
— manufactured in Miraj (read paper)
This article demonstrates the theoretical and practical construction of microtonal scales for the intonation of North Indian ragas, using tools available with Bol Processor (BP3) + Csound.
It comes as a complement to pages Microtonality and Just intonation, a general framework and The Two-vina experiment. Nonetheless, its understanding does not require a preliminary study of these related pages.
This exercise on raga intonation demonstrates the ability of BP3 to deal with sophisticated models of micro-intonation and support a fruitful creation of music embodied by these models.
Theory versus practice
To summarize the background, the framework for constructing “just-intonation” scales is a deciphering of the first six chapters of Nāṭyaśāstra, a Sanskrit treatise on music, dance and drama dating back to a period between 400 BCE and 200 CE. For convenience we call it “Bharata’s model” although there is no historical record of a single author bearing this name.
Using exclusive information driven from the text and its description of the Two-vina experiment provides an infinite set of valid interpretations of the ancient theory as shown in A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra (Bel 1988a). Among these, the one advocated by many musicologists — influenced by western acoustics and scale theories — states that the frequency ratio of the harmonic major third would be 5/4. This is equivalent to fixing the frequency ratio of the syntonic comma to 81/80.
Even though this interpretation yields a consistent model for just-intonation harmony — read Just intonation, a general framework — it would be far-fetched to stipulate that the same holds for raga intonation. Accurate measurements of raga performance using our Melodic Movement Analyzer (MMA) in the early 1980s revealed that melodic structures inferred from statistics (using selective tonagrams, read below) often differ significantly from scales predicted by the “just-intonation” interpretation of Bharata’s model. Part of the explanation may be the strong harmonic attraction effect of drones (tanpura) played in the background of performances of raga.
Talking about grama-s (scale frameworks) in the ancient Indian theory, E.J. Arnold wrote (1982 p. 40):
Strictly speaking the gramas belong to that aspect of nada (vibration) which is anahata (“unstruck”). That means to say that the “grama” can never be heard as a musical scale [as we did on page Just intonation, a general framework]. What can be heard as a musical scale is not the grama, but any of its murcchanas.
As soon as electronic devices such as the Shruti Harmonium (1979) and the Melodic Movement Analyzer (1981) became available, the challenge of research on raga intonation was to reconcile two methodologies: a top-down approach checking hypothetical models against data, and a data-driven bottom-up approach.
The “microscopic” observation of melodic lines (now rendered easy by software like Praat) confirmed the importance of note treatment (ornamentation, alankara) and time-driven dimensions of raga which are not taken into account by scale theories. For instance, long discussions have been held on the rendering of note “Ga” in raga Darbari Kanada (Bel & Bor 1984; van der Meer 2019) and typical treatment of notes in other ragas (e.g. Rao & Van der Meer 2009; 2010). The visual transcription of a phrase of raga Asha makes it evident:


In order to extract scale information from this melodic continuum, a statistical model was implemented to display the distribution of pitch across one octave. The image shows the tonagram of a 2-minute sketch (chalana) of raga Sindhura taught by Pandit Dilip Chandra Vedi.

The same melodic data was processed again after a filtering of 3 windows attempting to isolate “stable” parts of the line. The first window, typically 0.1 seconds, would eliminate irregular segments, the second one (0.4 s.) would discard segments outside a rectangle of 80 cents in height, and the third one was used for averaging. The outcome is a “skeleton” of the tonal scale displayed as a selective tonagram.
These results often would not match scale metrics predicted by the “just-intonation” interpretation of Bharata’s model. Proceeding further in this data-driven approach, we produced the (non-selective) tonagrams of 30 ragas (again chalana-s) to compute a classification based on their tonal material. Dissimilarities between pairs of graphs (calculated with Kuiper’s algorithm) were approximated as distances, from which a 3-dimensional classical scaling was extracted:

This experiment suggests that contemporary North-Indian ragas are amenable to meaningful automatic classification on the sole basis of their (time-independent) intervalic content. This approach is analogous to techniques of human face recognition able to identify related images with the aid of limited sets of features.

at the National Centre for the Performing Arts (Mumbai) in 1983
This impressive classification has been obtained via statistical analyses of static representations of raga performance. This means that the same result would be achieved after playing the sound file in reverse direction, or even slicing it to segments reassembled in a random order… Music is a dynamic phenomenon that cannot be reduced to tonal “intervals”. Therefore, subsequent research in the representation of melodic lines of raga — once it could be efficiently processed by 100% digital computing — led to the concept of Music in Motion, i.e. synchronising graphs with sounds so that visuals reflect the music as it is being heard, arguably the only suitable “notation” for raga (Van der Meer & Rao 2010; Van der Meer 2020).
This graph model is probably a great achievement as an educational and documentary tool, indeed the environment I was dreaming of when designing the Melodic Movement Analyzer. However, promoting it as a theoretical model is the continuation of a western selective bias. To the amount of my knowledge, no Indian music master ever attempted to describe the intricacies of raga via hand-drawn melograms, although they could. The fascination of technology — and western ‘science’ at large — is not an indication of its relevance to ancient Indian concepts.
Music is appreciated by ears; therefore, a theory of music should be evaluated on its ability to produce musical sounds via predictive model(s). Numbers, charts and graphics are mere tools for the interpretation and anticipation of sound phenomena. This approach is termed analysis by synthesis in Daniel Hirst’s book on speech prosody (Hirst, 2022, forthcoming, p. 137):
Analysis by synthesis involves trying to set up an explicit predictive model to account for the data which we wish to describe. A model, in this sense, is a system which can be used for analysis — that is deriving a (simple) abstract underlying representation from the (complicated) raw acoustic data. A model which can do this is explicit but it is not necessarily predictive and empirically testable. To meet these additional criteria, the model must also be reversible, that is it must be possible to use the model to synthesise observable data from the underlying representation.
This is the raison d’être of the following investigation.
Microtonal framework
The “flexible” model derived from the theoretical model of Natya Shastra (read The Two-vina experiment) discards the assertion of a precise frequency ratio for the harmonic major third classified as anuvadi (asonant) in ancient literature. This amounts to admitting that the syntonic comma (pramāņa ṣruti in Sanskrit) might take any value between 0 and 56.8 cents.
Let us look at graphic representations (by the Bol Processor) to illustrate these points.
The basic framework of musical scales, according to Indian musicology, is a set of 22 tonal positions in the octave named shruti-s in ancient texts. Below is the framework displayed by Bol Processor (microtonal scale “grama”) with a 81/80 syntonic comma. The names of positions “r1_”, “r2_” etc follow the constraints of lowercase initials and appending a underline character to distinguish octave numbers. Positions “r1” and “r2” are two options for locating komal Re (“Db” or “re bemol”) whereas “r3” and “r4” designate shuddha Re (“D” or “re”) etc.

