seen

how the F# Dorian is the invisible aspect of the triangle of keys. Here we have these same three

numbers uninvolved in the above equation, which is made up of mirror flow partners of the Vedic

Square.

The mirror four hexagrams either side of the central 4.5 are 9, 13, 50 and 54:

KEN = Keeping Still, Mountain, Self-Restraining, Discerning

TUNG JEN = Fellowship with Men, Associating

TING = The Caldron, Refining, Transforming

KUEI MEI = The Marrying Maiden, Immaturing

KEN+ TUNG JEN+ TING+ KUEI MEI = 244

244+189 = 433

This figure is very close to the 432.

The “marrying maiden” is quiet close to the idea of a “Dorian Pregnancy”, discussed in a previous

chapter. There is also the idea of transformation.

There are no real conclusions being drawn from these experiments. They hopefully make for

interesting observation. The theme does keep returning to the idea of marriage and unity. The

connection with a more mystical meaning could lead to some interesting experiments.

224

Chapter thirty three

The Tzolkin, the Vedic Square and scales

The Tzolkin grid, the Vedic square's nine number sequences, and the circle of tones mirror

structure are run in tandem, and it is seen that there is a perfectly symmetrical correspondence

between them.

The Tzolkin grid is a calendar constructed by the Mayans (also called the Mayan Calendar), and

covers the time period from August 11th 3114 BC, to the year December 21st 2012. There is so

much information available regarding this calendar, the Mayans and their history and beliefs, that

the reader again is invited to look around on the internet, where they will find info they require to

understand the Tzolkin.

The idea here is to find the same 4.5 axis at the center of the Tzolkin, to unite it with its mirror

Tzolkin, and later to apply other numbers and musical intervals through it, using the Tzolkin as the

basic layout. It will show that the two triangles of keys, making up this circle of tones mirror

structure, also fits in with the 20 *13 Tzolkin grid.

The 4.5 axis position will again be seen as the invisible point that supports, in a symmetrical

sense, the visible number pairs that are arranged within the Tzolkin grid. It is the Vedic Square

that endows this grid with its arrangement. Even though rather obvious once someone “gets it”, it

does present a relationship or interplay between symmetry and asymmetry, invisible and visible

like axis positions, and also the consistent triangle of notes/chords/keys structure that seems to

appear. It all flows from and back to this position, transferring to a visible axis along the way. The

Tzolkin is yet one more example of this mirror process.

The following page shows a diagram of the Tzolkin grid.

225

Tzolkin Grid (Mayan Calendar)

226

1 21 41 61 81 101 121 141 161 181 201 221 241

2 22 42 62 82 102 122 142 162 182 202 222 242

3 23 43 63 83 103 123 143 163 183 203 223 243

4 24 44 64 84 104 124 144 164 184 204 224 244 5 25 45 65 85 105 125 145 165 185 205 225 245

6 26 46 66 86 106 126 146 166 186 206 226 246

7 27 47 67 87 107 127 147 167 187 207 227 247

8 28 48 68 88 108 128 148 168 188 208 228 248

9 29 49 69 89 109 129 149 169 189 209 229 249

10 30 50 70 90 110 130 150 170 190 210 230 250

11 31 51 71 91 111 131 151 171 191 211 231 251

12 32 52 72 92 112 132 152 172 192 212 232 252

13 33 53 73 93 113 133 153 173 193 213 233 253

14 34 54 74 94 114 134 154 174 194 214 234 254

15 35 55 75 95 115 135 155 175 195 215 235 255

16 36 56 76 96 116 136 156 176 196 216 236 256

17 37 57 77 97 117 137 157 177 197 217 237 257

18 38 58 78 98 118 138 158 178 198 218 238 258

19 39 59 79 99 119 139 159 179 199 219 239 259

20 40 60 80 100 120

140 160 180 200 220 240 260

The left to right flow of this 20 * 13 grid is really comprised of the Vedic number sequence 1 3 5 7

9 2 4 6 8 throughout. Or, as Indig numbers, +1 +3 –4 –2 0 +2 +4 –3 –1. This will become very

plain when the grid is broken down to its single digit values.

See this grid as beginning from the top left and bottom right simultaneously, with the number 1

and its mirror partner the number 8 (which is the single digit sum of the number 260). This is the

beginning of the journey of opposing flows toward the center of the grid. The nine number

sequences of the Vedic Square, 1/8 2/7 3/6 4/5 5/4 6/3 7/2 8/1 9/9, are what guide this flow to the

center, and then swap-over from positive to negative, clockwise to counter-clockwise, before

commencing the journey again.

After the first pair (1/8), top left and bottom right, there is a move to 21=3 at the top left, whilst

there is a mirror move 240=6 at the bottom right. This contrary flow movement within the grid is

like two great cycles opposing each other about to be given different quantities of number without

breaking their symmetrical link with the number sequences from the Vedic square they belong to.

Here is part of the first and last row to confirm this:

Top left to right - 1 21(3) 41(5) 61(7) 81(9)

Bottom right to left - 260(8) 240(6) 220(4) 200(2) 180(9)

Asymmetry is guided by symmetry, through dual positional quality of movement.

