Musical Mathematics

on the art and science of acoustic instruments


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Part III: Indian Music


Ancient Beginnings


Section 11.20

          One of the oldest and most revered texts on Indian music is a work entitled Natyasastra, written by Bharata (early centuries a.d.). Although large portions of Bharata’s treatise recount performance practices of the theater and dance, Volume 2, Chapters 28–33, deal exclusively with music. In Natyasastra 28.21, Bharata begins his description of the classical 22-sruti scale by giving the names of seven svaras, translated below as notes, and also interpreted in this discussion as tones and scale degrees.

* * * *

Nat. 28.21:     The seven notes [svaras] are: Sadja [Sa], Rsabha [Ri], Gandhara [Ga], Madhyama [Ma], Pañcama [Pa], Dhaivata [Dha], and Nisada [Ni].[1] (Text in brackets mine.)

* * * *

Bharata then defines the musical qualities of four different kinds of sounds, and specifies the consonant and dissonant intervals contained in two different scales called Sadjagrama (Sa-grama) and Madhyamagrama (Ma-grama).

* * * *

Nat. 28.22:     [According] as they relate to an interval of [more or less] Srutis, they are of four classes, such as Sonant (vadin), Consonant (samvadin), Assonant (anuvadin), and Dissonant (vivadin). 

That which is an Amsa [note] anywhere, will in this connection, be called there Sonant (vadin). Those two notes which are at an interval of nine or thirteen Srutis from each other are mutually Consonant (samvadin), e.g., Sadja and Madhyama, Sadja and Pañcama, Rsabha and Dhaivata, Gandhara and Nisada in the Sadja Grama. Such is the case in the Madhyama Grama, except that Sadja and Pañcama are not Consonant, while Pañcama and Rsabha are so . . .

Nat. 23:     In the Madhyama Grama, Pañcama and Rsabha are Consonant while Sadja and Pañcama are so in the Sadja Grama [only]. 

The notes being at an interval of [two or] twenty Srutis are Dissonant, e.g., Rsabha and Gandhara, Dhaivata and Nisada

. . . As a note [prominently] sounds it is called Sonant; as it sounds in consonance [with another] it is Consonant; as it sounds discordantly [to another] it is Dissonant, and as it follows [another note] it is called Assonant. These notes become low or high according to the adjustment of the strings . . . of the Vina . . .[2] (Bold italics mine.)

* * * *

          With this general background information — which will prove crucial in constructing the scales — Bharata then quantifies these seven scale degrees according to how many srutis (from the Sanskrit sru, lit. to hear; sruti in music, an interval) are contained between each degree.

* * * *

Nat. 28.23:     . . . Now, there are two Gramas: Sadja and Madhyama. Each of these two (lit. there) include twenty-two Srutis in the following manner: 

Nat. 24:     Srutis in the Sadja Grama are shown as follows: three [in Ri], two [in Ga], four [in Ma], four [in Pa], three [in Dha], two [in Ni], and four [in Sa]. 

In the Madhyama Grama, Pañcama should be made deficient in one Sruti. The difference which occurs in Pañcama when it is raised or lowered by a Sruti and when consequential slackness or tenseness [of strings] occurs, will indicate a typical (pramana) Sruti.[3] (Bold italics and text in brackets mine.)

* * * *

          Next, Bharata describes a demonstration on two vinas, each equipped with seven strings, and tuned exactly alike to the Sa-grama. The tuning of one vina remains unchanged. Bharata gives directions for changing the tuning of the other vina in four separate steps. Each step requires the lowering of all seven degrees by increments of one sruti. Consequently, after the first step, or after lowering all the strings by 1 sruti, no two degrees of the two vinas match because the smallest interval of the Sa-grama consists of 2 srutis. After the second step, or after lowering Ga by 2 srutis, it will sound the same tone as Ri of the unchanged vina; and after lowering Ni by 2 srutis, it will sound the same tone as Dha of the unchanged vina. Similarly, after the third step, or after lowering Ri by 3 srutis, it will sound the same tone as Sa of the unchanged vina; and after lowering Dha by 3 srutis, it will sound the same tone as Pa of the unchanged vina. Finally, after the fourth step, or after lowering Ma by 4 srutis, it will sound the same tone as Ga of the unchanged vina; after lowering Pa by 4 srutis, it will sound the same tone as Ma of the unchanged vina; and after lowering Sa by 4 srutis, it will sound the same tone as Ni of the unchanged vina. In a passage translated by N. A. Jairazbhoy,[4] Bharata states, “ . . . lower again, in exactly this manner . . .” (punarapi tadvadevapakarsat),[5] which means that this experiment was intended to prove that all sruti intervals are exactly equal in size. Bharata implies that only controlled decreases by identical srutis will produce the scale degrees on the changed vina that exactly match the degrees of the unchanged vina. In this context, the unchanged vina represents a scientific control, or an aural reminder of the changed vina before it was lowered.