The 22 shruti-s can be listened to on page Just intonation, a general framework, keeping in mind (read above) that this is a framework and not a scale. No musician would ever attempt to play or sing these positions as “notes”!
What happens if the value of the syntonic comma is modified? Below is the same framework with a comma of 0 cent. In this case, any “harmonic position” — one whose fraction contained a multiple of 5 — slides to its nearest Pythagorean neighbour (only multiples of 3 and 2). The result is a “Pythagorean tuning”. On top of the circle the remaining gap is a Pythagorean comma. Positions are slightly blurred because of mismatches linked with a very small interval (the schisma).

The following is the framework with a syntonic comma of 56.8 cents (its upper limit):

In this representation, “harmonic major thirds” of 351 cents would most likely sound “out of tune” because the 5/4 ratio yields 384 cents. In fact, “g2” and “g3” are both distant by a quartertone between Pythagorean “g1” (32/27) and Pythagorean “g4” (81/64). Nonetheless, the internal consistency of this framework (counting perfect fifths in blue) makes it still eligible for the construction of musical scales.
Between these limits of 0 and 56.8 cents, the graphic representation of scales and their internal tonal structure remain unchanged if we keep in mind that the size of major-third intervals is decided by the syntonic comma.
Construction of scale types

The model extracted from Bharata’s Natya Shastra is not an evident reference for prescribing raga intonation because this musical genre started its existence a few centuries later.
Most of the background knowledge required for the following presentation is borrowed from Bose (1960) and my late colleague E. James Arnold who published A Mathematical model of the Shruti-Swara-Grama-Murcchana-Jati System (Journal of the Sangit Natak Akademi, New Delhi 1982). Arnold studied Indian music in Banaras and Delhi during the 1970s and the early 1980s.
Bose was convinced (1960 p. 211) that the scale named Kaishika Madhyama is equivalent to a “just-intonation” seven-grade scale of western musicology. In other words, he took for granted that the 5/4 frequency ratio (harmonic major third) should be equivalent to the 7-shruti interval, but this statement had no influence on the rest of his analysis.

Arnold (1982 p. 17) immediately used integer ratios to design intervals with the fixed syntonic comma (81/80), but as suggested above this has no impact on his model with respect to its structural description. He insisted on setting up a “geometrical model” rather than a speculative description based on numbers as many authors (e.g. Alain Daniélou) had attempted it. The most innovative aspect of Arnold’s study has been the use a circular sliding model to illustrate the matchings of intervals in transposition processes (murcchana-s) — read page The Two-vina experiment.
Indeed it would be more convenient to keep expressing all intervals in numbers of shruti-s in compliance with the ancient Indian theory, but a machine needs metrical data to draw graphics of scales. For this reason we show graphs using a 81/80 syntonic comma, keeping in mind the option of modifying this value at a later stage.

The 22-shruti framework offers the possibility of constructing 211 = 2048 chromatic scales, among which only 12 are “optimally consonant”, i.e. containing only one wolf major fifth (smaller by 1 syntonic comma = 22 cents).
The building blocks of the tonal system according to traditional Indian musicology are two seven-grade scales named Ma-grama and Sa-grama. Bose wrote (1960 p. 13): the Shadja Grāma developed from the ancient tetrachord in which the hymns of the Sāma Veda were chanted. Later on another scale, called the Madhyama Grāma, was added to the secular musical system. The two scales (Dorian modes, according to western terminology) differ by the position of Pa (“G” or “sol”) which may differ by a syntonic comma (pramāņa ṣruti). In the Sa-grama, interval Sa-Pa is a perfect fifth (13 shruti-s) whereas it is a wolf fifth (12 shruti-s) in the Ma-grama. Conversely, interval Pa-Re is a perfect fifth in Ma-grama and a wolf fifth in Sa-grama.
Bharata used the Sa-grama to expose his thought experiment (The Two vinas) aimed at determining the sizes of shruti-s. Then he introduced two additional notes: kakali Nishada (komal Ni or “Bflat”) and antara Gandhara (shuddh Ga or “E”) to get a nine-grade scale from which “optimally consonant” chromatic scales could be derived from modal transpositions (murcchana). The process of building these 12 chromatic scales, namely “Ma01″, “Ma02″… “Sa01″, “Sa20″ etc. is explained on page Just intonation, a general framework.
Selecting notes in each chromatic scale yields 5 to 7-note melodic types. In the Natya Shastra these melodic types were named jāti. These may be seen as ancestors of ragas even though their lineages and structures are only speculated (read on). The term thāṭ (pronounce ‘taat’) translated as “mode” or “parent scale” — has later been adopted, each thāṭ being called by the name of a raga (see Wikipedia). Details of the process, terminology and surveys of subsequent musicological literature will be found in publications by Bose and other scholars.
The construction of the basic scale types is explained by Arnold (1982 p. 37-38). The starting point is the chromatic Ma-grama in its basic position — namely “Sa_murcchana” in the “-cs.12_scales” Csound resource file. This scale can be visualized, using Arnold’s sliding model, by placing the S note of the inner wheel on the S of the outer crown :

This yields the following intervals:

“Optimal consonance” is illustrated by two features: 1) there is only one wolf fifth (red line) in the scale (between D and G), and 2) every note is connected with another one by a perfect fifth (blue line). This consonance is of prior importance to Indian musicians. Consonant intervals are casually placed in melodic phrases to enhance the “flavor” of their notes, and no wolf fifth should exist in the scale.
Note that the Ma-grama chromatic scale has all its notes in their lower enharmonic position.
The Ma-grama chromatic scale has been renamed “Sa_murcchana” in this occurrence because ‘S’ of the moving wheel is facing ‘S’ of the fixed crown. The names of notes have been (in a single click) converted to the Indian convention. Note that key numbers also have been (automatically) fixed to match exclusively labeled notes. In this way, the upper “sa” is assigned key 72 instead of 83 in the “Ma01″ scale showed on page Just intonation, a general framework. The tonal content of this “Sa_murchana” is exposed on this table:


Selecting only “unaltered” notes in “Sa_murcchana” — sa, re, gak, ma, pa, dha, nik — yields the “kaphi1″ scale type named after raga Kaphi (pronounced ‘kafi’). This may be associated to a D-mode (Dorian) in western musicology.
This scale type is stored under the name “kaphi1″ because there will be one more version of the Kaphi scale type.
In “Sa_murcchana” the selection of notes can differ in two ways:
- Select antara Gandhara (namely “ga”) in replacement of the scale’s Gandhara (namely “gak”), thereby raising it by 2 shruti-s. This yields a vikrit (modified) scale type, namely “khamaj1″ associated with raga Khamaj.
- Select both antara Gandhara and kakali Nishada (namely “ni” in replacement of “nik” raised by 2 shruti-s) which creates the “bilaval1″ scale type associated with raga Bilaval.