The number 9 also comes into the light now, as each pair equals 261 throughout (261 distills to

the number 9).

When the end of both the rows is reached the top row moves down to 242 whilst the bottom row

moves up to 19, and again flow in opposite directions to each other, always keeping the number

sequence partners of the Vedic square intact. It can only be describing a system of opposite flows,

like shades of light/dark or expansion/contraction. So where do the flows meet? Do they meet or

do they cross over a hidden axis point in the center? This is where the true cross over point

occurs:

110 130 150

* 111 131 151

227

130.5, whose single digit value is 4.5, is the very center, and uninvolved, yet surrounded by

symmetry, not through the quantity of distribution around itself (asymmetry), but through the dual

number pairs of the Vedic square, which are no more than a simple seed describing clockwise and

counter-clockwise cycles.

110 = 2 - 151 = 7

130 = 4 - 131 = 5 (The 4.5 position)

150 = 6 - 111 = 3

The cross over point is 130.5

It is rather conclusive that the clockwise and anti-clockwise cycles are crossing over at the 4.5

position, and that this 4.5 position is the perfectly symmetrical point that holds all the number

sequence pairs in symmetry. This is at the very heart of this mirror theory, or “both sides” theory, a

switching over into the mirror side at the 4.5.

Bearing in mind the perfect symmetry held at the invisible swap-over point of the dual cycles, one

can ascertain the position of one cycle, like a clockwise cycle, to its anti-clockwise counterpart.

When the grid is at 115 on one side, for example, it is at 146 the other. These two figures distill

down to a 7 and 2 when the digits are cross added, which are mirror flow partners of the Vedic

Square. This also occurs between numbers 155/106 ( a 2/7).

The center row also converges to its own center, from opposite ends, and crosses over at the 4.5

(130/131), bringing perfect symmetry to the dual cycles.

228

The first movement away from the center 130.5 is then a 4/5 number pair. Things are springing

out of this center like they do at the tri-tone/4.5 position of the major scales. The black columns

either side both commence with a 2/7 pair. This almost ushers the two waves into the center row.

106 = 7 and 146 = 2 115 = 7 and 155 = 2

The Vedic square number sequences dominate the Tzolkin as much as they have been found to

dominate the music scales.

Here is the complete Tzolkin grid broken down to single digit, showing the 4.5 crossover point at

the centre of the seventh column.

229

Single digit Tzolkin grid

This grid could also be represented using the Indig numbers, and the same symmetry would be

seen. The flow of the number sequences creates a cross shape. The mirroring occurs both

horizontally and vertically, commencing from the 4.5 in the very center. It will also occur from the

230

1 3 5 7 9 2 4 6 8 1 3 5 7

2 4 6 8 1 3 5 7 9 2 4 6 8

3 5 7 9 2 4 6 8 1 3 5 7 9

4 6 8 1 3 5 7 9 2 4 6 8 1 5 7 9 2 4 6 8 1 3 5 7 9 2

6 8 1 3 5 7 9 2 4 6 8 1 3

7 9 2 4 6 8 1 3 5 7 9 2 4

8 1 3 5 7 9 2 4 6 8 1 3 5

9 2 4 6 8 1 3 5 7 9 2 4 6

1 3 5 7 9 2 4 6 8 1 3 5 7

2 4 6 8 1 3 5 7 9 2 4 6 8

3 5 7 9 2 4 6 8 1 3 5 7 9

4 6 8 1 3 5 7 9 2 4 6 8 1

5 7 9 2 4 6 8 1 3 5 7 9 2

6 8 1 3 5 7 9 2 4 6 8 1 3

7 9 2 4 6 8 1 3 5 7 9 2 4

8 1 3 5 7 9 2 4 6 8 1 3 5

9 2 4 6 8 1 3 5 7 9 2 4 6

1 3 5 7 9 2 4 6 8 1 3 5 7

2 4 6 8 1 3 5 7

9 2 4 6 8

top left and bottom right, as mentioned. Or one could now begin at the top right and bottom left as

well.

The next examples use the 20*13 layout of the Tzolkin, but passes various numbers or musical

notes through it.

Number 11 cycled through Tzolkin grid

231

The above is a Tzolkin grid that has the number 11 cycled through it. Each new total is broken

down to single digit, and shown below the main total. This exposes the Vedic square number pairs

as they again flow to the center 4.5 position. This 4.5 occurs between the 8/1 number pair

(between 1430 and 1441). No matter what number is cycled through this grid, the center axis is

always 4.5

The number sequence from the Vedic square, 4 8 3 7 2 6 1 5 9, is occurring horizontally

throughout the grid, and in certain variations. This is quite obvious because any number when it is

cycled will produce one of nine number sequences.

There is also the Vedic square number sequence 2 4 6 8 1 3 5 7 9, and its variations, flowing

vertically.

