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, 2/1This 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(?).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.
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.
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, 2/1In 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.
References
[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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