          Bharata then summarizes

* * * *

Nat. 28.25–26:     In the Sadja Grama, Sadja includes four Srutis, Rsabha three, Gandhara two, Madhyama four, Pañcama four, Dhaivata three, and Nisada two. 

Nat. 27–28:     [In the Madhyama Grama] Madhyama consists of four Srutis, Pañcama three, Dhaivata four, Nisada two, Sadja four, Rsabha three, and Gandhara two Srutis. [Thus] the system of [mutual] intervals (antara) has been explained.[6]

* * * *

In the absence of clearly defined length ratios and interval ratios,[7] this mixture of numerical and verbal terms seems completely open to interpretation. However, a historically accurate analysis reveals that the musical possibilities contained in this text are extremely limited and point toward only one plausible explanation. Before we examine this interpretation of Bharata’s text, let us first eliminate two possibilities.

          In Natyasastra 28.24, Bharata distinguishes between the Sa-grama and the Ma-grama by stating that in the former scale, Pa contains 4 srutis, and in the latter scale, Pa contains only 3 srutis. He defines this difference based on a typical sruti, or a pramana sruti. Bharata goes on to describe his experiment with two vinas, which only works if the pramana sruti is a standard interval, or an interval of a constant size. All these formulations lead to one possibility, namely, that Bharata was contemplating a geometric division of the “octave” into 22 equal parts. To achieve such a division” requires the calculation of a complicated irrational number called a common ratio,[8] which, in this case, leads directly to 22-tone equal temperament. Recall that in 1584 and 1585, Chu Tsai-yü[9] and Simon Stevin,[10] respectively, solved for the “semitone,” or the common ratio of 12-tone equal temperament, when they independently discovered simplified solutions for the twelfth root of 2. They were able to calculate this complicated constant without logarithms because composite number 12 is a product of primes 3 × 2 × 2.[11] This factorization enabled Tsai-yü and Stevin to effectively extract the required root in two different ways:

A similar technique for the pramana sruti of 22-tone equal temperament does not yield favorable results because a factorization of composite number 22 yields prime numbers 11 and 2. Here a “simplified” solution would require the extraction of the eleventh root of the square root of 2:

Because 11 is a relatively large prime number, this equation cannot be solved without logarithms. Consequently, even though Bharata was under the impression that the pramana sruti represents a constant interval, the fact remains that a scientific or artistic experience of 22-tone equal temperament is impossible without advanced mathematics. No human being is able to accurately and consistently tune such a scale by simply listening and adjusting the tension of the strings on a vina.

          Although there is no evidence of irrational length ratios in ancient Indian music, the intriguing question remains, “Is there a mathematical theory Indian musicians could have contemplated that explains the origin of 22 divisions per “octave?” This question probably has no definitive answer. However, we may attempt an explanation by examining the rational interval ratio 32/31, which very closely approximates the irrational interval ratio :


Now, raise 32/31 to the 22nd power, and observe that twenty-two successive rational pramana srutis exceed one “octave” by only 9 ¢:

This discrepancy is less than half of the spiral of twelve “fifths,” which exceeds seven “octaves” by the ditonic comma, known as the comma of Pythagoras, or by 23.5 ¢.[12] 

          Unfortunately, to actually tune a scale through 22 powers of 32/31 is not any easier than through 22 powers of : both methods require the precision and control provided by a monochord or canon with moveable bridges. Again, no human being can accurately tune so many successive rational or irrational intervals by ear. Since there is no historical evidence of the construction of monochords in ancient India,[13] these “theories” could not have been realized in the tuning of ancient Indian instruments. The only difference is that the rational approximation could have been contemplated in numerical terms, whereas the irrational division could not have been contemplated in numerical terms. We conclude, therefore, that a literal interpretation of Bharata’s pramana sruti as an irrational interval ratio has no theoretical validity. A “theory” that cannot be realized (under any conditions) at a given moment in time must be distinguished from a theory that can be realized at a given moment in time. Whether ancient or not, Bharata’s “theory” of pramana sruti was at best an expression of his imagination. Finally, the seductive oversimplification provided by a pramana sruti eliminated the development of a rigorous integration of music and mathematics. As long as the pramana sruti existed, there was no need to describe subtle differences in intonation with length ratios and interval ratios. The pramana sruti prevented ancient Indian writers and musicians from thinking in ratios, and from intentionally tuning to ratios.


Section 11.21

          The vina in Bharata’s text is not the kind of stringed instrument that appeared in the second half of the first millennium a.d. The latter alapini vina is classified as a stick-zither because it consisted of a straight bamboo or wooden tube over which a single string was stretched.[14] This instrument evolved into the modern stick-zither called bin in North India, and the modern lute also called vina in South India, both of which are now equipped with frets, full gourd resonators, four playing strings, and numerous sympathetic strings. In contrast, Figure 11.21 shows that Bharata’s vina was an arched bow-harp.[15] The ancient vina consisted of a curved arm jointed to a hollow boat-shaped body that acted as a cavity resonator to amplify the strings. This wooden resonator was covered with a soundboard made of leather, through which the strings passed. Most vinas had seven, nine, ten, or fourteen strings.