Each number within this grid comprising of the number 11 cycled will now be represented by a

chromatic note. There are twelve chromatic notes spanning an octave, each separated by a semi-

tone interval. From one note on a piano to its next adjacent note is equivalent to a semi-tone

interval movement.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

C Db D Eb E F Gb G Ab A Bb B C Db etc

The next grid deals in these chromatic steps, where each octave is separated by twelve numbers,

so that C is 1, 13, 25, 37 etc. When we substitute the correct chromatic note for each number

within the grid we uncover the triangles belonging to the circle of tones structure. This is shown on

the next page.

232

The chromatics of the 11 cycled

Vertically the notes are descending one semi-tone/chromatic at a time.

See how the notes mirror perfectly from the central column, C 1441. Bear in mind the same

number pairs of the Vedic square flowing to and from the center. The triangles of the circle of

tones are emerging horizontally along the grid.

233

Let the Bb at 1331 be an axis point. From Bb up to D 1551 is a major 3rd, and from Bb down to F#

1111 is also a major 3rd. Every ascending movement in pitch is mirrored by a descending

movement in pitch of similar proportion. The notes are mirroring as the grid branches out from the

centre column. It is also mirroring from the 4.5, which is represented as 1435.5, the very centre of

the grid.

This, therefore, is like another Mode Box, but one that accommodates the actual Circle of Tones.

The whole seventh column is an axis point. From every note along this central column is

reflected one of the triangle of notes of the mode boxes, D F# Bb, C E Ab, G B Eb, Db F A.

These triangles appear again and again on the mirror side of many musical examples, whether it

is a chord or a scale, one finds that the information (formula) mirrors three times in all, all three

moves a major 3rd or the inversion, the minor 6th. This establishes a triangle of notes or keys or

chords. Study the above grid and notice that only these four triangles exist, with the seventh

column being the axis for them. Study the Mode Box of C Major and notice that the same Bb is

axis to these same triangles of notes flowing along the 45-degree angle on the mirror side.

Each octave is separated by 660 on the horizontal line. For example, Bb 11 to Bb 671 = 660.

The triangles cover four octaves. For example, the Bb top left and the Bb top right is separated by

four octaves.

There are the notes Bb Db E G flowing along one of the 45-degree angles. This again is the

Diminished chord, made up solely of minor 3rds, and another perfectly symmetrical chord, like the

triangle made up of three major 3rd moves. Therefore the diminished chord, with its four moves of

minor 3rds forms a square.

234

On the opposite 45-degree angle we see the circle of 5ths beginning at Bb, then F C G D A etc. All

the 45-degree angles that flow right to left show the circle of 5ths.

This grid then, is chromatic vertically, the triangles from the circle of tones horizontally, minor 3rds

along the 45-degree angles left to right, and the circle of 5ths along the 45-degree angles flowing

right to left. Other grids based on other numbers show similar four way relationships, but this grid

based on the number 11 is the most intriguing.

What follows is a series of 13*20 grids built on different numbers, in order to highlight the

consistency with which the triangles of keys, as found within the mode box, relate to the Tzolkin

grid. It will also show that any number cycled this way will produce the Vedic square number pairs

that evolve to and from the center 4.5 axis. In fact, as 4.5 is the mean number between every pair,

1/8 2/7 3/6 4/5 etc, there is always a link with this invisible-like axis position at every twin

coordinate within the grid, or any similar type of grid.

235

Number 17 cycled through the Tzolkin

Again the same four triangle relationships have emerged horizontally.

Twenty minor 2nds, for example, are the same as twenty chromatic semi-tones, which has already

been seen to go on to create the triangles horizontally across the grid.

236

To find out easily the underlying structure here one can take the first 13 numbers within the grid

and double them, like this:

C1 Db2 = Minor 2nd

Db2 Eb4 = Major 2nd

D3 F6 = Minor 3rd

Eb4 G8 = Major 3rd

E5 A10 = Perfect 4th

F6 B12 = Flat 5th (Tri-tone)

F#7 Db14 = Perfect 5th

G8 Eb16 = Minor 6th

Ab9 F18 = Major 6th

A10 G20 = Minor 7th

Bb11 A22 = Major 7th

B12 B24 = Octave

C13 Db26 = flat 9th

This doubling has created the interval table in proper running order. We can see that cycling the

number 6 is what produces the tri-tone, so we start with the 6 and cycle it twenty times in order to

determine the bottom note of the first column of the Tzolkin. This in effect is to begin at the note F

and to move by the interval of a tri-tone from there. Moving by a tri-tone can only produce two

different notes.

F, B, F, B, F, B, etc

This confirms very quickly that the Tzolkin grid based on the number 6 can only produce the F and

the B note. Determining the other intervals is not as easy as this, some harder than others to plot.

