Plato - Harmonia

8/9/2019 Plato - Harmonia

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Platos

Harmonia


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C 261.6C# 276.5

D 293.7

D# 310.4

E 328.6F 348.4

F# 369.1G 391.1

G# 414.3

A 438.9A# 465

B 492.7

C 523.2 Double the first C

The doubling of a pitch is a law that binds all harmonia. Pythagoras is credited with

discovering this ratio, and the arrangement of a music scale is based upon it. It is not disputable.

What is disputable is the frequency of each note and the distance between notes, which is largely

determined by what sound is pleasing to the ear. Modern frequencies are not etched in stone;

classical artists determined A at 432 hertzs, in eighteenth century France this note had a value of

376 hertzs, and in seventeenth century Germany it was set as high as 560 hertzs. There is no

set formulae.

The modern Chromatic scale divides the octave into twelve parts. Two successive

pitches, C to C# for example, are related to the previous pitch by a factor of the twelfth root of

2--a ratio of 1.05946309436. The intervals between these notes are called half-steps. This

measure of the music scale has its shortcomings. Some notes are not as pleasing to the ears as

others, and some do not sound pleasing when played with others.

The division of a scale into twelve parts is modern; little is understood of ancient music

scales, but what we do know is that seven and five note scales existed. What we dont know is

why we know so little. If we turn to Plato, the solution to this problem is obvious--this

knowledge was kept in the hands of the men who wielded control of the ancient religions;


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The seven portions do in fact represent seven frequencies, but, the scale is bound

numerically from the first pitch (1) to the last (27), and the problem of frequencies doubling is

solved when the intervals are filled in. Plato instructs us on how to do this:

Next he went on to fill up both the triple and double intervals, cutting off more

parts from the original mixture and placing them between the terms so that within

each interval there were two means, the one (harmonic) exceeding the one extreme

and being exceeded by the other by the same fraction of the extreme, the other

(arithmetic) exceeding the one extreme by the same number whereby it was

exceeded by the other. This gave rise to intervals of 3/2 and 4/3 and 9/8 within the

original intervals.1

Unfortunately, no one has done this the way Plato has instructed. Cornford is a perfect

example. To prove our point we will show both sides of the argument; Cornford starts with

seven notes:

1. (For example: if we take the extremes of 6 and 12, the harmonic mean is 8, exceeding

the one extreme (6) and exceeded by the other (12) by one-third of 12. The arithmetic mean is 9,

exceeding 6 and falling short of 12 by the same number.)


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Next he fills in the means:

Then he explains: Omitting the numbers in brackets, which occur in both series, we

obtain the single series:

If we now fill in the corresponding notes, the result is as follows:

Cornford continues: As the last sentence remarks, this gives rise to intervals of a fifth

3/2 or a fourth 4/3 or a tone 9/8 within the original intervals. The final step, taken in the next

sentence, is to fill up every tetrachord with two intervals of a tone 9/8 and a remainder 256/243

nearly equivalent to our semitone. And he went to fill up all the intervals of 4/3 (fourths) with

the interval 9/8, leaving over in each a fraction. This remaining interval of the fraction had its

terms in the numerical proportion of 256 to 243. If we take the first octave (two disjunct

tetrachords), the result can be illustrated (approximately as follows, though Plato would have

thought of the tetrachord in the shape of A G F E, rather than C D E F:


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Cornford goes on to say that the harmonic and arithmetic means have their place in music theory

as determining the intervals of a fourth and fifth, but little else in the way of this scheme seems

to pertain to music.

Yet it is obvious that Platos dialogue pertains entirely to music. Something is missing;

and so we must turn back to where we started--to when Plato first divided the whole into seven

portions. The secret of this scheme is that the whole has a numeric value. The following

numbers stand behind the double and triple proportion:

(1) 432(2) 864

(3) 1296(4) 1728

(9) 3888(8) 3456

(27) 11,664

The first four numbers of Platos proportion will be quickly recognized by most as related

to the Hindu yugas (ages). We also see this number and its components quite often in ancient

architecture. Moreover, the numeral 432 is twice the sum of the cubes of the sides of the famous

3-4-5 (3x3x3+4x4x4+5x5x5) Pythagorean triangle, and it is found in the doubletriangle of the

Hindus --six being the length of each side of each triangle (6x6x6), totaling the number--432.