          Since no specimens survived, two critical aspects about the internal construction of the harp-vina remain unknown. (1) We do not know whether the strings were attached to a strip of wood (called a string holder), which in turn was anchored to the inside walls of the resonator, or whether the curved arm extended all the way through the resonator and, thereby, provided a continuous structure that held the strings at both ends. (2) We also do not know by what mechanism the strings were tuned. Ancient sculptures and reliefs show no tuning pegs in the upper part of the arm. Therefore, the strings were probably adjusted with tuning-cords, very much like the modern saùng-gauk of Burma, also an arched harp with a boat-shaped resonator.[16] However, if the arm passed through the resonator, the remote possibility exists that tuning pegs were located in the lower part of the arm. In either case, adjusting the tension of the strings must have been tedious at best because neither design facilitates the demanding process of precision tuning. It is difficult to achieve fine incremental adjustments with tuning-chords because such lashings are subject to creep and slippage; similarly, it is difficult to manipulate tuning pegs inside the resonator because they are situated in a confined and awkward location. Finally, note carefully that the vina had no post to give it structural rigidity between the lower open end where the resonator terminates in a round corner, and the upper open end where the arm terminates in a scroll. This open semi-circular design severely limited the tuning possibilities of the instrument.

* * * *

It is nevertheless very difficult to understand how such harps . . . could be tuned or kept in tune; not so much because no tuning devices are to be seen in the representations, as because it would be impossible to make an instrument with a curved frame and no post, were the frame even of steel, so rigid that a change of tension in one string would not alter that of all the others.[17]

* * * *

We conclude, therefore, that it is extremely unlikely that the ancient vina was used to tune technically difficult scales. Bharata’s demonstration was at best a “thought experiment” designed to illustrate the distribution of sruti intervals. The slightest motion between open ends would have obliterated the subtle intonational differences of scale degrees lowered (or strings loosened) in 1-sruti increments.[18] As on most folk harps, the vina was probably tuned to “octaves,” “fifths,” “fourths,” and “thirds.”



Section 11.22

          Among modern Indian and European writers, the greatest controversy surrounding Bharata’s text consists of two fundamentally different interpretations regarding the distribution of sruti intervals in both the Sa-grama and the Ma-grama. A careful reading of Natyasastra 28.24–28 does not reveal whether these intervals come before (below) or after (above) the indicated tones. However, if we take into consideration Bharata’s description of consonant and dissonant intervals, only one correct interpretation emerges. I can categorically say that all the writers who advocate the incorrect interpretation never take Bharata’s interval descriptions in Natyasastra 28.22–23 into consideration; it is as if the original text does not exist. Table 11.15 gives the correct and incorrect interpretations of the Sa-grama and Ma-grama.[19]




In the correct version of the Sa-grama, the sruti intervals — 4, 3, 2, 4, 4, 3, 2 — come before the indicated tones, and in the incorrect version, after the indicated tones. Only the correct version complies with Bharata’s demonstration on the two vinas. For example, recall that in the fourth step, Bharata requires that after lowering Sa by 4 srutis, it will have the same tone as Ni of the unchanged vina. In the Sa-grama, such a retuning would be impossible if a 4-sruti interval did not precede Sa.

          In Natyasastra 28.22, Bharata specifically defines consonant intervals as containing either nine or thirteen srutis. Since 9 srutis + 13 srutis = 22 srutis, let us express 9 srutis as ratio 4/3, 13 srutis as ratio 3/2, and 22 srutis as ratio 2/1, because 4/3 × 3/2 = 2/1. Therefore, if Ma is a “fourth,” and if Pa is a “fifth,” then the interval between Ma and Pa is 3/2 ÷ 4/3 = 9/8. Now, since in the Sa-grama the interval between Ma and Pa contains 4 srutis, we conclude that all such intervals in the scale represent a “tone,” ratio 9/8. Refer to Figure 11.22(a), Row 1, and notice that Sa, Ma, Pa, and Sa' are vertically aligned with ratios 1/1, 4/3, 3/2, and 2/1, respectively, in Row 4; and that all 4-sruti intervals in Row 2 are vertically aligned with interval ratios 9/8 in Row 3. With these values, compute two svara ratios in Row 4: Ni = 2/1 ÷ 9/8 = 16/9, and Ga = 4/3 ÷ 9/8 = 32/27. Next, suppose that a 3-sruti interval represents a “small whole tone,” ratio 10/9, and that a 2-sruti interval represents a “semitone,” ratio 16/15. With the latter ratio, calculate one more svara ratio in Row 4: Dha = 16/9 ÷ 16/15 = 5/3. Finally, according to this interpretation, interval ratio 10/9 occurs between Sa–Ri and Pa–Dha by default; and interval ratio 16/15 occurs between Ri–Ga by default.

