Scale Divisions

34.1 Scales

"The usual divisions of the tetrachord in classical Greek music are, in the diatonic genus or scale: semi-tone, whole-tone, whole-tone.
chromatic: semi-tone, semi-tone, tone-and-a half.
enharmonic: quarter-tone, quarter-tone, di-tone".

Pythagorean Musical Theory

34.2 Scales before the composer, Bach (1685-1750) were "untempered".

Just Intonation was a scale used before Bach and corresponds to modern corn price cycles:

Ray Tomes Main Page

34.3 The 4th, 5th and Whole Tones

Interval- A-A (Octave), A-E (Fifth), A-D (Forth).
A-B (Whole), B-C (Whole), D-E (Whole), E-Fs (Whole).

Note	A	B	C	D	E	Fs	Gs	A2
Ratio	1	8/9	8/9 x 8/9 = 64/81	3/4	2/3	8/9 x 2/3 = 16/27	8/9 x 16/27 = 128/243	1/2
Frequency (cps)	440	9/8 x 440 = 495	81/64 x 440 = 556.875	4/3 x 440 = 586.667	3/2 x 440 = 660	27/16 x 440 = 742.5	243/128 x 440 = 535.312	2 x 440 = 880
Frequency 432 (cps)	432	9/8 x 432 = 486	81/64 x 432 = 546.75	4/3 x 432 = 576	3/2 x 432 = 648	27/16 x 432 = 729	243/128 x 432 = 820.125	2 x 432 = 864

34.4 Lydian Pythagorean Scale

Note	C4	D4	E4	F4	G4	A4	B4	C5
Frequency Ratio to C4	1	9/8	81/64	4/3	3/2	27/16	243/128	2
Number of Cents above C4	0	204	408	498	702	906	1110	1200
Interval between adjacent Notes in Cents	204	204	90	204	204	204	90	.

The Musicians Guide to Acoustics
By Campbell
ML 3805. C24 1987

34.5 Diagram. Ancient Scales

Ancient Scales, Midi sound files. Lucy Tuning and Pi

Greek Esoteric Music Theory

34.6 Diatonic Scale (s = sharp, f = flat)

Cs, Ds, E, Fs, Gs, A, B

34.7 Dorian Scale

C, Df, Ef, F, Gf, Af, Bf

Common Sense of Music
By Sigmund Spaetik

34.8 Diagrams. Modern Scales

34.9 Diagram. Harmonics of the Equal Tempered Scale

34.10 Turning Major to minor

"Any major scale can be turned into a minor scale by simply dropping the interval of the 3rd (step number three) half a tone lower, so that the progression involves only half a tone from two to three".

34.11 Two Musical Laws to Remember

"There are just two things to remember grimly and with everlasting determination:

The letters in music are constant, and represent always the same notes in their particular section of the keyboard.
The numbers of the intervals are merely relative, since a scale can start anywhere on the keyboard, and its starting point is always number "one" ".

34.12 Diagram. Diatonic Scale. Book: Music of the Spheres, 523. M938

"By the 18th century, the diatonic scale had pretty well crystallized into its modern form - its notes taking the numerical relationship C-4, D- 4 1/2, E-5, F-5 1/3, G-6, A-6 2/3, B-7 1/2, C'-8. Thus, C and C' make a perfect octave (4 : 8 = 1 : 2), C and G a perfect 5th (4 : 6 = 2 : 3), and C and F a perfect 4th (4 : 5-1/3 = 3 : 4) and so on. And the appealing 7th chord C, E, G Bflat, C' was discovered to derive its magic from the simple exactitude of its proportions 4 : 5 : 6 : 7 : 8! Just why this strange insistant musical "call" should find its most satisfying "answer" in a chord exactly a 4th higher",

34.13.1

F, A, C', F', is still not well understood. I think it will undoubtedly be explained by wave symmetry but, in the meantime, the 7th chord, seems certainly one of the most beautiful mysteries in nature.

The diatonic scale has thus served its purpose ideally within its own limits, even offering a few perfect intervals and triads in others keys than its own, as the accompanying table shows. Aristoxenos would probably have been impressed by the fact that all its tonal differences in frequency are multiples of the prime number 11",

34.14 Diagram. Diatonic Vibrational Differences in Diatonic Scale. Music of the Spheres

34.14.1

"And Pythagoras would surely applaud its plurality of means. E the arithmetic mean between C and G, F the harmonic mean of C and C', and G the geometric mean of C and D'!

Yet, as musicians well know, the diatonic scale failed utterly as a practical all around tool of music- for the same reason that Keplers circumscription of the spheres failed as a tool of astronomy. It just would not quite fit! It could do very nicely at certain points or within particular limitations but the mere act of pressing it into place in one region would always and inevitably force it out of line in another. Ill fitting notes were so common, in fact, in the early clavichords and pianos that they were given the name "wolves"- they howled so much. If a major diatonic scale were constructed starting with D instead of C, it could not possibly match up with all the notes of the C scale since several would be at the wrong pitch intervals. Instead, it would need 4 (See above correspondence of 4) new notes for its diatonic perfection. And, to provide for all the 12 musical keys, every octave would have to have 72 notes!".

(Correspondences, Fuller Synergetics, 1022.15, 974.03, Table 943.00).

34.15

Music of the Spheres
523.M938

Impossible Correspondence Index