Here is the minor 3rd interval cycled twenty times from the note D3. As you can see, this is only

twice as much work as in determining the tri-tone:

D, F, Ab, B, D, F, Ab, B, D, F, Ab, B, D, F, Ab, B, D, F, Ab, B

The result here is the Diminished chord, being made up of four minor 3rds. Yet with it finishing on

the note B we find that beginning the second column is the note D once more. Therefore the

number 3 cycled through the Tzolkin can be seen as an octaves grid horizontally. Here it is:

237

In order to determine the overall vedic square number sequence interplay within the grid it is a

question of finding the differences between the adjacent notes, horizontally, and across the two

45-degree angles. The above grid flows as the 369 number sequence of the Vedic square

vertically, 396 horizontally, only the single digit 3 along the 45-degree angle left to right, and 693

238

3 D

63 D

123 D

183 D

243 D

303 D

363 D

423 D

483 D

543 D

603 D

663 D

723 D

6 F

66 F

126 F

186 F

246 F

306 F

366 F

426 F

486 F

546 F

606 F

666 F

726 F

9Ab

69 Ab

129 Ab

189 Ab

249 Ab

309 A

369 Ab

429 Ab

489 Ab

549 Ab

609 Ab

669 Ab

729 Ab

12 B

72 B

132 B

192 B

252 B

312 B

372 B

432 B

492 B

552 B

612 B

672 B

732 B

15 D

75 D

135 D

195 D

255 D

315 D

375 D

435 D

495 D

555 D

615 D

675 D

735 D

18 F

78 F

138 F

198 F

258 F

318 F

378 F

438 F

498 F

558 F

618 F

678 F

738 F

21Ab

81 Ab

141Ab

201 Ab

261 Ab

321 Ab

381 Ab

441 Ab

501 Ab

561 Ab

621 Ab

681 Ab

741 Ab

24 B

84 B

144 B

204 B

264 B

324 B

384 B

444 B

504 B

564 B

624 B

684 B

744 B

27 D

87 D

147 D

207 D

267 D

327 D

387 D

447 D

507 D

567 D

627 D

687 D

747 D

30 F

90 F

150 F

210 F

270 F

330 F

390 F

450 F

510 F

570 F

630 F

690 F

750 F

33 Ab

93 Ab

153 Ab

213 Ab

273 Ab

333 Ab

393 Ab

453 Ab

513 Ab

573 Ab

633 Ab

693 Ab

753 Ab

36 B

96 B

156 B

216 B

276 B

336 B

396 B

456 B

516 B

576 B

636 B

696 B

756 B

39 D

99 D

159 D

219 D

279 D

339 D

399 D

459 D

519 D

579 D

639 D

699 D

759 D

42 F

102 F

162 F

222 F

282 F

342 F

402 F

462 F

522 F

582 F

642 F

702 F

762 F

45 Ab

105 Ab

165 Ab

225 Ab

285 Ab

345 Ab

405 Ab

465 Ab

525 Ab

585 Ab

645 Ab

705 Ab

765 Ab

48 B

108 B

168 B

228 B

288 B

348 B

408 B

468 B

528 B

588 B

648 B

708 B

768 B

51 D

111 D

171 D

231 D

291 D

351 D

411 D

471 D

531 D

591 D

651 D

711 D

771 D

54 F

114 F

174 F

234 F

294 F

354 F

414 F

474 F

534 F

594 F

654 F

714 F

774 F

57Ab

117 Ab

177 Ab

237 Ab

297 Ab

357 Ab

417 Ab

477 Ab

537 Ab

597 Ab

657 Ab

717 Ab

777 Ab

60 B

120 B

180 B

240 B

300 B

360 B

420 B

480 B

540 B

600 B

660 B

720 B

780 B

along the opposite 45-degree angle. All in all one can see that only the variations of 369 and 639

dominate this grid.

Here is a list that shows each interval cycled twenty times, and either seen to create or not to

create the triangles of notes as found in the mode boxes.

Chromatics – Yes

Major 2nds – Most definitely, because it would create the circle of tones throughout.

Minor 3rds – No. Produces octaves instead

Major 3rds – Yes. Three major 3rds form a triangle from the mode box

Fourths – Yes

Flat 5th (tritone) – No. Only F and B are produced

Perfect 5th – Yes. If the perfect 4th does, so does the 5th.

Minor 6th – Yes. If the major 3rd does, so does the minor 6th.

Major 6th – No. If the minor third produces octaves, so does the major 6th.

Minor 7th - Yes, the circle of tones, as in the major 2nds list.

Major 7th – Yes. If the minor 2nd does, so does the major 7th.

Octaves. You may find it hard to find more than one overall note here..:-)

The flat 9th is the minor 2nd of the next octave.

And so on.

Therefore any number can be cycled through the grid, and one can easily determine the note

content such a number will generate.

The next example is that of the number 9 cycled through the Tzolkin. This creates of vertical flow

of major 6th intervals throughout the grid. Horizontally there are only octaves

239

Sums 0f 9

240

Mirror worlds collided 2012

The Vedic Square is a grid comprised of nine number sequences, although only eight of them are

required, as the ninth is always implied in the other eight. The VS shows that the base 10 number

line is dominated by only these sequences, which hide mirror flows. It can be seen that the 1st

seq mirrors the flow of the 8th, the 2nd mirrors the flow of the 7th, the 3rd mirrors the 6th and the

4th mirros the 5th. At the 4.5 they switch over, and then the 5th mirrors the 4th and so on.

The Mayan calendar, however, was constructed on the logic of base 20. This is not so important

as the fact that the VS goes on to show how the number 2012 is what unites both the Mayan

calendar and the number line of base 10. The Mayans had no way of knowing about the base 10

system at the time, I assume. Yet one will see that the number 2012 is what pops out when we

bring the logic of the Vedic square to bear on the calendar.

