The Creation of Musical Scales, part II.

by Thomas Váczy Hightower.

My first search was to look at musical practice in ancient times, not only in

Europe but all over the world. There were several other musical scales than

the diatonic scale, where the semitones where located other places than me-

fa and si-do. In the Gregorian modals for instance the different placement of

the semi tones creates the specific modes.

Pentatonic Music

In the pentatonic folk music semitones do not exist. By practice the people

have found out, that the 5 notes scale gave the possibilities to play any key

without significant disharmony. Theorists would say that the scale was

composed of ascending and descending fifths only in two steps in each

direction. A pentatonic scale can be played by only using the black keys on

the piano.

EASTERN MUSIC

After a study of ancient main cultural music, mainly Chinese and Indian, I

realized how universal the concept of the octave was in every musical

culture.

According to Helmholtz, the Arabic and Persian scales, the Japanese and

the Pacific scales are too within an octave. The division of the octave differ

from culture to culture.

Arab music divides the octave into sixteen unequal intervals. The Persians

divided their octave into 24 steps, so they must have used quarter tones.

From excavated Egyptian flutes a seven note scale C, D, E, F#, G, A, B,

have been discovered, which is identical with the Syntolydian scale of

ancient Greece. Japanese music used mainly a pentatonic scale.

Chinese music

Music was the cornerstone in the Chinese civilization, which was the

longest living culture in history. It was considered to embody within its

tones, elements of the celestial order. The audible sound, including music,

was but one form of manifestation of a much more fundamental form of

Super physical Sound. The fundamental Primal Sound was synonymous to

that which the Hindus call OM. The Chinese believed that this Primal

Sound, Kung or "Huang Chung" (directly translated "yellow bell") was,

though inaudible, present everywhere as a Divine Vibration. Furthermore, it

was also divided into 12 lesser Sounds or Tones. These twelve Cosmic

Tones were emanations of, and an aspect of, the Primal Sound, but were

closer in vibration to the tangible, physical world. Each of the 12 Tones was

associated with one of the 12 zodiacal regions of the heavens.

Audible sound was conceived as being a physical level manifestation of the

12 tones. Sound on Earth was a kind of sub tone of the celestial vibration.

They were believed to contain a little part of the celestial tones' divine

power.

"As above, so below", as the Egyptian Hermes Thot said. In the Lords

prayer a similar wish is spoken.

For the ancient Chinese the alignment with the divine prime tone was the

emperor's most important task. The alignment of earth with heaven and man

with the Supreme was literally the purpose of life. The entire State affairs

and order was dependent upon the right tuning of the fundamental tone, the

yellow bell, or Kung.

As an ancient text warns: "If the Kung is disturbed, then there is

disorganization; the prince is arrogant".

If the Kung was out of tune, because the celestial realm has changed,

disorder and inharmonious behavior in the society became obvious. Every

instrument (also the measuring instruments) was tuned and utilized in

accordance with the holy tone.

The instrument, which could give to man the fundamental tone for a musical

scale, which was in perfect harmony with the universe, was the key to

earthly paradise, and essential to the security and evolution of the society.

It became the Chinese Holy Grail.

One legend tells of the amazing journey of Ling Lun, a minister of the

second legendary Chinese Emperor, Huang Ti. Ling Lun was sent like an

ancient Knight of King Arthur to search for a special and unique set of

bamboo pipes. These pipes was so perfect that they could render the precise

standard pitches to which all other instruments throughout the land could be

tuned.

That sacred tone, which relate to the Western modern pitch of F, was

considered as the fundamental cosmic tone. The Chinese was aware of the

slow changing cosmic influence and consequently the Kung has to change

accordingly. The emperor had the task of tuning the Kung so it was in

alignment with the cosmic tone.

Cousto has in his book, The Cosmic Octave, an interesting observation on

this matter. He relates the Kung to the frequency of the Platonic Year. The

duration of the Platonic Year, (The Pythagorean Great Year) is about 25,920

years and represents the amount of time the axis of the Earth takes to

complete a full rotation.

The vernal equinox is the point at which the equator (of Earth) intersects the

ecliptic (or zodiac), which is the position of the sun at the beginning of

spring - March 21st.

The vernal equinox takes an average of 2,160 years to travel through one

sign of zodiac. This period of time is known as an age. It is not possible to

state exactly when one age is ending and a new beginning, because the signs

overlap to a certain degree.

The journey of the vernal equinox through each of the 12 signs of the

Zodiac equals one great year of approx, 25,920 years. (Presently we are on

the cusp of Aquarius as the age of Pisces is ending.)

This amount of years is close to the high number of generating fifths when

we come into a cycle of 25,524 notes.

Cousto calculates the note of the Platonic year to be F in the Western Equal

Temperament pitch, which is found in the 48th octave with a frequency of

344.12 Hz. or in the 47th. octave to be 172.06 Hz. Note that the

corresponding a' has a frequency of 433.564 Hz. (Modern Western concert

pitch is 440 Hz.)

Calculation: 31 556 925.97(the tropical year in seconds) * 25,920 (Platonic year).

Since the length (of a vibrating string, or the period of time) is in reverse

proportionality to the frequency, the length of the Platonic year in seconds shall be the

denominator. The frequency is very low, so we will raise the frequency to the range of

hearing by multiplying with the necessary amount of octaves, e.g. 48 octaves so we

arrive to 344,12 Hz. (47 octaves will be the half, 172,06 Hz.)

If we want to reach the spectrum of light, we multiply with 89 octaves which leads us

to a frequency of 1/31 556 925.97 * 1/25 920 * 2 89

= 7,56 * 10 14

Hz. corresponding

to a wavelength of 0.396 micrometer, which we perceive as violet near the ultra

violet. This is the color of the Platonic Year. The complementary color to violet is

yellow. Their fundamental tone was called the yellow bell.

It is a wonder for me, how the ancient Chinese could be aware of their

sacred fundamental tone, Kung, to be in accordance with the Platonic Year

and choose the great rhythm of the Earth.

