(1) Ancient Indian musicians must have tuned their वीणाs (bow-harps) acoustically, using their ears as a guide (and without using any 'cyclic' principles). Hence their scales must have been largely reflective of our contemporary understanding of acoustic/harmonic principles.
(2) Only those processes (e.g. मूर्छनs) that were described by मुनि भरत could be used for further investigation.
Thus, I had postulated a 'acoustic natural' scale as a plausible model of the षड्ज-ग्राम and inferred that its मूर्छनs (modulations) automatically generated a 22-शृति gamut.
The Rise of Music in the Ancient World
This week I happened to stumble across Curt Sachs' "The Rise of Music in the Ancient World" [Sac1943] which has a section dealing with ancient Indian music. I was pleasantly surprised to find that Sachs had arrived at pretty much the same formulation as mine following an almost identical thought process. This post documents Sachs' major results and also indicates where they deviate from mine.
The following are verbatim quotes from Sachs' book (pages 166-167):
There has been much pondering over the puzzling problem of why and how the Hindus came to a division into twenty-two parts.
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Actually, the srutis were not units but, on the contrary, of three different sizes necessitated by the very nature of Indian scales.
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India's standard scales depended on the divisive principle; they had major whole tones of 204, minor whole tones of 182, and semitones of 112 Cents.
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The incessant readaptation of the octave required facilities for changing semitones or major whole tones into minor whole tones, of adding and cutting off adequate portions.
All permutations in these 'give-and-take' were feasible with only three elements: (a) twenty-two Cents or a 'comma,' the difference between the major and the minor whole tone (204-182 Cents); (b) seventy Cents, the difference between the minor whole tone and the semitone; (c) ninety Cents , the difference between the semitone and the comma.
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The give-and-take operation also indicates the exact sequence of the twenty-two shrutis:
D 112 70 E 22 90 F 22 70 22 90 G 22 90 70 22 A 90 22 70 B 22 90 C 22 70 112 D
The first and last steps of 112 Cents, minimum steps with which any model scale begins and ends, are not split in this operation.
Comparison with Sachs' Work
Similarities
The first similarity is that both Sachs and I identify the following three intervals as 1-शृति sized 'elements':
81:80 (Comma of Didymus, denoted c): ~021.51c
25:24 (Small Semitone, denoted s): ~070.67c
256:243 (Pythagorean Limma, denoted L): ~090.22c
The second similarity is that both Sachs and I partition the octave such that the notes at the 1st and the 21st शृति positions are indeterminate.
Differences
First of, Sachs doesn't clearly motivate how he arrived at the three different 'elements'. He states this as a 'matter-of-fact' with no caveats. (I am led to think that these were taken from Fox Strangways' earlier work [Str1914].)
Secondly, I identify the following interval as 0-शृति sized:
2048:2025 (Diaschisma, denoted d): ~019.55cBut there is no reference to this interval in Sachs' work (where it concerns ancient Indian music).
Finally, there is a slight difference in the exact partitioning of the octave. If the 16:15 semitone is denoted S, then Sachs' octave is partitioned into the interval sequence
"S s c L c s c L c L s c L c s c L c s S"whereas my octave is partitioned into the (palindromic) interval sequence
"S s c L c s c L c s d s c L c s c L c s S"Note that both sequences are identical in great measure. However, the interval between the 4/3 & 3/2 notes is partitioned by Sachs into the sequence "c L s c" with no obvious justification of why this particular order should be preferred over an alternative sequence, like "c s L c", for example, or indeed over any other combination of those four 'elements'.
On the other hand, my partitioning yields the (palindromic) sequence "c s d s c" between the same two notes. Now, s*d=L. Thus, by assimilating the d either into the preceding or the subsequent s, two different interval sequences can be arrived at: "c L s c" and "c s L c". Both are valid within the context of my model and there is no obvious reason to choose one over the other. It is possible that "c L s c" is favoured in ascending scales, while "c s L c" in descending scales. In any case, I've already argued that both these sequences may have been treated as 'equivalent' for all practical purposes since the d is 0-शृति sized interval (i.e. smaller than the प्रमाण शृति) [Kam2009a].
Similar Work By Other Authors
Fox Strangways too had followed a process similar to mine [Str1914]. He too had arrived at 20 notes in the octave; and in his scheme, 45/32 & 64/45 were two possibilities for the 11-शृति position. However, he had (artificially) introduced notes at the 1st & 21st शृतिs "by analogy" [Str1914, pg 114] even though those two शृतिs were never used in any जाती during मुनि भरत's time.
Final Thoughts
In retrospect, it is no surprise that multiple authors have arrived at conclusions similar to mine. Given the principle of a 'natural' scale (based on an overtone series) there are only a limited number of ways to generate a full gamut based on modulations of the parent scale.
References
[Kam2009a] Roshan Kamath, "An Acoustic Investigation into Ancient Indian Musical Tuning(s)", http://roshbaby.blogspot.com/2009/02/acoustic-investigation-into-ancient.html, Feb 2009.
[Sac1943] Curt Sachs, "The Rise of Music in the Ancient World", W. W. Norton & Company, Inc., 1943.
[Str1914] A. H. Fox Strangways, "The Music of Hindostan", Oxford University Press, 1914.
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