The Vedic Square relates to nine number sequences that every number into infinity is a member

of. The number 9 is implied in every sequence and doesn't have to be represented. This in turn

relates to the Tzolkien grid in an uncanny way. Simply bringing the mirror number pairs together

and applying basic symmetry to them we get this result:

2^7 + 7^2

1^8 + 8^1

3^6 + 6^3

4^5 + 5^4

= 2012

It is rather uncanny that 2012 should emerge. The number sequences themselves are a way of

expressing the entirety of number, so for them to fall on the 2012 is no mean feat.

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Chapter thirty four

Magic Squares

9*9 Magic Square

Apparently this is a magic square that is associated with the Moon.

The smiley faces represent the Invisible Axis at the 4.5.:-) Complimentary number pairs are

transmitted in both directions at once, vertically and at the 45 degree angle. From the 9, for

example, there is a flow outwards and a flow coming in along the 45 degree angles, that is always

in symmetry according to the number pairs of the Vedic Square. Around each number 9 there is a

1 and an 8, which are number partners. Then there is a 7 and a 2, again number partners.

Running in between the 9s are the 4/5 number partners, the mean between which is of course 4.5.

This is the same mean that is between the 1/8, or 2/7 etc. Therefore the number 9 is also

superimposed with a 4.5 axis at its very heart. This would correspond with the two 4.5 axis

positions of the Mode Boxes.

The fact that the opposite 45-degree angle shows orderly numbers, like 1111 or 2222 etc, is also

related to the Mode Boxes, where there it would show orderliness in notes, like CCCC or DDDD

242

etc along one of the 45-degree angles. Along the other 45-degree angle on the mirror side of a

Mode Box there is the twin triangle Circle of Tones structure. There is much correlation, therefore,

between both this magic square and a mode box.

And again, what at first seems only an asymmetrical arrangement of numbers, has a symmetrical

link at a deeper layer.

Here's an indig number representation of the 9*9 magic square. The mirroring around the axis is

horizontal, vertical and the 45-degree angle. One can imagine a figure of 8 infinity loop that

emanates from the axis.

243

Progressive mirror magic square pairs

These are 3 x 3 mirror magic squares. The first square can be seen to have a sum total of 15 (6),

and its mirror 12 (3). Together this adds up to 27. One will see that the initial 27 is always the

difference between the rest of the magic square pairs. Each sum total is written above the magic

square.

Original 3*3 Mirror 3*3

15 12

Mirror numbers have been dealt with in a previous chapter. The next set of nine numbers will

begin at the number 10, which breaks down to the number 1 if cross adding of digits is performed.

The number 11 will represent the number 2, the number 12 will represent the number 3, and so

on. Therefore, instead of the number 8, at the top left corner of the first magic square, the number

17 will be substituted for the top left of the next magic square shown below. For the number 1, the

number 10 will be substituted, and so on.

42 39

The number 17 mirrors to the number 10 in the mirror magic square. This is simply the next logical

8\1 relationship. Basically this is still a Vedic Square relationship, and indig number clockwise/anti-

clockwise flow too. The 15 and the 12 are also the next 6/3 relationship. These are the most

1 8 3

6 4 2

5 0 7

8 1 6

3 5 7

4 9 2

17 10 15

12 14 16

13 18 11

10 17 12

15 13 11

14 9 16

244

logical relationships, which is not to say there does not exist many more ways of combining mirror

numbers. The only defining rule is that of the number 9. Whatever way one combines the number

pairs, they will always equal a 9 when they are added together. The more magic squares are built

up this way the more it would become plain how many other ways number pairs can be created.

One may create a 2/25 relationship, because this is still a 2/7, but combined from different cycles

of the number 7’s evolution.

69 66

This is just to show that magic squares show a consistent result when they are mirrored. The

result is not mere unrelated gibberish, but a systematic number flow in reverse to the original. The

Mode Box also shows this on the mirror side.

One does discover discrepancies when magic squares greater than 3*3 are drawn. But perhaps

even these have an explanation in an otherwise perfectly mirrored counterpart.

26 19 24

21 23 25

22 27 20

19 26 21

24 22 20

23 18 25

245

This next example is magic squares that are claimed to be based on the four elements, Fire – Air –

Water – Earth. I have simply mirrored them:

MIRROR

Mirror Fuego Mirror Aire

Mirror Aqua Mirror Tierra

The top right number of each non-mirror side magic square is 4 2 6 8. The mirror responds 5 7 3

1. This in effect creates the 2nd number sequence of the Vedic Square…2 4 6 8 1 3 5 7 9. The 9 ,

as always is implied in the pairs coming together from both sides of the mirror. The top left of each

magic square also creates this sequence. In fact this number sequence also runs through the

Tzolkin/Mayan calendar.