          To confirm these assumptions, return to Natyasastra 28.22 and observe that Bharata insists that in the Sa-grama, intervals Sa–Ma, Sa–Pa, Ri–Dha, and Ga–Ni are consonant. A sequence of solid brackets in Figure 11.22(a) confirms that these tones span interval ratios 4/3, 3/2, 3/2, and 3/2, respectively. In the last sentence of Natyasastra 28.22, and in the first sentence of Natyasastra 28.23, Bharata juxtaposes the Ma-grama and Sa-grama and observes the following opposite conditions: in the Ma-grama, Sa–Pa [or C–G'] is not consonant while Pa–Ri [or G–D] is consonant, and in the Sa-grama, Sa–Pa [or C–G] is consonant while Pa–Ri [or G–D'] is not consonant. (With respect to the Ma-grama, in Figure 11.22(b) the tone G' is immediately above F', and with respect to the Sa-grama, in Figure 11.22(a) the tone D' is immediately above C'.) Although Bharata does not explicitly define Pa–Ri as a dissonant interval in the Sa-grama, we may deduce this description based on a logical analysis of his juxtaposition. In Figure 11.22(a) the dashed bracket shows that the inversion of Pa–Ri — interval Ri–Pa — is a dissonant “sharp fourth,” as in 3/2 ÷ 10/9 = 27/20 = 519.6 ¢, which means that Pa–Ri is a dissonant “flat fifth,” as in 20/9 ÷ 3/2 = 40/27 = 680.4 ¢.




          Further evidence that these distributions of srutis in the Sa-grama and Ma-grama are authentic may be found in a text entitled Dattilam, written by Dattila (early centuries a.d). Bharata refers to Dattila as an authority on music (Natyasastra 1.26), but Dattila does not mention Bharata. Two English translations of the Dattilam exist, one by E. Wiersma-Te Nijenhuis,[20] and the other by Mukund Lath.[21] The latter states, “The whole testimony shows that Dattila was at least as ancient as Bharata and that the Dattilam is almost certainly his authentic creation.”[22] In Dattilam 12–14, the author explicitly states that in the Sa-grama, Sadja is the first degree, and that three srutis higher, Rsabha is the second degree, etc. Similarly, in the Ma-grama, Madhyama is the first degree, and three srutis higher, Pañcama is the second degree, etc. The Nijenhuis translation reads

* * * *

Dat. 12:     The sound (dhvani), which is indicated by the term Sadja is [the starting point] in the Sadjagrama. From this one the third [sruti] upwards is, no doubt, Rsabha

Dat. 13:     From this one the second [sruti] is Gandhara, from this one the fourth [sruti] is Madhyama. From Madhyama in the same way Pañcama; from this one the third [sruti] is Dhaivata

Dat. 14:     From this one the second [sruti] is Nisada; from this one the fourth [sruti] is Sadja. In the Madhyamagrama, Pañcama is the third [sruti] from Madhyama.[23]

* * * *

With respect to the Madhyamagrama, the Lath translation reads

* * * *

Dat. 14:     In the Madhyama-grama, Pañcama is the third higher [sruti] commencing with Madhyama.[24]

* * * *

          We turn now to the Ma-grama, which Bharata describes in Natyasastra 28.24 as having a Ma–Pa interval reduced by one pramana sruti. Given that in the Sa-grama, the interval between Ma–Pa contains 4 srutis, in the Ma-grama it therefore contains 3 srutis. Since 4 srutis represents ratio 9/8, and 3 srutis, ratio 10/9, calculate the difference between these two intervals by dividing the larger ratio by the smaller ratio: 9/8 ÷ 10/9 = 81/80 = 21.5 ¢. In Western tuning theory, this discrepancy is called the syntonic comma, or the comma of Didymus.[25] Unfortunately, 81/80 is less than half the size of 32/21 [55.0 ¢], the closest rational approximation of the common ratio of 22-TET. If Bharata either heard or thought of 81/80 (or some similar microtonal interval) as a pramana sruti, it is understandable given the structural imperfections and mathematical limitations of the ancient harp-vina. In any case, the same shift that decreases the Ma–Pa interval by one sruti, increases the Pa–Dha interval by one sruti; therefore, in the Ma-grama, the interval Pa–Dha contains 4 srutis.

          Refer to Figure 11.22(b), and note that it shows the Ma-grama based on the same organizational principles as the Sa-grama in Figure 11.22(a). Again, to verify the authenticity of these ratios, return to the second paragraph of Natyasastra 28.22. Here, Bharata clearly indicates that in the Ma-grama, intervals Sa–Ma or inversion Ma–Sa, Ri–Dha or inversion Dha–Ri, Ga–Ni or inversion Ni–Ga, and Pa–Ri are consonant. The solid brackets in Figure 11.22(b) confirm that these tones span interval ratios 3/2, 4/3, 4/3, and 3/2, respectively. And in same sentence, Bharata observes that the interval Sa–Pa is not consonant. In Figure 11.22(b), the dashed bracket shows that this interval is a dissonant “sharp fourth,” ratio 27/20.