This numeral is the base of Platos scale.

Now we will complete the scheme the correct way. The first step is to insert the

harmonic and arithmetic means between the terms of the double and triple proportions:

Next he went on to fill up both the triple and double intervals, cutting off more

parts from the original mixture and placing them between the terms so that within

each interval there were two means, the one (harmonic) exceeding the one extreme

and being exceeded by the other by the same fraction of the extreme, the other


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(arithmetic) exceeding the one extreme by the same number whereby it was

exceeded by the other. These links gave rise to intervals of 3/2 and 4/3 and 9/8

within the original intervals.

The arithmetic and harmonic means inserted between the terms of the double proportion:

Ratios 1:1 4:3 3:2 2:1

Frequencies 432 576 648 8649/8

Ratios 2:1 4:3 3:2 4:1

Frequencies 864 1152 1296 17289/8

Ratios 4:1 4:3 3:2 8:1Frequencies 1728 2304 2592 3456

9/8

The arithmetic and harmonic means inserted between the terms of the triple proportion:

Ratios 1:1 3:2 2:1 3:1 4.5:1 6:1 9:1 13.5 18 27

Frequencies 432 648 864 1296 1944 2592 3888 5832 7776 11,6644/3 4/3 4/3

The insertion of the arithmetic and harmonic means produced intervals of 4/3, 3/2 and

9/8 within the original intervals. Exactly what the philosopher stated (Cornfords version

produced other intervals, 9/2, 16/3, 27/2). Plato is very specific on what to do next:And he

went on to fill up all the intervals of 4/3 with the interval of 9/8, leaving over in each a

fraction. This remaining interval of the fraction had its terms in the numerical proportion

256 to 243. Cornford made the assumption of thinking 4/3 meant fourths or tetrachords. He

was wrong. It means exactly what it says. The next step is to fill up the intervals of the 4/3s in

both the double and triple proportions:


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Ratios 1:1 4:3 3:2 2:1

Frequencies 432 486 512 576 648 864 >DoubleIntervals 9/8 256/243 9/8 9/8 9/8

Ratio 3:2 2:1Frequencies 648 729 768 864 >TripleIntervals 9/8 256/243 9/8

Thus the combination of the double and triple proportion produces seven notes. Cornford

is wrong again; a scale can be constructed using the double and triple proportions.

The final step is to complete the proportions. The tonic (first note of a scale) in the Greek

music scale was A, and so, in our re-construction of Platos harmonia we will ascribe A as being

equal to 432 hertzs or vibrations per second. The sequence of notes that follow are derived with

A as the tonic. Starting with the double proportion:

Notes A G F E D C B A

Ratios 1:1 9:8 32:27 4:3 3:2 27:16 16:9 2:1Frequencies 432 486 512 576 648 729 768 864

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

Notes A G F E D C B ARatios 2:1 9:8 32:27 4:3 3:2 27:16 16:9 4:1Frequencies 864 972 1024 1152 1296 1458 1536 1728

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

Notes A G F E D C B ARatios 4:1 9:8 32:27 4:3 3:2 27:16 16:9 8:1

Frequencies 1728 1944 2048 2304 2592 2916 3072 3456

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


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frequency to 6912 in the triple proportion. The same thing will be found to happen with each

note.

The total number of notes and frequencies in Platos scale:

A--- 432, 864, 1728, 3456, 6912

G--- 486, 972, 1944, 3888, 7766

F--- 512, 1024, 2048, 4096, 8748

E--- 576, 1152, 2304, 4608, 9216

D--- 648, 1296, 2592, 5184, 10,368

C--- 729, 1458, 2916, 5832, 11,664

B--- 768, 1536, 3072, 6144

Thirty four. Seven notes, twenty seven frequencies, and all the systems of notes that

evolve from this scheme:

But when you have grasped, my dear friend, the number and nature of the

intervals formed by high pitch and low pitch in sound, and the notes that bound

those intervals, and all the systems of notes that result from them, the systems

which we have learned, conformably to the teaching of the men of old days who

discerned them, and (called) themscales... (Philebus 16E)

Intervals formed by high pitch and low pitch... systems of notes...notes that bound

intervals... our rendering of Platos harmonia fits exactly. Others have not.