Creation of a scale

It might be a surprise that the diatonic scale was the foundation for the

ancient Chinese's and the Indians music, though the musical theory and

practices differ from the Western.

For the old Chinese, their musical scale was developed by the circle of

perfect fifths up to 60 degrees or keys, the 60 Lü, though they usually only

used the first 5 fifths in their pentatonic music, because they knew that they

represents the limit of consonance in modal music. In addition, the ancient

Chinese saw a symbolic representation of the pentatonic scale rooted from

their belief of music being the representation of the relationship between

heaven and earth the five elements).

The Chinese was Centuries ago well aware of the existents of our modern

Equal Temperament. They dismissed such a tempered scale not only for its

badly false notes, but mainly because the tuning was not in alignment

with the cosmic tone.

According to the book by David Taime, "The Secret Power of Music", 3

was the symbolic numeral of heaven and 2 that of the earth; sounds in the

ratio of 3:2 will harmonize heaven and earth. As a way to apply that

important concept, the Chinese took the foundation note, "Huang Chung",

and from it produced a second note in the ratio of 3:2.

A more in depth explanation made by Alain Daniélou in his "Music and the

Power of Sound":

"Music, being the representation of the relationship between heaven and

earth, must quite naturally have this confirmation of a center or tonic (gong)

surrounded by four notes assimilated to the four directions of space, the four

perceptible elements, the four seasons, and so on.

The pentatonic scale thus presents a structure that allows it to be an adequate

representation of the static influence of heaven on earth. But a static

representation of a world in motion could not be an instrument of action

upon that world. It is necessary to evolve from the motionless to the moving,

from the angular to the circular, from the square to the circle. To express the

movements of the universe, the sounds will have to submit to the cyclic laws

that, in their own field, are represented by the cycle of fifths."

The spiral of fifths

As we have already seen, the fifth is the third sound of the

series of harmonics, the first being the fundamental and the

second its octave. According to the formula of the Tao-te

ching, "One has produced two, two has produced three, three

has produced all the numbers", we can understand why the third sound, the

fifth, must necessarily produce all the other sounds by its cyclic repetitions.

The first to be produced will be the four principal sounds, which form

comparatively simple ratios with the tonic.

For the sake of convenience we will use Western notes:

I, C

II, G = 3/2

III, D = 9/8 = (3/2)2 * ½ (lower an octave)

IV, A+ ( a comma sharp) = 27/16 = (3/2)3 * ½ (lower an octave)

V, E+ (a comma sharp) = 81/64 = (3/2)4 * ¼ (lower 2 octaves)

To these five primary sounds, whose disposition represents the elementary

structure of the perceptible world, the pentatonic scale, there can be added

two auxiliary sounds:

VI, B+, (a comma sharp) = 243/128 = (3/2)5 * ¼ (lower 2 octaves)

VII, L+F# (sharpen a major half tone) = 729/512 = (3/2)6 * 1/8 (lower 3

octaves)

The seven-notes Chinese scale

C D E+ (F)¤ L+F# G A+ B+ C’

1/1 9/8 81/64 (4/3) 729/512 3/2 27/16 243/128 2/1

The notation is Western for the sake of convenience. See Chinese & Western Music.

The two auxiliary sounds - 243/128 and 739/512 - should not be used as

fundamentals, though they are needed for transpositions, because they

belong to the scale of invisible worlds, and therefore we can neither perceive

their accuracy nor build systems upon them without going out of tune.

¤ Let us note here that the most striking difference between the system of

fifths and that of harmonic relations to a tonic resides in the perfect fourth,

which is an essential interval in the scale of proportions. The scale of fifths

has an augmented fourth as its sixth fifth, (3/2)6.

Instead of starting from C, we could had begun one fifth below, that is to

say, from F, and we would have obtained this essential note without

changing anything in our scale, except that, since we begin with a masculine

interval instead of a feminine interval, the character of the whole system is

modified.

The five successive fifth, whether in an ascending or a descending series,

represents the limit of consonance in modal music too. Beyond this limit, no

interval can appear harmonious, nor can it be accurately recognized. A rule

originating from the same principle was also known in medieval Europe,

where the tritone was prohibited as diabolical, that is, as connected with

forces that are supernatural and therefore uncontrollable. Folk music in its

pentatonic form had understood this too by only using the span of two fifths

up and down.

After these seven notes, the next five notes generated by the series of fifths

are:

VIII, bDb lowered a minor half tone, IX, bAb lowered a minor half tone, X, bEb a minor half tone lower, XI, bBb a minor half tone lower, XII, F+ a

comma sharp.

We now have twelve sounds, which divide the octave chromatically into

twelve half tones.

The twelfth fifth (note 13) in a 7 octave span brings us back to the

fundamental, but with a slight difference.

It is higher than the fundamental by one comma, the Pythagorean comma

(312 / 219 = 531,441/524,288, (5.88 savarts or 23.5 cents). It is, therefore, in

our notation, C+, one comma sharp.

In this way, successive series of twelve fifths will be placed one above the

other at one-comma intervals, up to the 52nd fifth (note 53)which fill the

octave.

The Chinese continued the cycle of fifths up to 25,524 notes with a basic

intervals of 0.0021174 savarts. This cycle is very near to that of precession

of the equinoxes, or the Pythagorean Great Year, which is of 25,920 solar

years. Why the Chinese continued so many octaves in the cycle of fifths,

could have something to do with their reference tone, Kung.

In practice, for reasons that are symbolic as well as musical, after the 52nd

fifth (53rd note) the Chinese follow the series only for the next seven

degrees, which place themselves above those of the initial seven-note scale,

and they stop the series at the 60th note. The reason given is that 12 (the

number of each cycle) * 5 (the number of the elements) = 60.

The scale of 60 Lü

The Chinese scale being invariable, constitute in effect a single mode. Every

change in expression will therefore depend upon modulation, a change of

tonic.

Firstly the choice of gender: fifths whose numbers in the series are even, are

feminine. The odd numbered fifths are masculine.

The choice of tonic are depended of complicated rules and rituals, which

main purpose is to be in accordance with celestial as well as earthly influx

or circumstances. Accordingly, the Chinese has to choose the right key for

the hour of the day and the month. Even during a performance.

