Author: Bernard Bel

This arti­cle demon­strates the the­o­ret­i­cal and prac­ti­cal con­struc­tion of micro­ton­al scales for the into­na­tion of North Indian ragas, using tools avail­able with the Bol Processor (BP3).

It is intend­ed to com­ple­ment the pages Microtonality and Just into­na­tion, a gen­er­al frame­work and The Two-vina exper­i­ment. However, its under­stand­ing does not require a pri­or study of these relat­ed pages.

This raga into­na­tion exer­cise demon­strates BP3's abil­i­ty to han­dle sophis­ti­cat­ed mod­els of micro-intonation and to sup­port the fruit­ful cre­ation of music embody­ing these models.

Theory versus practice

To sum­marise the back­ground, the frame­work for con­struct­ing 'just into­na­tion' scales is a deci­pher­ing of the first six chap­ters of the Nāṭyaśāstra, a Sanskrit trea­tise on music, dance and dra­ma dat­ing from a peri­od between 400 BC and 200 AD. For con­ve­nience, we call it "Bharata's Model", although there is no his­tor­i­cal record of a sin­gle author by that name.

Using exclu­sive infor­ma­tion dri­ven from the text and its descrip­tion of the Two-vina exper­i­ment, an infi­nite num­ber of valid inter­pre­ta­tions of the ancient the­o­ry are pos­si­ble, 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 — is that the fre­quen­cy ratio of the har­mon­ic major third would be 5/4. This is equiv­a­lent to set­ting the fre­quen­cy ratio of the syn­ton­ic com­ma at 81/80.

Although this inter­pre­ta­tion pro­vides a con­sis­tent mod­el for just into­na­tion har­mo­ny - see Just into­na­tion, a gen­er­al frame­work — it would be a stretch to claim that the same applies to raga into­na­tion. Accurate assess­ment of raga per­for­mance using our Melodic Movement Analyser (MMA) in the ear­ly 1980s revealed that melod­ic struc­tures derived from sta­tis­tics (using selec­tive tona­grams, see below) often dif­fer sig­nif­i­cant­ly from the scales pre­dict­ed by the "just into­na­tion" inter­pre­ta­tion of Bharata's mod­el. Part of the expla­na­tion may be the strong har­mon­ic attrac­tion of drones (tan­pu­ra) played in the back­ground of raga performances.

Speaking of 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.

Once elec­tron­ic devices such as the Shruti Harmonium (1979) and the Melodic Movement Analyser (1981) became avail­able, the chal­lenge for raga into­na­tion research was to rec­on­cile two method­olo­gies: a top-down approach, test­ing hypo­thet­i­cal mod­els against data, and a data-driven bottom-up approach.

The "micro­scop­ic" obser­va­tion of melod­ic lines (now eas­i­ly ren­dered by soft­ware such as Praat) has con­firmed the impor­tance of note treat­ment (orna­men­ta­tion, alankara) and tem­po­ral dimen­sions of raga that are not tak­en into account by scale the­o­ries. For exam­ple, the ren­der­ing of the note 'Ga' in raga Darbari Kanada (Bel & Bor 1984; van der Meer 2019) and the typ­i­cal treat­ment of notes in oth­er ragas (e.g. Rao & Van der Meer 2009; 2010) have been dis­cussed at length. The visu­al tran­scrip­tion of a phrase from raga Asha illus­trates this:

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

The same melod­ic data was processed again after fil­ter­ing through 3 win­dows that attempt­ed to iso­late 'sta­ble' parts of the line. The first win­dow, typ­i­cal­ly 0.1 sec­onds, would elim­i­nate irreg­u­lar seg­ments, the sec­ond (0.4 sec­onds) would dis­card seg­ments out­side a rec­tan­gle of 80 cents height, and the third was used for aver­ag­ing. The result is a "skele­ton" of the tonal scale, dis­played as a selec­tive tona­gram.

These results would often not match the scale met­rics pre­dict­ed by the 'just into­na­tion' inter­pre­ta­tion of Bharata's mod­el. Continuing with 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 (com­put­ed using Kuiper's algo­rithm) were approx­i­mat­ed as dis­tances, from which a 3-dimensional clas­si­cal scal­ing was extracted:

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

This impres­sive clas­si­fi­ca­tion has been obtained by sta­tis­ti­cal analy­sis of sta­t­ic rep­re­sen­ta­tions of raga per­for­mance. This means that the same result would be obtained by play­ing the sound file in reverse, or even by slic­ing it into seg­ments reassem­bled in a ran­dom order… Music is a dynam­ic phe­nom­e­non that can­not be reduced to tonal "inter­vals". Therefore, sub­se­quent research into the rep­re­sen­ta­tion of the 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 the visu­als reflect the music as it is being heard, arguably the only appropriate"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 dreamed of when design­ing the Melodic Movement Analyser. However, to pro­mote it as a the­o­ret­i­cal mod­el is the con­tin­u­a­tion of a west­ern selec­tive bias. As far as I know, no Indian music mas­ter has ever attempt­ed to describe the intri­ca­cies of raga using hand-drawn mel­o­grams, although they could. The fas­ci­na­tion with tech­nol­o­gy — and west­ern 'sci­ence' in gen­er­al — is no indi­ca­tion of its rel­e­vance to ancient Indian concepts.

Music is judged by ears. Numbers, charts and graphs are mere­ly tools for inter­pret­ing and pre­dict­ing sound phe­nom­e­na. Therefore, a the­o­ry of music should be judged by its abil­i­ty to pro­duce musi­cal sounds via pre­dic­tive model(s). This approach is called 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 for the fol­low­ing investigation.

Microtonal framework

The "flex­i­ble" mod­el derived from the the­o­ret­i­cal mod­el of Natya Shastra (see The Two-vina exper­i­ment) rejects the claim of a pre­cise fre­quen­cy ratio for the har­mon­ic major third clas­si­fied in ancient lit­er­a­ture as anu­va­di (aso­nant). This amounts to admit­ting that the syn­ton­ic com­ma (pramāņa ṣru­ti in Sanskrit) could take any val­ue between 0 and 56.8 cents.

Let us look at some graph­i­cal rep­re­sen­ta­tions (from 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 called shru­ti-s in ancient texts. Below is the frame­work dis­played by the Bol Processor (micro­ton­al scale "gra­ma") with a 81/80 syn­ton­ic com­ma. The names of the posi­tions "r1_", "r2_", etc. fol­low the con­straints of low­er case ini­tials and the addi­tion of an under­score to dis­tin­guish octave num­bers. Positions "r1" and "r2" are two ways of locat­ing komal Re ("Db" or "re bemol"), while "r3" and "r4" denote shud­dha Re ("D" or "re"), etc.

These 22 shru­ti-s can be heard on the page Just into­na­tion, a gen­er­al frame­work, bear­ing in mind (see 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 changed? Below is the same frame­work with a com­ma of 0 cent. In this case, any "har­mon­ic posi­tion" — one whose frac­tion con­tains a mul­ti­ple of 5 — moves to its near­est Pythagorean neigh­bour (only mul­ti­ples of 3 and 2). The result is a "Pythagorean tun­ing". At the top of the cir­cle, the remain­ing gap is a Pythagorean com­ma. The posi­tions are slight­ly blurred because of the mis­match­es asso­ci­at­ed with a very small inter­val (the schis­ma).

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

In this rep­re­sen­ta­tion, "har­mon­ic major thirds" of 351 cents would most like­ly sound "out of tune" because the 5/4 ratio yields 384 cents. In fact, "g2" and "g3" are both dis­tant by a quar­ter tone between Pythagorean "g1" (32/27) and Pythagorean "g4" (81/64). Nevertheless, the inter­nal con­sis­ten­cy of this frame­work (count­ing per­fect fifths in blue) makes it suit­able for con­struct­ing musi­cal scales.

Between these lim­its of 0 and 56.8 cents, the graph­ic rep­re­sen­ta­tion of the scales and their inter­nal tonal struc­ture remain unchanged, bear­ing in mind that the size of the major-third inter­vals is deter­mined by the syn­ton­ic comma.

Construction of scale types

The mod­el tak­en from Bharata's Natya Shastra is not an obvi­ous ref­er­ence for pre­scrib­ing raga into­na­tion, as this musi­cal genre came into 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 in the 1970s and the ear­ly 1980s.

Bose was con­vinced (1960 p. 211) that the scale called Kaishika Madhyama was equiv­a­lent to a "just-intonation" seven-degree scale of west­ern musi­col­o­gy. In oth­er words, he took it for grant­ed that the 5/4 fre­quen­cy ratio (har­mon­ic major third) should be equiv­a­lent to the 7-shru­ti inter­val, but this state­ment had no influ­ence on the rest of his analysis.

Arnold (1982 p. 17) imme­di­ate­ly used inte­ger ratios to con­struct inter­vals with the fixed syn­ton­ic com­ma (81/80), but, as sug­gest­ed above, this does not affect his mod­el in terms of its struc­tur­al descrip­tion. He insist­ed on set­ting up a "geo­met­ric mod­el" rather than a spec­u­la­tive descrip­tion based on num­bers, as many authors (e.g. Alain Daniélou) had attempt­ed. The most inno­v­a­tive aspect of Arnold's study was the use of a cir­cu­lar slid­ing mod­el to illus­trate the match­ing of inter­vals in trans­po­si­tion process­es (murcchana-s) - see page The Two-vina exper­i­ment.

In real­i­ty, it would be more con­ve­nient to con­tin­ue to express all inter­vals in num­bers of shruti-s, in accor­dance with the ancient Indian the­o­ry, but a machine needs met­ric data to draw graph­ics of scales. For this rea­son, we show graphs with a syn­ton­ic com­ma of 81/80, keep­ing in mind the pos­si­bil­i­ty of chang­ing this val­ue at will.

The 22-shru­ti frame­work offers the pos­si­bil­i­ty of con­struct­ing 2¹¹ = 2048 chro­mat­ic scales, of which only 12 are "opti­mal­ly con­so­nant", i.e. con­tain only one wolf fifth (small­er by 1 syn­ton­ic com­ma = 22 cents).

The build­ing blocks of the tonal sys­tem accord­ing to tra­di­tion­al Indian musi­col­o­gy are two seven-degree scales called Ma-grama and Sa-grama. Bose (1960 p. 13) writes: 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, in west­ern ter­mi­nol­o­gy) dif­fer in the posi­tion of Pa ("G" or "sol") which may dif­fer by a syn­ton­ic com­ma (pramāņa ṣru­ti). In the Sa-grama, the inter­val Sa-Pa is a per­fect fifth (13 shru­ti-s) where­as in the Ma-grama it is a wolf fifth (12 shru­ti-s). Conversely, the 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. He then intro­duced two addi­tion­al notes: kakali Nishada (komal Ni or "Bflat") and antara Gandhara (shud­dh Ga or "E") to obtain a nine-degree scale from which "opti­mal­ly con­so­nant" chro­mat­ic scales could be derived from modal trans­po­si­tions (mur­ccha­na). The process of con­struct­ing these 12 chro­mat­ic scales, which we named "Ma01", "Ma02"… "Sa01", "Sa20", etc., is explained on the page Just into­na­tion, a gen­er­al frame­work.

