An Acoustic Investigation into Ancient Indian Musical Tuning(s) - II

Some time ago I'd pondered the question of why the ancient Indian musical octave was considered to be a 22-step (22-शृति) interval [Kam2009a]. In my investigation I had followed primarily two cardinal principles:

(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.
...
Actually, the srutis were not units but, on the contrary, of three different sizes necessitated by the very nature of Indian scales.
...
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.
...
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.
...
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): ~019.55c

But 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.

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References

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[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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An Acoustic Investigation into Ancient Indian Musical Tuning(s)

Ancient Indian music theory was based primarily on the tuning of the (ancient) वीणा, a Bow-Harp [Bha1984]. The topic of tuning of the वीणा leads to much discussion and disagreement in musicological circles even today. Based on the information which has reached us via Greek physicists who 'reverse-engineered' Greek tuning from Greek musical scales, and presumably Greek music being inherited from an Eastern musical practice [Dan1995, Day1891], it is quite possible that Indian tunings were also founded on acoustic/harmonic principles.

मुनि भरत's treatise, the नाट्यशास्त्र, which became the basis for subsequent musicological works, identifies three different interval sizes in the two parent tunings (षड्ज-ग्राम and मध्यम-ग्राम) and assigns them certain weights or शृति values [Jai1975]. More importantly, the scales of मुनि भरत's time were heptatonic (i.e. employing 7 स्वरs) and based on मूर्छनs (modes) of a given ग्राम (parent tuning). This is central to what follows.

(Note: It is more correct to translate ग्राम as 'pitch collection'; however, the phrase 'parent tuning' is used in this article since tuning is the topic under discussion.)

The purpose of this article is to start with an acoustically determined scale, and then see if the basis of the नाट्यशास्त्र cannot be recovered using our current understanding of the state-of-the-art in those days. Primarily, it seeks to understand why the ancient Indian octave was presumably divided into 22 steps (शृतिs). This article is also motivated somewhat by earlier work on contemporary Indian scales which ended up with a non-coarse grouping of 22 equivalence note classes [Kam2008]. However, it is in no way a defense of the claim that contemporary Indian intonation too is based upon a set of 22 fixed शृतिs.


An 'Acoustic Natural' Scale for the षड्ज-ग्राम

A 'natural' scale based on acoustic principles and which satisfies the necessary conditions of the नाट्यशास्त्र would be:

1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1

This scale employs three different intervals; namely, 9:8 (T), 10:9 (t), and 16:15 (S) much in the line of the scales documented by मुनि भरत. Specifically, the scale above is intended to represent the tuning known as षड्ज-ग्राम which was composed of two tetrachords internally bounded by the Just Fourth (4/3) and the Perfect Fifth (3/2). Note however that in NO way does this imply that मुनि भरत's षड्ज ग्राम was exactly the scale mentioned above; neither is it the point of this article.

The Sixth in the षड्ज-ग्राम is Pythagorean (27/16) and NOT the 'Just' Sixth (5/3) since tetrachordal symmetry was observed in the tuning of the षड्ज-ग्राम्. The scale that used the 'Just' Sixth instead was known as the मध्यम-ग्राम. More about it below.

The षड्ज-ग्राम can be represented in terms of intervallic values as "T t S T T t S".

(Open Question: As the वीणा was a bow-harp, it is hard to defend the stand that the वीणा was tuned to an 'acoustically natural' scale. Historically, instruments like these were tuned by a 'cyclic' process of ascending/descending fifths/fourths which would, without tempering, lead to only two interval sizes, not three as required by मुनि भरत's description. Clearly there's a discrepancy somewhere; but not if one allows the possibility that ancient Indian musicians may have tuned their harps by exploiting 'overtonal consonance' instead of following a 'cyclic' process.)


The मूर्छनs (Modes) of the षड्ज-ग्राम

By rooting the above 'Acoustic Natural' षड्ज-ग्राम on each note, different मूर्छन (modes) of the same basic scale are obtained.

Mode of the First

This is "T t S T T t S" and yields 1/1, 9/8, 5/4, 4/3, 3/2, 27/16, 15/8, 2/1
This mode was also a जाती named षड्जी.