7 2 3

0 4 8

5 3 1

5 0 7

6 4 2

1 8 3

1 6 5

8 4 0

3 2 7

3 8 1

2 4 6

7 0 5

246

Chapter thirty five

The Alphabet and Overtones

An axis can be placed at the 13.5 position of the English alphabet, as this is its very centre.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

This 13.5 is also a representation of the 4.5 axis. On looking at that gap between the M and the N

it suddenly occurred to me to run the alphabet in tandem with the naturally occurring overtone

system. This is done by adding the naturally occurring overtone in the series to the naturally

evolving alphabet. One cycle per second is established as the overtone C:

A = 1 H = 8 O = 15 V = 221 (cps) = note C 8 = C 15 = B 22 = F#

B = 2 I = 9 P = 16 W - 232 = note C 9 = D 16 = C 23 = F# qt

C = 3 J = 10 Q = 17 X - 243 = note G 10 = E 17 = C# 24 = G

D = 4 K = 11 R = 18 Y = 254 = note C 11 = F# 18 = D 25 = G qt

E = 5 L = 12 S = 19 Z = 265 = note E 12 = G 19 = Eb 26 = Ab

F = 6 M = 13 T = 20 6 = note G 13 = Ab 20 = E

G = 7 N = 14 U = 21 7 = note Bb 14 = Bb 21 = F

Overtones series - C C G C E G Bb C D E F# G Ab Bb B C C# D Eb E F F# F#qt G Gqt Ab A

Alphabet - A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Just like the number/note grids discussed earlier, this can be seen as an alphabet/overtone/ grid.

No one is suggesting here that the pitch C, for example, is always the sound of the letter A when

we speak it. Obviously we can say an A to any pitch we like. But here the letter A is the first letter

of the alphabet, and the note C is the overtone of 1 cps.

247

The alphabet is shown in black, and the overtones are in red. The letter ‘A’ of the English

alphabet, for example, is aligned with the fundamental pitch of the Overtone series commencing

on the note C. The letter B is aligned with the first overtone of the series, which is an octave

higher than the first C, at a 2:1 ratio. The letter C is accompanied with the 3:1 ratio, which

becomes the overtone G, and so on. Using the Lambdoma chart in this book to confirm the

relationships should prove sufficient.

Some of the tones are sounding outside our ear’s frequency range. Also each tone is really an

approximation. The “qt” stands for quarter-tone, which is about half a semitone. The interval

between each pitch is becoming narrower the higher the number of cycles per second.

If we arrange the overtones so that they produce the scale of C Major (the doh reh meh fah sol lah

teh doh scale), we have:

Scale of C Major - C D E F G A B Overtone - 1 9 5 21 3 27 15 = 81

These numbers are taken from the above alphabet/overtone list. The number 9, for example,

signifies the overtone position for the pitch D, the ratio being seen as 9:1 from the fundamental. As

the 27th overtone is the note A, and we are also at the end of the alphabet, one could view the 27th

letter as becoming A again, an octave higher. Half of 27 is 13.5.

If we then replace each letter of the alphabet back onto the same overtone numbers as those

under the scale of C major above we have:

1 9 5 21 3 27 15 A I E U C A O

This equates to all the vowels plus the note or letter C. Quite a feat for a simple C major scale,

actually capturing all the vowels in this way.

The note A is 13.5 cycles per second and would fall between the letters M and N. This is dead

centre of the alphabet, spelling the word MAN. The A here would be a form of vibration,

suggesting that this centre of MAN is vibration, surrounded by what could be termed language.

ALPHABET/OVERTONES cycled through C MAJOR

=

All the VOWELS + FUNDAMENTAL

I discovered some four years later that 13.5 hertz is regarded as a switch over point between

Alpha and Beta waves of the brain. This was a very intriguing insight, because the alphabet also 248

shows that a switch over point exists at this 13.5 (any 4.5 type axis is a switch-over point). Alpha

waves are those that exist during the brains meditative state. Whereas Beta waves are the state of

mind that is more outwardly expressive. Therefore for the MAN to be at the very centre here,

made up of sound and language, would imply in some ways a drawing from a person's inner state,

and a switch to the outward expression through this central 13.5 position, standing. In meditation

the five vowels have always been considered an important element, in chanting or toning, for

example.

Of course one cannot really prove anything here, yet it has been an interesting enough result to

log. There is definitely the overtone of “A” at the centre of the alphabet, which is also a 4.5 in the

shape of 13.5 hertz. Alpha/Beta and Alphabet correlate perfectly here. For those who believe in a

Creation, then it opens up the idea of which standpoint would be the best in order to utter such big

commands like “Let there be Life”! Perhaps at this command the electrons obey the Will uttered

and form their clockwise and anti-clockwise, masculine and feminine, Light and Dark cycles, and

bring about into manifestation the evolution that leads to life.

The alphabet can even be shown with the relative indig numbers. The example overleaf shows it

as two touching triangles, with the word MAN at the centre.

249

The B and the Y, for example, equal the 2/7 mirror number pair of the Vedic square.

250

Chapter thirty six

Working Conclusion

It is the Dorian mode axis that is responsible for the access points to the other side of the mirror.