The selec­tion of notes in each chro­mat­ic scale pro­duces 5 to 7 note melod­ic types. In the Natya Shastra these melod­ic types are called jāti. These can be seen as the ances­tors of ragas, although their lin­eages and struc­tures are only spec­u­la­tive (read on). The term thāṭ (pro­nounced 'taat') — trans­lat­ed as 'mode' or 'par­ent scale' — was lat­er adopt­ed, each thāṭ being called by the name of a raga (see Wikipedia). Details of the process, ter­mi­nol­o­gy and sur­veys of sub­se­quent musi­co­log­i­cal lit­er­a­ture can 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 "-to.12_scales" tonal­i­ty resource file. This scale can be visu­alised, using Arnold's slid­ing mod­el, by plac­ing the S note of the inner wheel on the S of the out­er crown :

This yields the fol­low­ing intervals:

"Optimal con­so­nance" is illus­trat­ed by two fea­tures: (1) there is only one wolf fifth (red line) in the scale — between D and G —, and (2) each note is con­nect­ed to anoth­er note by a per­fect fifth (blue line). This con­so­nance is of para­mount impor­tance to Indian musi­cians. Consonant inter­vals are casu­al­ly placed in melod­ic phras­es to enhance the "flavour" of their notes, and there should be no wolf fifth in the scale.

Note that the Ma-grama chro­mat­ic scale has all its notes in their low­er enhar­mon­ic positions.

The Ma-grama chro­mat­ic scale has been renamed "Sa_murcchana" here, because 'S' of the mov­ing wheel is oppo­site the 'S' of the fixed crown. The note names have been con­vert­ed (with a sin­gle click) to the Indian con­ven­tion. Note that the key num­bers have also been (auto­mat­i­cal­ly) fixed to match only the labelled notes. In this way, the upper "sa" is assigned key 72 instead of 83 in the "Ma01" scale shown on the Just into­na­tion, a gen­er­al frame­work page. The tonal con­tent of this "Sa_murchana" is shown in this table:

Selecting only "unal­tered" notes in "Sa_murcchana" — sa, re, gak, ma, pa, dha, nik — results in the "kaphi1" scale type named after the raga Kaphi (pro­nounced 'kafi'). This can be asso­ci­at­ed with a D-mode (Dorian) in west­ern musicology.

This scale type is saved under the name "kaphi1" because there will be anoth­er ver­sion of the Kaphi scale type.

In the "Sa_murcchana" the selec­tion of notes can be done in two dif­fer­ent ways:

• Select antara Gandhara (name­ly "ga") in place of the scale's Gandhara (name­ly "gak"), there­by rais­ing it by 2 shru­ti-s. This will result in a vikrit (mod­i­fied) scale type, name­ly "khamaj1", asso­ci­at­ed with raga Khamaj.
• Select both antara Gandhara and kakali Nishada (name­ly "ni" in place of "nik" raised by 2 shru­ti-s) which cre­ates the "bilaval1" scale type asso­ci­at­ed with raga Bilaval.

This "bilaval1" scale type is one of three ver­sions of the Bilaval cre­at­ed by the mur­ccha­na process. Although it cor­re­sponds to the scale of the white keys on a key­board instru­ment, it is not the usu­al "just into­na­tion" dia­ton­ic scale, because of a wolf fifth between "sa" and "pa".

An alter­na­tive Bilaval scale type called "bilaval3" (extract­ed from the "Ni1_murcchana", see below) cor­re­sponds to Giozeffo Zarlino's "nat­ur­al" scale — see Just into­na­tion: a gen­er­al frame­work. This is not to be con­fused with Zarlino's mean­tone tem­pera­ment dis­cussed on the Microtonality page.

A fourth option: rais­ing "nik" to "ni" and keep­ing "gak", would pro­duce a scale type in which "ni" has no con­so­nant rela­tion­ship with any oth­er note in the scale. This option is there­fore dis­card­ed from the model.

Each mur­ccha­na of the Ma-grama chro­mat­ic scale pro­duces at least three types of scale by select­ing unal­tered notes, antara Gandhara or both antara Gandhara and kakali Nishada.

For exam­ple, to cre­ate the "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 "-to.raga" tonal­i­ty resource file. Images are avail­able with a sin­gle click and scale struc­tures are com­pared on the main page.

The entire process is sum­ma­rized in the fol­low­ing table (Arnold 1982 p. 38):

Step	Ma-grama chro­mat­ic mur­ccha­na start­ing from	Shuddha gra­ma	Vikrit gra­ma (antara)	Vikrit gra­ma (antara + kakali)
1	Sa	kaphi1	khamaj1	bilaval1
2	Ma1	khamaj2	bilaval2	kalyan1
3	Ni1	bilaval3	kalyan2	marva1
4	Ga1	kalyan3	marva2	purvi1
5	Dha1	marva3	purvi2	todi1
6	Re1	purvi3	todi2
7	Ma3	todi3		lalit1 bhairao1
8	Ni3			lalit2 bhairao2 bhairavi1
9	Ga3			todi4 bhairavi2
10	Dha3		bhairavi3	asavari1
11	Re3	bhairavi4	asavari2	kaphi2
12	Pa3	asavari3	kaphi3	khamaj3

The use of this table deserves a graph­i­cal demon­stra­tion. For exam­ple, let us cre­ate a 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 place of "gak" in the Ma-grama scale, and "kakali Nishada" which is "ni" in place of "nik" in the Ma-grama scale. This process is clear in the mov­ing wheel model:

To make this selec­tion and export the "kalyan1" scale type, fill in the form on the "Ma1_murcchana" page as shown in the image.

Below is the result­ing scale type.

Remember that note posi­tions expressed as whole-number fre­quen­cy ratios are only a mat­ter of con­ve­nience for read­ers famil­iar 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' is increased 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 that do not belong to the orig­i­nal (7-degree) Ma-grama, tak­en from its "chro­mat­ic ver­sion": Dha1, Re1, Ma3, Ni3, Ga3. This exten­sion is nec­es­sary to cre­ate scale types for Todi, Lalit and Bhairao that include aug­ment­ed sec­onds.

In his 1982 paper (p. 39-41) Arnold linked his clas­si­fi­ca­tion of scale types to 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 the Ma-grama on one of its shud­dha swara-s (basic notes).

Every jāti is asso­ci­at­ed with a note of relax­ation (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 the names of the shud­dha jatis are in fact tied to their nyasa swaras, this also sug­gests that they should be tied to the mur­ccha­nas belong­ing to those nyasa swaras (Arnold 1982 p. 40).

In oth­er pub­li­ca­tions (notably Arnold & Bel 1985), Arnold used the cycle of 12 chro­mat­ic scales to sug­gest that the enhar­mon­ic posi­tions of the notes could express states of ten­sion or release linked to the chang­ing ambi­ence of the cir­ca­di­an cycle, there­by pro­vid­ing an expla­na­tion for the per­for­mance times assigned to tra­di­tion­al ragas. Low enhar­mon­ic posi­tions would be asso­ci­at­ed with dark­ness and high­er ones with day­light. Thus, ragas con­struct­ed using the Sa mur­ccha­na of the Ma-grama chro­mat­ic scale (all low posi­tions, step 1) could be inter­pret­ed as being near mid­night, while those that mix low and high posi­tions (step 7) would car­ry the ten­sions of sun­rise and sun­set. Their sequence is a cycle because in the table 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 called sadja-sadharana in musi­co­log­i­cal lit­er­a­ture (Shringy & Sharma 1978).

A list of 85 ragas with per­for­mance times pre­dict­ed by the mod­el can be found in Arnold & Bel (1985). This hypoth­e­sis is indeed inter­est­ing — and it does hold for many well-known ragas — but we have nev­er found the time to under­take a sur­vey of musi­cians' state­ments about per­for­mance times which might have assessed its validity.

Practice

Given scale types stored in the "-to.raga" tonal­i­ty resource file, the Bol Processor 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 every con­ceiv­able direction…

Choice of a raga

We will take up the chal­lenge of match­ing one of 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 19^th cen­tu­ry lit­er­a­ture on music.

Joep Bor (1999)

This choice is chal­leng­ing for a num­ber of rea­sons. Firstly, the four vari­ants of "todi" scales are derived from a (ques­tion­able) exten­sion of the grama-murcchana sys­tem. Then, the notes "ni" and "rek", "ma#" and "dhak" are close to the ton­ic "sa" and the dom­i­nant "pa" and could be "attract­ed" by the ton­ic and dom­i­nant, thus dis­rupt­ing the "geom­e­try" of the the­o­ret­i­cal scales in the pres­ence of a drone.

Finally, and most impor­tant­ly, the performer's style and per­son­al choic­es are expect­ed to be at odds 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 up the Todi chal­lenge with an analy­sis of eight occur­rences using the Melodic Movement Analyser (Bel 1988b). The analyser had pro­duced streams of accu­rate pitch mea­sure­ments which, after being fil­tered as selec­tive tona­grams, were sub­ject­ed to sta­tis­ti­cal analy­sis (Bel 1984; Bel & Bor 1984). The events includ­ed 6 per­for­mances of raga Todi and 2 exper­i­ments in tun­ing the Shruti Harmonium.

The MMA analy­sis revealed a rel­a­tive­ly high con­sis­ten­cy of note posi­tions, with stan­dard devi­a­tions bet­ter than 6 cents for all notes except "ma#", for which the devi­a­tion rose to 10 cents, still an excel­lent sta­bil­i­ty. Comparison of these results with the 'flex­i­ble' grama-murcchana mod­el showed less than 4 cent stan­dard devi­a­tion of inter­vals for 4 dif­fer­ent scales in which the syn­ton­ic com­ma (pramāņa ṣru­ti) would be set at 6, 18, 5 and 5 cents respec­tive­ly. In dis­cussing tun­ing schemes, Wim van der Meer even sug­gest­ed 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

Working with the Shruti Harmonium nat­u­ral­ly led us to meet Kishori Amonkar (1932-2017) in 1981. She was a lead­ing expo­nent of Hindustani music, hav­ing devel­oped a per­son­al style that claimed to tran­scend clas­si­cal schools (gha­ranas).

Most inter­est­ing­ly, she per­formed accom­pa­nied by a swara man­dal (see pic­ture), a zither that she tuned for each raga. Unfortunately we were not equipped to mea­sure these tun­ings with suf­fi­cient accu­ra­cy. So we used the Shruti Harmonium to pro­gramme the inter­vals accord­ing to her instructions.

This exper­i­ment did not go well for two rea­sons. A tech­ni­cal one: on that day, a fre­quen­cy divider (LSI cir­cuit) on the har­mo­ni­um was defec­tive; until it was replaced, some pro­grammed inter­vals were inac­ces­si­ble. A musi­cal one: the exper­i­ment showed that this pre­cise har­mo­ni­um was not suit­able for tun­ing exper­i­ments with Indian musi­cians. The fre­quen­cy ratios had to be entered on a small key­board, a use too far removed from the prac­tice of string tun­ing. This was a major incen­tive to design and build our "micro­scope for Indian music", the Melodic Movement Analyser (MMA) (Bel & Bor 1984).

In the fol­low­ing years (1981-1984) MMA exper­i­ments took up all our time and revealed the vari­abil­i­ty (but 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 the Bol Processor BP3… if only the expert musi­cians of that time were still alive!

Choice of a scale type

We need to decide between the four "todi" scale types pro­duced by the mur­ccha­na-s of the Ma-grama chro­mat­ic scale. For this we can use the mea­sure­ments of the Melodic Movement Analyser (Bel 1988b p. 15). Let us take aver­age mea­sure­ments and those of a per­for­mance by Kishori Amonkar. These are note posi­tions (in cents) against the ton­ic "sa".

Note	Average	Standard devi­a­tion	Kishori Amonkar
rek	95	4	96
gak	294	4	288
ma#	606	10	594
pa	702	1	702
dhak	792	3	792
(dhak)	806	3	810
ni	1107	6	1110

For the moment we will ignore "dhak" in the low­er octave as it will be dealt with sep­a­rate­ly. Let us com­pare Kishori Amonkar's results with the four scale types:

Note	Kishori Amonkar	todi1	todi2	todi3	todi4
rek	96	89	89	89	112
gak	288	294	294	294	294
ma#	594	590	590	610	610
pa	702	702	702	700	702
dhak	792	792	792	792	814
ni	1110	1088	1109	1109	1109

There are sev­er­al ways of find­ing the best match for musi­cal scales: either by com­par­ing scale inter­vals or by com­par­ing note posi­tions in rela­tion to the base note (ton­ic). Because of the impor­tance of the drone, we will use the sec­ond method. The choice is sim­ple here. Version "todi1" can be dis­card­ed because of "ni", the same with "todi3" and "todi4" because of "ma#". We are left with "todi2" which has a very good match, even with the mea­sure­ments of per­for­mances by oth­er musicians.