It is a extensive scheme but to get an idea we can say that they corresponds

to political matters, seasons, hour of the day, elements, color, geographic

direction, planets and moon.

This scale of fifths, perfect for transposition because of its extreme

accuracy, also allows the study of astrological correspondences and of

terrestrial influx in their Tone Zodiac.

We notice that the Chinese scale is much identical with the Pythagorean

tuning, which also was produced by generating a perfect fifth (3: 2). How

the Chinese derived their scale goes back to 3000 BC, when the European

stone-age man still was beating wood logs. The prevalent opinion in the

West about our music superiority should hereby be moderated.

The Indian music system

The ancient Indians had a less formalized approach to their music than the

Chinese. Generally speaking they emphasized the personal inner

contemplation more than the outward organized rituals. One can say that

they sought the inner alignment with the divine supreme by means of the

sounds AUM or OM, which were (are) the earthly sound of the prime

creator, Brahman.

For the Hindus as the Chinese the spoken or chanted words were the carrier

of some of the creating energy and composed by the prime Creator.

Pronounced correctly it was believed that special words were able to alter

humans thoughts and feelings and literally change and form physical matter.

Raga is the basic form in classical Indian music. There is a whole system of

Raga's, which differ respectively from the North India or the South.

Originally there were only 7 Raga's. These may have been the remnant of an

ancient reference to the seven Cosmic Tones. The seven principal notes, or

savaras, connected with the seven main planets, and two secondary notes

corresponding to the nodes of the moon. This brings the total number of

notes in the scale to nine principal notes, which is related to the nine groups

of consonants of the Sanskrit alphabet.

The Raga system grants the musicians freedom of expression within the

limitations of a certain inviolable mode. Since music was so important a

force in altering phenomena upon Earth, they considered it would be

unwise, dangerous, and perhaps even suicidal in the long run to allow

musicians to perform whatever they wished.

The Indian solution was then to apply a system of rules which, while

effectively determining what type of music was performed and even its

spiritual atmosphere and the period of the day, did not indicate the notes

themselves. This was a convincingly successful solution to the problem

which the music of ancient civilizations always came up against.

The Chinese's had a more rigid system. They created variations by use of

instruments and especially in the expression of the single note. The

dimensions of tone color, or timbre was highly developed in the East. The

ear had to learn to distinguish subtle nuances. The same note, produced on a

different string, has a different timbre. The same string, when pulled by

different fingers had a different timbre, etc. Further more and very

important, the whole spiritual being of the musician himself was crucial.

That applies also to Indian music.

As in the Western diatonic scale, the Indian scale was based on 7 main

notes: SA, RE, GA, MA, PA, DHA and NI. If we goes back to the most

ancient texts on music, the scales was divided into two tetra chords, similar

to the ancient Greeks, and later put together with a whole tone (9/8) between

Ma - Pa so a full octave was completed.

The Indian notes relate coarsely to the Western ratios, though the tuning is

very harmonious and create a world of difference. We have to emphasize

that the use of harmony as we know it was and is not musically practiced.

Here is a crucial point. The Indian music is modal. There is a strong

relationship to the tonic. When a third is played it always relate to the third

degree as in Western harmonious tradition the third has a relative position,

because it can be the root, the fifth or third of a chord.

Eastern listeners often make remarks such as "Beethoven symphonies are

interesting, but why have all those chords been introduced, spoiling the

charm of the melodies".

The modal music of India is 'horizontal' as the Western is 'vertical'. The

vertical, harmonious system, in which the group of related sounds is given at

once, might be more direct though also less clear. The accurate

discrimination of the different elements that constitute a chord is not usually

possible.

The modal, horizontal system, on the other hand, allows the exact

perception and immediate classification of every note, and therefore permits

a much more accurate, powerful and detailed outlining of what the music

express.

One can say that the attention span in the Eastern musical language has to be

much longer since, in time, the different and distinct sounds adding up in the

listeners mind creates the chords or the whole musical idea. Only then, by

remembering with attention all the elements that constitute the musical

image, the full meaning finally can be understood.

The Indian musical system operates with a combination of immutable and fixed pitch

so the key can be recognized and variable notes. The 2nd ,3rd,4th, 6th and 7th notes

are variable, but the 1st (Sa or Do) and the perfect 5th (Pa or Sol) are immutable and

of a fixed pitch. The drone is accordingly often Do-Sol (Sa-Pa), which becomes the

ultimate open chord containing all other notes within it as a series of subtle

harmonics.

This drone (a constant note or tonic), whether actually played on an instrument like

the tampura or simply heard within oneself as the Om sound, is the constant reference

without which no Indian musician would play.

One must not be confused by the vast use of micro intervals, sliding or

bending the notes, prominent in Indian music. The musicians can freely use

these microtones as private points, often moving freely between two notes

as a kind of infinitely exploitable space, eventually returning home to the

tonic of the Raga. The musician has a freedom to play tones as his

inspiration demands so long as he obeys the sacred rules of types and its

mood.

The 22 Shrutis (degrees)

Musical intervals can be defined in two ways, either by numbers (string

lengths, frequencies) or by their psychological correspondences, such as

feelings and images they necessarily evokes in our minds. There is no sound

without a meaning, so the Indians consider the emotions that different

intervals evoke as exact as sound ratios. The feeling of the shrutis depends

exclusively on their position in relation to the tonic and indicate the key for

the ragas.

The 22 different keys or degrees encompass what the Indians consider the

most common feelings and reflection of the human mind. They was aware

of the division of the octave into 53 equal parts, the Pythagorean Comma,

and its harmonic equivalent, the comma diesis, (the syntonic comma, the

difference between the major and the minor tones).

However, they choose the 22nd division of the octave based on the limit to

differentiate the keys as well as psychological and meta physical

reason. The symbolic correspondences of the number 22 and 7, (7 strings

and main notes), could also play a part since the relationship between the

circle and the diameter is expressed as the approximate value of Pi, 22/7.