          If we now assign Sa, ratio 1/1, to C, Figure 11.22(a), Row 5, shows that in Western music theory, the Sadjagrama is a kind of minor scale, which includes a Pythagorean “minor seventh” [996 ¢], ratio 16/9 [Bb], that sounds a “fifth” below Ma' [8/3 ÷ 3/2 = 16/9], and a Pythagorean “minor third” [294 ¢], ratio 32/27 [Eb], that sounds a “fifth” below Ni [16/9 ÷ 3/2 = 32/27].[26] In contrast, if we assign Ma, ratio 1/1, to F, Figure 11.22(b), Row 5, shows that the Madhyamagrama is a kind of major scale, which includes a 5-limit “major third” [386 ¢], ratio 5/4 [A], and the “minor seventh,” ratio 16/9 [Eb] as before.

          Finally, let us reflect on the meaning of some ancient Indian terms. Although Western musicians may think of Sa-grama and Ma-grama as scales, the word grama in the purest sense of the word does not mean scale. The abstract concept of scale” remained unknown in India until the 19th century a.d. (See Section 11.32.) We should understand the word grama to mean tone-system, or tuning. A grama refers to the tuning of a musical instrument, or to the general distribution and quantification of tones (svaras) and intervals (srutis) of the Sa-grama and Ma-grama. As such, these tone-systems serve no musical purpose. Musicians in ancient India considered organizations of tones and intervals as musical entities only when they were endowed with technical properties called laksana, and performance qualities called rasa. We will consider these two terms in Sections 23 and 27, respectively.


Section 11.23

          One of the most important discussions in Bharata’s book occurs in Natyasastra 28.38–149. Here Bharata describes a highly organized system of melodic modes called jatis. At the end Chapter 28, and almost as an aside, Bharata explains the functions of the jatis:

* * * *

Nat. 28.150–151:     These are the Jatis with their ten characteristics [laksana]. These should be applied in the song (pada) with dance movements (Karanas) and gestures suitable to them (lit. their own). I shall now speak of their distinction in relation to the Sentiments (rasa) . . .[27] (Text in brackets mine.)

* * * *

Dattila confirms the monumental significance of the jatis in a single unequivocal statement:

* * * *

Dat. 97:     . . . anything which is sung is based on the jatis.[28]

* * * *

          Bharata divides his jatis into two different technical categories: suddha (pure) and vikrta (modified). The following description of the suddha jatis is from a major treatise entitled Sangitaratnakara by Sarngadeva (1210–1247). (See Sections 30–31.) 

* * * *

San. Ch. I, Sec. 7, A. (i) (b):     The definition of suddhata: To define suddhata, it is stated that the jatis, which have their denominative note [descriptive note] as the [1] final note (nyasa), [2] the semi-final note (apanyasa), [3] the fundamental note (amsa) [prominent note], and [4] the initial note (graha), which do not have the final note in the high register and which are complete [i.e., heptatonic] are known as suddha jatis.[29] (Bold italics and text in brackets mine.)

* * * *

Dattilam 62 gives a similar description. The text (translation?) of Bharata is less precise because it excludes the Semi-final note. Therefore, in the next passage we will assume that a single tone — which acts as Initial, Prominent, Semi-final, and Final note — is the denominative note of a given suddha jati.[30] (See Figure 11.24.)

Quote I

* * * *

Nat. 28.44:     . . . In the Sadja Grama the pure (Jatis) are Sadji [after Sadja], Arsabhi [after Rsabha], Dhaivati [after Dhaivata] and Naisadi [after Nisada] and in the Madhyama Grama they are Gandhari [after Gandhara], Madhyama [after Madhyama] and Pañcami [after Pañcama]. ‘Pure’ (suddha) in this connection means having Svaramsa ( = Amsa), Graha, and Nyasa consisting of all the seven notes (lit. not deficient in notes). When some of these Jatis lack two or more of the prescribed characteristics except the Nyasa, they are called ‘modified’ (vikrta) . . .[31] (Text in brackets mine.)

* * * *

Here, as in Dattila’s and Sarngadeva’s texts, the terms amsa, graha, nyasa, and apanyasa belong to the laksana, or to ten technical characteristics, which, when uniquely applied to the jatis, give each mode its distinctive technical property. Bharata describes the laksana thus:

* * * *

Nat. 28.74:     Ten characteristics [laksana] of the Jatis are: Graha [Initial note], Amsa [Prominent note], Tara [High register], Mandra [Low register], Nyasa [Final note], Apanyasa [Semi-final note] . . . Alpatva [Rare note] . . . Bahutva [Copious note] . . . Sadava [Hexatonic mode], and . . . Audava [Pentatonic mode].[32] (Text in brackets mine.)

* * * *

          The seven suddha jatis are by definition heptatonic modes. However, in Natyasastra 28.103–149, Bharata describes eighteen more jatis. Seven of these have the same names as the suddha jatis but, due to their modified nature, are here classified as vikrta jatis; the remaining eleven have unique names and are classified as sankara jatis, or jatis comprised from a mixture of suddha jatis.[33] Therefore, it is best to think of Bharata’s jatis in three separate categories: suddha, vikrta, and sankara. Bharata divides these eighteen jatis into the following mode-types:

* * * *

Nat. 28.56:     Of these, four [are] heptatonic (saptasvara) . . . ten [are] pentatonic (pañcasvara) . . . and . . . four [are] hexatonic (satsvara).[34] (Text in brackets mine.)