Adjustment of the scale

The largest devi­a­tions are on "rek" which was sung 7 cents high­er than the pre­dict­ed val­ue and "gak" which was sung 6 cents low­er. Even a 10-cent devi­a­tion 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 greater 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 due to its depen­dence on note treat­ment: if the note's approach is 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 scale type "todi2" cor­re­sponds to the per­for­mance. However, let us show how the mod­el can be mod­i­fied 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 the num­bers on its table), we can see that all the note posi­tions except "ma#" are Pythagorean. The series to which a note belongs is indi­cat­ed by the colour of its point­er: blue for Pythagorean and green for harmonic.

This means that chang­ing the size of the syn­ton­ic com­ma — in strict accor­dance with the grama-murcchana mod­el — will only adjust "ma#". To change the posi­tion of "ma#" from 590 to 594 cents (admit­ted­ly a ridicu­lous adjust­ment) we need to reduce 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 change con­firmed by the new image.

A table on the "todi-ka" page shows that the "rek-ma#" inter­val is still con­sid­ered a "per­fect" fifth, even though it is small­er by 6 cents.

It may not be obvi­ous whether the syn­ton­ic com­ma needs to be increased or decreased to fix the posi­tion of "ma#", but it is easy to try the oth­er way in case the direc­tion was wrong.

Other adjust­ments devi­ate from the "pure" mod­el. These result in chang­ing fre­quen­cy ratios in the table on the "todi-ka" page. An increase in "rek" from 89 to 96 cents requires an increase of 7 cents, cor­re­spond­ing to a ratio of 2(7/1200) = 1.00405. This changes the posi­tion of "rek" from 1.053 to 1.057.

In the same way, a reduc­tion in "gak" from 294 to 288 cents requires a reduc­tion of 6 cents, giv­ing a ratio of 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 for this is that when­ev­er a fre­quen­cy ratio is mod­i­fied by its floating-point val­ue, the machine checks whether the new val­ue is close to an inte­ger ratio of the same series. For exam­ple, chang­ing "rek" back to 1.053 would restore its ratio to 256/243. Accuracy bet­ter than 1‰ is required for this adjustment.

A tun­ing scheme for this scale type is sug­gest­ed by the machine. The graph­i­cal 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 just show­ing what to do when necessary.

The remain­ing adjust­ment will be that of the "dhak" in the low­er octave. To do this, we will dupli­cate the pre­vi­ous scale, renam­ing it "todi_ka_4" to indi­cate that it is designed for the 4^th octave. In the new scale, called "todi_ka_3", we raise "dhak3" by 810 -792 = 18 cents.

This rais­es its posi­tion from 1.58 to 1.597. Note that this puts it exact­ly in 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 to match the 17.5 cents com­ma. This is cer­tain­ly sig­nif­i­cant in the melod­ic con­text of this raga, which is one rea­son why all the musi­cians made the same size adjust­ment in their tun­ing experiments.

This case is a sim­ple illus­tra­tion of raga into­na­tion as a trade-off between har­monic­i­ty with respect to the drone and the need for con­so­nant melod­ic inter­vals. It also shows that the Shruti Harmonium could not fol­low the prac­tice of the musi­cians because its scale ratios were repli­cat­ed in all octaves.

Choice of a recording

We don't have the record­ing on which the MMA analy­sis was made. One prob­lem with old tape record­ings is the unre­li­a­bil­i­ty of the speed of tape trans­port. Also, on a long record­ing, the fre­quen­cy of the ton­ic can change slight­ly due to vari­a­tions in room tem­per­a­ture, which affects the instru­ments — includ­ing the dila­tion of the tape…

In order to try to com­pare scales a with real per­for­mances, and to exam­ine extreme­ly small "devi­a­tions" (which have lit­tle musi­cal sig­nif­i­cance, in any case), it is there­fore safer to work with dig­i­tal record­ings. This was the case with Kishori Amonkar's Todi, record­ed in London in ear­ly 2000 for the Passage to India col­lec­tion, and avail­able free of copy­right (link on Youtube). The fol­low­ing is based on this recording.

Setting up the diapason

Let us cre­ate the fol­low­ing "-gr.tryRagas" gram­mar:

-se.tryRagas
-to.raga

S --> _scale(todi_ka_4,0) sa4

In "-se.tryRagas" the note con­ven­tion should be set to "Indian" so that "sa4" etc. is accept­ed even when no scale is specified.

The gram­mar calls "-to.raga", which con­tains the def­i­n­i­tions of all the scale types cre­at­ed by the pro­ce­dure described above. Unsurprisingly, it does not play the note "sa" at the fre­quen­cy of the record­ing. We there­fore need to mea­sure the ton­ic in order to adjust the fre­quen­cy of "A4" (dia­pa­son) in "-se.tryRagas" accord­ing­ly. There are sev­er­al ways to do this with increas­ing accuracy.

A semi­tone approx­i­ma­tion can be obtained 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 would there­fore have to adjust the fre­quen­cy of the dia­pa­son (in "-se.tryRagas") to:

440 x r x 2^(9/12) = 440 x 2^((9-3)/12) = 311 Hz

A much bet­ter approx­i­ma­tion is obtained by extract­ing a short occur­rence of "sa4" at the very begin­ning of the performance:

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

This sam­ple con­tains 9 cycles for a dura­tion of 38.5 ms. The fun­da­men­tal fre­quen­cy is there­fore 9 x 1000 / 38.5 = 233.7 Hz. Consequently, adjust the dia­pa­son in "-se.tryRagas" to 233.7 x 2^(9/12) = 393 Hz.

The last step is a fine tun­ing by com­par­ing the pro­duc­tion of the notes in the gram­mar by ear with the record­ing of "sa4" played in a loop. To do this, 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 spec­i­fied in the instru­ment "Vina" called by "-.raga" and the Csound orches­tra file "new-vina.orc".

Listening to the sequence may not reveal any pitch dif­fer­ences, but these will be appar­ent to a trained ear when super­im­posed on the recording:

One of the four occur­rences sounds best in tune. Let us assume 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 x 2^(10/1200) = 395.27 Hz.

In fact the best fre­quen­cy is 393.22 Hz, which means that the sec­ond eval­u­a­tion (which gave 393 Hz) was fair — and the singers' voic­es very reli­able! Now we can check 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 actu­al­ly be sum­marised by the third: 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 too demand­ing in terms of accu­ra­cy for the analy­sis of a vocal per­for­mance, but it will be notice­able when work­ing with a long-stringed instru­ment such as the rudra veena. We will demon­strate this 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 in the recording.

👉  The fol­low­ing demo uses the BP3's inter­face to Csound, which until recent­ly was the only way to cre­ate micro­ton­al inter­vals. The same can now be done using MIDI micro­tonal­i­ty, both in real time and with MIDI files. It is pos­si­ble to cap­ture MIDI mes­sages from a key­board and send them to a MIDI device with cor­rec­tions made by a micro­ton­al scale. In this way, there is no need for the cre­ation of gram­mars for these tests.

First we try a sequence with the empha­sis on "rek". The fol­low­ing sequence of notes 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 (posi­tion 0'50") then repeat­ed in super­posed form with the sequence pro­duced by the grammar:

In this exam­ple, the scale "todi_ka_3" has been used because of the occur­rence of short instances of "dhak3". The posi­tion of "rek" is iden­ti­cal in the 3^d and 4^th octaves. The blend­ing of the voice with the plucked instru­ment is notable in the last held note.

In the next sequence (loca­tion 1'36") the posi­tion of "gak4" is esti­mat­ed. The gram­mar is as follows:

S --> KishoriAmonkar2
KishoriAmonkar2 --> Scale {137/100, sa4 rek4 gak4 rek4} {31/10, rek4} {18/10, gak4} {75/100,rek4} {44/10, sa4}
Scale --> _scale(todi_ka_4,0)

This time, the scale "todi_ka_4" was cho­sen, even though it had no effect on the into­na­tion since "dhak" is missing.

A word about build­ing the gram­mar: we looked at the sig­nal of the record­ed phrase and mea­sured the (approx­i­mate) dura­tion of the notes: 1.37s, 3.1s, 1.8s, 7.5s, 4.4s. We then con­vert­ed these dura­tions into inte­ger ratios — frac­tions of the basic tem­po whose peri­od is exact­ly 1 sec­ond, as spec­i­fied in "-se.tryRagas": 137/100, 31/10 etc.

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

No we try a phrase with a long pause on "dhak3" (loca­tion 3'34"), which proves that the scale "todi_ka_3" per­fect­ly match­es this occur­rence of "dhak":

S --> KishoriAmonkar3
KishoriAmonkar3 --> scale(todi_ka_3,0) 11/10 {19/20, ma#3 pa3} {66/10,dhak3} {24/10, ni3 dhak3 pa3 }{27/10,dhak3} 12/10 {48/100,dhak3}{17/10,ni3}{49/10,dhak3}

Early occur­rence of "ma#4" (loca­tion 11'38"):

S --> KishoriAmonkar4
KishoriAmonkar4 --> _scale(todi_ka_4,0) 4/10 {17/10, ni3}{26/100,sa4}{75/100,rek4}{22/100,gak4}{17/10,ma#4}{16/100,gak4}{34/100,rek4}{56/100,sa4}{12/100,rek4}{84/100,gak4}{27/100,rek4}{12/10,sa4}

Reaching "dhak4" (loca­tion 19'46"):

S --> KishoriAmonkar5
KishoriAmonkar5 --> _scale(todi_ka_4,0) 13/10 {16/10,ma#4}{13/10,gak4}{41/100,ma#4}{72/100,ma#4 dhak4 ma#4 gak4 ma#4}{18/10,dhak4}{63/100,sa4}{90/100,rek4}{30/100,gak4}{60/100,rek4}{25/100,sa4}{3/2,rek4}

With a light touch of "pa4" (loca­tion 23'11"):

S --> KishoriAmonkar6
KishoriAmonkar6 --> _scale(todi_ka_4,0) 28/100 {29/100,ma#4}{40/100,dhak4}{63/100,ni4 sa5 ni4}{122/100,dhak4}{64/100,pa4}{83/100,ma#4}{44/100,pa4}{79/100,dhak4}

Pitch accu­ra­cy is no sur­prise in Kishori Amonkar's per­for­mances. With a keen aware­ness of "shru­ti-s", she would sit on the stage and pluck her swara man­dal, care­ful­ly tuned for each raga.

A test with the rudra veena

Asad Ali Khan (1937-2011) was one of the last per­form­ers of the rudra veena at the end of the 20^th 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 on this instru­ment is such that we have been able to iden­ti­fy tiny vari­a­tions that are con­trolled and sig­nif­i­cant in the con­text of the raga. Read for exam­ple Playing with Intonation (Arnold 1985). To mea­sure vibra­tions below the audi­ble range, we occa­sion­al­ly placed a mag­net­ic pick­up near the last string.

Below are the sta­tis­tics of the Melodic Movement Analyser mea­sure­ments of the raga Miyan ki Todi inter­pret­ed by Asad Ali Khan in 1981. The sec­ond col­umn con­tains mea­sure­ments of his tun­ing of the Shruti Harmonium dur­ing an exper­i­ment. The columns on the right show the pre­dict­ed note posi­tions accord­ing to the grama-murchana mod­el with a syn­ton­ic com­ma of ratio 81/80. As shown in Kishori Amonkar's per­for­mance above, "dhak" can take dif­fer­ent val­ues depend­ing on the octave.

