"The 7 intervals (piano white keys) between successive notes of a diatonic scale are not all equal. Some are twice as large as others...

The usual divisions of the tetrachord in classical Greek music are, in the diatonic genus or scale, semitone, whole tone, whole tone; in the chromatic, semitone, semitone, tone and a half; in the enharmonic, quarter tone, quarter tone, ditone.... and found in the footnote:

According to modern usage, the scales are here considered as ascending series...
Thus the tetrachords are e, f, g, a (diatonic), e, f, g flat, a (chromatic), and
e, e+, f = (g double flat), a (enharmonic)".

(Note: Philosophic arguments between ancient philosophers about music is so convoluted and confusing that I will try to keep their statements to a minimum. Only by empherical experiment with frequencies and intervals can we know who was correct as to the frequency of matter, and the frequencies necessary to change matter.

(See Music Theory and Ancient Cosmology)

31.2 Archytas

"According to this approach (of Archytas), the diatonic tetrachord has the structure: tone, tone, semitone; The chromatic, tone and a half, semitone, semitone; and the enharmonic, ditone, diesis, diesis... in the footnote was found these figures:

(4 : 3) : (9 : 8) : (9 : 8) = 256 : 243 (the remainder).

The interval Parhypate (28 : 27) of Archytas, with the Hypate, which is one whole-tone lower (9 : 8), yields the pleasing interval of the diminished third (7 : 6). This very interval, Parhypate-Hyperhypate frequently occurs in the fragment of the music for Euripides' Orestes; so Archytas apparently took as his point of departure the practical application of enharmonics".

Lore and Science.

31.3 Ancient heptachord

synemmenon- E,F,G,A,B flat,C,D.

31.4 Nicomachus

Neate (Moon)- D.

Paraneate (Aphrodite)-C.

Paramese (Hermes)- B flat.

Mese
(Sun)- A.

Hypermese (Ares)- G.

Parhypate (Zeus)- F.

Hypate
(Cronus) E.

31.5 Terpander

E, F, G, A, C, D, E'.

31.6 Nicomachus

E, F, G, A(Intercalcate trite), B sharp, C, D, E'.

31.7 Pythagoras

E, F, G, A, B flat(Point of conjunction), C, D.

31.7.1

31.8 Plato

"In the Timaeus... Plato, by using the arithmetic, geometric and harmonic progressions, constructed a series of notes by which the World-Soul was divided into harmonic intervals. In so doing he showed that the interval of a fourth consisted of two whole tones, each in the sesquioctave proportion of 9 : 8 and a residue or fraction which has its terms in the numerical proportion, 256 : 243.

31.8.1

(1)....4/3....3/2....(2)....8/3....3....(4)....16/3......6......(8).

(1)....3/2......2.......(3)....9/2....6....(9)....27/2....18....(27).

Combining both he obtained:

(1)....4/3....3/2....(2)....8/3....(3)....(4)....9/2....16/3....6....(8)....(9).... 27/2....18....(27).

Concentrating on the first four terms in parentheses- 1, 2, 3, 4-

which now yield two octaves comprising the smaller consonances, fourths and fifths, the following musical pitches are fixed:

1(E)....4/3(A)....3/2(B)....2(E')....8/3(A')....3(B')....4(E")

The notes falling within the first octave were then ascertained by Plato to be two disjunct tetrachords or a diatonic scale":

1(E)....9/8(Fsharp)....81/64(Gsharp)....4/3(A)....3/2(B)....27/16(Csharp)
....243/128(Dsharp)....2(E')

31.9.1

31.10 Diagram. Musical Scales

31.11 Plato

"Plato... subtracted two whole tones from the fourth and found the remaining interval or leimma to be":

256/243 : 4/3 / 9/8 x 9/8 or 4/3 - 81/64.

(Correspondences, Fuller Synergetics Table 963.10).

31.12 Plato's Means

"What lay behind Plato's elaboration of the diatonic scale is a procedure whereby the harmonic and arithmetic means are inserted between each of two successive terms of a series derived from the progression:

1, 2, 3, 4, 8, 9, 27,

a combination of the two geometric progressions-- 1, 2, 4, 8 and 1, 3, 9, 27-- each of which has in common the number 1, the ratio between their terms being 2 : 1 and 3 : 1 respectively. The 7 terms thus contain the ratios for the octave (2 : 1), the octave and a fifth (3 : 1), the double octave (4 : 1), the triple octave (8 : 1), the whole tone (9 : 8), the entire compass from 27",

31.12.1

"to 1 being four octaves and a major sixth"....