* * * *

An analysis of the text shows that all four heptatonic modes belong to the sankara jatis, six of the ten pentatonic modes belong to the vikrta jatis, and therefore only one of the four hextonic modes to the vikrta jatis.

          We will not discuss the eleven sankara jatis in full detail,[35] but we will analyze Bharata’s seven suddha jatis and seven vikrta jatis. To do this, we must first familiarize ourselves with an extremely important development in ancient Indian music: the utilization of auxiliary notes called svarasadharana, translated into English as Overlapping notes. Bharata defines sadharana and svarasadharana as

Quote II

* * * *

Nat. 28.34:     . . . The Overlapping (sadharana) means the quality of a note rising between two [consecutive] notes [in a Grama]. 

Nat. 35:     . . . [1] The Kakali and [2] the transitional note (antarasvara) are the Overlapping notes (svarasadharana). Now if two Srutis are added to Nisada, it is called Kakali Nisada and not Sadja; as it is a note rising between the two (pure Nisada and Sadja), it becomes Overlapping. Similarly, [the two Srutis being added to it] Gandhara becomes transitional Gandhara { Antara Gandhara } and not Madhyama, because it is a transitional note (antarasvara) between the two (Madhyama and Gandhara). Thus the Overlapping notes [occur].[36] (Numbers in brackets, bold italics, and text in braces mine.)

* * * *

And again, Dattila gives a terse description:

* * * *

Dat. 16:     Nisada is called Kakali, when [the note] is raised by two srutis. Similarly, Gandhara is called Antarasvara.[37]

* * * *

          For convenience, we will simply refer to Antara Gandhara as “An,” and to Kakali Nisada as “Ka.” With respect to the Sa-grama, calculate the svara ratio of An by first increasing the interval between Ri and Ga from 2 srutis to 4 srutis, and then multiplying the svara ratio of Ri times a 4-sruti interval, which gives: An = 10/9 × 9/8 = 5/4. Similarly, calculate the svara ratio of Ka by first increasing the interval between Dha and Ni from 2 srutis to 4 srutis, and then multiplying the svara ratio of Dha times a 4-sruti interval, which gives: Ka = 5/3 × 9/8 = 15/8. With respect to the Ma-grama, calculate the svara ratio of Ka by first increasing the interval between Dha and Ni from 2 srutis to 4 srutis, and then multiplying the svara ratio of Dha times a 4-sruti interval, which gives: Ka = 5/4 × 9/8 = 45/32. Similarly, calculate the svara ratio of An by first increasing the interval between Ri and Ga from 2 srutis to 4 srutis, and then multiplying the svara ratio of Ri times a 4-sruti interval, which gives: An = 5/3 × 9/8 = 15/8. (See Figure 11.23.)

          Bharata describes the general implementation of the svarasadharana in Natyasastra 28.36–37. Because the meaning of this passage in the Ghosh translation is not clear, consider now an alternate translation found in the commentary of Mukund Lath’s A Study of Dattilam. (See Section 11.22.) Lath translated this passage from a different source and identifies it as Natyasastra 28.35–36:

* * * *

Nat. 28.35–36:     The antara-svara [An] should always be associated (with the jati) when making an ascending movement; its use should be exceedingly spare and never in making descending movements. If the antara-svara be used in descending movements, whether sparingly or with profusion, it destroys the sruti and the jati-raga.[38] (Text in brackets mine.)

* * * *

Lath then adds the following interpretation of this passage based on a famous commentary of the Natyasastra by Abhinava Gupta, (fl. c. a.d. 1000):

* * * *

Abhinava points out that the word antara-svara in these verses denoted not only the auxiliary ga [An] but also the kakali ni [Ka] and the maxim applies equally to both the auxiliary notes: “antarasvarasabdena catra kakalyapi samgrhita iti krtopyayameva kramah” (A.B. on N.S. 28, 36).[39] (Text in brackets mine.)

* * * *

          Bharata describes the specific implementation of the svarasadharana with respect to the jatis in Natyasastra 28.38. Again, consider this translation by Lath, which he cites as Natyasastra 28.37: 

Quote III

* * * *

Nat. 28.37:     There are three jatis which are connected with the use of the sadharana svaras (i.e., antara ga and kakali ni), namely, Madhyama, Pañcami, and Sadjamadhya.[40]

* * * *

Madhyama and Pañcami are classified as both suddha and vikrta jatis, and Sadjamadhya, as a sankara jati.


Section 11.24

          Before we begin a ratio analysis of the suddha jatis and the vikrta jatis, consider first a tuning sequence designed to produce the Sa-grama on an ancient harp-vina with nine open strings. In the illustration below, a progression of white notes represents the required scale degrees, and black notes indicate previously tuned degrees in the sequence. Notice that this procedure only requires tuning “octaves,” “fifths,” “fourths,” and one “major third.”