The modal or Harmonic division of the octave

Indian music is essentially modal, which means that the intervals on which

the musical structure is built are calculated in relation to a permanent tonic.

That does not mean that the relations between other than the tonic are not

considered, but that each note will be established first according to its

relation to the fixed tonic and not, as in the case of cycle of fifths by any

permutations of the basic note.

The modal structure can therefore be compare to the proportional division of

the string (straight line) rather than to the periodic movement of the spiral of

fifths.

All the notes obtained in the harmonic system are distinct from those of the

cyclic system, which is based on different data. Though the notes are

theoretically distinct and their sequence follow completely different rules, in

practice they lead to a similar division of the octave into fifty three intervals.

The scale of proportions is made of a succession of syntonic commas, 81/80,

which divide the octave into 53 intervals. Among those, 22 notes was

chosen for their specific emotional expressions:

Note

degree Interval

Value in

cents Interval Name Expressive qualities

1 1/1 0 unison marvelous, heroic, furious

2 256/243 90.22504 Pythagorean limma comic

3 16/15 111.7313 minor diatonic semitone love

4 10/9 182.4038 minor whole tone comic, love

5 9/8 203.9100 major whole tone compassion

6 32/27 294.1351 Pythagorean minor third comic, love

7 6/5 315.6414 minor third love

8 5/4 386.3139 major third marvelous, heroic, furious

9 81/64 407.8201 Pythagorean major third comic

10 4/3 498.0452 perfect fourth marvelous, heroic, furious

11 27/20 519.5515 acute fourth comic

12 45/32 590.2239 tritone love

13 729/512 611.7302 Pythagorean tritone comic, love

14 3/2 701.9553 perfect fifth love

15 128/81 792.1803 Pythagorean minor sixth comic, love

16 8/5 813.6866 minor sixth comic

17 5/3 884.3591 major sixth compassion

18 27/16 905.8654 Pythagorean major sixth compassion

19 16/9 996.0905 Pythagorean minor seventh comic

20 9/5 1017.596 just minor seventh comic, love

21 15/8 1088.269 classic major seventh marvelous, heroic, furious

22 243/128 1109.775 Pythagorean major seventh comic, love

The ancient Egyptians

The ancient Egyptians had similar beliefs as the Chinese and Hindus. In

their "Book of the Dead" and other sources it is stated, that God or his lesser

servant gods, created everything, by combining visualization with utterance.

First the god would visualize the thing that was to be formed; then he would

pronounce its name: and it would be.

From as late as the reign of Alexander II, a text dating from about 310 BC

still has the God of Creation, Ra, declaring: "Numerous are the forms from

that which proceeded from my mouth." The god Ra was also called Amen-

Ra, with the prefix "Amen". The Egyptian priesthood understood well the

word Amen, or AMN, and it was equated with the Hindu OM.

Egyptian music as Greek have most probably had its roots in Indian music,

or at least in that universal system of modal music of which the tradition has

been fully kept only by the Indians.

The pyramid can easy be a symbolic representation of Earth with its four

perceptible elements and all its characteristics are regulated by the number

four, the four seasons, four direction of space, etc. Especially the projection

of the single into the multiple.

WESTERN MUSIC

Pythagoras

The Greek philosopher Pythagoras (570 - 490 BC) spent 22 years in Egypt

mainly with the high priest in Memfis, where he became initiated to their

secret knowledge of Gods. When the Persians conquered Egypt, he was kept

in captivity in Babylon for 16 years before he could return to Greek and

begin his teaching.

I began to study the theory of the Pythagorean and their esoteric schools.

Very little is known of them. Pythagoras demanded silence about the

esoteric work. This historic school was founded in the Greek colony,

Kroton, in southern Italy about 2500 years ago.

I realized after dozens of books about the matter what an outstanding role

that school had for the creation of the western civilization. He created an

entirely new concept. Any person - man or woman - who had a sincere wish

for knowledge could enter the school stepwise, with a number of initiations.

The tradition of priesthood monopoly of knowledge of God was broken.

Pythagoras' study of the moving string and his discovery of the harmonic

progression of simple whole numbers was the first real scientific work and

the creation of modern science. But his vision went far beyond present

science in his deep understanding of the integration of the triad: A-science,

B-work on being, C-love and study of God. Something modern science

could learn from.

Nicomachus of Gerasa

Nicomachus the Pythagorean (second century B.C.) was the first who wrote

about Pythagoras legendary encounter with "the harmonious blacksmith"

and the weights of the 4 different hammers being 12, 9, 8 and 6, that

determined the variation in the pitches Pythagoras heard.

This story illustrate how the numerical proportions of the notes were

discovered. His methodical measuring of the hammers and how the sound

was produced and related (collecting data), then making experiments with

strings, their tension and lengths (repeating the findings and, with

mathematic, formulating them into a law), was the first example of the

scientific method.

We will not dwell with the question about the force of the impact or the

tension of the strings, which later was discovered as the square root of the

force, but just stick to the proportion of weights and the pitches he heard,

which led him to his discovery.

Pythagoras' experiments led to the combination of two tetra chords, (two

fourths), separated with a whole tone, 9/8, which constitute an octave. He

changed the traditional unit in Greek music, the tetra chord, into the octave

by an octachord.

In the time of Pythagoras the tradition was strongly based on the seven strings of the

lyre, the heptachord. The Greeks considered the number 7 sacred and given by the

god, Hermes, who handed down the art of lyre playing to Orpheus. The seven strings

lyre was also related to the 7 planets among other things the ancients venerated.

The lyre often, but not always, consisted of 7 strings comprising two tetra chords each

one spanning the most elementary concord, the fourth, both joint together on the note

mese.

According to legend, a son of Apollo, Linos, invented the four stringed lyre with 3

intervals, a semi-tone, whole tone and a whole tone comprising a fourth;

the fourth, "the first and most elementary consonance" as Nicomathus calls it, and

from which all the musical scales of ancient Greek music eventually developed.

Trepander of Antissa on Lesbos, born about 710 B.C., assumed a mythological status

for his musically genius. His most lasting contributions was perhaps his

transformation of the four stringed lyre to the instrument which became

institutionalized by tradition to the heptachord.