This “bilaval1″ scale type is one among three versions of Bilaval created by the murcchana procedure. Although it matches the scale of white keys on a western keyboard instrument, it is not the common “just intonation” diatonic scale because of a wolf fifth between “sa” and “pa”.
An alternate Bilaval scale type named “bilaval3″ (extracted from “Ni1_murcchana”, see below) does match Giozeffo Zarlino’s “natural” scale — read Just intonation: a general framework. This should not be confused with Zarlino’s meantone temperament discussed on page Microtonality.

A fourth option: raising “nik” to “ni” and keeping “gak”, would produce a scale type in which “ni” does not have any consonant relation with another note of the scale. This option is therefore discarded from the model.
Every murcchana of the Ma-grama chromatic scale produces at least three scale types by selecting unaltered notes, antara Gandhara or both antara Gandhara and kakali Nishada.

Practically, to create for instance “Ni1_murcchana”, open the “Sa_murcchana” page and enter “nik” (i.e. N3) as the note to be placed on “sa”.
Raga scale types are stored in the “-cs.raga” Csound resource file. Images are available in a single click and scale structures are compared on the main page.
The entire process is summarized in the following table (Arnold 1982 p. 38):
Step | Ma-grama chromatic murcchana starting from | Shuddha grama | Vikrit grama (antara) | Vikrit grama (antara + kakali) |
1 | Sa | kaphi1 | khamaj1 | bilaval1 |
2 | Ma1 | khamaj2 | bilaval2 | kalyan1 |
3 | Ni1 | bilaval3 | kalyan2 | marva1 |
4 | Ga1 | kalyan3 | marva2 | purvi1 |
5 | Dha1 | marva3 | purvi2 | todi1 |
6 | Re1 | purvi3 | todi2 | |
7 | Ma3 | todi3 | lalit1 bhairao1 | |
8 | Ni3 | lalit2 bhairao2 bhairavi1 | ||
9 | Ga3 | todi4 bhairavi2 | ||
10 | Dha3 | bhairavi3 | asavari1 | |
11 | Re3 | bhairavi4 | asavari2 | kaphi2 |
12 | Pa3 | asavari3 | kaphi3 | khamaj3 |
Usage of this table deserves a graphic demonstration. Let us for instance create scale type “kalyan1″ based on the “Ma1_murcchana”. The table says that both “antara and kakali” should be selected. This means “antara Gandhara” which is “ga” in replacement of “gak” in the Ma-grama scale, and “kakali Nishada” which is “ni” in replacement of “nik” in the Ma-grama scale. This process is clear on the movable wheel model:


To execute this selection and export the “kalyan1″ scale type, fill the form on page “Ma1_murcchana” as indicated on the picture.
Below is the resulting scale type.

Keep in mind that note positions expressed as integer frequency ratios are just a matter of convenience for readers acquainted with western musicology. It would be more appropriate to follow the Indian convention of counting intervals in numbers of shruti-s. In this example, the interval between “sa” and “ma” raised from 9 shruti-s (perfect fourth) to 11 shruti-s (tritone).
Arnold’s model is an extension of the murcchana system described in Natya Shastra because it accepts murcchana-s starting from notes which do not belong to the original (7-grade) Ma-grama, taken from its “chromatic version”: Dha1, Re1, Ma3, Ni3, Ga3. This extension is necessary for creating scale types for Todi, Lalit and Bhairao which include augmented seconds.
In his 1982 paper (p. 39-41) Arnold connected his classification of scale types with the traditional list of jāti-s, the “ancestors of ragas” described in Sangita Ratnakara of Śārṅgadeva (Shringy & Sharma, 1978). Seven jāti-s are cited (p. 41), each of them being derived from a murcchana of Ma-grama on one of its shuddha swara-s (basic notes).
Every jāti is assigned a note of tension release (nyasa swara). In contemporary ragas, the nyasa swara is often found at the end of a phrase or a set of phrases. In Arnold’s interpretation, the same should define the murcchana from which the melodic type (jāti) is born. Since, in fact, the names of the shuddha jatis are tied to their nyasa swaras, this too suggests that they should be tied to the murcchanas belonging from those nyasa swaras (Arnold 1982 p. 40).

In other publications (notably Arnold & Bel 1985), Arnold used the cycle of 12 chromatic scales to suggest that enharmonic positions of the notes might express tensions or release states bound to the changing ambience of the circadian cycle, thereby providing an explanation of performance times assigned to traditional ragas. Low enharmonic positions would be associated with darkness and higher ones with day light. In this way, ragas constructed with the aid of the Sa murcchana of Ma-grama chromatic scale (all low positions, step 1) might be interpreted near midnight whereas the ones mixing low and high positions (step 7) would carry the tensions of sunrise and sunset. Their succession is a cycle because, in the table shown above, it is possible to jump from step 12 to step 1 by lowering all note positions by one shruti. This circularity is implied by the process named sadja-sadharana in musicological literature (Shringy & Sharma 1978).
A list of 85 ragas with performance times predicted by the model is available in Arnold & Bel (1985). This hypothesis is indeed interesting — and it does hold for many well-known ragas — but we never found time to embark on a survey of musicians’ statements about performance times that might have assessed its validity.
Practice
Given scale types stored in the “-cs.raga” Csound resource file, Bol Processor + Csound can be used to check the validity of scales by playing melodies of ragas they are supposed to embody. It is also interesting to use these scales in musical genres unrelated with North Indian raga and distort them in any imaginable direction…
Choice of a raga

Public domain
We will take the challenge of matching one among the four “todi” scales with two real performances of raga Todi.
Miyan ki todi is presently the most important raga of the Todi family and therefore often simply referred to as Todi […], or sometimes Shuddh Todi. Like Miyan ki malhar it is supposed to be a creation of Miyan Tansen (d. 1589). This is very unlikely, however, since the scale of Todi at the time of Tansen was that of modern Bhairavi (S R G M P D N), and the name Miyan ki todi first appears in 19th century literature on music.
Joep Bor (1999)
This choice is a challenge for several reasons. Among them, the four variants of “todi” scales have been driven from a (questionable) extension of the grama-murcchana system. Then, notes “ni” and “rek”, “ma#” and “dhak” are close to the tonic “sa” and the dominant “pa” and might be “attracted” by the tonic and dominant, thereby disrupting the “geometry” of theoretical scales in the presence of a drone.
Finally, and most important, the performer’s style and personal options are expected to come in contradiction with this theoretical model. As suggested by Rao and van der Meer (2010 p. 693):
[…] it has been observed that musicians have their own views on intonation, which are handed down within the tradition. Most of them are not consciously aware of academic traditions and hence are not in a position to express their ideas in terms of theoretical formulations. However, their ideas are implicit in musical practice as musicians visualize tones, perhaps not as fixed points to be rendered accurately every time, but rather as tonal regions or pitch movements defined by the grammar of a specific raga and its melodic context. They also attach paramount importance to certain raga-specific notes within phrases to be intoned in a characteristic way.
We had already taken the Todi challenge with an analysis of eight occurrences using the Melodic Movement Analyzer (Bel 1988b). The analyzer had produced streams of accurate pitch measurements which were submitted to a statistical analysis after being filtered as selective tonagrams (Bel 1984; Bel & Bor 1984). Occurrences included 6 performances of raga Todi and 2 experiments of tuning the Shruti Harmonium.

