The first six chapters of *Natya Shastra*, a Sanskrit treatise on music, dance and drama dating back to a period between 400 BCE and 200 CE, contain the premises of a scale theory which for long caught the attention of scholars in India and Western countries. Early interpretations by Western musicologists followed the “discovery” of this text in 1794 by philologist William Jones. Hermann Helmholtz’s theory of “natural consonance” gave way to many comparative speculations based on phenomena which Indian authors had earlier observed as inherent to the “self-production” (*svayambhū*) of musical notes (Iyengar 2017 p. 8).

Suvarnalata Rao and Wim van der Meer (2009) published a detailed record of attempts to elucidate the ancient theory of musical scales in musicological literature, coming back to the notions of *ṣruti* and *swara* which changed over time up to present-day musical practice.

In the second part of the 20^{th} century, experimental work with frequency meters led to contradictory conclusions drawn from the analysis of small samples of music performance. It was only after 1981 that systematic experiments were conducted in India by the ISTAR team (E.J. Arnold, B. Bel, J. Bor and W. van der Meer) with an electronic programmable harmonium (the *Shruti Harmonium*) and later on a “microscope” for melodic music, the *Melodic Movement Analyser* (MMA) (Arnold & Bel 1983, Bel & Bor I985) feeding accurate pitch data to a computer to process hours of music selected from historical recordings.

After several years of experimental work, it had become clear that, even though the intonation of Indian classical music is far from a random process, it would be hazardous to assess an interpretation of ancient scale theory with the aid of today’s musical data. There at least three reasons for this:

- There are infinitely valid interpretations of the ancient theory, as we will show.
- The concept of raga, i.e. the basic principle of Indian classical music, appeared first in the literature circa 900 CE in Matanga’s
*Brihaddeshi*, and it underwent a gradual development until 13^{th}century, when Sharangadeva enlisted 264 ragas in his*Sangitratnakara*. - Drones were not in use at the time of
*Natya Shastra*; the influence of the drone on intonation is considerable, if not predominant, in contemporary music performance.

The ancient Indian theory of scales remains useful for its insight into early melodic classification (the *jāti* system) which later might have engendered the raga system. Therefore, it may be best envisaged as a *topological* description of tonal structures. Read Raga intonation for a more detailed account of theoretical and practical issues.

The topic of this page is an interpretation of the experiment of the two vinas described in Chapter XXVIII.24 of the *Natya Shastra*. An analysis of the underlying model has been published in *A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra* (Bel 1988) which the following presentation will render more comprehensive.

## The historical context

Bharata Muni, the author(s) of *Natya Shastra* might have heard about theories of musical scales attributed to “ancient Greeks”. At least, Indian scholars were in position to borrow these models and expand them considerably because of their genuine knowledge of calculus.

Readers of C.K. Raju — notably his outstanding work *Cultural Foundations of Mathematics* (2007) — are aware that Indian mathematicians/philosophers are not only famous for inventing positional notation which took six centuries to be adopted in Europe… They also laid out the foundations of calculus and infinitesimals, later exported by Jesuit priests from Kerala to Europe and borrowed/appropriated by European scholars (Raju 2007 pages 321-373).

The calculus first developed in India as a sophisticated technique to calculate precise trigonometric values needed for astronomical models. These values were precise to the 9

^{th}place after the decimal point; this precision was needed for the calendar, critical to monsoon-driven Indian agriculture […]. This calculation involved infinite series which were summed using a sophisticated philosophy of ratios of inexpressed numbers [today called rational functions…].Europeans, however, were primitive and backward in arithmetical calculations […] and barely able to do finite sums. The decimal system had been introduced in Europe by Simon Stevin only at the end of the 16

C. K. Raju (2013 p. 161- 162)^{th}c., while it was in use in India since Vedic times, thousands of years earlier.

This may be cited in contrast with the statements of Western historians, among which:

The history of mathematics cannot with certainty be traced back to any school or period before that of the Greeks […] though all early races knew something of numeration […] and though the majority were also acquainted with the elements of land-surveying, yet the rules which they possessed […] were neither deduced from nor did they form part of any science.

W. W. Rouse Ball,A Short Account of the History of Mathematics. Dover, New York, 1960, p. 1–2.

Therefore, it may seem paradoxical, given such an intellectual baggage, to write an entire chapter on musical scales without a single number! In *A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra* I showed a minimal reason: Bharata’s description leads to an infinite set of solutions that should be formalized with algebra rather than a finite set of numbers.

## The experiment

The author(s) of *Natya Shastra* invite(s) the reader to take two *vina*-s (plucked stringed instruments) and tune them on the same scale.