Trepander did before Pythagoras extend the heptachord from its minor seventh limits

to a full octave, but without having to add the forbidden eight string.

He removed the Bb string, the trite of the conjunct tetra chord, and add the octave

string, E1 yielding a scale of E F G A C D E

1.

This arrangement left a gap of a minor third between A and C, and seemed to have

enhanced the Dorian character of Trepander's composition.

Harmonia

Only Pythagoras escaped censure for adding an eighth string to the ancient

and venerated lyre because of his position as a great master and religious

prophet. His purpose was to teach man the unifying principle and immutable

laws of harmonia by appealing to his highest powers - the rational intellect

and not to his untrustworthy and corruptible senses. Pythagoras altered the

heptachord solely to engage man's intellect in proper "fitting together" -

harmonia - of the mathematical proportions.

Plutarch (44-120 B.C.) stats that for Pythagoras and his disciples, the word

harmonia meant "octave" in the sense of an attunement which manifests

within its limits both the proper fitting together of the concordant intervals,

fourth and fifth, and the difference between them, the whole tone.

Moreover, Pythagoras proved that whatever can be said of one octave can be

said of all octaves. For every octave no matter what pitch range it

encompasses, repeats itself without variation throughout the entire pitch

range in music. For that reason, Pythagoras considered it sufficient to limit

the study of music to the octave.

This means that within the framework of any octave, no matter what its

particular pitch range, there is a mathematically ordained place for the forth,

the fifth, and for the whole tone. It is a mathematical matter to show, that all

of the ratios involved in the structure of the octave are comprehended by the

single construct, which is 12-9-8-6.

For the Pythagoreans, this construct came to constitute the essential

paradigm - of unity from multiplicity.

The arithmetic- and harmonic mean

We see that 12:6 express the octave, 2:1; 9 is the arithmetic mean, which is

equal to the half of the sum of the extremes, (12 + 6)/2 = 9.

Further, 8 is the harmonic mean of 12:6, being superior and inferior to the

extremes by the same fraction.

Expressing this operation algebraically, the harmonic mean is 2ac/a+c, or in

this series, 2*12*6/12+6 = 8.

Among the peculiar properties of the harmonic proportion is the fact, that

the ratio of the greatest term to the middle is greater than the middle to the

smallest term: 12:8 >8:6. It is this property that made the harmonic

proportion appear contrary to the arithmetic proportion.

In terms of

musical

theory, these

two

proportions are basic for division of the octave since the fifth, 3/2, is the

arithmetic mean of an octave and the fourth, 4/3, is the harmonic mean of an

octave.

The principal of dividing the string by an arithmetical proportion is done by

the formula: a:b is divided by 2a:(a+b) and (a+b):2b.

The ancient Greeks did presumably such division in their studies of the

singing string by the monochord

The semi tone

We have already seen that in the diatonic genus each tetra chord was divided

into two full tones and one semi tone. A full tone derives from a fifth minus

a fourth, 3/2 - 4/3 = 9/8. The semi tone will be 4/3 - (9/8 + 9/8), or 4/3 -

81/64 = 256/243.

This semi tone is called leimma, and is somewhat smaller than the half tone

computed by dividing (for musical ratios dividing means the square root) the

whole tone in half: (9/8)½ = 3/2*2½ .

The square root of 2 was for the Pythagoreans a chocking fact because their

concept of rational numbers was scattered. (For me it represent the beauty of

real science, because it revealed the flaws in the Pythagorean paradigm of

numbers). Their own mathematic proved with the Pythagoreans doctrine of

the right angle triangle, (the sum of the squares of the two smaller sides of a

right-angled triangle is equal to the square of the hypotenuse), that in music

as in geometry there are fractions, m/n, that are incommensurables such as

square root 2, which can not be expressed with whole numbers or fractions,

the body of rational numbers, but with irrationals numbers not yet

developed.

This discovery was hold as a secret among the Pythagoreans and led to the

separation of algebra and geometry for centuries until Descartes in the 17th

century united them again.

For music it meant that there were no center of an octave, no halving of the

whole tone, no perfect union of opposites, no "rationality" to the cosmos.

Semi tone could be the "door" to other dimensions!

My errand here is to give some clues to the meta-physical functions of

semitones, which seem to involve the potential of shifting to a different

world or to enter another dimension. The key to attain a different spiritual

world exists in the search for the exact right tone, that resonates with that

particular "door" to other dimensions and worlds. The human being contains

more dimensions than just 3 spatial dimensions.

Philolaus

We have to bear in mind that Pythagoras himself left no written record of

his work; it was and is against esoteric principles. Either did those few

students, who survived the pogrom of Pythagoras. It is one in the next

generation of Pythagoreans, Philolaus (ca.480- ? B.C.), who broke the

precept of writing down the masters teaching. However, Philolaus' records

are lost, so it is Nicomachus fragments of his writing, in his Manual of

Harmonics, that actually is the only source the posterity has.

According to Nicomachus / Philolaus, the whole tone, 9/8, was divided

differently than the Pythagoreans by representing the whole tone with 27,

the cube of 3, a number highly esteemed by the Pythagoreans. Philolaus

divided the whole tone in two parts, calling the lesser part of 13 units a

"diesis", and the greater part of 14 units, "apotome". Philolaus had in effect

anticipated Plato's calculations in the Timaeus!

Timaeus by Plato

Plato (427-347 B.C.) gave in his work Timaeus a new meaning to the

Pythagorean harmonic universe by - in a purely mathematical method -

enclosing it within the mathematically fixed limits of four octaves and a

major sixth. It was determined by the numbers forming two geometrical

progressions of which the last term is the twenty seventh multiple of the first

term:

27 = 1+2+3+4+8+9

The two geometric progressions in which the ratios between the terms is 2:1 and 3:1,

respectively:

1-2-4-8 and 1-3-9-27. Combining this two progressions, Plato produced the seven-

termed series: 1-2-3-4-8-9-27. The numbers in this series contains the octave, the

octave and a fifth, the double octave, the triple octave, the fifth, the fourth and the

whole tone. The entire compass from one to twenty-seventh multiple comprises

therefore four octaves and a major sixth. In numerical terms it contains four octaves,

16:1 * 3:2 (a fifth) * 9:8 (a whole-tone) equals 27:1.