Mode of the Second

This is "t S T T t S T" and yields 1/1, 10/9, 32/27, 4/3, 3/2, 5/3, 16/9, 2/1
This mode was also a जाती named अर्षभी.

Mode of the Third

This is "S T T t S T t" and yields 1/1, 16/15, 6/5, 27/20, 3/2, 8/5, 9/5, 2/1

Mode of the Fourth

This is "T T t S T t S" and yields 1/1, 9/8, 81/64, 45/32, 3/2, 27/16, 15/8, 2/1

Mode of the Fifth

This is "T t S T t S T" and yields 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1

Mode of the Sixth

This is "t S T t S T T" and yields 1/1, 10/9, 32/27, 4/3, 40/27, 128/81, 16/9, 2/1
This mode was also a जाती named धैवती.

Mode of the Seventh

This is "S T t S T T t" and yields 1/1, 16/15, 6/5, 4/3, 64/45, 8/5, 9/5, 2/1
This mode was also a जाती named निषादी.


The Modal Gamut

Collect the individual notes that result from the above मूर्छनs and the following gamut results. The notes of the (default) षड्ज-ग्राम are indicated with bold letters.

1/1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 81/64, 4/3, 27/20, 45/32, 64/45, 40/27, 3/2, 128/81, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 2/1.

The S (16:15) interval

The S interval between the 5/4 and 4/3 notes is observed to be 'split' into two sub-intervals by the introduction of the 81/64 note. Hence, one is inclined to consider the S as a 2-step interval.

The t (10:9) interval

The t interval between the 9/8 and 5/4 notes is observed to be 'split' into three sub-intervals by the introduction of 32/27 and 6/5 notes. This is also true for the t interval between the 27/16 and 15/8 notes. Thus, the t may be considered as a 3-step interval.

The T (9:8) interval

The T interval between the 1/1 and 9/8 notes is 'split' into three sub-intervals by the 16/15 and 10/9 notes. However, since the S (16:15) is already determined to be a 2-step interval or since the t (10:9) is already determined to be a 3-step interval, this makes the T a 4-step interval. An identical reasoning applies to the T interval between the 3/2 and 27/16 notes.

What about the T between the 4/3 and 3/2 notes?

At first glance this interval seems to be split into five sub-intervals. However, this shall be addressed later below once we assign 'weights' (शृतिs) to each note in the complete gamut.


शृति values for each note in the gamut

'Weights' can be assigned to each note in the gamut by adding up the weights (i.e. step-sizes) of the individual intervals that make up the note. For e.g., the 5/4 composed of a T and t shall get assigned a weight of 4+3=7.

These weights are termed शृतिs following मुनि भरत's terminology.

00: 1/1
01: -/-
02: 16/15
03: 10/9
04: 9/8
05: 32/27
06: 6/5
07: 5/4
08: 81/64
09: 4/3
10: 27/20
11: 45/32
11: 64/45
12: 40/27
13: 3/2
14: 128/81
15: 8/5
16: 5/3
17: 27/16
18: 16/9
19: 9/5
20: 15/8
21: -/-
22: 2/1

(Note the conspicuous absence of notes at शृति positions 1 and 21. This may be the underlying reason why none of the जातीs defined by मुनि भरत employed notes at those positions. Also note that both the 45/32 and 64/45 notes end up with the same शृति value 11.)

The first observation is that we end up with a 22 शृति octave. मुनि भरत termed the interval between the 16th and 17th शृति positions as the प्रमाण शृति. This comes out to be the Comma of Didymus (81:80, ~21.51c). It is thought that मुनि भरत considered this as the smallest musically relevant interval, and any interval smaller that this could very easily have been identified with the unison (1:1). Now, the various intervallic distances in the above gamut are:

2048:2025 (Diaschisma): ~019.55c, शृति value 0(?).
81:80 (Comma of Didymus): ~021.51c, शृति value 1.
25:24 (Small Semitone): ~070.67c, शृति value 1.
256:243 (Pythagorean Limma): ~090.22c, शृति value 1.
16:15 (Just Semitone): ~111.73c, शृति value 2.