The second column is the standard deviation on intervals, and the third column the standard deviation on positions relative to the tonic
The MMA analysis revealed a relatively high consistency of note positions showing standard deviations better than 6 cents for all notes except “ma#” for which the deviation rose to 10 cents, still an excellent stability. Matching these results against the grama-murcchana “flexible” model revealed less than 4 cent standard deviation of intervals for 4 different scales in which the syntonic comma (pramāņa ṣruti) would be set to 6, 18, 5 and 5 cents respectively. In discussing tuning schemes, Wim van der Meer even envisaged that musicians could “solve the problem” of a “ni-ma#” wolf fifth by tempering fifths over the “ni-ma#-rek-dhak” cycle (Bel 1988b p. 17).
Our conclusion was that no particular “tuning scheme” could be taken for granted on the basis of “raw” data. It would be more realistic to study a particular performance by a particular musician.
Choice of a musician

Credit সায়ন্তন ভট্টাচার্য্য - Own work, CC BY-SA 4.0
Work with the Shruti Harmonium naturally incited us to meet Kishori Amonkar (1932-2017) in 1981. She was a foremost exponent of Hindustani music, having developed a personal style that claimed to transcend classical schools (gharanas).
Most interesting, she used to perform with the accompaniment of a swara mandal (see picture), a zither which she would tune for each individual raga. Unfortunately, we were not equipped for measuring these tunings with sufficient accuracy. Therefore we brought the Shruti Harmonium to program intervals as per her instructions.
This experiment did not work well for two reasons. A technical one: that day, a frequency divider (LSI circuit) was defective on the harmonium; until it was replaced some programmed intervals were inaccessible. A musical one: the experiment revealed that this accurate harmonium was unfit for tuning experiments with Indian musicians. Frequency ratios needed to be typed on a small keyboard, a usage too remote from the practice of string tuning. This was a major incentive for designing and constructing our “microscope for Indian music”, the Melodic Movement Analyzer (MMA) (Bel & Bor 1984).
During the following years (1981-1984) MMA experiments took our entire time, revealing the variability (yet not the randomness) of raga intonation. For this reason we could not return to tuning experiments. Today, a similar approach would be much easier with the help of Bol Processor BP3… if only the expert musicians of that period were still alive!
Choice of a scale type
We need to decide between the four “todi” scale types produced by murcchana-s of the Ma-grama chromatic scale. To this effect we may use measurements by the Melodic Movement Analyzer (Bel 1988b p. 15). Let us pick up average measurements and the ones of a performance of Kishori Amonkar. These are note positions (in cents) against the tonic “sa”.
Note | Average | Standard deviation | Kishori Amonkar |
rek | 95 | 4 | 96 |
gak | 294 | 4 | 288 |
ma# | 606 | 10 | 594 |
pa | 702 | 1 | 702 |
dhak | 792 | 3 | 792 |
(dhak) | 806 | 3 | 810 |
ni | 1107 | 6 | 1110 |
For the moment we ignore “dhak” in the lower octave as it will be dealt with separately. Let us match Kishori Amonkar’s results with the four scale types:
Note | Kishori Amonkar | todi1 | todi2 | todi3 | todi4 |
rek | 96 | 89 | 89 | 89 | 112 |
gak | 288 | 294 | 294 | 294 | 294 |
ma# | 594 | 590 | 590 | 610 | 610 |
pa | 702 | 702 | 702 | 700 | 702 |
dhak | 792 | 792 | 792 | 792 | 814 |
ni | 1110 | 1088 | 1109 | 1109 | 1109 |

There are several ways of finding the best match for musical scales: either comparing scale intervals or comparing note positions with respect to the base note (tonic). Due to the importance of the drone we opt for the second method. The selection is easy here. Version “todi1″ may be discarded because of “ni”, the same with “todi3″ and “todi4″ because of “ma#”. We are left with “todi2″ which has a very good matching, including with the measurements of performances by other musicians.
Adjustment of the scale
The largest deviations are on “rek” which was performed 7 cents higher than the predicted value and “gak” 6 cents lower. Even a 10-cent variation is practically impossible to measure on a single note sung by a human, including a high-profile singer like Kishori Amonkar; the best resolution used in speech prosody is larger than 12 cents.
Any “measurement” of the MMA is an average of values along the rare stable melodic steps. It may not be representative of the “real” note because of its dependency on note treatment: if the approach of the note lies in a range on the lower/higher side, the average will be lower/higher than the target pitch.
Therefore it would be acceptable to declare that the “todi2″ scale type matches the performance. Nonetheless, let us demonstrate ways of modifying the model to reflect the measurements more accurately.

First we duplicate “todi2″ to create “todi-ka” (see picture). Note positions are identical in both versions.
Looking at the picture of the scale (or figures on its table) we notice that all note positions except “ma#” are Pythagorean. The series which a note belongs to is marked by the color of its pointer: blue for Pythagorean and green for harmonic.

This means that modifying the size of the syntonic comma — in strict compliance with the grama-murcchana model — will only adjust “ma#”. In order to change “ma#” position from 590 to 594 cents (admittedly a ridicule adjustment) we need to decrease the size of the syntonic comma by the same amount. This can be done at the bottom right of the “todi-ka” page, changing the syntonic comma to 17.5 cents, a modification which is confirmed by the new picture.
A table on the “todi-ka” page indicates that the “rek-ma#” interval is still evaluated as a “perfect” fifth even though it is smaller by 6 cents.
It may not be evident whether the syntonic comma needs to be increased or decreased to fix the position of “ma#”, but it is easy to try the other way in case the direction was wrong.

Other adjustments will depart from the “pure” model. These lead to changing frequency ratios in the table of the “todi-ka” page. Raising “rek” from 89 to 96 cents requires a raising of 7 cents amounting to ratio 2(7/1200) = 1.00405. This brings the position of “rek” from 1.053 to 1.057.
In the same way, lowering “gak” from 294 to 288 cents requires a lowering of 6 cents amounting to ratio 2(-6/1200) = 0.9965. This brings the position of “gak” from 1.185 to 1.181.