A word of caution to clarify the context: this chapter of *Natya Shastra* may be read as a thought experiment rather than a process involving physical objects. There is no certitude that these two *vina*-s ever existed — and even that “Bharata Muni”, the author/experimenter, was a unique person. His/their approach is a validation (* pramāņa*) resorting to empirical proof, in other words driven by the physically manifest (

*pratyak*) rather than inferred from “axioms” constitutive of a theoretical model. This may be summed up as “preferring physics to metaphysics”.

*ş*aConstructing and manipulating *vina*-s in the manner indicated by the experimenter seems an insurmontable technological challenge. This has been a subject of discussion among a number of authors — read Iyengar (2017 pages 7-sq.) Leaving aside the possibility of a practical realization is not a denial of physical reality, as formal mathematics would systematically mandate. Calling it a “thought experiment” is a way of asserting the link with the physical model. In the same manner, using circular graphs to represent tuning schemes and algebra to describe relations between intervals are aids to understanding which do not reduce the model to specific, idealistic interpretations similar to speculations on integer numbers cherished by Western scholars. These graphs aim at facilitating a computational design of instruments modeling these imagined instruments — read Raga intonation and Just intonation, a general framework.

Let us follow Bharata’s instructions and tune both instruments on a scale called “*Sa-grama*” about which the author declares:

The seven notes [svaras] are: Şaḍja [Sa], Ṛşbha [Ri], Gāndhāra [Ga], Madhyama [Ma], Pañcama [Pa], Dhaivata [Dha], and Nişāda [Ni].

It is tempting to identify this scale as the conventional Western seven-grade scale *do, re, mi, fa, sol, la, si* (“C”, “D”, “E”, “F”, “A”, “B”) which some scholars have done despite the faulty interpretation of intervals.

Intervals are notated in *shruti*-s which may, for a start, be taken as an ordering device rather than a unit of measurement. The experiment will confirm that a four-*shruti* interval is larger than a three-*shruti*, a three-*shruti* larger than two-*shruti* and the latter larger than a single *shruti*. In different contexts the word “*shruti*” designates note positions instead of intervals between notes. This ambiguity is also a source of confusion.

The author writes:

Śrutis in the Şaḍja Grāma 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].

Bharata enlists 9-*shruti* (consonant) intervals: “Sa-Pa”, “Sa-Ma”, “Ma-Ni”, “Ni-Ga” and “Re-Dha”. In addition, he defines another scale named “*Ma-grama*” in which “Pa” is one *shruti* lower than “Pa” in the *Sa-grama*, so that “Sa-Pa” is no longer consonant whereas “Re-Pa” is consonant because it is made of 9 *shruti*-s.

Intervals of 9 or 13 *shruti*-s are declared “consonant” (*samvadi*). Leaving out the octave, the best consonance in a musical scale is the perfect fifth with a frequency ratio close to 3/2. When tuning stringed instruments, a ratio differing from 3/2 generates beats indicating that a string is *out of tune*.

If frequency ratios are expressed logarithmically with 1200 cents representing one octave, and further converted to angles with a full octave on a circle, the description of *Sa-grama* and *Ma-grama* scales may be summarized on a circular diagram (see picture).

Two cycles of fifths have been highlighted in red and green colors. Note that both the “Sa-Ma” and “Ma-Ni” intervals are perfect fifths, discarding the association of *Sa-grama* with the conventional Western scale: the “Ni” should be mapped to “B flat”, not to “B”. Further, the “Ni-Ga” perfect fifth implies that “Ga” is also “E flat” rather than “E”. The *Sa-grama* and *Ma-grama* scales are therefore “D modes”. For this reason, “Ga” and “Ni” appear underlined on the diagrams.

Authors eager to identify *Sa-grama* and *Ma-grama* as a Western scale claimed that when the text says that there are *“3 shruti-s in Re”* it should be understood between Re and Ga. Yet this interpretation is inconsistent with the second lowering of the moveable *vina* (see below).

We must avoid premature conclusions about intervals in these scales. The two cycles of fifths are unrelated except that the “distance” between the “Pa” of *Ma-grama* and that of *Sa-grama* is “one-*shruti*”:

The difference which occurs in Pañcama when it is raised or lowered by a Śruti and when consequential slackness or tenseness [of strings] occurs, will indicate a typical (pramāņa) Śruti. (XXVIII, 24)

In other words, the size of this * pramāņa ṣruti* is not specified. It would therefore be misleading to postulate its equivalence with the syntonic comma (frequency ratio 81/80). Doing so reduces Bharata’s model to “just intonation”, indeed with interesting properties in its application to Western harmony (read page), but with a questionable relevance to the practice of Indian music. As claimed by Arnold (1983 p. 39):

The real phenomenon of intonation in Hindustani Classical Music as practised is much more amorphous and untidy than any geometry of course, as recent empirical studies by Levy (1982), and Arnold and Bel (1983) show.