Note	Asad Ali Khan per­form­ing	Asad Ali Khan tun­ing	todi1	todi2	todi3	todi4
rek	99	100	89	89	89	112
gak	290	294	294	294	294	294
ma#	593	606	590	590	610	610
pa	702	702	702	702	700	702
dhak3	795	794	792	792	792	814
dhak2	802
ni	1105	1108	1088	1109	1109	1109

Again, the best match would be the "todi2" scale with a syn­ton­ic com­ma of 17.5 cents. We cre­at­ed two scales, "todi_aak_2" and "todi_aak_3" for the 2^nd and 3^rd octaves.

The scale con­struct­ed dur­ing the Shruti Harmonium exper­i­ment is less rel­e­vant because of the influ­ence of the exper­i­menter play­ing the scale inter­vals with a low-attracting drone (pro­duced by the machine). In his attempt to resolve the 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!

The scale "todi_aak_2" (in the low octave) con­tains inter­est­ing inter­vals (har­mon­ic major thirds) which allows us to antic­i­pate effec­tive melod­ic move­ments. The tun­ing scheme sum­maris­es these relationships.

We now take frag­ments of Asad Ali Khan's per­for­mance of Todi (2005) avail­able on Youtube (fol­low this link).

The per­for­mance begins in the low­er octave, so with the scale "todi_aak_2". The fre­quen­cy of Sa was mea­sured at 564.5 Hz using the method described above.

Let us start with a sim­ple melod­ic phrase repeat­ed twice, the sec­ond time super­im­posed on the note sequence pro­duced by the grammar.

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 char­ac­ter '&', which is used to con­cate­nate sound objects (or notes) across the bound­aries of poly­met­ric expres­sions (between curly brack­ets). This makes it pos­si­ble to play the final "rek3" and "sa3" as con­tin­u­ous notes. This con­ti­nu­ity can be seen in the graph below:

It is time to make sure that fine tun­ing and adjust­ing scales are more than just 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 over­lay twice, once with "todi_aak_3" and the sec­ond time with "todi_aak_2":

To check the dif­fer­ence between these two ver­sions of "dhak2", we can play them one after the oth­er and then super­im­pose them:

S --> _tempo(1/2) _scale(todi_aak_3,0) dhak2 _scale(todi_aak_2,0) dhak2 {_scale(todi_aak_3,0) dhak2, _scale(todi_aak_2,0) dhak2}

With fun­da­men­tal fre­quen­cies of 132.837 Hz and 133.341 Hz, the beat fre­quen­cy (of the sine waves) would be 133.341 - 132.837 = 0.5 Hz. The per­ceived beat fre­quen­cy is high­er because of the inter­fer­ence between the 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 out­side the range of the Csound instru­ment "Vina", it is played here as "rek2" with a pitch­bend cor­rec­tion of one semitone.

The ren­der­ing of phras­es in the low­er octave is very approx­i­mate because of the pre­dom­i­nance of meend (pulling the string). Some effects can be bet­ter imi­tat­ed using per­for­mance con­trols — see Sarasvati Vina, for exam­ple — but this requires a mas­tery of the real instru­ment in order to design pat­terns of musi­cal "ges­tures" rather than sequences of sound events… Imitating the melod­ic intri­ca­cy of a raga is not the sub­ject of this page; we are mere­ly check­ing the rel­e­vance of scale mod­els to the "tonal skele­ton" of ragas.

Accidental notes

Raga scales extract­ed from mur­chana-s of the Ma-grama chro­mat­ic scale (see above) con­tain only notes that are pre­sumed to belong to the raga. They can­not accom­mo­date acci­den­tal notes or the scales used in the com­mon prac­tice of mix­ing ragas.

Let us take, for exam­ple, a frag­ment of the pre­vi­ous exam­ple, which was poor­ly rep­re­sent­ed by the sequence of notes pro­duced by the gram­mar. (We learn from our mis­takes!) We might be tempt­ed to replace the expres­sion {38/10, pa2 gak2 pa2 dhak2 _ pa2 _} with {38/10, pa2 ga2 pa2 dhak2 _ pa2 _}, using "ga2" which does not belong to the scale "todi_aak_2". Unfortunately, this results in 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 figure.

To solve this prob­lem we can recall that the scale "todi2" was extract­ed from "Re1_murcchana". The lat­ter con­tains all the notes of a chro­mat­ic scale in addi­tion to those extract­ed. 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 a key num­ber to each note 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 be aware of key map­pings when using cus­tom names for "notes".

Another prob­lem is that in "todi_aak_2" note "dhak" has been raised from 792 to 810 cents, which is not its val­ue in "Re1_murcchana". This can be fixed by cre­at­ing anoth­er vari­ant of the scale with this cor­rec­tion, or sim­ply using the pitch­bend to mod­i­fy "dhak2" — in which case the same pitch­bend could have been used to raise "gak2" in the first place.

Finally, the best way 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 notes will nev­er be used.

To conclude…

This whole dis­cus­sion has been 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 it is need­ed at all — 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}

Listen to Asad Ali Khan's actu­al per­for­mance of raga Todi to appre­ci­ate its expres­sive power!

For a more con­vinc­ing demo, instead of Csound, I rec­om­mend using MIDI micro­tonal­i­ty in real time to cap­ture notes played on a key­board and send them to a MIDI device with cor­rec­tions made by the micro­ton­al scale.

Attempting to fol­low the intri­ca­cies of alankara (note treat­ment) with a sim­plis­tic nota­tion of melod­ic phras­es demon­strates the dis­con­nec­tion between 'model-based' exper­i­men­tal musi­col­o­gy and the real­i­ty of musi­cal prac­tice. This explains why we have relied on descrip­tive mod­els (e.g. auto­mat­ic nota­tion) cap­tured by the Melodic Movement Analyser or com­put­er tools such as Praat, rather than attempt­ing to recon­struct melod­ic phras­es from the­o­ret­i­cal mod­els. Experiments with scales deal with the "skele­tal" nature of into­na­tion, which is a nec­es­sary but not suf­fi­cient para­me­ter for describ­ing melod­ic types.

All exam­ples shown on this page are avail­able in the bp3-ctests-main.zip sam­ple set shared on GitHub. Follow the instruc­tions on Bol Processor ‘BP3’ and its PHP inter­face to install BP3 and learn its basic operation.

Bernard Bel — Dec. 2020

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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, 71^e (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 (13^th 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 began in 1980 with the found­ing of the International Society for Traditional Arts Research (ISTAR) in New Delhi, India. We had pro­duced joint arti­cles and pro­pos­als which enabled us (Arnold and Bel) to receive a grant from the International Fund for the Promotion of Culture (UNESCO). A book­let of ISTAR projects was then print­ed in Delhi, and a larg­er team received sup­port from the Sangeet Research Academy (SRA, Calcutta/Kolkata), the Ford Foundation (USA) and lat­er the National Centre for the Performing Arts (NCPA, Bombay/Mumbai).

The fol­low­ing are extracts from let­ters of sup­port received dur­ing this ini­tial peri­od — after the con­struc­tion of the Shruti Harmonium and dur­ing the design of the Melodic Movement Analyser. (ISTAR book­let, 1981 pages 20-22)

In fact, the full poten­tial of this approach can only be realised now, tak­ing advan­tage of the (vir­tu­al­ly unlim­it­ed) dig­i­tal devices that are replac­ing the hard­ware we cre­at­ed for this pur­pose 40 years ago!

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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…
— Iégor 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.
— Dr. 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.

Just into­na­tion (into­na­tion pure in French) is a word used by com­posers, musi­cians and musi­col­o­gists to describe var­i­ous aspects of com­po­si­tion, per­for­mance and instru­ment tun­ing. They all point to the same goal of "playing/singing in tune" — what­ev­er that means. Implementing a gener­ic abstract mod­el of just into­na­tion in the Bol Processor is a chal­lenge beyond our cur­rent com­pe­tence… We approach it prag­mat­i­cal­ly by look­ing at some musi­cal tra­di­tions that pur­sue the same goal with the help of reli­able the­o­ret­i­cal models.

A com­plete and con­sis­tent frame­work for the con­struc­tion of just-intonation scales - or "tun­ing sys­tems" - was the grama-murcchana mod­el elab­o­rat­ed in ancient India. This the­o­ry has been exten­sive­ly com­ment­ed on and (mis)interpreted by Indian and Western schol­ars: for a detailed review see Rao & van der Meer 2010. We will show that an arguably accept­able inter­pre­ta­tion yields a frame­work of chro­mat­ic scales that can be extend­ed to Western clas­si­cal har­mo­ny and eas­i­ly han­dled by the Bol Processor — either pro­duc­ing Csound scores or real-time MIDI micro­tonal­i­ty.

This page is a con­tin­u­a­tion of Microtonality but can be read independently.

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.

Historical background

Methods of tun­ing musi­cal instru­ments have been doc­u­ment­ed in var­i­ous parts of the world for over 2000 years. For prac­ti­cal and per­son­al rea­sons we will con­cen­trate on work in Europe and the Indian subcontinent.

Systems described as "just into­na­tion" are attempts to cre­ate a tun­ing in which all tonal inter­vals are con­so­nant. There is a large body of the­o­ret­i­cal work on just into­na­tion - see Wikipedia for links and abstracts.

Models are amenable to Hermann von Helmholtz's notion of con­so­nance which deals with the per­cep­tion of the pure sinu­soidal com­po­nents of com­plex sounds con­tain­ing mul­ti­ple tones. According to the the­o­ry of con­so­nance, the fre­quen­cies of these upper par­tials are inte­ger mul­ti­ples of the fun­da­men­tal fre­quen­cy of the vibra­tion. In mechan­i­cal musi­cal instru­ments, this is close to real­i­ty when long strings are gen­tly struck or plucked. However, this har­mo­ny is lack­ing in many wind instru­ments, espe­cial­ly reed instru­ments such as the sax­o­phone or the Indian shehnai, and even in per­cus­sion instru­ments or bells which com­bine sev­er­al modes of vibration.

Therefore, if just into­na­tion is invoked to tune a musi­cal instru­ment, it must be anal­o­gous to a zither, a swara man­dal, a harp­si­chord, a piano or a pipe organ, includ­ing elec­tron­ic devices that pro­duce sim­i­lar sounds.

Perhaps because of their late "dis­cov­ery" of cal­cu­lus — actu­al­ly "bor­rowed" from Indian, Persian and Arabic sources — Europeans cul­ti­vat­ed a fas­ci­na­tion with num­bers strong­ly advo­cat­ed by priests as an image of "God's per­fec­tion". We may recall Descarte's claim that the length of a curve is "beyond human under­stand­ing" — because π can­not be writ­ten as an inte­ger ratio…

In real life, musi­cians devel­oped pro­ce­dures for tun­ing their instru­ments by lis­ten­ing to inter­vals and pick­ing out the ones that made sense to their ears — see The two-vina exper­i­ment page. After the devel­op­ment of musi­cal acoustics, attempts were made to inter­pret these pro­ce­dures in terms of fre­quen­cy ratios. This was a risky ven­ture, how­ev­er, because the dream of per­fec­tion led to the sim­plis­tic pro­mo­tion of "per­fect ratios".

Seeking the kind of per­fec­tion embod­ied in num­bers is the best way to pro­duce bland music. Although just into­na­tion — inter­vals with­out beats — is now pos­si­ble on elec­tron­ic instru­ments, it is based on a nar­row con­cept of tonal­i­ty. This can be ver­i­fied by lis­ten­ing to ancient Western music played in dif­fer­ent tem­pera­ments — see page Comparing tem­pera­ments — and even to Indian clas­si­cal music — see page Raga into­na­tion.

