Raga intonation

Tanpura: the drone of Indian musi­cians (man­u­fac­tured in Miraj)

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 1988). 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 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.

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

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) — see 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)
1Sakaphi1khamaj1bilaval1
2Ma1khamaj2bilaval2kalyan1
3Ni1bilaval3kalyan2marva1
4Ga1kalyan3marva2purvi1
5Dha1marva3purvi2todi1
6Re1purvi3todi2
7Ma3todi3lalit1
bhairao1
8Ni3lalit2
bhairao2
bhairavi1
9Ga3todi4
bhairavi2
10Dha3bhairavi3asavari1
11Re3bhairavi4asavari2kaphi2
12Pa3asavari3kaphi3khamaj3
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 could nev­er 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

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 1988). 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 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 would be adjust­ed to 6, 18, 5 and 5 cents. In dis­cussing tun­ing schemes we even envis­aged that musi­cians might “solve the prob­lem” of a “ni-ma#” wolf fifth by tem­per­ing fifths over the “ni-ma#-rek-dhak” cycle.

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

Working with the Shruti Harmonium nat­u­ral­ly incit­ed us to meet Kishori Amonkar 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. We were not equipped for mea­sur­ing these tun­ings with a suf­fi­cient accu­ra­cy. Therefore we brought the Shruti Harmonium to pro­gram inter­vals as per her instructions.

This 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 to tun­ing exper­i­ments with Indian musi­cians. Frequency ratios need­ed to be typed on a small key­board, a usage too remote from the con­text of per­for­mance. This was a major incen­tive for design­ing and con­struct­ing a “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 1988 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
rek95496
gak2944288
ma#60610594
pa7021702
dhak7923792
(dhak)8063810
ni110761110
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
rek96898989112
gak288294294294294
ma#594590590610610
pa702702702700702
dhak792792792792814
ni11101088110911091109
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. 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 match­ing 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 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 their per­for­mances or 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

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 music and sequence of plucked notes separately

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:

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 music and sequence of plucked notes separately

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 was one of the last per­form­ers of the rudra veena in the end of the 20th cen­tu­ry and a very sup­port­ive par­tic­i­pant in sci­en­tif­ic research on raga into­na­tion. 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. Again in this raga, “dhak” may take dif­fer­ent val­ues in the peer­for­mance, depend­ing on the octave.

NoteAsad Ali Khan
per­form­ing
Asad Ali Khan
tun­ing
todi1todi2todi3todi4
rek99100898989112
gak290294294294294294
ma#593606590590610610
pa702702702702700702
dhak3795794792792792814
dhak2802
ni110511081088110911091109

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 music and sequence of plucked notes separately

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 music and sequence of plucked notes separately

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.

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.

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