Raga intonation

Tanpura: the drone of Indian musi­cians
— 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 assess­ment 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:

A brief phrase of raga Asha tran­scribed by the MMA and in west­ern con­ven­tion­al notation
Non-selective tona­gram of raga Sindhura sung by Ms. Bhupender Seetal

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 record­ing of Sindhura on a selec­tive tonagram

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:

A map of 30 North-Indian ragas con­struct­ed by com­par­ing tona­grams of 2-minute sketch­es (cha­lana-s) of sung per­for­mances (Bel 1988b)

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.

Setup of Bel’s Melodic Movement Analyzer MMA2 (black front pan­el) on top of Fundamental Pitch Extractor
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 “gra­ma” scale dis­play­ing 22 shruti-s accord­ing to the mod­el of Natya Shastra, with a 81/80 syn­ton­ic comma

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 “gra­ma scale” of 22 shruti-s with a syn­ton­ic com­ma of 0 cent.

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

The “gra­ma scale” of 22 shruti-s with a syn­ton­ic com­ma of 56.8 cents.

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

Manuscript of the descrip­tion of Zarlino’s “nat­ur­al” scale

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 (right) and Bel (left) demon­strat­ing shruti-s at the inter­na­tion­al East-West music con­fer­ence, Bombay 1983

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.

Sa-grama and Ma-grama accord­ing to Natya Shastra. Red and green seg­ments indi­cate perfect-fifth con­so­nance. Underlined note names indi­cate ‘flat’ positions.

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 :

The Ma-grama chro­mat­ic scale in its basic posi­tion named “Sa_murcchana’

This yields the fol­low­ing intervals:

The Ma-grama chro­mat­ic scale in its basic posi­tion and with notes labeled in English

“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:

Tonal con­tent of “Sa_murcchana”
Scale type named “kaphi1”

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.
A scale type named “bilaval3” match­ing Zarlino’s “nat­ur­al” scale

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.

An incom­plete­ly con­so­nant scale type

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):

StepMa-grama chro­mat­ic
mur­ccha­na start­ing from
Shuddha gra­maVikrit gra­ma (antara)Vikrit gra­ma
(antara + kakali)
Scale types of the extend­ed grama-murcchana series (Arnold 1982)

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:

Selecting notes to cre­ate the “kalyan1” scale type from the “Ma1_murcchana” of chro­mat­ic Ma-grama. “M1” is placed on “S”. Then the stan­dard inter­vals are picked up from the Ma-grama mov­ing wheel, replac­ing G1 with G3 and N1 with N3 as indi­cat­ed in the table.

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.

The “kalyan1” 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).

Performance times asso­ci­at­ed with murcchana-s of the Ma-grama, accord­ing to Arnold (1985)

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.


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

Todi Ragini, Ragamala, Bundi, Rajasthan, 1591
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 four “best” tun­ing schemes for raga Todi (Bel 1988b p. 16)
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

Kishori Amonkar per­form­ing raga Lalit
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”.

NoteAverageStandard devi­a­tionKishori Amonkar
The “dhak” between brack­ets is a mea­sure­ment on the low octave

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:

NoteKishori Amonkartodi1todi2todi3todi4
Scale type “todi2”, the best match to a per­for­mance of Kishori Amonkar

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.

Modified “todi2” scale match­es the mea­sured “ma#”

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. 

Final ver­sion of “todi2” adjust­ed to Kishori Amonkar’s per­for­mance in the medi­um octave (4)

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 “todi2” scale type with “dhak” adjust­ed for the low octave (3)

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:


S --> _scale(todi_ka_4,0) sa4

Adjusting note con­ven­tion in “-se.tryRagas”

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:

A short occur­rence of “sa4” in the begin­ning of Kishori Amonkar’s raga Todi

Then select a seem­ing­ly sta­ble seg­ment and expand the time scale to get a vis­i­ble signal:

Expansion of a very brief “sta­ble” occur­rence of “sa4”

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:

Recording on “sa4” super­posed with a sequence of “sa4” at slight­ly increas­ing pitch­es. Which occur­rence is in tune?
➡ 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

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:

A phrase with empha­sis on “rek” sung by Kishori Amonkar, then repro­duced in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_3”
➡ 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)

A phrase tar­get­ing “gak” repeat­ed in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_4”

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.

Signal of the pre­ced­ing record­ed phrase

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

Pianoroll of the note 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}

A phrase rest­ing on “dhak3” repeat­ed in super­po­si­tion with the sequence of notes pro­duced by the gram­mar using scale “todi_ka_3”
Pianoroll of the note sequence pro­duced by the gram­mar with a rest on “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}

Early occur­rence of “ma#4”

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}

Hitting “dhak4”…

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}

A light touch of “pa”

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 play­ing 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.

NoteAsad Ali Khan
Asad Ali Khan

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.

Adjustments of the “todi2” scale for Asad Ali Kan’s per­for­mance on the rudra veena. Low octave on the left and medi­um on the right.

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!

Tuning scheme for “todi_aak_2”

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.

A phrase of raga Todi by Asad Ali Khan repeat­ed 2 times, the sec­ond time in super­po­si­tion with the sequence of notes 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:

The end of the phrase, show­ing “rek3” and “sa3” as con­tin­u­ous notes

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″:

The same “dhak2” with a note pro­duced using “todi_aak_3” and the sec­ond time “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}

The two ver­sions of “dhak2” in sequence then superimposed

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.

Low-octave phrase repeat­ed with attempt­ed super­im­po­si­tion of a note sequence

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 melod­ic phrase is repeat­ed 2 times to check its super­im­po­si­tion with the sequence of notes pro­duced by the gram­mar
➡ 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


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.

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