To actually physically gain entrance through this access point may not be as easy as verifying that

it exists on paper. Even so, most humans with an imagination can start thinking on setting up

possible experiments that may prove to be a way to manipulate these access points. And to aid in

the experiment it will be the whole circle of tones mirror structure that can be seen as relating to

the frequencies of the Dorian mode, or a set of symmetrical like numbers, represented as

frequencies, that would be first on the list when thinking in terms of swapping information from one

side of the mirror to the other. The divisions that were shown to contain mirror flows would also be

high on the list. And the all important 45-degree angle must also be seen as the carrier of this

access point.

Regardless of future idea for experimentation, it is enough to be aware that this mirroring

phenomena does carry a justified result . And that it exists is a simple matter of logic and reason,

because such things as Mode Boxes work as a whole unit, and so verify the mirror side as an

essential aspect.

4.51) 4.5 swaps sharps for flats in the circle of 5th

2) 4.5 swaps minor for major in mode box

3) Dorian is a 2/2 position mirror mode pair.

4) 4.5 is also a tri-tone position

5) 4.5 is a Dorian position.

6) Dorian is seen as center of the light/dark tonality distribution in circle of 5ths

7) Subsequent circles of 5ths are linked by the tri-tone/4.5

8) 4.5 dominates the number line, in terms of mirror number pairs

9) 4.5 is swap-over point of indig numbers and Vedic square/9*9 maths table sequences

10) number pyramid examples show 4.5 to be symmetry point amongst single digit totals.

11) dual flows of Fibonacci sequences shows 4.5 as swap over point.

12) 4.5 is linked to the 45 degree angle

251

Flower of Life

One can take each cycle of the major scale (one of the modes) and show it this way:

This shape is known as the Seed of Life, and it is a derivative of the Flower of Life.

There is also a mirror circle of modes. It will look no different to the above, but it links in through

the Phrygian mode being in the center, interfacing with the Ionian, then rotating counter-clockwise,

and creating the Ion/Phr, Dor/Dor/ Phr/Ion, Lyd/Loc etc modal pairs.

Each mode does individually cycle, and will display a tonal colour when brought into a

composition. The seed of life can be seen, in this sense, to also move in shades of dark and light.

These Modes conjure emotions, from country pub floral dancing to dark shred metal, they create

the mood. Music is also related to the transcendental, beyond emotional pull. For humans it

delivers on many levels.

252

The flower of life is also equated with the Egg of Life:

The shape of the Egg of Life is said to be the shape of a multi-cellular embryo in its first hours of

creation

A basic one dimensional depiction of the “Tube Torus shape is formed by ratcheting the Seed of

Life and duplicating the lines in its design. In Physics, the Tube Torus is considered a basic

structure in the study of Vortex forms. Some say the Tube Torus contains a code of vortex energy

that describes light and language in a unique way, perhaps as something of an Akashic Record

Taken from a wikipedia explanation of the flower of life.

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This tube torus is in many ways closer to the result of the four triangles of the Circle of Tones

mirror structure. This example shows it as four triangles moving in and out of the mirror:

1 = C

12 = G123 = B1234 = Eb12345 = G

123456 = B

1234567 = D12345678 = F#123456789 = Bb1234567891 = D

12345678912 = F#

123456789123 = Bb

1234567891234 = D

12345678912345 = F123456789123456 = A1234567891234567 = Db12345678912345678 = F

123456789123456789 = A

1234567891234567891 = Db

12345678912345678912 = F

123456789123456789123 = A

1234567891234567891234 = C12345678912345678912345 = E123456789123456789123456 = Ab

Here one sees that the evolution of the triplets/triangles are 1) G B Eb, 2) D F# Bb, 3) F A Db,

4) C E Ab. Numbers 1 and 3 constitute one of the interlocking triangle/circle of tones, as well as 2

and 4.

Taken as is, these four triangles of frequencies can be made to resemble a tube torus. And more

importantly, it also shows how the triangles are moving in and out of the mirror, with one triangle

on one side of the mirror yielding to another triangle on the other side. The chapter on opposing

forces also shows this.

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Merkaba

Because of the nature of the examples, being natural cycles and also based on frequencies, a

mention should be made to the symbol known as a Merkaba. In an uncanny way, both what has

been said about the Merkaba and the results exposed through mirroring do closely match. The

symbol itself is gained as a result of this mirroring of natural cycles.

All in all Balance is what these triangles of keys, that look like a Merkaba symbol when plotted on

a circle, are signifying. It is not too unreasonable to include the duality of masculine and feminine

into the mix. There are a few reasons why this should be considered; the major and minor

connection, the clockwise and anti-clockwise connection, the positive and negative, expansion

and contraction connections; all these have long been associated with masculine and feminine.

And in that light they are seen to evolve from and back to a marriage point within natural cycles.

The Merkaba is viewed as a dual spinning force of light, of the masculine and feminine forces in

unity, a symbol for consciousness itself. It may be too unscientific in some ways to associate these

forces with sexuality, but it is also quite plain that sexuality does play a role within nature, and here

we see the symmetry that would be evident in such a role.