31.13 Philolaus

Philolaus... may be summarized as follows:

"The interval of a fourth, calculated to be in the proportion, 4 : 3, such as obtains between the notes C and F, for example, may be represented in the ratio 256 : 192. Starting from 192",

31.13.1

"which represents the note C, if one adds to it the interval of a whole tone, or the sesquioctave proportion of 9 : 8, the resulting number, productive of the note D, will be 216. Thus 216 : 192 = 9 : 8. Again, if one adds to 216 another whole tone in the proportion of 9 : 8, the resulting number, productive of the note E, will be 243. And 243 : 216 =9 : 8.The notes between C,represented by 192, and F, represented by 256, are then found to be in the sequence:

.C.............D............E.............F.

192..........216..........243..........256.

31.13.2

"The sesquioctave",

31.13.3

"of 192 is that number which contains 192 once and its eighth besides, or 24. The sesquioctave of 216 is 243 which contains 216 once and its eighth, or 27. These numbers, 24 and 27",

31.13.4

"represent also the difference between 216 and 192 and 243 and 216 respectively. But the difference between 256 and 243, being 13, and representing the interval between E and F, a semitone, is neither half of 24 nor half of 27, this demonstrating that the whole tone cannot be divided into equal halves".

From the book Lore and Science.."According to the ordinary system of Greek music , the basic tetrachord (e-a) is considered by another, either synemmenon (a-d') or diezeugumenon (b-c'). In one case the Trite is b flat, in the other c', but in Philoaus it is b, a whole tone from the Mese and a forth away from the Nete. it is no wonder that, according to Nicomachus, many accused Philolaus of an error . This very fact, however, makes it improbable that some forger has injected an artificial archaism; the purpose of an artificial patina is to arouse confidence, not mistrust. So, before deciding that this is a mistake stemming from sheer stupidity, we should try to interpret the name in a way that it makes sense.

It is certain that the lyre, for a long time, had 7 strings and the number of strings were gradually increased in the fifth and forth centuries B.C., but next to nothing is known about how these 7 strings were tuned. From the time of the Aristotelian Problemata the theory is attested that the 7 stringed lyre embodied the synemmenon system, and that the diezeugmenon was introduced later; Nichomachus attributes this step to Pythagoras".

Lore and Science

182.2 B9174l

31.14 Ptolemy

"We can get further by consideration of the inconsistency in the system of Archytas which Ptolemy mentions. According to Archytas, the pattern of the chromatic genus is,

Mese (a)

....................................>32 : 27

Lichanos (g flat)

....................................>243 : 224

Parhypate (g)

....................................>28 : 27

Hypate (e)

In the effort to explain these remarkable figures, scholars have pointed out that the two lower intervals together make a whole tone , or that the interval Lichanos-Paramese is a pure forth. In the footnote was found these figures:

(243 : 224) x (28 : 27) = 9 : 8. Still, the series 6 : 5 (minor third), 15 : 14, 28 : 27 would be more "beautiful"; here the lower tones taken together make a whole tone (15 : 14) x (28 : 27) = 10 : 9.

But the rationale offered by Ptolemy has not been much noticed: "Archytas obtains the second tone in the chromatic genus (g flat)...with the help of the tone that occupied the same position in the diatonic genus (g), for, he says the second highest tone in the chromatic genus stands in the ratio 256 : 243 to the corresponding tone in the diatonic genus". This explanation is so odd that we cannot attribute it to an intermediary source or to Ptolemy himself.. Who could have gotten the idea, instead of using the obvious relationships pointed out by modern scholars, of introducing a calculation so complicated and based upon a different genus, the diatonic? We have no alternative but to recognize the derivation Ptolemy gives as that of Archytas. He found the highest interval in the chromatic tetrachord not by harmonic division and not by reference to the natural conchords, but by the extrinsic addition of two previously known values, that of the diatonic whole tone (9 : 8) and the ratio 256 : 243 "remainder" when two whole tones are subtracted from the forth . Thus Archytas is presupposing two things: a music theory which builds its scales by the addition and subtraction of intervals, and a calculation of the diatonic scale of the numeric values found in the Timaeus.

This music theory may be identified by the one rejected by Plato, which sought to identify the smallest interval, as a standard of measurement; and we may conclude from Plato, as Frank does, that this theory was better known than the Pythagorean. The Aristoxenian conception of the tonal continuum is in any case primary; both the language of professional musicians and the beginnings of musical theory are couched in its terms. According to this approach, the diatonic tetrachord has the structure tone, tone, semitone, the chromatic, tone and a half, semitone, semitone; and the enharmonic, ditone, diesis, diesis".