Fortunately, these calculations are done by the machine: use the “MODIFY NOTE” button on the scale page.
The picture shows that the information of “rek” and “gak” belonging to Pythagorean series (blue line) is preserved. The reason is that whenever a frequency ratio is modified by its floating-point value, the machine verifies whether the new value comes close to an integer ratio of the same series. For instance, changing back “rek” to 1.053 would restore its ratio 256/243. Accuracy better than 1‰ is required for this matching.

A tuning scheme for this scale type is suggested by the machine. The graphic representation shows that “ni” is not consonant with “ma#” as their interval is 684 cents, close to a wolf fifth of 680 cents. Other notes are arranged on two cycles of perfect fifths. Interestingly, raising “rek” by 7 cents brought the “rek-ma#” fifth back to its perfect size (702 cents).
Again, these are meaningless adjustments for a vocal performance. We are only showing how to proceed when necessary.

The remaining adjustment will be that of “dhak” in the lower octave. To this effect we duplicate the preceding scale after renaming it “todi_ka_4″, indicating that it is designed for the 4th octave. In the new scale named “todi_ka_3″, we raise “dhak3” by 810 -792 = 18 cents.
This raises its position from 1.58 to 1.597. Note that this brings it exactly to a position in the harmonic series since the syntonic comma is 17.5 cents.
In addition, “dhak-sa” is now a harmonic major third — with a size of 390 cents fitting the 17.5 cents comma. This is certainly meaningful in the melodic context of this raga, a reason why an adjustment of the same size had been done by all musicians in tuning experiments.
This case is a simple illustration of raga intonation as a trade-off between harmonicity with respect to the drone and the requirement of consonant melodic intervals. It also indicates that the Shruti Harmonium could not follow musicians’ practice because its scale ratios were replicated in all octaves.
Choice of a recording
We don’t have the recording on which the MMA analysis had been done. A problem with old tape recordings is the unreliability of speed in tape transportation. On a long recording, too, the frequency of the tonic may change a little due to variations of room temperature influencing instruments — including tape dilation…
To try matching scales a with real performances and examine extremely small “deviations” (which have little musical significance, in any) it is therefore safer to work with digital recordings. This was the case with Kishori Amonkar’s Todi recorded in London in the early 2000 for the Passage to India collection and available free of copyright (link on Youtube). The following is based on that recording.
Setting up the diapason
Let us create the following “-gr.tryRagas” grammar:
-se.tryRagas
-cs.raga
S --> _scale(todi_ka_4,0) sa4

In “-se.tryRagas” the note convention should be set to “Indian” so that “sa4” etc. is accepted even when no scale is specified.
The grammar calls “-cs.raga” containing the definitions of all scale types created by the procedure described above. Unsurprisingly, it does not play note “sa” at the frequency of the recording. We therefore need to measure the tonic to adjust the frequency of “A4” (diapason) in “-se.tryRagas” accordingly. There are several ways to achieve this with increasing accuracy.
A semitone approximation may be achieved by comparing the recording with notes played on a piano or any electronic instrument tuned with A4 = 440 Hz. Once we have found the key that is closest to “sa” we calculate its frequency ratio to A4. If the key is F#4, which is 3 semitones lower than A4, the ratio is r = 2(-3/12) = 0.840. To get this frequency on “sa4” we therefore would need to adjust the frequency of the diapason (in “-se.tryRagas”) to:
440 * r * 2(9/12) = 440 * 2((9-3)/12) = 311 Hz
A much better approximation is achieved by extracting a short occurrence of “sa4” at the very beginning of the performance:
Then select a seemingly stable segment and expand the time scale to get a visible signal:

This sample contains 9 cycles for a duration of 38.5 ms. The fundamental frequency is therefore 9 * 1000 / 38.5 = 233.7 Hz. Consequently, adjust the diapason in “-se.tryRagas” to 233.7 * 2(9/12) = 393 Hz.
The last step is a fine tuning comparing by ear the production of notes in the grammar with the recording of “sa4” played in a loop. To this effect we produce the following sequence:
S --> _pitchrange(500) _tempo(0.2) Scale _pitchbend(-15) sa4 _pitchbend(-10) sa4 _pitchbend(-5) sa4 _pitchbend(-0) sa4 _pitchbend(+5) sa4 _pitchbend(+10) sa4 _pitchbend(+15) sa4 _pitchbend(+20) sa4

These are eight occurrences of “sa4” played at slightly increasing pitches adjusted by the pitchbend. First make sure that the pitchbend is measured in cents: this is indicated in instrument “Vina” called by “-cs.raga” and Csound orchestra file “new-vina.orc”.
Listening to the sequence may not reveal pitch differences, but these will appear to a trained ear when superposed with the recording:
➡ This is a stereo recording. Use earphones to hear the song and sequence of plucked notes on separate channels
One of the four occurrences sounds best in tune. Suppose that the best match is on _pitchbend(+10). This means that the diapason should be raised by 10 cents. Its new frequency would therefore be 393 * 2(10/1200) = 395.27 Hz.
In fact the best frequency is 393.22 Hz, which amounts to saying that the second evaluation (yielding 393 Hz) was fair — and the singers’ voices very reliable! Now we can verify the frequency of “sa4” on the Csound score:
; Csound score
f1 0 256 10 1 ; This table may be changed
t 0.000 60.000
i1 0.000 5.000 233.814 90.000 90.000 0.000 -15.000 -15.000 0.000 ; sa4
i1 5.000 5.000 233.814 90.000 90.000 0.000 -10.000 -10.000 0.000 ; sa4
i1 10.000 5.000 233.814 90.000 90.000 0.000 -5.000 -5.000 0.000 ; sa4
i1 15.000 5.000 233.814 90.000 90.000 0.000 0.000 0.000 0.000 ; sa4
i1 20.000 5.000 233.814 90.000 90.000 0.000 5.000 5.000 0.000 ; sa4
i1 25.000 5.000 233.814 90.000 90.000 0.000 10.000 10.000 0.000 ; sa4
i1 30.000 5.000 233.814 90.000 90.000 0.000 15.000 15.000 0.000 ; sa4
i1 35.000 5.000 233.814 90.000 90.000 0.000 20.000 20.000 0.000 ; sa4
s
These methods could in fact be summarized by the third one: use the grammar to produce a sequence of notes in a wide range to determine an approximate pitch of “sa4” until the small range for the pitchbend (± 200 cents) is reached. Then play sequences with pitchbend values in increasing accuracy until no discrimination is possible.
In a real exercise it would be safe to check the measurement of “sa4” against occurrences in several parts of the recording.
This approach is indeed too demanding on accuracy for the analysis of a vocal performance, but it will be appreciable when working with a long-stringed instrument such as the rudra veena. We will show it with Asad Ali Kan’s performance.
Matching phrases of the performance
We are now ready to check whether note sequences produced by the model would match similar sequences of the recording.
We first try a sequence with emphasis on “rek”. The following note sequence is produced by the grammar:
S --> KishoriAmonkar1
KishoriAmonkar1 --> Scale _ {2, dhak3 sa4 ni3 sa4} {7, rek4} _ {2, dhak3 sa4 ni3 dhak3} {2, dhak3 _ ni3 sa4} {5, rek4}
Scale --> _scale(todi_ka_3,0)
Below is the phrase sung by the musicians (location 0′50″) then repeated in superposition with the sequence produced by the grammar:
➡ This is a stereo recording. Use earphones to hear the song and sequence of plucked notes on separate channels
In this example, scale “todi_ka_3″ has been used because of the occurrence of brief instances of “dhak3”. The position of “rek” is identical in the 3d and 4th octaves. The blending of voice with the plucked instrument is remarkable in the final held note.
In the next sequence (location 1′36″) the position of “gak4” will be appreciated. The grammar is the following:
S --> KishoriAmonkar2
KishoriAmonkar2 --> Scale {137/100, sa4 rek4 gak4 rek4} {31/10, rek4} {18/10, gak4} {75/100,rek4} {44/10, sa4}
Scale --> _scale(todi_ka_4,0)
This time, the scale “todi_ka_4″ was selected, even though it had no incidence on the intonation since “dhak” is absent.
A word about building the grammar: we looked at the signal of the recorded phrase and measured the (approximate) durations of notes: 1.37s, 3.1s, 1.8s, 7.5s, 4.4s. Then we converted these durations to integer ratios — fractions of the basic tempo whose period is exactly 1 second as per the setting in “-se.tryRagas”: 137/100, 31/10 etc.