          Refer now to Figure 11.23(a), Rows 1 and 2, which give the ratios and note names, respectively, of the Sa-grama from the lowest tone Sa, ratio 1/1, on String 1 to the highest tone Ri', ratio 20/9, on String 9. Now, if we were to tune this vina to the Ma-grama as illustrated in Figure 11.22(b), Row 5, we would (1) change the frequency of the lowest tone from C4 to F4, and (2) retune all the remaining strings. However, given the structural instability of the vina, such an increase in tension would pose significant mechanical difficulties. It is far more likely that musicians in ancient India rendered the Ma-grama as a modulated scale on the vina. One may produce a modulated Ma-grama by retuning only a single string of the original Sa-grama tuning. As explained below, such a modulation shifts ratio 1/1 from Sa on String 1 to Ma on String 4.

          Recall from the discussion in Section 11.22 that the principal difference between the Sa-grama and the Ma-grama is that the Ma–Pa interval contains 4 srutis in the former tuning, and 3 srutis in the latter tuning. An arrow that points from Figure 11.23(a) to Figure 11.23(b) indicates this difference. To reduce this interval, and thereby produce a modulated Ma-grama tuning based on the original Sa-grama tuning, requires two simple steps. (1) In Figure 11.23(b), Row 1, identify the tone of String 5 of the Ma-grama tuning by multiplying the tone of String 4 of the Sa-grama tuning times a 3-sruti interval: String 5 of Ma-grama = 4/3 × 10/9 = 40/27. (2) Tune the new Pa, ratio 40/27 on String 5 as a “fourth,” ratio 4/3, above String 2; that is: 10/9 × 4/3 = 40/27 = 680.4 ¢. Figure 11.23(b), Row 3, shows that such a simple retuning modulates the old tonic Sa, ratio 1/1, on String 1 to the new tonic Ma, ratio 1/1, on String 4. However, this shift does not render the modulated Ma-grama as a continuous heptatonic scale from the tonic, ratio 1/1 [F4], to the “minor seventh,” ratio 16/9 [Eb5], because the latter string is missing; but since String 3 sounds a 16/9 [Eb4] one “octave” below, note that the entire scale does exist on the vina. To confirm that 32/27 in Figure 11.23(b), Row 1, is mathematically equivalent to 16/9 in Figure 11.23(b), Row 3, verify the proportion[41] 

          In Figure 11.23(a), I arbitrary chose C4 as the fundamental frequency of the Sa-grama tuning. Furthermore, observe that the tones in Figure 11.23(b) and 11.23(c), Rows 1, represent modes of the modulated Ma-grama tuning in Rows 3. Do not confuse these Ma-grama modes with the original Sa-grama tuning. For example, in Figure 11.23(b) and 11.23(c), Rows 1, the Sa–Pa intervals are not 3/2’s, but 40/27’s. Since a mode may begin on any degree of a given tuning, one could also construct various Ma-grama modes in Figures 11.23(b) and 22(c) by assigning ratio 1/1 to Ri, Ga, Pa, etc.

          In Quote I, Bharata explains that the suddha jatis called Madhyama and Pañcami are derived from the Ma-grama, and in Quote III, that these two jatis “. . . are connected with the use of the sadharana svaras . . .” In other words, Bharata does not associate the two Overlapping notes with suddha jatis derived from the Sa-grama. Now, in Quote II, Bharata effectively states that to tune the svarasadharana, one must increase the Ri–Ga interval by two srutis to obtain An, and the Dha–Ni interval by two srutis to obtain Ka. Two arrows that point from Figure 11.23(b) to Figure 11.23(c) indicate these intervalic changes. In Figure 11.23(c), Rows 1 and 3, calculate the ratios of the svarasadharana as discussed in Section 11.23. Finally, in the context of the Ma-grama mode in Row 1, tune the new sadharana An, ratio 5/4, on String 3 as a 5/4 interval above String 1, and the new sadharana Ka, ratio 15/8, on String 7 as a 3/2 interval above String 3. With respect to the modulated Ma-grama in 7Row 3, the latter two ratios are mathematically equivalent to ratios 15/8 [An] and 45/32 [Ka], respectively.




[1]Ghosh, M., Translator (Vol. 1, Ch. 1–27, 1950; Vol. 2, Ch. 28–36, 1961). The Natyasastra, by Bharata, Volume 2, p. 5. Bibliotheca Indica, The Asiatic Society, Calcutta, India.



[2]Ibid., pp. 5–7.



[3]Ibid., p. 7.



[4]Jairazbhoy, N.A. (1975). An interpretation of the 22 srutis. Asian Music VI, Nos. 1–2, pp. 38–59.



[5]Ibid., p. 41.



[6]The Natyasastra, Volume 2, pp. 8–9.



[7]See Chapter 3, Section 12, and Chapter 9, Section 2.



[8]See Chapter 9, Sections 1 and 8.



[9]See Section 11.11.



[10]See Chapter 10, Sections 32–33



[11]See Chapter 10, Section 1.