The designation of the smallest interval as “* pramāņa ṣruti* ” is of major epistemic relevance and deserves a brief explanation. The semantics of “

*slackness or tenseness*” clearly belongs to “

*pratyakṣa pramāṇa*”, the means of acquiring knowledge by perceptual experience. More precisely,

*“pramāṇa (प्रमाण) refers to “valid perception, measure and structure””*(Wisdom Library), a notion of proof shared by all Indian traditional schools of philosophy (Raju 2007 page 63). We will get back to this notion in the conclusion.

An equivalent way of connecting the two cycles of fifths would be to define a 7-*shruti* interval, for instance “Ni-Re”. If the *pramāņa ṣruti* were a syntonic comma then this interval would be a harmonic major third with ratio 5/4. As evoked in Just intonation, a general framework, the invention of the major third as a

*consonant*interval dates back to the early 16

^{th}century in Europe. In

*Natya Shastra*this 7-

*shruti*interval had been rated “assonant” (

*anuvadi*).

In all writings referring to the ancient Indian theory of scales, I occasionally used “*pramāņa ṣruti*” and “syntonic comma” as equivalent terms. This is acceptable if one accepts that the syntonic comma is allowed to take values other than 81/80. Consequently, the “harmonic major third” should not automatically be assigned frequency ratio 5/4.

Picture above represents the two *vina*-s tuned identically on *Sa-grama*. Matching notes are marked by yellow spots. The inner part of the blue circle will be the movable *vina* in the following transposition processes, and the outer part the fixed *vina*.

### First lowering

Bharata writes:

The two Vīņās with beams (danḍa) and strings of similar measure, and with similar adjustment of the latter in the Şaḍja Grāma should be made [ready]. [Then] one of these should be tuned in the Madhyama Grāma by lowering Pañcama [by one Śruti]. The same (Vīņā) by adding one Śruti (lit. due to the adding of one Śruti) to Pañcama will be tuned in the Şaḍja Grāma.

In brief, this is a procedure for lowering all notes of the movable *vina* by one *pramāņa ṣruti*. First lower its “Pa” — e.g. make it consonant with “Re” of the fixed

*vina*— to obtain

*Ma-grama*on the movable

*vina*. Then readjust its whole scale to obtain

*Sa-grama*. Note that lowering “Re” and “Dha” implies appreciating again the size of a

*while preserving the “Re-Dha” consonant interval. The result is as follows:*

*pramāņa ṣruti*The picture illustrates the fact that there are no more matching notes between the two *vina*-s.

This situation can be translated to algebra. Let “a”, “b”, “c” … “v” be the unknown sizes of *shruti*-s in the scale (see picture on the side). A metrics “translating” Bharata’s model will be necessary for checking it on sound structures produced by an electronic instrument — the computer. The scope of this translation remains valid as long as no extra assertion has been stated which is not rooted in the original model.

Using symbol “#>” to indicate that two notes are not matching, this first lowering may be summarized by the following set of inequations:

s + t + u + v > m a + b + c > m d + e > m f + g + h + i > m n + o + p > m q + r > m | Sa #> Ni Re #> Sa Ga #> Re Ma #> Ga Dha #> Pa Ni #> Dha |

### Second lowering

The next step is again a lowering by one *shruti* with a different procedure.

Again due to the decrease of a Śruti in another [Vīņā], Gāndhāra and Nişāda will merge with Dhaivata and Ṛşbha respectively, when there is an interval of two Śrutis between them.

Note that it is no longer possible to rely on a lowered “Pa” to evaluate a *pramāņa ṣruti* for the lowering. The instruction is to lower the tuning of the movable vina until either “Re” and “Ga” or “Dha” and “Ni” are merged, which is claimed to be the same because of the final lowering of two

*shruti*-s (from the initial state):

Now we get an equation reporting that the two-*shruti *intervals are equal in size:

q + r = d + e

and five more inequations indicating the non-matching of other notes:

f + g + h + i > d + e a + b + c > d + e s + t + u + v > d + e n + o + p > d + e j + k + l + m > d + e | Ma #> Ga Re #> Sa Sa #> Ni Dha #> Pa Pa #> Ma |

We should keep in mind that the author is describing a physical process, not an abstract “movement” by which the moveable wheel (or *vina*) would “jump in space” from its initial to final position. Therefore we pay attention to things happening and not happening during the tuning of the vina, or *rotation* of the wheel, looking at the trajectories of dots representing note positions (along the blue circle). Things *not happening* (non-matching notes) yield inequations required for making sense of the algebraic model.