The “Greek” approach

Models of vibrat­ing strings attrib­uted to the "ancient Greeks" sug­gest that fre­quen­cy ratios of 2/1 (the octave), 3/2 (the major fifth) and 5/4 (the major third) pro­duce con­so­nant inter­vals, while oth­er ratios pro­duce a cer­tain degree of dis­so­nance.

The prac­tice of poly­phon­ic music on fixed-tuned instru­ments has shown that this per­fect con­so­nance is nev­er achieved with 12 notes in an octave — the con­ven­tion­al chro­mat­ic scale. In Western clas­si­cal har­mo­ny, it would require retun­ing the instru­ment accord­ing to the musi­cal genre, the piece of music and the har­mon­ic con­text of each melod­ic phrase or chord.

Imperfect tonal inter­vals pro­duce unwant­ed beats because their fre­quen­cy ratio can­not be reduced to sim­ple 2, 3, 4, 5 frac­tions. A sim­ple thought exper­i­ment, myth­i­cal­ly attrib­uted to Pythagoras of Samos, shows that this is inher­ent in arith­metic and not a defect in instru­ment design. Imagine the tun­ing of ascend­ing fifths (ratio 3/2) by suc­ces­sive steps on a harp with an octave shift to keep the result­ing note with­in the orig­i­nal octave. The fre­quen­cy ratios would be 3/2, 9/4, 27/16, 81/64 and so on. At this stage, the note appears to be a major third although its actu­al ratio (81/64 = 1.265) is high­er than 5/4 (1.25). The 81/64 inter­val is called the Pythagorean major third, which may sound "out of tune" in a con­ven­tion­al har­mon­ic con­text. The fre­quen­cy ratio (81/80 = 1.0125) between the Pythagorean and har­mon­ic major thirds is called the syn­ton­ic com­ma.

Whoever devised the so-called "Pythagorean tun­ing" went fur­ther in their inten­tion to describe all musi­cal notes by cycles of fifths. Going fur­ther up, 243/128, 729/512… etc. effec­tive­ly pro­duces a full chro­mat­ic scale: C - G - D - A - E - B - F♯ - C♯ - G♯… etc. But in addi­tion to the harsh sound of some of the result­ing inter­vals, things get bad if one hopes to end the cycle on the ini­tial note. If the series start­ed on 'C', it will end on 'C' (or 'B#'), but with a ratio of 531441/524288 = 1.01364, slight­ly high­er than 1. This gap is called the Pythagorean com­ma, which is con­cep­tu­al­ly dif­fer­ent from the syn­ton­ic com­ma (1.0125), although their sizes are almost iden­ti­cal. This para­dox is a mat­ter of sim­ple arith­metic: pow­ers of 2 (octave inter­vals) nev­er equal pow­ers of 3.

The attri­bu­tion of this sys­tem to the "ancient Greeks" is, of course, pure fan­ta­sy, since they (unlike the Egyptians) didn't have any use for fractions!

Despite the com­ma prob­lem, tun­ing instru­ments by series of per­fect fifths was com­mon prac­tice in medieval Europe, fol­low­ing the organum which con­sist­ed of singing/playing par­al­lel fifths or fourths to enhance a melody. One of the old­est trea­tis­es on "Pythagorean tun­ing" was pub­lished around 1450 by Henri Arnault de Zwolle (Asselin 2000 p. 139). In this tun­ing, major "Pythagorean" thirds sound­ed harsh, which explains why the major third was con­sid­ered a dis­so­nant inter­val at the time.

Because of these lim­i­ta­tions, Western fixed-pitch instru­ments using chro­mat­ic (12-tone) scales nev­er achieve the pitch accu­ra­cy dic­tat­ed by just into­na­tion. For this rea­son just into­na­tion is described in the lit­er­a­ture as "incom­plete" (Asselin 2000 p. 66). Multiple divi­sions (more than 12 per octave) are required to pro­duce all "pure" ratios. This has been unsuc­cess­ful­ly attempt­ed on key­board instru­ments, although it remains pos­si­ble on a computer.

The Indian approach

The grama-murcchana mod­el was described in the Natya Shastra, a Sanskrit trea­tise on the per­form­ing arts writ­ten in India some twen­ty cen­turies ago. Chapter 28 con­tains a dis­cus­sion of the "har­mon­ic scale", which is based on a divi­sion of the octave into 22 shru­ti-s, while only sev­en swara-s (notes) are used by musi­cians: "Sa", "Re", "Ga", "Ma", "Pa", "Dha", "Ni". These can be mapped onto con­ven­tion­al Western music nota­tion "C", "D", "E", "F", "G", "A", "B" in English, or "do", "re", "mi", "fa", "sol", "la", "si" in Italian/Spanish/French.

This 7-swara scale can be extend­ed to a 12-degree (chro­mat­ic) scale by means of diesis and flat alter­ations, which raise or low­er a note by a semi­tone. Altered notes in the Indian sys­tem are com­mon­ly called "komal Re", "komal Ga", "Ma tivra", "komal Dha" and "komal Ni". The word "komal" can be trans­lat­ed as "flat" and "tivra" as "diesis".

The focus of 20^th cen­tu­ry research in Indian musi­col­o­gy has been to 'quan­ti­fy' shruti-s in a sys­tem­at­ic way and to assess the rel­e­vance of this quan­tifi­ca­tion to the per­for­mance of clas­si­cal raga.

A strik­ing point in the ancient Indian the­o­ry of musi­cal scales is that it does not rely on numer­i­cal ratios, be they fre­quen­cies or lengths of vibrat­ing strings. This point was over­looked by 'colo­nial musi­col­o­gists' because of their lack of insight into Indian math­e­mat­ics and their fas­ci­na­tion with a mys­ti­cism of num­bers inher­it­ed from Neopythagoreanism.

As report­ed by Jonathan Barlow (per­son­al com­mu­ni­ca­tion, 3/9/2013, links my own):

The ustads in India from way back con­sid­ered that they fol­lowed Pythagoras, but ear­ly on they made the dis­cov­ery that try­ing to tune by num­bers was a los­ing game, and Ibn Sina (Avicenna) (980-1037 AD), who was their great philoso­pher of aes­thet­ics, said in plain terms that it was wis­er to rely on the ears of the experts. Ahobala tried to do the num­bers thing (and Kamilkhani) but they are rel­e­gat­ed to a foot­note of 17^th C musicology.

Bharata Muni, the author(s) of the Natya Shastra, may have heard of "Pythagorean tun­ing", a the­o­ry that Indian sci­en­tists could have expand­ed con­sid­er­ably, giv­en their exper­tise in the use of cal­cu­lus.. Despite this, not a sin­gle num­ber is quot­ed in the entire chap­ter on musi­cal scales. This para­dox is dis­cussed on my page The Two-vina exper­i­ment. In A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra, I showed a min­i­mal rea­son: Bharata's descrip­tion leads to an infi­nite set of solu­tions, which should be for­malised with some alge­bra, rather than a set of inte­ger ratios.

How (and why) should the octave be divid­ed into 22 micro-intervals when most Indo-European musi­cal sys­tems only name 5 to 12 notes? Some naive eth­no­mu­si­col­o­gists have claimed that Bharata's mod­el must be a vari­ant of the Arabic "quar­ter­tone sys­tem", or even a tem­pered scale with 22 inter­vals… If so, why not 24 shruti-s? Or any arbi­trary num­ber? The two-vina exper­i­ment pro­duces shruti-s of unequal sizes. No sum of micro­tones of 54.5 cents in a 22-degree tem­pered scale would pro­duce an inter­val close to 702 cents — the per­fect fifth that gives con­so­nance (sam­va­di) to musi­cal scales.

In the (thought?) exper­i­ment described in Natya Shastra (chap­ter 28), two vina-s — stringed instru­ments sim­i­lar to zithers — are tuned iden­ti­cal­ly. The author sug­gests low­er­ing all the notes of one instru­ment by "one shru­ti" and he gives a list of notes that will match between the two instru­ments. The process is repeat­ed three more times until all the match­es have been made explic­it. This gives a sys­tem of equa­tions (and inequa­tions) for the 22 unknown vari­ables. Additional equa­tions can be derived from a pre­lim­i­nary state­ment that the octave and the major fifth are "con­so­nant" (sam­va­di), thus fix­ing ratios close to 2/1 and 3/2. (Read the detailed pro­ce­dure on my page The Two-vina Experiment and the math­e­mat­ics in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra.)

However, a new equa­tion is need­ed, which Bharata's mod­el does not pro­vide. Interestingly, in Natya Shastra the major third is clas­si­fied as "asso­nant" (anu­va­di). Setting its fre­quen­cy ratio to 5/4 is there­fore a reduc­tion of this mod­el. In fact, it is a dis­cov­ery of European musi­cians in the ear­ly 16^th cen­tu­ry — when fixed-pitch key­board instru­ments had become pop­u­lar (Asselin 2000 p. 139) — that many musi­col­o­gists take for grant­ed in their inter­pre­ta­tion of the Indian mod­el. […] thirds were con­sid­ered inter­est­ing and dynam­ic con­so­nances along with their inverse, sixths, but in medieval times they were con­sid­ered dis­so­nances unus­able in a sta­ble final sonor­i­ty (Wikipedia).

The reduc­tion of Bharata's mod­el does not fit with the flex­i­bil­i­ty of into­na­tion schemes in Indian music — see page Raga into­na­tion. Experimental work on musi­cal prac­tice is not "in tune" with this inter­pre­ta­tion of the the­o­ry. The shru­ti sys­tem should be inter­pret­ed as a "flex­i­ble" frame­work in which the vari­able para­me­ter is the syn­ton­ic com­ma, name­ly the dif­fer­ence between a Pythagorean major third and a har­mon­ic major third. Adherence to the two-vina exper­i­ment only implies that the com­ma takes its val­ue between 0 and 56.8 cents (Bel 1988a).

➡ Measurements in "cents" refer to a log­a­rith­mic scale. Given a fre­quen­cy ratio 'r', its cent val­ue is 1200 x log(r) / log(2). The octave (ratio 2/1) is 1200 cents, and each semi­tone is about 100 cents.

The con­struc­tion and eval­u­a­tion of raga scale types based on this flex­i­ble mod­el is explained on my page Raga into­na­tion.

Extending the Indian model to Western harmony

One incen­tive for apply­ing the Indian frame­work to Western clas­si­cal music is that both tra­di­tions have giv­en pri­or­i­ty to the con­so­nance of per­fect fifths asso­ci­at­ed with a 3/2 fre­quen­cy ratio. In addi­tion, let us agree to fix the har­mon­ic major third to the inter­val 5/4 (384 cents), result­ing in a syn­ton­ic com­ma of 81/80 (close to 21.5 cents). The sys­tem of equa­tions derived from the two-vina exper­i­ment is com­plete, and it yields two addi­tion­al sizes of shruti-s: the Pythagorean lim­ma (256/243 = about 90 cents) and the minor semi­tone (25/24 = about 70 cents).

These inter­vals were known to Western musi­col­o­gists who were try­ing to find just into­na­tion scales that could be played on key­board instru­ments (12 degrees per octave). Gioseffo Zarlino (1517-1590) is a well-known con­trib­u­tor to this the­o­ret­i­cal work. His "nat­ur­al scale" was an arrange­ment of the three nat­ur­al inter­vals yield­ing the fol­low­ing chro­mat­ic scale — named "just intonation" in "-to.tryScales":

➡ This should not be con­fused with Zarlino's mean­tone tem­pera­ment (image), read pages Microtonality and Comparing tem­pera­ments.

