0002 Pythagoras

7/31/2019 0002 Pythagoras

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Chapter 2: A Condensed History2

We are told that Pytharoras experimented with an

device called theMonochord(which literally means

one string) by his studentPhilolaus. This was a

single stringed instrument with a moveable bridge

and by positioning the bridge in different positions itwas possible to play different notes on the string.

His reported aim in his analysis of the vibrations of

the strings was to define the music of the spheres

and was thus an attempt to understand the heavens.

He, along with other Greeks, were of the opinion

that the heavenly bodies moved in a form of musical

precision and that by analysing music one gained an

insight into the movement of the heavens. As early

as Ancient Babylon, mathemeticians believed thatthe heavens were governed by ratios of integers and

it is perhaps from here that Pythagoras found the

inspiration to experiment with the instrument.

Whatever the inspiration, there is little evidence that

any theory was used in the tuning of musical scales

prior to the life of Pythagoras. Conversely, the start

of music theory being important to the tuning of

instruments almost certainly begins with the words

of Philolaus. It is likely that prior to the advent of

the Pythagorean ratios that musical scales were veryvaried and perhaps even unique to the individual.

By using at least two monochords, Pythagoras was

trying to measure the lengths at which the notes

from the strings would ring together in what could

be considered perfect harmony. It is highly likely

that he believed that these notes would be found at

whole number ratios along the length of a string. It

is also quite likely that he was happy to find that

this appeared to be the case.

Through his experimentation, he discovered that

strings at a length ratio of 2:1 provided a consonant

sound, an interval the Greeks called a diapason (dia

means across, between or through in Greek). The

second consonance was found at a length ratio of

3:2, which became named the diapente. The final

consonance was found at a length ratio of 4:3 and

was called a diatessaron. These ratios, when stated

together, formed the ratio 1:2:3:4 which will nodoubt have reinforced Pythagoras beliefs.

Music theory in Ancient Greece was based around

the Tetrachord(in Greek, tetra means four). The

four notes were tuned to notes in a descending

order. The first and fourth notes were separated by

the interval of a diatessaron. The two other stringswere tuned to one of a number of intervals, the size

of which depended on the musical scale.

It was soon noticed that the two intervals 3:2 and

4:3 could be multipled together to become the 2:1

ratio. For example:

3 x 4 = 12 = 2

2 x 3 = 6 = 1

It was also noted that the interval between the notes

could be found by dividing one by the other. This

interval was known as the Whole Tone and is

defined as the ratio 9:8. The calculation would be:

3 x 3 = 9

2 x 4 = 8

This definition of a tone persists for centuries and

does not fully disappear from music theory until as

late as the eighteenth century. When two tetrachordsare placed within an octave and separated by a tone

the resulting scale is known as aDiatonic scale. The

two tetrachords in a diatonic scale are often called

diatonic tetrachords.

It is widely believed that Pythagoras constructed a

tuning system which he based on the interval ratios

that he discoved using the monchord. The exact

method he used for achieving this is a matter for

debate between music historians, but the belief mostcommonly held is that the intervals are calculated

using the 3:2 ratio.

As the note found at two thirds of the string length

is consonant with the note found at the full length,

then the note found by a further shortening of the

string is consonant with that found at the two thirds

point. By extending the string by a third, we find the

consonant note suggested by the 4:3 ratio. If this

longer string is then extended by a third of its newlength the resulting note is consonant.

Creating The Pythagorean Scale


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Metal In Theory 3

We can continue the process of both shortening and

lengthening the string by the 2:3 ratio to find a

series of steps which denote musically related notes.

Mathematics at the time saw no reason to go further

than a cubed number as only three dimensions were

visible to the naked eye. Due to this opinion,

Pythagoras probably achieved his seven note scale

by multiplying the ratios by themselves to create a

squared and a cubed ratio - thus making 3 powers of

the ratio 3:2 in each direction.

For the 2:3 , the squared ratio is:

2 x 2 = 4

3 x 3 = 9

And the cubed ratio is:

4 x 2 = 89 x 3 = 27

For the 3:2, the squared ratio is:

3 x 3 = 9

2 x 2 = 4

And the cubed ratio is:

9 x 3 = 274 x 2 = 8

These fractions describe a set of notes which span

from under half the length of the original string to

almost four times its length. Using the 2:1 ratio to

shorten or lengthen the string, we can place all of

the calculated notes in the same range. The range we

chose is between the original length of the string

and a point described by the 2:1 ratio itself. i.e. half

the original string length.

So, the calculation for the first power of the 3:2

ratio would be:

3 x 1 = 3

2 x 2 = 4

This gives us the third of the perfect intervals -

that found by the 3:4 ratio - and this is within the

range we require. When the result is still outside therange the 2:1 ratio is applied a second time

Fig 2.01 shows the calculations, the adjustment

ratios and the adjusted lengths of a string. Seven

was an especially important number to the Greeks

as it denoted the number of heavenly bodies, not

including stars, that they were aware of. These were

the Sun, the Moon, the Earth, Mercury, Venus, Mars

and Jupiter. To Pythgoras, the idea that his newly

defined scale had seven notes was very appealingand this fact was perhaps the reason that the scale

became an accepted part of music theory.