This step of the experiment confirms that it is wrong to locate Sa at the position of Ni for the sake of identifying *Sa-grama* with the Western scale. In this case, matching notes would no be Re-Ga and Dha-Ni, but Ga-Ma and Ni-Sa.

### Third lowering

Bharata writes:

Again due to the decrease of a Śruti in another [Vīņā], Ṛşbha and Dhaivata will merge with Şaḍja and Pañcama respectively, when there is an interval of three Śrutis between them.

This leads to equation

n + o + p = a + b + c

and inequations:

s + t + u + v > a + b + c f + g + h + i > a + b + c j + k + l + m > a + b + c | Sa #> Ni Ma #> Ga Pa #> Ma |

### Fourth lowering

The procedure:

Similarly the same [one] Śruti being again decreased, Pañcama, Madhyama and Şaḍja will merge with Madhyama, Gāndhāra and Nişāda respectively when there is an interval of four Śrutis between them.

This yields 2 equations:

j + k + l + m = f + g + h + i

s + t + u + v = f + g + h + i

## Algebraic interpretation

After eliminating redundant equations and inequations, constraints are summarized as follows:

(S1) d + e > m

(S2) a + b + c > d + e

(S3) f + g + h + i > a + b + c

(S4) j + k + l + m = f + g + h + i

(S5) s + t + u + v = f + g + h + i

(S6) n + o + p = a + b + c

(S7) q + r = d + e

The three inequations illustrate the fact that numbers of *shruti*-s denote an ordering of the sizes of intervals between notes.

Still, we have 22 variables and only 4 equations. These variables can be “packed” to a set of 8 variables which represent the “macro-intervals”, i.e. the steps of the *grama*-s. In this approach, *shruti*-s are sort of “subatomic” particles which these “macro-intervals” are made of… Now we need only 4 auxiliary equations to determine the scale. These may be provided by acoustic information, with intervals measured in cents. First we express that the sum of the variables, the octave, is equal to 1200 cents:

(S8) (a + b + c) + (d + e) + (f + g + h + i) + (j + k + l) + m + (n + o + p) + (q + r) + (s + t + u + v) = 1200

Then we interpret all *samvadi* relationships as perfect fifths (ratio 3/2 = 701.9 cents):

(S9) (a + b + c) + (d + e) + (f + g + h + i) + (j + k + l) + m = 701.9 (Sa-Pa)

(S10) (j + k + l) + m + (n + o + p) + (q + r) + (s + t + u + v) = 701.9 (Ma-Sa)

(S11) (d + e) + (f + g + h + i) + (j + k + l) + m + (n + o + p) = 701.9 (Re-Dha)

(S12) (f + g + h + i) + (j + k + l) + m + (n + o + p) + (q + r) = 701.9 (Ga-Ni)

including the “Re-Pa” perfect fifth in *Ma-grama*:

(S13) m + (n + o + p) + (q + r) + (s + t + u + v) + (a + b + c) = 701.9

S1O, S11 and S12 can all be derived from S9. These equations may therefore be discarded. We still need one more equation to solve the system. At this stage there are many options associated with tuning procedures. As suggested above, setting the harmonic major third to ratio 5/4 (386.3 cents) would provide the missing equation. This amounts to setting variable “m” to 21.4 cents (syntonic comma). However, this major third can take any value up to the Pythagorean third (81/64 = 407.8 cents) for which we would get m = 0.

Beyond this range, the two-vina experiment is no longer valid, but it leaves a great amount of possibilities including the temperament of some intervals which musicians might achieve spontaneously in parallel melodic movements. A set of solutions is exposed in *A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra* and a few of them have been tried on the Bol Processor to check musical examples for which they might provide adequate scales — read Raga intonation.

## Extensions of the model

In order to complete his system of scales, Bharata had to introduce two new notes in the basic *grama*-s: *antara Gandhara* and *kakali Nishada*. The new “Ga” is defined as “G” raised by 2 *shruti*-s. Similarly, *kakali Ni* is “N” raised by 2 *shruti*-s.

In order to position “Ni” and “Ga” correctly we must investigate the behavior of the new scale in all transpositions (*murcchana*-s), including those starting with “Ga” and “Ni”, and infer equations corresponding to an optimal consonance of the scale. We end up with 11 equations for only 10 variables, which means that this perfection cannot be achieved. One constraint must be released.