Below is a pianoroll of the sequence produced by the grammar:

No we try a phrase with a long rest on “dhak3” (location 3′34″) proving that scale “todi_ka_3″ matches perfectly this occurrence of “dhak”:
S --> KishoriAmonkar3
KishoriAmonkar3 --> scale(todi_ka_3,0) 11/10 {19/20, ma#3 pa3} {66/10,dhak3} {24/10, ni3 dhak3 pa3 }{27/10,dhak3} 12/10 {48/100,dhak3}{17/10,ni3}{49/10,dhak3}

Early occurrence of “ma#4” (location 11′38″):
S --> KishoriAmonkar4
KishoriAmonkar4 --> _scale(todi_ka_4,0) 4/10 {17/10, ni3}{26/100,sa4}{75/100,rek4}{22/100,gak4}{17/10,ma#4}{16/100,gak4}{34/100,rek4}{56/100,sa4}{12/100,rek4}{84/100,gak4}{27/100,rek4}{12/10,sa4}

Hitting “dhak4” (location 19′46″):
S --> KishoriAmonkar5
KishoriAmonkar5 --> _scale(todi_ka_4,0) 13/10 {16/10,ma#4}{13/10,gak4}{41/100,ma#4}{72/100,ma#4 dhak4 ma#4 gak4 ma#4}{18/10,dhak4}{63/100,sa4}{90/100,rek4}{30/100,gak4}{60/100,rek4}{25/100,sa4}{3/2,rek4}

With a light touch of “pa4” (location 23′11″):
S --> KishoriAmonkar6
KishoriAmonkar6 --> _scale(todi_ka_4,0) 28/100 {29/100,ma#4}{40/100,dhak4}{63/100,ni4 sa5 ni4}{122/100,dhak4}{64/100,pa4}{83/100,ma#4}{44/100,pa4}{79/100,dhak4}

Pitch accuracy is no surprise in performances by Kishori Amonkar. With a strong awareness of “shruti-s”, she would sit on the stage plucking her swara mandal carefully tuned for each raga.
A test with the rudra veena

Asad Ali Khan (1937-2011) was one of the last performers of the rudra veena in the end of the 20th century and a very supportive participant in our scientific research on raga intonation.
➡ An outstanding presentation of Asad Ali Khan and his idea of music is available in a film by Renuka George.
Pitch accuracy is such on this instrument that we could identify tiny variations controlled and significant in the context of the raga. Read for instance Playing with Intonation (Arnold 1985). In order to measure vibrations below the range of audible sounds, we occasionally fixed a magnetic pickup near the last string.
Below are the statistics of measurements by the Melodic Movement Analyzer of raga Miyan ki Todi interpreted by Asad Ali Khan in 1981. The second column contains the measurements of his tuning of the Shruti Harmonium during an experiment. Columns on the right display predicted note positions according to the grama-murcchana model with a syntonic comma of ratio 81/80. As previously pointed out in Kishori Amonkar’s performance, “dhak” may take different values depending on the octave.
Note | Asad Ali Khan performing | Asad Ali Khan tuning | todi1 | todi2 | todi3 | todi4 |
rek | 99 | 100 | 89 | 89 | 89 | 112 |
gak | 290 | 294 | 294 | 294 | 294 | 294 |
ma# | 593 | 606 | 590 | 590 | 610 | 610 |
pa | 702 | 702 | 702 | 702 | 700 | 702 |
dhak3 | 795 | 794 | 792 | 792 | 792 | 814 |
dhak2 | 802 | |||||
ni | 1105 | 1108 | 1088 | 1109 | 1109 | 1109 |
Again, the best match would be the “todi2″ scale with a syntonic comma of 17.5 cents. We created two scales, “todi_aak_2″ and “todi_aak_3″ for the 2nd and 3th octaves.

The scale constructed during the Shruti Harmonium experiment is of lesser relevance because of the influence of the experimenter playing scale intervals with a low-attracting drone (produced by the machine). In his attempt to resolve dissonance in the scale — which always contained a wolf fifth and several Pythagorean major thirds — Khan saheb ended up with a tuning identical to the initial one but one comma lower. This was not a musically significant situation!