Platos music scale bears a resemblance to the Pythagorean scale that emerged in the

post-Christian era, as it contains five tones (9/8) and two half tones (256/243) of the same


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proportions. The Pythagorean scale also determined the ratios of the fourth (4:3) and fifth (3:2)

in the same manner. But this scale (which is undoubtedly an inaccurate production of the true

Pythagorean scale) has capital flaws; moreover, its origin is quite obviously found in the

mistaken interpretation of Platos harmonia (look closely at Cornfords finished product and you

will find the Pythagorean scale).

Platos scale, on the other hand, has no deficiencies. It is a unity. A system of notes that

come into existence when the art of measure embraces the unlimited. It is the whole of music.

The use of the double and triple proportions in the construction of the music scale places

a limit to that which is unlimited-- the high and low in pitch. Plato spoke of the law and order

that are marked by limit in Philebus (25, 26). Then, speaking of the family of unlimited to

Protarchus:

Socrates: And now, as the next step, combine with it the family of the limit.

Protarchus: What family? Please explain.

Socrates: That of equal and double and any other kind that puts and end to the

conflict of opposites with one another, making them well proportioned and harmonius by the

introduction of number..and in the case of high and low in pitch, or of swift and slow, which are

unlimited, does not the introduction of these same elements at once produce limit and establish

the whole art of music in full perfection?

Platos scale does just that. Establishes limit. The design of the scale insures it. The

double proportion is a tool used in the scales construction, but the triple proportion is thefinished

product. All the notes of the double proportion are contained within it. It has a beginning and an

end. It cannot be extended in either direction to form another octave without breaking down. It

is a closed system. This cannot be said of any other music scale.

But is it perfect? For the last two thousand years mankind has been trying to equalize the

octave. It cant be done. Why? It is a creature of proportion.


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A proportion exists by means of lesser and greater parts. For music to sound in tune,

the different notes being sounded simultaneously need to be related in a way which is pleasing to

the ear. This is where proportion comes into play. The insertion of the harmonic and arithmetic

means between the terms produced a center:

432 576 648 864(9/8)

72 parts separate this center. If we obtain seven notes there will be seven tones. If the

tones were equal to the center, 72 parts, then the parts would exceed the term (432)--504 parts, or

72 extra parts. In order for this to work, five of the parts must equal 72 and the other two parts

need be cut in half--36 parts. This is the reason there are five tones and two semi-tones in an

octave. However, if we divide the tones and half-tones with parts of 72 and 36 it doesnt work.

The intervals between notes are erratic, the half tones are all on one side of the octave, and the

relationship between notes is tenuous:

432 468 504 576 648 720 792 86436 36 72 72 72 72 72

13/12 14/13 8/7 9/8 10/9 11/10 12/10

This is why we were instructed by Plato to fill up all the intervals of the terms with 9/8

(the tone determined by the harmonic and arithmetic means); the end result:

432 486 512 576 648 729 768 864

54 26 64 72 81 39 96

Balance. The tones and half-tones on the left side of the scale add up to 144 parts. The

tones and half-tones on the right side of the scale add up to 216 parts. Both sides are in


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proportion to each other, and to the center. And of course, the notes stand in proportion to each

other--they are not arbitrarily divided--they are related by means of proportion.

This music scale is mathematically superior to modern scales. The fact that Plato

concealed it in his scheme indicates its extreme value. As it now stands, the unveiling of a

music scale existing over two thousand years ago, constructed in a manner heretofore unheard of,

is a discovery that should send a shock to scholastic communities the world over. It will certainly

deal a fatal blow to the idea that our ancestors were primitive, and likewise, open modern minds

to other secrets left behind by the ancients for us to decipher.

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Plato - Harmonia