An option is to release constraints on major thirds, fifths or octaves, leading to a form of temperament. For instance, stretching the octave by 3.7 cents generates perfect fifths (701.9 cents) and harmonic major thirds close to equal temperament (401 cents) with a comma of 0 cents. This tuning technique was advocated by Serge Cordier (Asselin 2000 p. 23; Wikipedia).

Another option is to come as close as possible to “just intonation” without modifying perfect fifths and octaves. This is possible if the comma (variable “m”) is allowed an arbitrary value between 0 and 56.8 cents. Limits are imposed by the inequations derived from the two-vina experiment.

These “just systems” are calculated as follows:

a + b + c = j + k + l = n + o + p = Maj - C

d + e = h + i = q + r = u + v = L + C

f + g = s + t = Maj - L - C

m = C

where L = 90.25 cents (limma = 256/243), Maj = 203.9 cents (major wholetone = 9/8)

and 0 < C < 56.8 (*pramāņa ṣruti* or syntonic comma)

This leads to the 53-grade scale named “grama” which we use as a framework for consonant chromatic scales eligible for pure intonation in Western harmony when the syntonic comma is sized 81/80. Read Just intonation, a general framework:

In BP3, the just-intonation framework has been extended so that any value of the syntonic comma (or the harmonic major third) can be set on a given scale structure. This feature is demonstrated on page Raga intonation.

## The relevance of circular representations

It is safe to classify the two-vina experiment as thought experiment because of the unlikelihood that it could be worked with mechanical instruments. Representing it on a circular graph (a moveable wheel inside a fixed crown) achieves the same goal without resorting to imaginary devices.

Circular representations belong to Indian traditions of various schools, among which the description of rhythmic cycles (*t āl*-s) used by drum players. These graphs are meant to outline the rich internal structure of musical constructions that cannot be reduced to “beat counting” (Kippen 2020).

For instance, the image above was used to describe the *ţhekkā* (cycle of quasi-onomatopoeic syllables representing the drum strokes) of *tāl Pañjābi* which reads as follows:

Unfortunately, early printing press technology may have rendered uneasy the publication and transmission of these aids to learning.

If contemporaries of Bharata ever used similar circular representations for reflecting on musical scales, we guess that archeological traces might not be identified properly as their drawings could be mistaken for *yantra*-s, astrological charts and the like!

## Return to epistemology

Bharata’s experiment is a typical example of the preference for facts inferred from empirical observations over a proclaimed universal logic aimed at establishing “irrefragable demonstrations”.

Empirical proofs are universal,

notmetaphysical proofs; eliminating empirical proofs is contrary toallsystems of Indian philosophy. Thus elevating metaphysical proofs above empirical proofs, as formal mathematics does, is a demand to reject all Indian philosophy as inferior. Curiously, like Indian philosophy, present-day science too uses empirical means of proof, so this is also a demand to reject science as inferior (to Christian metaphysics).Logic is not universal either as Western philosophers have foolishly maintained: Buddhist [quasi truth-functional] and Jain [three-valued] logics are different from those currently used in formal mathematical proof. The theorems of mathematics would change if those logics were used. So, imposing a particular logic is a means of cultural hegemony. If logic is decided empirically, that would, of course, kill the philosophy of metaphysical proof. Further, it may result in quantum logic, similar to Buddhist logic […].

C. K. Raju (2013 p. 182-183)

The two-vina experiment can be likened to the (more recent) physical proof of the “Pythagorean theorem”. This theorem (Casey 1885 p. 43) was known in India and Mesopotamia long before the time of its legendary author (Buckert 1972 p. 429, 462). In the Indian text *Yuktibhāşā* (ca. 1530 CE), a figure of a right-angle triangle is drawn on a palm leaf with squares on its two sides and its hypothenuse. Then the figure is cut and rotated in a way highlighting that the areas are equal.

Clearly, the proof of the “Pythagorean theorem” is

C. K. Raju (2013 p. 167)veryeasy if one is either (a) allowed to make measurements, or, equivalently (b) allowed to move figures about in space.

This process takes place on several steps of *moving figures* in a way similar to *moving scales* (or figures representing scales) in the two-vina experiment. The 3 single-*shruti* tonal intervals may be likened to the areas of the 3 squares in *Yuktibhāşā*. The following remark would therefore apply to Bharata’s procedure:

The details of this rationale are not our immediate concern beyond observing that drawing a figure, carrying out measurements, cutting, and rotation are all empirical procedures. Hence, such a demonstration would today be rejected as invalid solely on the ground that it involves empirical procedures that

C. K. Raju (2007 p. 67)oughtnot to be any part of mathematical proof.

Bernard Bel — Dec. 2020

## References

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