In 1974, E. James Arnold, inspired by the French musi­col­o­gist Jacques Dudon, designed a cir­cu­lar mod­el to illus­trate the trans­po­si­tion of scales (mur­ccha­na) in Bharata's mod­el. Below is the sequence of inter­vals (L, C, M…) over an octave as derived from the two-vina exper­i­ment.

Positions R1, R2 etc. are labelled with abbre­vi­a­tions of names Sa, Re, Ga, Ma, Pa, Dha, Ni. For exam­ple, Ga ("E" in English) can have four posi­tions, G1 and G2 being enhar­mon­ic vari­ants of komal Ga ("E flat" = "mi bémol"), while G3 and G4 are the har­mon­ic and Pythagorean posi­tions of shud­dha Ga ("E"= "mi") respectively.

The notes of the chro­mat­ic scale have been labelled using the Italian/Spanish/French con­ven­tion "do", "re", "mi", "fa"… rather than the English con­ven­tion to avoid con­fu­sion: "D" is asso­ci­at­ed with Dha ("A" in English, "la" in Italian/French) and not with the English "D" ("re" in Italian/French).

Frequency ratios are illus­trat­ed by pic­tograms show­ing how each posi­tion can be derived from the base note (Sa). For exam­ple, the pic­togram near N2 ("B flat" = "si bémol") shows 2 ascend­ing per­fect fifths and 1 descend­ing major third.

Cycles of per­fect fifths have been marked with red and green seg­ments. The red series is gen­er­al­ly called "Pythagorean" — con­tain­ing G4 (81/84) — and the green one "har­mon­ic" — con­tain­ing G3 (5/4). The blue arrow shows a har­mon­ic major third going from S ("C" = "do") to G3. Both cycles are iden­ti­cal, with the har­mon­ic and Pythagorean posi­tions dif­fer­ing by 1 syn­ton­ic comma.

Theoretically, the har­mon­ic series could also be con­struct­ed in a "Pythagorean" way, by extend­ing the cycles of per­fect fifths. Thus, after 8 descend­ing fifths, G3 ("E" = "mi") would be 8192/6561 (1.248) instead of 5/4 (1.25). The dif­fer­ence is a schis­ma (ratio 1.001129), an inter­val beyond human per­cep­tion. It is there­fore more con­ve­nient to show sim­ple ratios.

The frame­work imple­ment­ed in Bol Processor deals with inte­ger ratios, which allows for high accu­ra­cy. Nevertheless, it delib­er­ate­ly eras­es schis­ma dif­fer­ences. This is the result of approx­i­mat­ing cer­tain ratios, e.g. replac­ing 2187/2048 with 16/15.

Note that there is no trace of schis­ma in the clas­si­cal Indian the­o­ry of musi­cal scales; there wouldn't be even if Bharata's con­tem­po­raries had con­struct­ed them via series of ratio­nal num­bers, because of their deci­sion to dis­re­gard infin­i­tes­i­mals as "non-representable" enti­ties (cf. Nāgārjuna's śūniyavā­da phi­los­o­phy, Raju 2007 p. 400). If 2187/2048 and oth­er com­plex ratios of the same series were deemed imprac­ti­cal, the Indian mathematician/physicist (fol­low­ing the Āryabhaţīya) would replace them all with "16/15 āsan­na (near val­ue)"… This is an exam­ple of Indian math­e­mat­ics designed for cal­cu­la­tion rather than proof con­struc­tion. In the Western Platonic approach, math­e­mat­ics aimed at "exact val­ues" as a sign of per­fec­tion, which led its pro­po­nents to face seri­ous prob­lems with "irra­tional" num­bers and even with the log­ic under­ly­ing for­mal proof pro­ce­dures (Raju 2007 p. 387-389).

This dia­gram, and the mov­ing gra­ma wheel that will be intro­duced next, could be built with any size of the syn­ton­ic com­ma in the range 0 to 56.8 cents (Bel 1988a). The two-vina exper­i­ment implies that L = M + C. Thus, the syn­ton­ic com­ma is also the dif­fer­ence between a lim­ma and a minor semi­tone. To build a frame­work for the flex­i­ble mod­el, sim­ply allow all har­mon­ic posi­tions to move by the same amount in the direc­tion of their Pythagorean enhar­mon­ic variants.

While major thirds would be 1 com­ma larg­er if a Pythagorean inter­val (e.g. G4, ratio 81/64) were cho­sen instead of a har­mon­ic one (G3, ratio 5/4), major fifths also dif­fer by 1 com­ma, but the Pythagorean fifth (P4, ratio 3/2) is larg­er than the har­mon­ic one (P3, ratio 40/27). The lat­ter has been called the wolf fifth because its use in melod­ic phras­es or chords is said to sound "out of tune", with a negative/evil mag­i­cal connotation.

No posi­tion on this mod­el requires more than 1 ascend­ing or descend­ing major third. This makes sense to instru­ment tuners who know that tun­ing per­fect fifths by ear is an easy task that can be repeat­ed in sev­er­al steps — here a max­i­mum of 5 or 6 up and down. Tuning a har­mon­ic major third, how­ev­er, requires a lit­tle more atten­tion. It would there­fore be unre­al­is­tic to imag­ine a pre­cise tun­ing pro­ce­dure based on a sequence of major thirds —although this can be achieved with the aid of elec­tron­ic devices.

At the top of the pic­ture (posi­tion "fa#") we notice that nei­ther of the two cycles of fifths clos­es on itself due to the pres­ence of a Pythagorean com­ma. The tiny dif­fer­ence (schis­ma, ratio 1.001129) between Pythagorean and syn­ton­ic com­mas is illus­trat­ed by two pairs of posi­tions: P1/M3 and P2/M4.

Another pecu­liar­i­ty at the top of the pic­ture is the appar­ent dis­rup­tion of the sequence L-C-M. However, remem­ber­ing that L = M + C, the reg­u­lar­i­ty is restored by choos­ing between P1/M3 and P2/M4.

Approximations have no effect on the sound of musi­cal inter­vals, since no human ear would appre­ci­ate a schis­ma dif­fer­ence (2 cents). However, oth­er dif­fer­ences must remain explic­it, since inte­ger ratios indi­cate the tun­ing pro­ce­dure by which the scale can be con­struct­ed. Thus the replace­ment inte­ger ratio may be more com­plex than the "schis­mat­ic" one, as in the case of R1, ratio 256/243 instead of 135/128, because the lat­ter is built with a sim­ple major third above D4 instead of belong­ing to the Pythagorean series.

Tuning Western instruments

The prob­lem of tun­ing fixed pitch instru­ments (harp­si­chord, pipe organ, pianoforte…) has been well doc­u­ment­ed by the organ/harpsichord play­er, builder and musi­col­o­gist Pierre-Yves Asselin (Asselin 2000). In his prac­ti­cal approach, just into­na­tion is a back­ground mod­el that can only be approx­i­mat­ed on 12-degree scales by tem­pera­ment — com­pro­mis­ing the pure inter­vals of just into­na­tion to meet oth­er require­ments. Temperament tech­niques applied to the Bol Processor are dis­cussed on the Microtonality and Comparing tem­pera­ments pages.

The col­umn at the cen­tre of this pic­ture, with notes with­in ellipses, is a series of per­fect fifths which Asselin called "Pythagorean".

Series of fifths are infi­nite. Selecting sev­en of them (in the mid­dle col­umn) cre­ates a scale called the "glob­al dia­ton­ic frame­work" (milieu dia­tonique glob­al, see Asselin 2000 p. 59). In this exam­ple, the frame­works are those of "C" and "G" ("do" and "sol" in French).

Extending series of per­fect fifths beyond the sixth step pro­duces com­pli­cat­ed ratios that can be approx­i­mat­ed (with a schis­ma dif­fer­ence) to those pro­duced by har­mon­ic major thirds (ratio 5/4). Positions on the right (major third up, first order) are one syn­ton­ic com­ma low­er than their equiv­a­lents in the mid­dle series, and posi­tions on the left (major third down, first order) are one syn­ton­ic com­ma higher.

It is pos­si­ble to cre­ate more columns on the right ("DO#-2", "SOL#-2" etc.) for posi­tions cre­at­ed by 2 suc­ces­sive jumps of a har­mon­ic major third, and in the same way on the left ("DOb#+2", "SOLb#+2" etc.), but these second-order series are only used for the con­struc­tion of tem­pera­ments — see page Microtonality.

This mod­el pro­duces 3 to 4 posi­tions for each note, a 41-degree scale, which would require 41 keys (or strings) per octave on a mechan­i­cal instru­ment! This is one rea­son for the tem­per­ing of inter­vals on mechan­i­cal instru­ments, which amounts to select­ing the most appro­pri­ate 12 posi­tions for a giv­en musi­cal repertoire.

This tun­ing scheme is dis­played on scale "3_cycles_of_fifths" in the "-to.tryTunings" tonal­i­ty resource of Bol Processor.

Series of names have been entered, togeth­er with the frac­tion of the start­ing posi­tion, to pro­duce cycles of per­fect fifths in the scale. Following Asselin's nota­tion, the fol­low­ing series have been cre­at­ed (trace gen­er­at­ed by the Bol Processor):

1. From 4/3 up: FA, DO, SOL, RE, LA, MI, SI, FA#, DO#, SOL#, RE#, LA#
2. From 4/3 down: FA, SIb, MIb, LAb, REb, SOLb
3. From 320/243 up: FA-1, DO-1, SOL-1, RE-1, LA-1, MI-1, SI-1, FA#-1, DO#-1, SOL#-1, RE#-1, LA#-1
4. From 320/243 down: FA-1, SIb-1, MIb-1, LAb-1, REb-1, SOLb-1
5. From 27/20 up: FA+1, DO+1, SOL+1, RE+1, LA+1, MI+1, SI+1, FA#+1, DO#+1, SOL#+1, RE#+1, LA#+1
6. From 27/20 down: FA+1, SIb+1, MIb+1, LAb+1, REb+1, SOLb+1

This was more than enough to deter­mine the 3 or 4 posi­tions of each note, since sev­er­al notes can reach the same posi­tion at a schis­ma dis­tance. For exam­ple, "REb" is in the same posi­tion as "DO#-1". The IMAGE link shows this scale with (sim­pli­fied) fre­quen­cy relationships:

Compared to the mod­el advo­cat­ed by Arnold (1974, see fig­ure above), this sys­tem accepts har­mon­ic posi­tions on either side of the Pythagorean posi­tions, which means that Sa ("C" or "do"), like all unal­tered notes, can take three dif­fer­ent posi­tions. In Indian music, Sa is unique because it is the fun­da­men­tal note of every clas­si­cal per­for­mance of a raga, fixed by the drone (tan­pu­ra) and tuned to suit the singer or instru­men­tal­ist. However, we will see that trans­po­si­tions (murcchana-s) of the basic Indian scale(s) pro­duce some of these addi­tion­al positions.

A tun­ing scheme based on three (or more) cycles of per­fect fifths is a good grid for con­struct­ing basic chords in just into­na­tion. For exam­ple, a "C major" chord is made up of its ton­ic "DO", its dom­i­nant "SOL" a per­fect fifth high­er, and "MI-1" a major har­mon­ic third above "DO". The first two notes can belong to a Pythagorean series (blue marks on the graph) and the last one to a har­mon­ic series (green marks on the graph). Minor chords are con­struct­ed in a sim­i­lar way, which will be explained later.

This does not com­plete­ly solve the prob­lem of play­ing tonal music with just into­na­tion. Sequences of chords must be cor­rect­ly aligned. For exam­ple, should one use the same "E" in "C major" and in "E major"? The answer is "no", but the rule must be made explicit.

How is it pos­si­ble to choose the right one among the 3⁷ * 4⁵ = 2 239 488 chro­mat­ic scales shown in this graph?