Plato then proceeded first to locate in each of the octaves the harmonic mean, the

fourth, then the arithmetic mean, the fifth. By inserting the harmonic and the

arithmetic means respectively between each of the terms in the two geometric

progressions, Plato formulated mathematical everything Pythagoras had formulated

by collecting acoustic data.

Plato did, however, independently of the Pythagoreans compute the semi

tone in the fourth, which consist of two whole tones plus something, which

is less than the half of a whole tone, namely 256:243, the "leimma".

According to Flora Levin in her commentary of Nicromachus' "The Manual

of Harmonics", Plato went further than Pythagoras by completing all the

degrees in a diatonic scale:

1 9/8 81/64 4/3 3/2 27/16 243/128 2

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

Plato's calculations led to the inescapable fact of no center to the octave, no

halving of the whole tone with rational numbers, no rationality of the

cosmos. Nicomachus did his part of covering up the secret by

misrepresenting Plato and putting off some of the shattering discoveries of

irrational numbers to some future time.

The semi tones in the different modes

Pythagoras had practiced music long before he transformed the heptachord

into an octachord that led him to discover the mathematical laws

determining the basic structure of an octave. He had fully understood the

therapeutic value of music in healing the body and soul. Most of all he knew

the set of conditions for the melody. He recognized strongly that every tetra

chord on which melody was based embodies the "natural" or physical

musical progression of whole tone-whole tone-semi-tone.

He maintained the fundamental structure of both tetra chords in his scale

and for musical reason he understood that this distribution of intervals had

to be maintained for all melodic purposes with their configurations and

inversions.

This was the foundation of the ancient Greek music, which further

developed into The Greater Perfect System

The confusion of systems

The Greek music has an inherent confusion of musical systems. A mix of

the cyclic system of perfect fifths (Pythagorean tuning), and the modal

system (tetra chords). We can only get a very faint idea of what ancient

Greek music really was about because European theorists through time have

made errors and misunderstandings.

In reality, the Arabs and the Turks happened to receive directly the

inheritance of Greece. In many cases the works of Greek philosophers and

mathematicians reached Europe through the Arabs. Most serious study on

Greek music were written by Arabs scholars such as al-Färäbi in the tenth

century and Avicenna a little later, while Westerners - Boethius in particular

- already had made the most terrible mistakes.

It is the Arabs who maintained a musical practice in conformity with the

ancient theory, so to get an idea of ancient Greek music, we should turn to

the Arab music.

The Pythagorean Tuning

The musical scale, said to be created by Pythagoras, was a diatonic musical

scale with the frequency rate as:

1, 9:8, 81:64, 4:3, 3:2, 27:16, 243:128, 2.

This scale is identical to the cyclic scale of fifths as the Chinese, if we take F

as the tonic.

It has 5 major tones (9/8) and 2 semi tones, limma (256/243), in the mi-fa

and si-do interval.

The third, 81/64, is a cyntonic comma sharper than the harmonic third, 5/4.

The Pythagorean scale was based on the three prime intervals: the octave,

the perfect 5th and the perfect 4th. "Everything obeys a secret music of

which the "Tetractys" is the numerical symbol"(Lebaisquais).

By generating 12 perfect fifths in the span of 7 octaves, 12 tones were

produced. In order to place the tones within one octave, the descending

perfect 4th (the subdominant) was used, and a 12 notes chromatic scale was

made.

He discovered what later was called the Pythagorean comma, the

discrepancy between 12 fifths and 7 octaves gives (3:2)12 > (2:1)7.

Calculated through, it is: 129.74634 : 128 = 1.014. Or in cents: 23.5. See

more about Pythagoras' Comma.

Do not mistake Pythagoras' Comma for the syntonic comma, equal to 22 cents, which

is derived from the difference between the major tone and the minor tone in the Just

Diatonic Scale, or discrepancy between the Pythagorean third and the third in the

harmonic series which is 5:4.

As far back as 2,500 years ago the Pythagorean figured out that it was

impossible to derive a scale in which the intervals could fit precisely into an

octave. The ancient Greeks explained this imperfection - the comma - as an

example of the condition of mortal humans in an imperfect world.

This fundamental problem with the 3 prime ratios: 2:1, 3:2, 4:3, - which can

be formulated in mathematical terms as interrelated prime numbers which

have no common divisor except unity - has been compromised in a number

of different temperaments of the diatonic scale up to out time.

In ancient Greek music several other modes were used based on the tetra

chords with a span of the perfect fourth. Later two tetra chords were put

together with a full tone in between so an octave was established. A number

of different modes were used in practical music performance. The different

placement of the two half tones made the different modes.

An account of ancient Greek contributions to musical tuning would not be complete

without mentioning the later Greek scientist Ptolemy (2nd c. A.D.). He proposed an

alternative musical tuning system which included the interval of the major third based

on that between the 4th and 5th harmonics, 5 / 4. This system of tuning was ignored

during the entire Medieval period and only re-surfaced with the development of

polyphonic harmony.

Gregorian church music

From those ancient Greek modes the Christian Gregorian church derived

their music, though their names were a complete mix-up of the original

Greek names for their modes. What is important in this context is the

placement of the two semitone's in the octave. They were placed differently

in order to create different modes, that produced a special tonality or mood.

The interaction between tones and semitone's made each characteristic

mode.

The Gregorian church music from the late Middle Ages developed an

amazing beauty and spirituality. We owe the monks a debt of gratitude for

their part - singing to worship the refinement of the soul and Divinity. A

side effect was the healing power in the strong amount of higher harmonics,

which vigourating effect Alfred Tomatis has described in my page, The

Power of Harmonics.

As long as the musical practice mainly was monophonic the amount of

scales could be numerous. When the wish for harmonious polyphonic

singing was appearing, the elimination of scales began because only the

scales that were in agreement with the harmonics could be used.