Because there are seven notes, the scale is referred

to as a heptatonic scale - hepta means seven and

tonos means tone. Additionally, the eighth note

when ascending through the scale is defined by the

2:1 ratio and is called the octave (octa means eight).

Now, despite our extensive use of a twelve note

scale, the interval between two notes at a 2:1 ratio is

known by a name which has its origin in a seven

note scale which is more than 2,000 years old.

When the seven calculated string lengths are

reordered from longest to shortest, they define what

is referred to as the Pythagorean Heptatonic scale

(seeFig 2.02). It stands as the first scale built on a

basis in mathematics and as such begins a process of

mathematical analysis in music that lasts to this day.

RatioCalculated

RatioDecimal

Ratio

to Next

Roman

Letter

(2 / 3) ^ 0 1 / 1 1.0000 8 / 9 D

(2 / 3) ^ 2 8 / 9 0.8889 243 / 256 C

(3 / 2) ^ 3 27 / 32 0.8438 8 / 9 B

(3 / 2) ^ 1 3 / 4 0.7500 8 / 9 A

(2 / 3) ^ 1 2 / 3 0.6667 8 / 9 G

(2 / 3) ^ 3 16 / 27 0.5926 243 / 256 F

(3 / 2) ^ 2 9 / 16 0.5625 8 / 9 E

(1 / 2) ^ 1 1 / 2 0.5000 8 / 9 D

Fig 2.02 : Pythagorean Heptatonic Scale values in order

Ratio Calculation Adjustment Adjusted Ratio

(3 : 2) ^ 3 27 / 8 1 : 4 27 : 32

(3 : 2) ^ 2 9 / 4 1 : 4 9 : 16

(3 : 2) ^ 1 3 / 2 1 : 2 3 : 4

1 : 1 1 / 1 1 : 1 1 : 1

(2 : 3) ^ 1 2 / 3 1 : 1 2 : 3

(2 : 3) ^ 2 4 / 9 2 : 1 8 : 9

(2 : 3) ^ 3 8 / 27 2 : 1 16 : 27

Fig 2.01 : Pythagorean Series using the 3:2 ratio


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Chapter 2: A Condensed History4

It can be seen inFig 2.02 that many of the intervals

between notes are described as a ratio of 8:9, which

we have already seen was defined as a Whole Tone.

In this table is another ratio of a special significance

- the ratio 243:256 - known as the Semitone. We can

see how this figure is arived at by divining the ratio

between the second and third degrees of the scale.

27 / 8 = 27 x 9 = 243

32 / 9 = 32 x 8 = 256

The letters assigned to the notes inFig 2.02 are at

first counter-intuitive to the modern musician - the

letters seem to be the mirror image of what would

be expected in a modern scale. This is because the

Ancient Greeks listed their scales in descending

order as opposed to the ascending order which is

commonly used today. The letters themselves areactually derived from Roman music theory, although

the Romans used fifteen letters to describe their

scale to the modern systems seven.

Whilst the mathematical theory is of importance to

the modern musician, a full scale analysis of the

Greek theory system lies beyond the scope of this

book. That said, a short detour into the outlines of

the system is worthwhile as the concepts which

drive it also drive the theory behind the modernmusical system defined fifteen hundred years later.

The Greater Perfect System (Systma Teleion

Meizon) was a Greek scale that was built on a set of

four stacked tetrachords called theHypatn,Mesn,

Diezeugmenn andHyperbolain tetrachords. Each

of these tetrachords contains the two fixed notes that

bound it.Fig 2.03 shows these notes in the darker

shade of grey, whilst the tetrachords are highlighted

in a lighter shade.

The cousin of the Greater Perfect, the Lesser Perfect

System, was built on three stacked tetrachords - the

Hypatn, Mesn and Synmenn. The first two of

these are the same as the first two tetrachords of the

Greater Perfect, whilst the third tetrachord is placed

above the Mesn. When viewed together, with the

Synmenn tetrachord placed between the Mesn

and Diezeugmenn tatreachords, they make up the

Immutable System (Systma Ametabolon) which is

also referred to as the Unmodulating System.

Returning to the Pythagorean mathematics behind

the system, we can further analyse the intervals

between any two notes in the scale. The results of

which are found in ratio form in Fig 2.04. The titles

of the columns indicate the note from which the

interval should be measured whilst the titles of the

rows indicate the target notes. The ratios are stated

in the form:

Target Length : Initial Length

We can see from the table that all but one of the five

note intervals are defined by the ratio 2:3 (a true

harmonic fifth). The remaining interval between the

notes F and B can be found by analysis to be one

semitone smaller than a harmonic fifth.Fig 2.04

also shows us that all but one four note interval is

defined by the ratio 3:4 (a true harmonic fourth).