In the approach of Pierre-Yves Asselin (2000) — inspired by the work of Conrad Letendre in Canada — rules were derived from options val­i­dat­ed by lis­ten­ers and musi­cians. Conversely, the gra­ma frame­work exposed below is a "top-down" approach — from a the­o­ret­i­cal mod­el to its eval­u­a­tion by practitioners.

The grama framework

Using Bharata's mod­el — see page The two-vina exper­i­ment — we can con­struct chro­mat­ic (12-degree) scales in which each tonal posi­tion (out of 11) has two options: har­mon­ic or Pythagorean. This is one rea­son to say that the frame­work is based on 22 shru­ti-s. In Indian musi­co­log­i­cal lit­er­a­ture, the term shru­ti is ambigu­ous, as it can mean either a tonal posi­tion or an interval.

In Bol Processor BP3 this "gra­ma" frame­work is edit­ed as fol­lows in "-to.12_scales":

We use lower-case labels for R1, R2 etc. and append a '_' after labels to dis­tin­guish enhar­mon­ic posi­tions from octave num­bers. So, "g3_4" means G3 in the fourth octave.

Two options for each of the 11 notes yields a set of 2¹¹ = 2048 chro­mat­ic scales. Of these, only 12 are "opti­mal­ly con­so­nant", i.e. they con­tain only one wolf fifth (small­er by 1 syn­ton­ic com­ma). These 12 scales are the ones used in har­mon­ic or modal music to expe­ri­ence max­i­mum con­so­nance. The author(s) of Naya Shastra had this inten­tion in mind when they described a basic 12-tone "opti­mal" scale called "Ma-grama". This scale is called "Ma_grama" in tonal­i­ty resource "-to.12_scales":

Click on the IMAGE link on the "Ma_grama" page to obtain a graph­i­cal rep­re­sen­ta­tion of this scale:

In this pic­ture the per­fect fifths are blue lines and the (unique) wolf fifth between C and G is a red line. Note that posi­tions marked in blue ("Db", "Eb" etc.) are Pythagorean and har­mon­ic posi­tions ("D", "E" etc.) appear in green. Normally, a "Pythagorean" posi­tion on this frame­work is one where nei­ther the numer­a­tor nor the denom­i­na­tor of the frac­tion is a mul­ti­ple of 5. Multiples of 5 indi­cate jumps of har­mon­ic major thirds (ratio 5/4 or 4/5). This sim­ple rule is bro­ken, how­ev­er, when com­plex ratios are replaced by sim­ple equiv­a­lents at a dis­tance of one schis­ma. Therefore, the blue and green mark­ings on the Bol proces­sor images are main­ly used to facil­i­tate the iden­ti­fi­ca­tion of a posi­tion: a note appear­ing near a blue mark­ing could as well belong to the har­mon­ic series with a more com­plex ratio, bring­ing it close to the Pythagorean position.

It will be impor­tant to remem­ber that all the notes of the Ma-grama scale are in their low­est enhar­mon­ic posi­tions. Other scales are cre­at­ed by rais­ing a few notes by a comma.

This Ma-grama is the start­ing point for the gen­er­a­tion of all "opti­mal­ly con­so­nant" chro­mat­ic scales. This is done by trans­pos­ing per­fect fifths (upwards or down­wards). The visu­al­i­sa­tion of trans­po­si­tions becomes clear when the basic scale is drawn on a cir­cu­lar wheel which is allowed to move with­in the out­er crown shown above. The fol­low­ing is Arnold's com­plete mod­el, show­ing the Ma-Grama in the basic posi­tion, pro­duc­ing the "Ma01" scale:

This posi­tion­ing of the inner wheel on top of the out­er wheel is called a "trans­po­si­tion" (mur­ccha­na).

Intervals are shown on the graph. For exam­ple, R3 ("D" = "re") is a per­fect fifth to D3 ("A" = "la").

The "Ma01" scale pro­duced by this M1 trans­po­si­tion pro­duces the "A minor" chro­mat­ic scale with the fol­low­ing intervals:

C l Db c+m D c+l Eb c+m E c+l F c+m F# c+l G l Ab c+m A c+l Bb c+m B c+l C

• m = minor semi­tone = 70 cents
• l = lim­ma = 90 cents
• c = com­ma = 22 cents

This con­struc­tion of the "A minor" scale cor­re­sponds to the Western scheme for the pro­duc­tion of just into­nat­ed chords: the fun­da­men­tal "A" (ratio 5/3) is "LA-1" on the "3_cycles_of_fifths" scale, which is in the "major third upwards" series as well as its dom­i­nant "MI-1", while "C" (ratio 1/1) belongs to the "Pythagorean" series.

At first sight, the scale con­struct­ed by this M1 trans­po­si­tion also resem­bles a "C major" scale, but with a dif­fer­ent choice of R3 (har­mon­ic "D" ratio 10/9) instead of R4 (Pythagorean "D" ratio 9/8). To pro­duce the "C major" scale, "D" should be raised to its Pythagorean posi­tion, which amounts to R4 replac­ing R3 on Bharata's mod­el. This is done by using an alter­na­tive root scale called "Sa-Grama" in which P4 replaces P3.

P3 is called "cyu­ta Pa" mean­ing "Pa low­ered by one shru­ti" — here a syn­ton­ic com­ma. The wheel rep­re­sen­ta­tion sug­gests that oth­er low­ered posi­tions may lat­er be high­light­ed by the trans­po­si­tion process, name­ly cyu­ta Ma and cyu­ta Sa.

At the bot­tom of the "Ma01" page on "-to.12_scales", all the inter­vals of the chro­mat­ic scale are list­ed, with the sig­nif­i­cant inter­vals high­light­ed in colour. The wolf fifth is coloured red. Note that when the scale is opti­mal­ly con­so­nant, only one cell is coloured red.

A tun­ing scheme is sug­gest­ed at the bot­tom of page "Ma01". It is based on the (pure­ly mechan­i­cal) assump­tion that per­fect fifths are tuned first with­in the lim­it of 6 steps. Then har­mon­ic major thirds and minor sixths are high­light­ed, and final­ly Pythagorean thirds and minor sixths can also be tak­en into account.

We can use "Ma01" as a 23-degree micro­ton­al scale in Bol Processor pro­duc­tions because all the notes rel­e­vant to the chro­mat­ic scale have been labelled. However it is more prac­ti­cal to extract a 12-degree scale with only labelled notes. This can be done on the "Ma01" page. The image shows the expor­tat of the "Cmaj" scale with 12 degrees and a raised posi­tion of D.

Using "Cmaj" for the name makes it easy to declare this scale in its spe­cif­ic har­mon­ic con­text. In the same way, a 12-degree "Amin" can be export­ed with­out rais­ing the "D".

"D" ("re") is there­fore the sen­si­tive note when switch­ing between the "C major" scale and its rel­a­tive "A minor".

In all 12-degree export­ed scales it is easy to change the note con­ven­tion — English, Italian/Spanish/French, Indian or key num­bers. It is also pos­si­ble to select diesis in replace­ment of flat and vice ver­sa, as the machine recog­nis­es both options.

Producing the 12 chromatic scales

A PowerPoint ver­sion of Arnold's mod­el can be down­loaded here and used to check the trans­po­si­tions pro­duced by the Bol Processor BP3.

To cre­ate suc­ces­sive "opti­mal­ly con­so­nant" chro­mat­ic scales, the Ma-grama should be trans­posed by descend­ing or ascend­ing per­fect fifths.

For exam­ple, cre­ate "Ma02" by trans­pos­ing "Ma01" from a per­fect fourth "C to F" (see pic­ture). Nothing else needs to be done. All the trans­po­si­tions are stored in tonal­i­ty resource "-to.12_scales". Each of these scales can then be used to export a minor and a major chro­mat­ic scale. This pro­ce­dure is explained in detail on the page Creation of just-intonation scales.

Enharmonic shift of the tonic

An inter­est­ing point raised by James Arnold in our paper L'intonation juste dans la théorie anci­enne de l'Inde : les appli­ca­tions aux musiques modale et har­monique (1985) is the com­par­i­son of minor and major scales of the same ton­ic, for exam­ple mov­ing from "C major" to "C minor".

To get the "C minor" scale, we need to cre­ate "Ma04" by using four suc­ces­sive descend­ing fifths (or ascend­ing fourths). Note that writ­ing "C to F" on the form does not always pro­duce a per­fect fourth trans­po­si­tion because the "F to C" inter­val may be a wolf fifth! This hap­pens when going from "Ma03" to "Ma04". In this case, select, for exam­ple, "D to G".

From "Ma04" we export "Cmin". Here comes a surprise:

The inter­vals are those pre­dict­ed (see "A minor" above), but the posi­tions of "G", "F" and "C" have been low­ered by one com­ma. This was expect­ed for "G" because of the replace­ment of P4 by P3. The bizarre sit­u­a­tion is that both 'C' and 'F' are one com­ma low­er than what seemed to be their low­est (or only) posi­tion in the 22-shruti mod­el. The authors of Natya Shastra had antic­i­pat­ed a sim­i­lar process when they invent­ed the terms "cyu­ta Ma" and "cyu­ta Sa"…

This shift­ing of the base note can be seen by mov­ing the inner wheel. After 4 trans­po­si­tions, the posi­tion M1 of the inner wheel will cor­re­spond to the posi­tion G1 of the out­er wheel, giv­ing the fol­low­ing configuration:

This shift of the ton­ic was pre­sent­ed as a chal­leng­ing find­ing in our paper (Arnold & Bel 1985). Jim Arnold had done exper­i­ments with Pierre-Yves Asselin play­ing Bach's music on the Shruti Harmonium and both liked the shift of the ton­ic on minor chords.

Pierre-Yves him­self men­tions a one-comma low­er­ing of "C" and "G" in the "C minor" chord. However, this was one of two options pre­dict­ed by his the­o­ret­i­cal mod­el. He test­ed it by play­ing the Cantor elec­tron­ic organ at the University, and reports that musi­cians found this option to be more pun­gent — "déchi­rant" — (Asselin 2000 p. 135-137).

The oth­er option (red on the pic­ture) was that each scale be "aligned" in ref­er­ence to its base note "C" ("DO"). This align­ment (one-comma rais­ing) can be done click­ing but­ton "ALIGN SCALE" on scale pages wher­ev­er the basic note ("C") is not at posi­tion 1/1. Let us lis­ten to the "C major"/ "C minor" / "C major" sequence, first "non-aligned" then "aligned":

Clearly, the "non-aligned" ver­sion is more pun­gent than the "aligned" one.

This choice is based on per­cep­tu­al expe­ri­ence, or "pratyakṣa pramāṇa" in Indian epis­te­mol­o­gy — see The two-vina exper­i­ment. We take an empir­i­cal approach rather than seek­ing an "axiomat­ic proof". The ques­tion is not which of the two options is true, but which one pro­duces music that sounds right.

Checking the tuning system

Checking a chord sequence

The con­struc­tion of just into­na­tion using the grama-murcchana pro­ce­dure needs to be checked in typ­i­cal chord sequences such as the "I-IV-II-V-I" series dis­cussed by Pierre-Yves Asselin (2000 p. 131-135):

After try­ing out five options sug­gest­ed by his the­o­ret­i­cal mod­el, the author chose the one pre­ferred by all the musi­cians. This is the into­na­tion they spon­ta­neous­ly choose when singing, with­out any spe­cial instruc­tion. This ver­sion also cor­re­sponds to Zarlino's "nat­ur­al scale".

In the pre­ferred option, the ton­ics "C", "F" and "G" belong to the Pythagorean series of per­fect fifths, except "D" in the "D minor" chord which is one com­ma low­er than in "G major".

In the pic­ture, the tri­an­gles with the top point­ing to the right are major chords, and the one point­ing to the left is the "D minor" chord.