Polyphonic music

The development in musical practice from monophonic to polyphonic and

after the Renaissance (the end of 15th. century), to harmony, made it

necessary to have especially the third harmonized. The Pythagorean third

(81:64) is a syntonic comma larger than the harmonic third (5:4). The need

for harmonizing the third in the part-songs became imperative as the

polyphonic music became predominant.

Just Intonation - a scale of proportion

Since the major triad became the foundation of harmony in Western music,

the Pythagorean scale has largely been discarded in favor of the Just

Diatonic Scale, or the scale of Zarlino (1540-94).

The frequencies of the notes in a root position major triad are given by the

fourth, fifth and sixth harmonics in the harmonic series, i.e. the frequencies

should be in the ratio 4: 5: 6. (1-5:4-3:2).

The Major Triad as a generator

If we look at this triad as C, E, and G, the tonic, and associate it with its

dominant G, B, D and the tonic’s sub-dominant F, A, C, each of which has

one tone in common with the triad of the tonic, we obtain the complete

series of tones for the major scale of C:

1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2.

This scale consist of three different intervals: major tone 9/8, minor tone

10/9, and major half tone 16/15. Therefore when tonic is changed, we shall

obtain sharps and flats of different nature in order to keep the frame of the

scale, and the very notes of the original scale will in some cases have to be

raised or lowered by one comma (the difference between the major and the

minor tone).

The scale of Zarlino (Just Intonation) is basically a mix of notes generated

by fifths, which allows right transpositions and notes which make correct

harmonic intervals; so in practice, two different systems are used conjointly

with the result of awkward transpositions.

Those how are familiar with Rodney Collin's "The Theory of Celestial Influence",

will notice that the Just Intonation is the scale he applies to his great work on octaves

by multiplying with 24.

The Mean Tone.

In musical practice, especially when playing with key-instruments or the

simple modulation of keys, the Just Intonation causes many difficulties,

mainly due to the fact of the major and minor tones. The two different

intervals of a tone in this scale was for that reason modified during the 17th

century into a mean or average of the major and minor tone. Since these two

tones together equal a major third, the mean tone is equal to half of the

major third or 193 cents.

This temperament is not surprisingly called Mean tone temperament, or 1:4

comma mean tone. (The fifths are all equal, but have been tempered by 5.5

cents, a quarter of a syntonic comma) and was the most used temperament

in Baroque-music.

There were some problems with the enharmonic notes. The two diatonic

semitones do not add up to give a (full) tone. The Mean tone semitones are

117.5 cents. So if one wishes to play in more than six major and three minor

keys, there is trouble. This is because en-harmonically equivalent note will

not have the same frequency. Additionally this temperament has some real

false notes, called the "wolf notes", due to the 3.5 cents short fifth, so the

circle will fall short of closure by 12 x 3.5 cents = 42 cents.

Equal Temperament.

The ultimate compromise appears in Equal Temperament, which is a

circular temperament. The Pythagorean comma (as approximately 24 cents)

made the circle too large. If the 12 perfect fifths -702 cents- are equally

distributed but contracted with 2 cents each, the circle of fifths will be

complete into a circle.

In the late 17th and 18th centuries a

number of circular temperaments were

employed making use of this device. It is

often said that J.S. Bach's 48 Preludes

and Fugues were written to demonstrate

the effectiveness of Equal Temperament.

However, recent research (Barnes 1979),

has shown that he probably wrote them

for a circular temperament similar to one

devised by Werckmeister (known as

Werckmeister III), where the distribution

of fifths was unequal, some were 6 cents smaller, some were perfect.

The Equal Temperament as we know it is completely equally distributed,

slightly diminished fifths (700 cents), that in one blow eliminates the

question about different frequencies of the enharmonic notes and

modulation limitations. The octave is equally divided into 12 semitones of

100 cents. The frequency ratio for each of the semitones are the twelfth root

of an octave: (2/1) 1/12 = 1.059463094.../1.

This temperament has two scales, a major and a minor. The difference lies

in the third, sixth and the seventh, which are a half tone lower in the minor

scale. Note that the same intervals are present in the minor scale as in the

major scale, although the order is different. We will not deal with the harmonic minor scale or the melodic minor scale.

Let us make a comparison in cents of the above mentioned scales with the

Just Diatonic Scale (Just Intonation) as base:

C D E F G A B C'

Just Diatonic 0 0 0 0 0 0 0 0

Pythagorean 0 0 +22 0 0 +22 +22 0

Mean Tone 0 -

11

0 +5.5 -5.5

+5.5

-5.5 0

Equal

Temperament

0 -4 +14 +2 -2 +16 +12 0

In the western culture Equal Temperament is now so established and its

tonality so tuned in our ears, that it sounds just right, though the third and

the sixth ought to give problems because they are pretty much sharper than

the much pure and expressive Just Intonation.

The artificial Equal Temperament

The great German scientist from the 19th Century, Hermann von Helmholtz,

who also was a capable musician, made a strong stand up for the Just

Intonation scale. He claimed in his "On the Sensations of Tone"

that..."continual bold modulational leaps threaten entirely to destroy the

feeling for tonality". Further he states: "The music based on the tempered

scale must be considered as an imperfect music... If we suppose it or even

find it beautiful, it means that our ear has been systematically spoiled since

childhood".

Professor Helmholtz brought many examples of beautiful use of Just

Intonation in singing by use of the English system "Tonic Sol Faists", which

overcame the difficulties of modulation by using a different musical

notation system. Strings- and wind instruments could also perform this; so

can modern keyboards.

The discussions about Equal Temperament versus Just Intonation has continued up to

present time. Daniel White has on his web page Tuning & Music Scales Theory made

a in depth analyze of this matter concluding that ET sounds "sweeter" than JT.

Compare with the other scales we have gone over, the Equal Temperament

has no definite relations between the sounds since it has lost its relationship

with simple ratios. The more complicated the ratios are, the more dissonant

are the chords. We have been used to the muddy sounds, but for people in

the East who are trained in modal memory and clear harmonically relations,

they can not conceive the meaning of Western music.