The exception to the rule is that found between B

and F which is found to be a semitone larger than

the harmonic fourth.

These two exceptions when considered together

should, like the harmonic fourth and fifth, define the

ratio of an octave (1:2). If we analyse this we find:

729 * 512 = 1

1024 * 729 = 2

TetrachordRoman

NoteGreek Note Planet

Hyperbolain

P Nt Saturn

O Parant Jupiter

N Trit Mars

Diezeugmenn

M Nt Sun

L Parant Venus

K Trit Mercury

I Parames Moon

H Mes -

Mesn

G Likhanos Saturn

F Parhypat Jupiter

E Hypat Mars

Hypatn

D Likhanos Sun

C Parhypat Venus

B Hypat Mercury

A Proslambanomenos Moon

Fig 2.03: The Greater Perfect System of Ancient Greece


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Metal In Theory 5

When we consider the difference between the two

we would expect to find them identical - a fifth

minus a semitone should equal a fourth plus the

same semitone. What we find is somewhat different:

729 / 512 = 729 x 729 = 531441

1024 / 729 = 1024 x 512 = 524288

This ratio describes an interval fractionally over an

octave and demonstrates a fault in Pythagorean

mathematics. Known as thePythagorean Comma, it

was the focus of much analysis over the next fifteen

hundred years as mathematically inclined theorists

attempted to solve the problem it defined.

Intervals between two consecutive notes all measure

either a tone (8:9) and a semitone (243:256). Three

note intervals also have two possible ratios, of

which the larger is two tones (64:81) and is referred

to as aDitone in Greek theory. The smaller of the

two three note intervals in a tone and a semitone

(27:32). The four note interval found between the B

and the F (512:729) is three tones and as such is

known as a Tritone, as is the interval between the

notes F and B (729:1024). Because there are two

possible tritones, the interval was avoided.

The six note intervals in the table can be found in

two forms - four tones (81:128) or four tones plus a

semitone (16:27). Finally, the seven note intervals

also appear in two forms - five tones (9:16) or five

tones plus a semitone (128:243).

If the last step of both the ascending and descendingseries is ignored, the scale generated contains five

notes. This is a Pentatonic scale and can be found in

many styles of music, not least far eastern (Chinese,

Japanese et al.) and European folk music. The

Pentatonic version of the scale, shown inFig 2.05,

removes the two least correct members of the tonal

series. This version of the table also inverts interval

ratios to list the note letters in a modern context.

The use of a Pentatonic scale removes the Tritone

and hence removes the inaccuracy within the system

- all fourths are now perfect fourths and all fifths

are similarly perfect.

Our understanding of Ancient Greek music itself is

rather limited by the fact that no notational system

existed with which to record the note sequences.

However, a number of Ancient Greek scales have

become part of the lexicon of modern music, mostly

thanks to the work ofKathleen Schlesingerin her

1939 book The Greek Aulos (an aulos is a type of

flute). From archaeological evidence she lists a setof scales which form the basis of Ancient Greek

music. More recent research has since brought the

accuracy of her work into question, but the scales in

her book still persist. Whatever the truth of

Schlesingers analysis, her work helps to cement the

link between Ancient music theory and that of the

modern era. As with much of music theory, her

work will probably be augmented by the discoveries

of future generations rather than being lost.

RatioCalculated

RatioDecimal

Ratio

to Next

Roman

Letter

(2 / 3) ^ 0 1 / 1 1.0000 9 / 8 D

(2 / 3) ^ 2 9 / 8 1.1250 32 / 27 E

(3 / 2) ^ 1 4 / 3 1.3333 9 / 8 G

(2 / 3) ^ 1 3 / 2 1.5000 32 / 27 A

(3 / 2) ^ 2 16 / 9 1.7778 9 / 8 C

(1 / 2) ^ 1 2 / 1 2.0000 9 / 8 D

Fig 2.05: Pythagorean Pentatonic Scale values in order

D C B A G F E D

D 1 : 1 9 : 16 16 : 27 2 : 3 3 : 4 27 : 32 8 : 9 1 : 2

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

B 27 : 32 243 : 256 1 : 1 9 : 16 81 : 128 729 : 1024 3 : 4 27 : 32

A 3 : 4 27 : 32 8 : 9 1 : 1 9 : 16 81 : 128 2 : 3 3 : 4

G 2 : 3 3 : 4 64 : 81 8 : 9 1 : 1 9 : 16 16 : 27 2 : 3

F 16 : 27 2 : 3 512 : 729 64 : 81 8 : 9 1 : 1 128 : 243 16 : 27

E 9 : 16 81 : 128 2 : 3 3 : 4 27 : 32 243 : 256 1 : 1 9 : 16

D 1 : 2 9 : 16 16 : 27 2 : 3 3 : 4 27 : 32 8 : 9 1 : 1

Fig 2.04: Pythagorean Heptatonic Scale intervals

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