What exactly would these dual spinning forces of light be? Are they simply the positive and

negative forces? We know that light has a clockwise and anti-clockwise nature. We also see that

these dual flows have their mirror motions traced and kept in perfect symmetry by a symbol that

does look like a star of David, or a Merkaba, as if all the mirror pairs evolve from and back toward

its perfect point of unity. A non-dual symbol is quite an uncanny one to unearth using simple

number and music cycles, and focusing on the position aspect around an axis, and finding a point

of non-duality at the 4.5, where this symbol is housed. And perhaps the 4.5 houses the balance

tones that can bring a similar unity to the human.

By considering the connection between the mirror structure that resembles a Merkaba and the

idea of a Merkaba within ancient writings, we see that there is little difference between the “story”

both tell of. It is a story where the masculine will unite with the feminine. Carl Jung helped make

this view palatable for the western world. The view here is that a time will come when the ego is

unseated, and the conscious and sub-conscious aspects within a human will unite once more.

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This is the marriage that spiritual people talk about. The ego that cycles within a duality, only to

eventually meet non-duality.

If the Merkaba is a real force, then the results here are made for experimenting along those lines,

that may contribute to its understanding.

The various frequency responses of individual areas of the body were shown. These frequencies

were seen to be one of the triangles of the circle of tones, G Eb B. It is reasonable to suggest that

this relationship is like an interface between what happens outwardly within nature, inwardly within

the human form, and further inwardly to the point of unity within the human form, and an access

way to the Spirit form itself perhaps. And because there is the phenomena of 'in and out of the

mirror', one can speculate that there is another triangle of frequencies that another body is

responding to , as our mirror self. Or perhaps anti-matter self.

It is only when a whole mirror unit is represented that one can take in the overall picture and seek

to analyze the results. And it is maintained that only one result emerges throughout the many

different approaches and natural cycles used. The result is very simple in the end, and almost

obvious. But there is the prize of viewing this same contrary spinning field of light structure that is

the Merkaba/star of David symbol. It is no mean feat to actually find two corresponding contrary

field of light symbols that look exactly like each other, based on natural cycles including light, and

one being objective in result and the other being a statement made by so called mystics when

talking of the inter-dimensional vehicle they called a Merkaba.

Light waves obey the same principles inherent within sound waves, that is, they both combine to

create summation waves and divide to create difference waves. The relationships found with

music cycles adhere to the behavior of light waves. The use of the octave is the common sense

behind this relationship. By the 40th octave the light spectrum has reached the colour range. A

few octaves higher is the infra red light domain, and on and on toward the x-ray waves and

gamma ray waves. What is established within one octave will be true for the proceeding octaves.

The same is true for simple numbers.

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Conjecture:

In order for the most possible information to emerge with the least effort, nature employs cycles ,

sequences which can be thought of as seeds. Theses seeds then allow bigger systems to grow

out from them. Such is the nature of the 4.5, and the nine number sequences of the Vedic Square.

We have also seen that every other base system possible boils down to only nine separate

sequences. This compactness can give rise to infinite variety.

With the information shown in this book one may wish to get a little “sci-fi” and imagine the

creation of a teleportation machine based on it. Because the circle of tones structures swap-over

through either side of the mirror a machine can be built that brings together the right frequencies in

order to cause, through summation tones, the opposite circle of tones structure to emerge.

Blending all six tonal combinations together and placing an object in the chamber that is oscillating

these frequencies, causes the object to be transferred to the mirror side. The Dorian mode and the

4.5 would naturally be access points to the other side. This is because each of the notes

comprising a circle of tones are 4.5 intervals of each other (the tri-tone interval), and are also

Dorian mode partners of each other . This would be a sci-fi experiment that can be performed in

reality.

The Mode Box, or Fibonacci mode box is the sound chamber, so to speak. The reason for thinking

this is the fact that the whole box is needed in order to see the mirror structure of the triangles

emerge. Some of the reasons for wishing to create a living Mode box, for example, is to analyse

the effect it may have on consciousness.

The Mode box does not need to be created in the audible range. One can experiment with octaves

and also temperament.

Is the dual triangle structure, in terms of the results exposed through mirroring cyclic formulas, a

means to travel between the dimensions, using the 'continual triangles', as shown in a previous

chapter?

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Parallel universe theory

After many set backs, string theorists were finally able to create what is termed a theory of

everything. By adding an 11th dimension into their calculations it was seen how each universe is a

thin membrane that ripples like a wave through this 11th dimension. Every now and again the

ripples from two adjacent universes touch, and this is what causes a big bang to occur. That is it in

a nutshell.

Can it be that a big bang is actually a 4.5 moment? What this would imply is that the law of cycles,

headed by the mirror activity within the Vedic square is responsible for the moment that each

ripple will make contact with another universe. It would be defined as the 4.5 cross-over moment,

and the tonal fountain would then produce the clockwise and anti-clockwise pairs that define

matter.

If it were true, then access would be possible into each universe by following the logic of the 4.5

tritone axis cross-over, as was shown in the 144 major scale grid.

Well, it is nice to speculate! Hopefully you have found some of the examples in this book

interesting, and may even decide to take up the thread yourselves.

The second volume will share some ideas on how to use the mirror structure in one's

compositions, as well as the more crazy conclusions that this phenomena implies.

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