Scale “todi_aak_2″ (in the low octave) contains interesting intervals (harmonic major thirds) which lets us anticipate effective melodic movements. The tuning scheme summarizes these relations.
We are now taking fragments of Asad Ali Khan’s performance of Todi (2005) available on Youtube (follow this link).
The performance began in the low octave, therefore with scale “todi_aak_2″. The frequency of Sa was measured at 564.5 Hz with the method explained earlier.
Let us start with a simple melodic phrase repeated two times, the second time in superposition with the note sequence produced by the grammar.
➡ This is a stereo recording. Use earphones to hear the song and sequence of plucked notes on separate channels
S --> AsadAliKhan1
AsadAliKhan1 --> _scale(todi_aak_2,0) 45/100 {69/10,sa3} {256/100,dhak2} {78/10,dhak2} {12/10,sa3 ni2 rek3&} {48/10,&rek3} {98/100,sa3 ni2 sa3&} {27/10,&sa3}
This grammar contains an unusual sign ‘&’ used to concatenate sound-objects (or notes) beyond the borders of polymetric expressions (between curled brackets). This makes it possible to play the final “rek3” and “sa3” as continuous notes. This continuity is clear on the following graph:

It is time to make sure that accurate tunings and adjustments of scales are more than an intellectual exercise… After all, the main difference between scales “todi_aak_2″ and “todi_aak_3″ is that “dhak” is 7 cents higher in “todi_aak_2″, which means a third of a comma! To check the effect of the fine tuning, listen to the superimposition two times, once with “todi_aak_3″ and the second time with “todi_aak_2″:
To check the difference between these two versions of “dhak2” we can play them in sequence, then superimposed:
S --> _tempo(1/2) _scale(todi_aak_3,0) dhak2 _scale(todi_aak_2,0) dhak2 {_scale(todi_aak_3,0) dhak2, _scale(todi_aak_2,0) dhak2}
With fundamental frequencies 132.837 Hz and 133.341 Hz, the beat frequency (of sine waves) would be 133.341 - 132.837 = 0.5 Hz. The perceived beat frequency is higher because of the interference between higher partials. This suggests that a difference of 7 cents is not irrelevant in the context of notes played by a long-stringed instrument (Arnold 1985).
More in the lower octave:
S --> AsadAliKhan2
AsadAliKhan2 --> scale(todi_aak_2,0) _volume(64) _pitchrange(500) _pitchcont 93/100 {81/10,pa2}{38/10,pa2 gak2 pa2 dhak2 pa2 }{19/10,gak2}{43/10, _pitchbend(0) rek2 _pitchbend(-100) rek2&} _volumecont _volume(64) {2, _pitchbend(-100) &rek2} _volume(0) _volume(64) {23/10,ni2__ dhak2}{103/100,sa3&}{4,&sa3} 15/10 _volume(64) {38/10,sa3} _volume(0)
As “sa2” is out of range of the Csound instrument “Vina”, it is performed here as “rek2” with a pitchbend correction of one semitone.

The rendering of phrases in the low octave is very approximative because of the predominance of meend (pulling the string). Some effects could be better imitated with the aid of performance controls — see for instance Sarasvati Vina — but this requires a mastery of the real instrument to design patterns of musical “gestures” rather than sequences of sound events… Imitating the melodic intricacy of raga is not the topic of this page; we are merely checking the relevance of scale models to the “tonal skeleton” of ragas.
Accidental notes
Raga scales extracted from murcchanas of the Ma-grama chromatic scale (see above) contain exclusively notes presumably belonging to the raga. They cannot accommodate accidental notes nor the scales used by mixing ragas, a common practice.
Let us take for instance a fragment of the preceding example which was poorly rendered by the sequence of notes produced by the grammar. (We learn from our mistakes!) We may feel like replacing expression {38/10, pa2 gak2 pa2 dhak2 _ pa2 _} with {38/10, pa2 ga2 pa2 dhak2 _ pa2 _} making use of “ga2” which does not belong to the “todi_aak_2″ scale. Unfortunately, this produces an error message:
ERROR Pitch class ‘4’ does not exist in _scale(todi_aak_2). No Csound score produced.

This amounts to saying that scale “todi2″ contains no mapping of key #64 to “ga” — nor key # 65 to “ma”, see picture.
To solve this problem we may recall that scale “todi2″ has been extracted from “Re1_murcchana”. The latter contains all grades of a chromatic scale in addition to the extracted ones. Therefore it is sufficient to replace “_scale(todi_aak_2,0)” with “_scale(Re1_murcchana,0)” in this section:
_scale(Re1_murcchana,0) {38/10, pa2 ga2 pa2 dhak2 _ pa2 _} _scale(todi_aak_2,0) etc.
The scale editor takes care of assigning each note a key number based on the chromatic scale if a standard English, Italian/French or Indian note convention is used. In other cases this mapping should be done by hand. Designers of microtonal scales should stay aware of key mappings if they use customized names for “notes”.
Another problem arises because in “todi_aak_2″ note “dhak” had been raised from 792 to 810 cents, which is not its value in “Re1_murcchana”. This may be fixed by creating another variant of the scale with this correction, or simply use the pitchbend to modify “dhak2” — in which case the same pitchbend could have been used in the first place to raise “gak2”.
Finally, the best approach to avoid this problem would be to use the source chromatic scale “Re1_murcchana”, a murcchana of Ma-grama, to construct raga scales even though some grades will never be used.
To conclude…
This whole discussion was technical. There is no musical relevance in trying to associate plucked notes with very subtly ornamented melodic movements. The last excerpt (2 repetitions) will prove — if at all necessary — that the intonation of Indian ragas is much more than a sequence of notes in a scale, whatever its accuracy:
S --> AsadAliKhan3
AsadAliKhan3 --> scale(todi_aak_3,0) 94/100 {26/10,sa3}{23/10,sa3 rek3 gak3}{195/100,ma#3}{111/100,rek3}{24/10,rek3 sa3}{33/10,sa3 sa3}{71/100,rek3}{76/100,gak3}{71/100,dhak3 ma#3}{176/100,dhak3}{75/100,sa4}{27/10,dhak3__ sa4}{620/100,sa4 dhak3 ma#3 dhak3 ma#3 gak3 _ ma#3 dhak3 dhak3&}{266/100,&dhak3}{672/100,pa3____ pa3_ pa3 pa3 pa3__}{210/100,pa3 ma#3 pa3 dhak3}{222/100,dhak3}{163/100,gak3 ma#3}{426/100,gak3_ rek3____}{346/100,sa3}
➡ This is a stereo recording. Use earphones to hear the song and sequence of plucked notes on separate channels