[12]See Chapter 10, Section 22.



[13]Asian Music VI, 1975, p. 41.



[14]Sadie, S., Editor (1984). The New Grove Dictionary of Musical Instruments, Volume 3, pp. 729–730. Macmillan Press Limited, London, England.



[15]Coomaraswamy, A.K. (1930). The parts of a vina. Journal of the American Oriental Society 50, No. 3, pp. 244–253.



[16]The New Grove Dictionary of Musical Instruments, Volume 3, pp. 304–305.



[17]Journal of the American Oriental Society 50, No. 3, p. 250.



[18]To avoid significant cumulative errors in tuning scales with many small intervals requires sophisticated instruments that are physically stable and acoustically accurate. In contrast, tuning scales with only a few large intervals on simple instruments may produce errors that are less objectionable.



[19]Bhandarkar, R.S.P.R. (1912). Contribution to the study of ancient Hindu music. The Indian Antiquary XLI, pp. 157–164, 185–195, 254–265.


     In this remarkably thorough study, Bhandarkar traces the inaccuracies found in works on ancient Indian music by both Asian and European writers. He also stresses correct interpretations of sruti distributions and grama constructions.



[20]Nijenhuis, E.W., Translator (1970). Dattilam: A Compendium of Ancient Indian Music. E. J. Brill, Leiden, Netherlands.


     This translation of the Dattilam spans pp. 17–61, whereas Nijenhuis’ commentary spans pp. 62–425. Throughout the commentary, Nijenhuis includes many translated excerpts from the works of Bharata, Narada, Matanga, and Sarngadeva. On several occasions, footnotes refer the reader to these latter translations in Nijenhuis’ Dattilam.



[21]Lath, M., Translator (1978). A Study of Dattilam: A Treatise on the Sacred Music of Ancient India. Impex India, New Delhi, India.



[22]Ibid., p. x.



[23]Dattilam, p. 19.



[24]A Study of Dattilam, p. 218.



[25]See Chapter 10, Section 27.



[26]See Chapter 10, Section 22.



[27]The Natyasastra, Volume 2, p. 28.



[28]Dattilam, p. 33.



[29]Shringy, R.K., and Sharma, P.L., Translators (Vol. 1, Ch. 1, 1978; Vol. 2, Ch. 2–4, 1989). Sangitaratnakara, by Sarngadeva, Volume 1, p. 267. Volume 1, Motilal Banarsidass, Delhi, India; Volume 2, Munshiram Manoharlal, New Delhi, India.



[30]Rowell, L. (1981). Early Indian musical speculation and the theory of melody. Journal of Music Theory 25.2, pp. 217–244.


     The following quotations appear on pp. 232–235 in Rowell’s excellent article:


     “The most vital choices are those which direct the course of a melody into one of the prescribed jatis, the ancestors of the modern concept of raga. The word jati is one of those bland words so useful in musical terminology; it is a past passive participle of the verbal root jan [cognate with the Greek word genesis] meaning “to be born, arise,” and thus its developed meaning: “kind, type, species.” A glance at the standard ten characteristics of jati discloses most of the familiar standards by which a mode is recognized in Medieval Western theory — incipit, final, confinal, and ambitus (high and low): 1. graha, initial; 2. amsa, prominent, usually called “sonant” by Indian authors; 3. tara, high; 4. mandra, low; 5. sadava, hexatonic; 6. auduvita, pentatonic; 7. alpatva, scarce, weak; 8. bahutva, copious; 9. nyasa, final; 10. apanyasa, confinal, an internal cadence tone.

     . . . Amsa, according to the Natyasastra, had its own list of ten laksanas: it is the generating tone, it determines not only the low tone but the interval between low and high tones, it is the tone most frequently heard, it determines the initial, the final, the three types of confinals, and is the tone which all the others follow. Amsa, one gathers, is no trivial concept.

     . . . Vadi or “sonance” is treated formally as a subtonic of grama (scale), but the concept first becomes operational when applied to the structure of an individual jati. Here are four possibilities: 1. vadi, sonant, “ruling note” (amsa); 2. samvadi, consonant, harmonic affinity; 3. vivadi, dissonant, distorted; 4. anuvadi, neutral. The commentator Abhinava quotes an old analogy: Vadi is the king, samvadi is the minister who follows him, vivadi is like the enemy and should be sparingly employed, and anuvadi denotes the retinue of the followers.’”



[31]The Natyasastra, Volume 2, pp. 15–16.



[32](a) The Natyasastra, Volume 2, pp. 18–19.


(b) Dattilam, p. 177.



[33]Dattilam, p. 168.



[34]The Natyasastra, Volume 2, p. 17.



[35]Because the sankara jatis have no recognizable symmetries and no internal patterns, they have little theoretical or practical value for tracing the

evolution of Indian music.



[36]The Natyasastra, Volume 2, p. 13.



[37]Dattilam, p. 19.



[38]A Study of Dattilam, p. 227.






[40]Ibid., p. 228.



[41]See Chapter 3, Section 6.