The only interval smaller than the Comma of Didymus is the Diaschisma which occurs exactly once in the gamut as the intervallic distance between the 45/32 and 64/45 notes. Thus, it is possible that the 45/32 and 64/45 notes were considered to be identical given the state of measurement and accuracy that must have been available to मुनि भरत. This is also obliquely reflected in the same शृति value that gets assigned to both notes. This in itself may have been sufficient justification for मुनि भरत to identify those two notes. On the other hand, the 45/32 note did not occur in any जाती of the षड्ज-ग्राम while the 64/45 note did not occur in any जाती of the मध्यम-ग्राम (see below) so it is conceivable that this confusion never arose.

Note that three different intervals (81:80, 25:24, and 256:243) are considered to be a 1-शृति interval.

Back to the T between the 4/3 and 3/2 notes

If the 45/32 and 64/45 notes truly get identified with each other, it leaves 4 sub-intervals between the 4/3 and 3/2 notes. Even without this identification, we have only 13-9=4 शृति values between the 4/3 and 3/2 notes anyway; and both the 45/32 and 64/45 notes occupy the same शृति position. Thus, the T interval between the 4/3 and 3/2 notes too can be argued to be a 4-step interval consistent with its step-size in other positions.


The मूर्छनs (Modes) of the मध्यम-ग्राम

A complementary tuning, the मध्यम-ग्राम, was also in vogue during मुनि भरत's time. The primary difference between the two tunings was that the Sixth in the मध्यम-ग्राम was 'Just' (5/3) compared to the Pythagorean (27/16) of the षड्ज-ग्राम.

1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1

In terms of intervallic values, this scale can be represented as "T t S T t T S". Following the same procedure as done above for the षड्ज-ग्राम and collecting the various notes yields a subset of the gamut that has already been determined.

1/1, 16/15, 10/9, 9/8, 32/27, 6/5, 5/4, 4/3, 27/20, 45/32, 64/45, 40/27, 3/2, 8/5, 5/3, 27/16, 16/9, 9/5, 15/8, 2/1.

No new notes are 'discovered'.

As an aside, of the seven modes of the मध्यम-ग्राम, the following three were also जातीs in use:

गाँधारी (mode of the Third) - This is "S T t T S T t" and yields 1/1, 16/15, 6/5, 4/3, 3/2, 8/5, 9/5, 2/1.

मध्यम (mode of the Fourth) - This is "T t T S T t S" and yields 1/1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8, 2/1.

पंचमी (mode of the Fifth) - This is "t T S T t S T" and yields 1/1, 10/9, 5/4, 4/3, 3/2, 5/3, 16/9, 2/1.


Conclusions

An octave with 22 शृति divisions naturally arises from the modulations of an 'acoustically natural' scale. However, these 22 शृतिs provide only 20 notes since the notes at the 1 & 21 positions are indeterminate. Inspite of the 22 शृतिs, it must not be forgotten that (ancient) Indian music itself was based only on 7 स्वरs (notes). Thus, the 22 शृतिs that arise from the modes of the two ग्रामs are of theoretical interest only and serve no practical purpose otherwise. However, a knowledge of these शृतिs may be handy in comprehending the subsequent evolution of Indian music.

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References

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[Bha1984] Pt. Vishnu Narayan Bhatkhande, "Music Systems in India: A Comparative Study of Some of the Leading Music Systems of the 15th, 16th, 17th, & 18th Centuries", South Asia Books, 1984.

[Dan1995] Alain Danielou, "Music and the Power of Sound: The Influence of Tuning and Interval on Consciousness", Inner Traditions International, 1995.

[Day1891] ^^ ... the historian Strabo says that among the Greeks those who regard all Asia as far as India as a country sacred to Dionysius, "attribute to that country the invention of nearly all the science of music."^^, Capt. Charles Russell Day, "The Music and Musical Instruments of Southern India and the Deccan", Novello, Ewer & Co., 1891, page 19.

[Jai1975] Nazir Ali Jairazbhoy, "An Interpretation of the 22 Srutis", Asian Music, Vol. 6, No. 1/2 (Perspectives on Asian Music: Essays in Honor of Dr. Laurence E. R. Picken), 1975, pp. 38-59.

[Kam2008] Roshan Kamath, "A Model for Implied Intonations in Classical हिन्दूस्तानी (Hindustani) Music", http://roshbaby.blogspot.com/2008/07/model-for-implied-intonations-in.html, 2008.

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