Lore and Science

(Note: As the arguments go back and forth between learned men we cannot tell yet who
is correct but I'm betting on Archytas' interval ratios. Yet, the discourse give
an important hint as to what is correct. This authors discourse gives the hint).

It begins: "If nothing can with certainty be proven about the genuine antiquity of the Philolaic terminology provided by Nichomachus, such is not the case with regard to Philolaus' alleged division of the whole tone into two dieseis, to which Nicomachus refers. Although Nicomachus does not provide the particulars of Philolaus' analyses of these dieseis in the Enchiridion, this information is found in the extracts of his longer works. Moreover, it is the source of Boethius' explanation of Philolaus' division of the whole tone. According to Nicomachus, Philolaus postulated the division of the tetrachord and found it to consist of two whole tones and a diesis (Philolaus' term for a semi-tone). The method of ascertaining this division, detailed by Nicomachus and attributed to Philolaus by Boethius, may be summarized as follows:

31.15 The Intervals of a 4th

The interval of a forth calculated to be in proportion, 4 : 3, such as obtains between the note C and F, for example, may be represented by the ratio 256 : 192. Starting from 192",

31.15.1

"which represents the note C, if one adds to it the interval of a whole tone, or the sesquioctave proportion of 9 : 8, the resulting number, productive of the note D, will be 216. Thus 216 : 192 = 9 : 8. Again, if one adds to 216 another whole tone in the proportion 9 : 8, the resulting number productive of the note E, will be 243. And 243 : 216 = 9:8. The notes between C, represented by 192, and F, represented by 256, are then found to be in the sequence:

C...192, D...216, E...243, F...256

31.1.5.2

"The sesquioctave",

31.15.3

"of 192 is that number which contains 192 once and its eighth besides, or 24. The sesquioctave of 216 is 243 which contains 216 and its eighth, or 27. These numbers, 24 and 27",

31.1.5.4

"represent also the difference between 216 and 192 and 243 and 216 respectively. But the difference between 256 and 243, being 13, and representing the interval between E and F, a semi-tone, is neither half of 24 nor half of 27, this demonstrating that a whole tone cannot be divided into equal halves.

Boethius has Philolaus approaching the process from the number 27, the cube of the first uneven number 3. This number 27 is to 24 as 9 is to 8, the difference between it and 24 being 3 or the eighth part of 24. These two numbers, 24 and 27, are thus the units calculated by Nicomachus to represent the whole-tones between the ratio 243 : 216 and 216 : 192. According to Boethius, Philolaus concentrated on the number 27, dividing it into two parts, the small semi-tone or or diesis of 13 units and the apotome of 14 units. The difference between these, the number 1, he called comma and a half comma he called schisma.

The most interesting aspect of these numerical circumlocutions is the manifestly non-Pythagorean disregard for the representation of string lengths through numerical ratios, the division by this technique of the forth into two equally large whole-tones and a small semi-tone being purely mechanical. But most damaging to Nicomachus' claim for the antiquity of Philolaus' determination is the fact that the calculations such as the above are slavishly dependent upon Platos a priori analyses in the Timaeus. As Frank has demonstrated with clarity, the ratio 256 : 243, determined by Plato to be the semi-tone, seems to have spawned an elaborate array of numerical speculations, all of which are ultimately based on the accidental fact that the difference between its terms is 13 and that between the terms of the following whole-tone. 243 : 216, is 27.

Since 27 has great cosmic significance in the Timaeus and also in the Republic... in which 272",

31.15.5

"is the number of days and nights of the year, it became the focal point in the division of the whole tone. This type of numerical speculation is acoustically and musically absurd, however, in its assumption that musical intervals can be calculated in terms of the sums and differences between ratios".

(Note: As you can see the author has reacted against the "absurdity of numerical speculation".....later
you will see the importance of these corresponding connecting numbers and the interval
ratios of which Archytas spoke).

31.16 Pythagoras

"The Pythagoreans held that the tone cannot be divided into two equal parts, because there is not a rational mean between 8 and 9: they accordingly distributed it into a minor semi-tone or 243 / 256, and a major semi-tone or 2048 / 2187; of which two is the product 8 / 9. The Pythagorean limma is slightly less than the 'natural' semi- tone which is 15 / 16 or 24 / 256".

© Copyright. Robert Grace. 1999