Listen to Asad Ali Khan’s actual performance of raga Todi to appreciate its expressive power!
Trying to follow the intricacy of alankara (note treatment) with a simplistic notation of melodic phrases shows the disruption between “model-based” experimental musicology and the reality of musical practice. This explains why we resorted to descriptive models (e.g. automatic notation) captured by the Melodic Movement Analyzer or computer tools such as Praat, rather than attempting to reconstruct melodic phrases from theoretical models. Experiments on scales deal with the “skeletal” nature of intonation, which is a necessary yet not sufficient parameter for describing melodic types.
All examples shown on this page are available in the sample set bp3-ctests-main.zip shared on GitHub. Follow instructions on Bol Processor ‘BP3’ and its PHP interface to install BP3 and learn its basic operation. Download and install Csound from its distribution page.
Bernard Bel — Dec. 2020
References
Arnold, E.J.; Bel, B. L’intonation juste dans la théorie ancienne de l’Inde : ses applications aux musiques modale et harmonique. Revue de musicologie, JSTOR, 1985, 71e (1-2), p.11-38.
Arnold, E.J. A Mathematical model of the Shruti-Swara-Grama-Murcchana-Jati System. Journal of the Sangit Natak Akademi, New Delhi 1982.
Arnold, E.J.; Bel, B. A Scientific Study of North Indian Music. NCPA Quarterly Journal, vol. XII Nos. 2 & 3, Bombay 1983.
Arnold, W.J. Playing with Intonation. ISTAR Newsletter Nr. 3-4, June 1985 p. 60-62.
Bel, B. Musical Acoustics: Beyond Levy’s “Intonation of Indian Music”. ISTAR Newsletter Nr 2, April 1984.
Bel, B. A Mathematical Discussion of the Ancient Theory of Scales according to Natyashastra. Note interne, Groupe Représentation et Traitement des Connaissances (CNRS), March 1988a.
Bel, B. Raga : approches conceptuelles et expérimentales. Actes du colloque “Structures Musicales et Assistance Informatique”, Marseille 1988b.
Bel, B.; Bor, J. Intonation of North Indian Classical Music: working with the MMA. National Center for the Performing Arts. Video on Dailymotion, Mumbai 1984.
Bharata. Natya Shastra. There is no currently available English translation of the first six chapters of Bharata’s Natya Shastra. However, most of the information required for this interpretation has been reproduced and commented by Śārṅgadeva in his Sangita Ratnakara (13th century AD).
Bor, J.; Rao, S.; van der Meer, W.; Harvey, J. The Raga Guide. Nimbus Records & Rotterdam Conservatory of Music, 1999. (Book and CDs)
Bose, N.D. Melodic Types of Hindustan. Bombay, 1960: Jaico.
Hirst, D. Speech Prosody - Chapter 8. Modelling Speech Melody. Preprint, 2022.
Rao, S.; Van der Meer, W. The Construction, Reconstruction, and Deconstruction of Shruti. Hindustani music: thirteenth to twentieth centuries (J. Bor). New Delhi, 2010: Manohar.
Shringy, R.K.; Sharma, P.L. Sangita Ratnakara of Sarngadeva: text and translation, vol. 1, 5: 7-9. Banaras, 1978: Motilal Banarsidass. Source in the Web Archive.
Van der Meer, W.; Rao, S. Microtonality in Indian Music: Myth or Reality. Gwalior, 2009: FRSM.
Van der Meer, W. Gandhara in Darbari Kanada, The Mother of All Shrutis. Pre-print, 2019.
Van der Meer, W.; Rao, S. MUSIC IN MOTION. The Automated Transcription for Indian Music (AUTRIM) Project by NCPA and UvA, 2010.
Van der Meer, W. The AUTRIM Project, Music in Motion, 2020.
Initial feedback
This project started in 1980 with the foundation of the International Society for Traditional Arts Research (ISTAR) in New Delhi (India). We had shared articles and proposals which allowed us (Arnold and Bel) to be recipients of a grant from the International Fund for the Promotion of Culture (UNESCO). A brochure of ISTAR projects was then printed in Delhi, owing to which a larger team received support from the Sangeet Research Academy (SRA, Calcutta), the Ford Foundation (USA) and the National Centre for the Performing Arts (NCPA, Bombay).
The following are excerpts of letters of support received during this initial period — after the construction of the Shruti Harmonium and during the design of the Melodic Movement Analyzer. (ISTAR brochure, 1981 pages 20-22)
Indeed, the full potential of this approach may only be achieved now, taking advantage of (virtually unlimited) digital devices replacing the hardware we had created 40 years ago to this effect!
The work of Mr. Arnold and Mr. Bel, as much from the theoretical point of view as from the point of view of the practical realization, appears to be one of the best of these last years, as concerns the musical analysis of the classical music of India…
— Igor REZNIKOFF, Director, UER of Philosophy, History of Art and Archeology, Mathematics, University of Paris X - Nanterre.
I consider that this work presents the greatest interest and is capable of considerably advancing the understanding of the problem of the use of micro-intervals in the music of India, and more generally, that of the intervals found in different modal musics.
— Gilbert ROUGET, Director of Research at CNRS, in charge of the Department of Ethnomusicology at the Musée de l’Homme, Paris.
The ideas and conceptions of Mr. Arnold and Mr. Bel seem tome to have the utmost interest musically because they rest not just on pure theories; but on a profound understanding of melodic and modal music, etc. The project which Mr. Bel presented to me could bring about a realization much more interesting and effective than that of the various “melographs” which have been proposed…
— Émile LEIPP, Director of Research at the CNRS, Director of Laboratoire d’Acoustique, University of Paris VI.
The project entitled “A Scientific study of the modal music of North India” undertaken by E. James Arnold and Bernard Bel is very interesting and full of rich potentials. This collaboration of mathematics and physical sciences as well as engineering sciences on the one hand, and Indology and Indian languages, musicology, as well as applied music on the other hand can be reasonably expected to yield fascinating results.
— Dr. Prem Lata SHARMA, Head of the Department of Musicology and Dean of the Faculty of Performing Arts, Banaras Hindu University.
Mr. Arnold’s work on the logic of the grama-murcchana system and its ‘applications’ to current Indian music is a most stimulating and original piece of investigation. Mr. Arnold’s research and he and his partner (Mr. Bel)‘s work have immense implications for music theory and great value for theoretical study of Indian music.
— Bonnie C. WADE, Associate Professor of Music, University of California
Looking forward into the future, it (the Shruti harmonium) opens up a new field to composers who wish to escape from the traditional framework in which they are trapped, by virtue of the multiplicity of its possibilities for various scales, giving hence a new material.
— Ginette KELLER, Grand Prize of Rome, Professor of Musical Analysis and Musical Aesthetics, ENMP and CNSM, Paris.
I was astonished to listen to the “shrutis” (microtones) produced by this harmonium which they played according to my suggestion, and I found the ‘gandhars’, ‘dhaivats’, ‘rikhabs’ and ‘nikhads’ (3rds, 6ths, 2nds and 7ths) of ragas Darbari Kanada, Todi, Ramkali and Shankara to be very correctly produced exactly as they could be produced on my violin.
— Prof. V.G. JOG, Violinist, recipient of the Sangeet Natak Akademi Award.
Once again, bravo for your work. When you have a precise idea about the cost of your analyzer, please let me know. I shall be able to propose it to research institutions in Asian countries, and our own research institution, provided that it can afford it, might also acquire such an analyzer for our own work.
— Tran Van KHE, Director of Research, CNRS, Paris.
The equipment which Mr. E.J. Arnold and B. Bel propose to construct in the second stage of the research which they have explained to me seems to be of very great interest for the elucidation of the problems concerning scales, and intonation, as much from the point of view of their artistic and musicological use, as from the theory of acoustics.
— Iannis XENAKIS, Composer, Paris.