Asselin (2000 p. 137) con­cludes that the minor mode is one syn­ton­ic com­ma low­er than the major mode. Conversely, the major mode should be one syn­ton­ic com­ma high­er than the minor mode.

This is ful­ly con­sis­tent with the mod­el con­struct­ed by grama-murcchana. Since minor chro­mat­ic scales are export­ed from trans­po­si­tions of Ma-grama with all its degrees in the low­est posi­tion, their base notes are also dri­ven to the low­est posi­tions. However this requires a scale "adjust­ment" in the cas­es of "Ma10", "Ma11" and "Ma12" so that no posi­tion is cre­at­ed out­side the basic Pythagorean/harmonic scheme of the Indian sys­tem. Looking at Asselin's draw­ing (above), this means that no posi­tion would be picked up in the 2^nd-order series of fifths in the right­most col­umn with two suc­ces­sive ascend­ing major thirds result­ing in a low­er­ing of 2 syn­ton­ic com­mas. This process is explained in more detail on the page Creation of just-intonation scales.

Let us lis­ten to the pro­duc­tion of the "-gr.tryTunings" gram­mar:

S --> Temp - Just
Temp --> Cmaj Fmaj Dmin Gmaj Cmaj
Just --> _scale(Cmaj,0) Cmaj _scale(Fmaj,0) Fmaj _scale(Dmin,0) Dmin _scale(Gmaj,0) Gmaj _scale(Cmaj,0) Cmaj
Cmaj --> {C3,C4,E4,G4}
Fmaj --> {F3,C4,F4,A4}
Dmin --> {D3,D4,F4,A4}
Gmaj --> {G3,B3,D4,G4}

First we hear the sequence of chords in equal tem­pera­ment, then in just intonation.

The iden­ti­ty of the last occur­rence with Asselin's favourite choice is marked by fre­quen­cies in the C-sound score: "D4" in the third chord (D minor) is one com­ma low­er than "D4" in the fourth chord (G major), while all oth­er notes (e.g. "F4") have the same fre­quen­cies in the four chords.

Show Csound score (just intonation)

To sum­marise, the ton­ic and dom­i­nant of each minor chord belong to the "low­er" har­mon­ic series of per­fect fifths appear­ing in the right-hand col­umn of Asselin's draw­ing repro­duced above. Conversely, the ton­ic and dom­i­nant of each major chord belong to the "Pythagorean" series of per­fect fifths in the mid­dle column.

Checking note sequences

The rules for deter­min­ing the rel­a­tive posi­tions of major and minor modes (see above) deal only with the three notes that define a major or minor chord. Transpositions (murcchana-s) of the Ma-grama pro­duce basic notes in the same posi­tions, but these are also chro­mat­ic (12-degree) scales. Therefore, they also estab­lish the enhar­mon­ic posi­tions of all the notes that would be played in that har­mon­ic context.

Do these com­ply with just into­na­tion? In the­o­ry, yes, because the 12 chro­mat­ic scales obtained by these trans­po­si­tions are "opti­mal­ly con­so­nant": each of them con­tains no more than a wolf fifth.

In 1980, James Arnold con­duct­ed exper­i­ments to ver­i­fy this the­o­ret­i­cal mod­el using my Shruti har­mo­ni­um, which pro­duced pro­grammed inter­vals to an accu­ra­cy of 1 cent. Pierre-Yves Asselin played clas­si­cal pieces while Jim manip­u­lat­ed switch­es on the instru­ment to select enhar­mon­ic variants.

Listen to three ver­sions of an impro­vi­sa­tion based on Mozart's musi­cal dice game. The first one is equal-tempered, the sec­ond one uses Serge Cordier's equal-tempered scale with an extend­ed octave (1204 cents, see Microtonality) and the third uses sev­er­al dif­fer­ent scales to repro­duce a just into­na­tion. To this end, vari­ables point­ing to scales based on the har­mon­ic con­text have been insert­ed in the first gram­mar rules:

S --> _vel(80) Ajust Bjust
Ajust --> Cmaj A1 A2 Gmaj A3 Cmaj A4 Dmaj A5 Cmaj A6 Gmaj A7 A8 Cmaj A1 A2 Gmaj A3 Cmaj A4 Dmaj A5 Cmaj A6 Gmaj A7 A'8
Bjust --> Gmaj B1 Cmaj B2 Dmaj B3 Cmaj B4 Fmaj B5 B6 Gmaj B7 Cmaj B8 Gmaj B1 Cmaj B2 Dmaj B3 Cmaj B4 Fmaj B5 B6 Gmaj B7 Cmaj B8
Cmaj --> _scale(Cmaj,0)
Dmaj --> _scale(Dmaj,0)
Fmaj --> _scale(Fmaj,0)
Gmaj --> _scale(Gmaj,0)
… etc.

Scale comparison

At the bot­tom of the pages "-to.12_scales" and "-to.Mozart", all scales are com­pared for their inter­val­ic con­tent. The com­par­i­son is based on frac­tions where these have been declared, or on floating-point fre­quen­cy ratios otherwise.

The com­par­i­son con­firms that the "Amin" chro­mat­ic scale is iden­ti­cal to "Fmaj".

By rais­ing "D" in "Ma01" we have cre­at­ed "Sa01", the first trans­po­si­tion of the Sa-grama scale. From "Sa01" we can make "Sa02" etc. by suc­ces­sive trans­po­si­tions (one fourth upwards). But the com­para­tor shows that "Sa02" is iden­ti­cal to "Ma01".

Similarly, the trans­po­si­tions "Ma13", "Ma14" etc. are iden­ti­cal to "Ma01", "Ma02" etc. The series of chro­mat­ic scales is (as expect­ed) cir­cu­lar, since "Ma13" returns to "Ma01".

For more details on fre­quen­cies, block keys, etc., see the Microtonality page.

Is this perfect?

This entire page is devot­ed to tonal sys­tems defined in terms of whole-numbered ratios (i.e. ratio­nal num­bers) mea­sur­ing tonal inter­vals. There were at least two strong incen­tives for the idea that any "pure" tonal inter­val should be treat­ed as a ratio of two whole num­bers, such as 2/1 for the octave, 3/2 for a "per­fect" fifth, 5/4 for a "har­mon­ic" major third, etc.

The his­to­ry of music (in the West) goes back to ideas attrib­uted to the Greek philoso­pher "Pythagoras" (see above) , who believed that all things were made of [ratio­nal] num­bers. This approach stum­bled upon the impos­si­bil­i­ty of mak­ing the octave cor­re­spond to a series of "per­fect fifths"…

As we found out — read above and The Two-vina exper­i­ment — this approach was not fol­lowed in India despite the fact that Indian sci­en­tists were sig­nif­i­cant­ly more advanced than the Greeks in the field of cal­cu­lus (Raju C.K., 2007).

Another incen­tive to the use of ratio­nal num­bers was Hermann von Helmholtz's notion of con­so­nance (1877) which became pop­u­lar after the peri­od of Baroque music in Europe, fol­low­ing the ini­tial claim of a "nat­ur­al tonal sys­tem" by Jean-Philippe Rameau in his Traité de l'harmonie réduite à ses principes naturels (1722). The devel­op­ment of key­board stringed instru­ments such as the pipe organ and the pianoforte had made it nec­es­sary to devel­op a tun­ing sys­tem that met the require­ments of (approx­i­mate­ly) tune­ful har­mo­ny and trans­po­si­tion to sup­port oth­er instru­ments and the human voice. It was there­fore log­i­cal to aban­don a wide vari­ety of tun­ing sys­tems, espe­cial­ly those based on tem­pera­ment, and to adopt equal tem­pera­ment as the stan­dard. By this time, com­posers were no longer explor­ing the sub­tleties of melodic/harmonic inter­vals; har­mo­ny involv­ing groups of singers and/or orches­tra paved the way for musi­cal innovation.

Looking back to the Baroque peri­od, many musi­col­o­gists tend to believe that the tun­ing sys­tem advo­cat­ed by J.S. Bach in The Well-tempered Clavier must have been equal tem­pera­ment… This belief can be dis­proved by a sys­tem­at­ic analy­sis of this cor­pus of pre­ludes and fugues on an instru­ment using all the tun­ing sys­tems en vogue dur­ing the Baroque peri­od — read the page The Well-tempered Clavier.

Composers and instru­ment mak­ers did not tune "by num­bers", as tun­ing pro­ce­dures were not doc­u­ment­ed (see Asselin P-Y., 2000). Rather, they tuned "by ear" in order to achieve a per­ceived reg­u­lar­i­ty of sets of inter­vals: the tem­pera­ment in gen­er­al. This was indeed a break with the "Pythagorean" mys­tique, because these tem­pera­ments can­not be reduced to fre­quen­cy inter­vals based on inte­ger ratios.

For instance, Zarlino’s mean­tone tem­pera­ment — read this page — is made of 12 fifths start­ing from “E♭” (“mi♭”) up to “G#” (“sol#”) dimin­ished by 2/7 of a syn­ton­ic com­ma (ratio 81/80). The fre­quen­cy ratio of each fifth is therefore

\[\ \frac{3}{2}\left(\frac{80}{81}\right)^{\frac{2}{7}}=\ 1.5\ x\ 0.99645\dots\ =\ 1.4946\dots\ \left(or\ 695.81\dots\ cents\right)\]

which can­not be reduced to an inte­ger ratio. In the same way, the twelve inter­vals of the equal tem­pera­ment scale are expressed in terms of irra­tional fre­quen­cy ratios.

Overture

The goal of just into­na­tion is to pro­duce "opti­mal­ly con­so­nant" chords and sequences of notes, a legit­i­mate approach when con­so­nance is the touch­stone of the high­est achieve­ment in art music. This was indeed the case in sacred music, which aimed for a "divine per­fec­tion" guar­an­teed by the absence of "wolf tones" and oth­er odd­i­ties. In a broad­er sense, how­ev­er, music is also the field of expec­ta­tion and sur­prise. In an artis­tic process, this can mean depart­ing from "rules" — just as poet­ry requires break­ing the seman­tic and syn­tac­tic rules of a language…

Even when chords are per­fect­ly con­so­nant and con­form to the rules of har­mo­ny (as per­ceived by the com­pos­er), note sequences may devi­ate from their the­o­ret­i­cal posi­tions in order to cre­ate a cer­tain degree of ten­sion or to make a bet­ter tran­si­tion to the next chord.

When the Greek-French com­pos­er Iannis Xenakis - known for his for­malised approach to tonal­i­ty - heard Bach's First Prelude for Well-Tempered Clavier played on the Shruti Harmonium in just into­na­tion, he declared his pref­er­ence for the equal-tempered ver­sion! This made sense for a com­pos­er whose music had been praised by Tom Service for its "deep, pri­mal root­ed­ness in rich­er and old­er phe­nom­e­na even than musi­cal his­to­ry: the physics and pat­tern­ing of the nat­ur­al world, of the stars, of gas mol­e­cules, and the pro­lif­er­at­ing pos­si­bil­i­ties of math­e­mat­i­cal prin­ci­ples" (Service T, 2013).

Bernard Bel — Dec. 2020 / Jan. 2021

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, 71^e (1-2), p.11-38.

Asselin, P.-Y. Musique et tem­péra­ment. Paris, 1985, repub­lished in 2000: Jobert. Soon avail­able in English.

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.

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.

Raju, C. K. Cultural foun­da­tions of math­e­mat­ics : the nature of math­e­mat­i­cal proof and the trans­mis­sion of the cal­cu­lus from India to Europe in the 16^th c. CE. Delhi, 2007: Pearson Longman: Project of History of Indian Science, Philosophy and Culture : Centre for Studies in Civilizations.

Service, T. A guide to Iannis Xenakis's music. The Guardian, 23 April 2013.