The Equal Temperament has in spite of its obvious weakness made it

possible for great composers to create beautiful music with extraordinary

numbers of new chords and modulations.

In the twentieth Century the tendency to move away from simple ratios of

the notes by sound ratios even far away from the Equal Temperament,

appeared in the atonal music.

Modern Dodecaphonic music

In modern times a number of atonal scales has been developed to serve

the new dodecaphonic music (Schönberg, Berg, Webern), where classical

notion of harmony and rhythm is dissolved. Basically the ancient diatonic

scale with its five whole tone and two semi tones, has been replaced with

a pure chromatic scale which is a main factor in the change from melodic

tonal music to atonal dodecaphonic music.

Though I am very fond of non figurative art, the modern atonal music is

still difficult for me to enjoy spontaneously. Educated people assure me

of the new beauty in contemporary music, which I can hear with my

head, but not with my heart.

I have, however, observed a certain indifference in the mainstream of

classical music, and find myself attracted to the early European music and

folk (World) music.

My main objection to the atonal dodecaphonic music underlies in the

detachment to the physical world. The scale belongs to the invisible

realm because it is created by ratios far away from the small numbers,

which are related to the perceptible world and basic emotions.

In the ancient musical systems we have seen how keen the musical scale

had to be related to the perceptible world represented mainly as small

numbered ratios (low number of generating fifths in the cyclic system or

simple harmonic ratios in the modal system)

Cyclic and modal numbers

In this world of five elements in which we live, no prime number higher

than five can enter into a system of sounds representing melodic or

harmonic relations. The Chinese system of cyclic fifths even refuse to get

beyond the number there; all its intervals are expressed in terms of

powers of two or three. The number for cyclic systems is 3.

Some modern theorists are using the terms 3 limit scale, 5 limit scale etc http://sonic-

arts.org/dict/just.htm

The introduction of the factor of five brings us to the harmonic modal

scale, of which the characteristic intervals are the harmonic major sixth,

5/3, the harmonic major third, 5/4, the minor third, 6/5, the major half

tone, 16/15 (24 /3*5), the minor half tone, 25/24, (5

2 /3*2

3 ) the syntonic

comma, 81/80, (34 / 2

4 * 5), and so forth.

Comparing with the Equal Temperament, the tempered half tone is

something like 1,059,463,094 / 1,000,000,000 against the major harmonic

half tone 16/15.

The number five "humanizes" the music. It makes the music an

instrument of expression of the tangible reality. The introductions of

higher prime numbers such as seven, would take us beyond this reality

into regions, that are not within the scope of our normal perceptions and

understanding.

Seven is considered the number of heavenly as well as infernal regions.

We have actually no means to know to which side it may led us!

In my opinion you can only touch humans deeply, if you play harmonious

or tonal music, because these tones belongs to the real world and the man

who walks the Earth. The scale has to be more or less in accordance with

the lower harmonics in the series. The way we hear and analyze sound is

actually much the same as the standing wave in a string. The basilar

membrane in the inner ear behave like a “string” and the “software” in

the brain is designed to look for the harmonic series. It is the most

agreeable - and most basic. What it all comes down to is, that the only

measure for all phenomena is the human.

Reference tone

Before a concert begins a reference tone, the concert pitch, is played so the

instruments can tune their middle a'. In modern time the pitch is set to 440

Hz. by the second International Standard Pitch Conference in London 1938.

It is a high pitch compared to the older concert pitch of 435 Hz. which was

introduced by the French government in 1859 in cooperation with musicians

such as Hector Berlioz, Meyerbeer and Rossini.

The concert pitch has vary earlier depending of Country and time. In the

book, "On the Sensation of Tone" by Helmholtz, a record of concert pitch in

Europe covers many pages. The characteristic for the Western music is, that

concert pitch is arbitrary. It has no relation to forces above man. There is no

reference to earthly or celestial influx, but only an artificial standard.

For the old Chinese the tuning of their fundamental tone, Kung, was a matte

of outmost importance for their civilization and had to be in alignment with

the Cosmic tone so the celestial influence could be channel into the society

by music.

We have earlier mentioned Cousto's calculations (in his book, The Cosmic

Octave). He relates the Kung to the frequency of the Platonic Year. The note

of the Platonic year to be F in the Western Equal Temperament pitch, which

is found in the 48th octave with a frequency of 344.12 Hz.

The Indians method had a character of meditation since the musicians not

only in the prelude had to tune his instrument to the keynote, he also attunes

himself to it, and gives the audience the opportunity to do so too. This long

introduction was essential since the musicians had to tune in to the "sadja",

the everlasting, never ceasing tone. According to Indian tradition it stands

for primordial vibration, which is called "nada" and express the universal

OM.

The OM sound, according to Cousto, corresponds approximately to the C

sharp in the small octave of the present day tuning system (136 Hz) and

corresponds to the 32nd octave tone of the Earth year. It means that by

lowering 136 Hz tone by 32 octaves, the resulting frequency will be as slow

as the amount of time it takes the Earth to circle the sun.

It is interesting to note that the Indians arrived at this tone, which we can

calculate mathematical, "simply" through intuition and meditation. (The calculations is: A day consist of 86,400 seconds. A tropical year has 365,242

days = 31,556,925,9747 seconds. The reciprocal value multiplied by 232

= 136,10221

Hz.) The concert pitch in western music, which is 440 Hz for the middle a, ought

to be 435,92 Hz based on the note corresponding to the average solar day,

according to Cousto.

"It don't mean a thing, if it ain't got that swing"

Those who are familiar with the jazz (swing) musician, Duke Ellington, will

"hear" Ella Fitzgerald sing this song. The reason I will end on that note is to

make clear, that music is more that scales and right tuning. Music contains

of 4 major elements:

melody, rhythm, harmony and interpretation or intention.

Having this in mind I will continue with The Sound of Silence, where I will

extend the law of octaves in to other realms than scales and tuning by an

elaboration on the metaphysical properties of sound and music.

Thomas Váczy Hightower © 2002