The two-vina experiment

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

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

Accurate set­tings of Bel’s Shruti Harmonium (1980)

In the sec­ond half of the 20th cen­tu­ry, exper­i­men­tal work with fre­quen­cy meters led to con­tra­dic­to­ry con­clu­sions from the analy­sis of small sam­ples of musi­cal per­for­mances. It was only after 1981 that sys­tem­at­ic exper­i­ments were car­ried out in India by the ISTAR team (E.J. Arnold, B. Bel, J. Bor and W. van der Meer) with an elec­tron­i­cal­ly pro­gram­ma­ble har­mo­ni­um (the Shruti Harmonium) and lat­er with a “micro­scope” for melod­ic music, the Melodic Movement Analyser (MMA) (Arnold & Bel 1983, Bel & Bor I985), which fed pre­cise pitch data into a com­put­er to process hours of music select­ed from his­tor­i­cal recordings.

After sev­er­al years of exper­i­men­tal work, it had become clear that although the into­na­tion of Indian clas­si­cal music is far from being a ran­dom process, it would be dan­ger­ous to judge an inter­pre­ta­tion of the ancient scale the­o­ry on the basis of today’s musi­cal data. There are at least three rea­sons for this:

  1. There are an infi­nite num­ber of valid inter­pre­ta­tions of the ancient the­o­ry, as we will show.
  2. The con­cept of raga, the basic prin­ci­ple of Indian clas­si­cal music, first appeared in lit­er­a­ture around 900 CE in Matanga’s Brihaddeshi and under­went grad­ual devel­op­ment until the 13th cen­tu­ry, when Sharangadeva list­ed 264 ragas in his Sangitratnakara.
  3. Drones were (prob­a­bly) not in use at the time of Natya Shastra; the influ­ence of the drone on into­na­tion is con­sid­er­able, if not dom­i­nant, in con­tem­po­rary music performance.

The ancient Indian the­o­ry of scales remains use­ful for its insight into ear­ly melod­ic clas­si­fi­ca­tion (the jāti sys­tem), which may lat­er have giv­en rise to the raga sys­tem. It is there­fore best thought of as a topo­log­i­cal descrip­tion of tonal struc­tures. Read Raga Intonation for a more detailed account of the­o­ret­i­cal and prac­ti­cal issues.

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

The historical context

Bharata Muni, the author(s) of the Natya Shastra, may have heard of the the­o­ries of musi­cal scales attrib­uted to the “ancient Greeks”. At any rate, Indian schol­ars were able to bor­row these mod­els and extend them con­sid­er­ably because of their real knowl­edge of arithmetic.

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

The cal­cu­lus first devel­oped in India as a sophis­ti­cat­ed tech­nique to cal­cu­late pre­cise trigono­met­ric val­ues need­ed for astro­nom­i­cal mod­els. These val­ues were pre­cise to the 9th place after the dec­i­mal point; this pre­ci­sion was need­ed for the cal­en­dar, crit­i­cal to monsoon-driven Indian agri­cul­ture […]. This cal­cu­la­tion involved infi­nite series which were summed using a sophis­ti­cat­ed phi­los­o­phy of ratios of inex­pressed num­bers [today called ratio­nal functions…].

Europeans, how­ev­er, were prim­i­tive and back­ward in arith­meti­cal cal­cu­la­tions […] and bare­ly able to do finite sums. The dec­i­mal sys­tem had been intro­duced in Europe by Simon Stevin only at the end of the 16th c., while it was in use in India since Vedic times, thou­sands of years earlier.

C. K. Raju (2013 p. 161- 162)

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

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

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

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

The experiment

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

A word of cau­tion to clar­i­fy the con­text: this chap­ter of Natya Shastra can be read as a thought exper­i­ment rather than a process involv­ing phys­i­cal objects. There is no cer­tain­ty that these two vina-s ever exist­ed — and even that “Bharata Muni”, the author/experimenter, was a unique per­son. His/their approach is one of val­i­da­tion (pramāņa) by empir­i­cal evi­dence, in oth­er words dri­ven by the phys­i­cal­ly man­i­fest (pratyakşa) rather than inferred from “axioms” con­sti­tu­tive of a the­o­ret­i­cal mod­el. This can be summed up as a “pref­er­ence for physics over metaphysics”.

Constructing and manip­u­lat­ing vina-s in the man­ner indi­cat­ed by the exper­i­menter appears to be an insur­mount­able tech­no­log­i­cal chal­lenge. This has been dis­cussed by a num­ber of authors — see Iyengar (2017 pages 7-sq.) Leaving aside the pos­si­bil­i­ty of prac­ti­cal real­i­sa­tion is not a denial of phys­i­cal real­i­ty, as for­mal math­e­mat­ics would sys­tem­at­i­cal­ly dic­tate. Calling it a “thought exper­i­ment” is a way of assert­ing the con­nec­tion with the phys­i­cal mod­el. Similarly, the use of cir­cu­lar graphs to rep­re­sent tun­ing schemes and alge­bra to describe rela­tion­ships between inter­vals are aids to under­stand­ing that do not reduce the mod­el to spe­cif­ic, ide­al­is­tic inter­pre­ta­tions sim­i­lar to the spec­u­la­tions about inte­gers cher­ished by Western sci­en­tists. These graphs are intend­ed to facil­i­tate the com­pu­ta­tion­al design of instru­ments that mod­el these imag­ined instru­ments — see Raga into­na­tion and Just into­na­tion, a gen­er­al frame­work.

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

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

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

Intervals are notat­ed in shru­ti-s, which can be thought of as an order­ing device rather than a unit of mea­sure­ment. Experiment will con­firm that a four-shru­ti inter­val is greater than a three-shru­ti, a three-shru­ti greater than a two-shru­ti and the lat­ter greater than a sin­gle shru­ti. In dif­fer­ent con­texts, the word “shru­ti” refers to note posi­tions rather than inter­vals between notes. This ambi­gu­i­ty is also a source of confusion.

The author writes:

Śrutis in the Şaḍja Grāma are shown as fol­lows: three [in Ri], two [in Ga], four [in Ma], four [in Pa], three [in Dha], two [in Ni], and four [in Sa].

Bharata uses 9-shru­ti (con­so­nant) inter­vals: “Sa-Pa”, “Sa-Ma”, “Ma-Ni”, “Ni-Ga” and “Re-Dha”. He also defines anoth­er scale called “Ma-grama” in which “Pa” is one shru­ti low­er than “Pa” in the Sa-grama, so that “Sa-Pa” is no longer con­so­nant where­as “Re-Pa” is con­so­nant because it is made up of 9 shru­ti-s.

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

Sa-grama and Ma-grama accord­ing to Natya Shastra. Red and green seg­ments indi­cate the two chains of per­fect fifths. Underlined note names denote ‘flat’ positions.

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

Two cycles of fifths are high­light­ed in red and green col­ors. Note that both the “Sa-Ma” and “Ma-Ni” inter­vals are per­fect fifths, which dis­cards the asso­ci­a­tion of Sa-grama with the con­ven­tion­al west­ern scale: the “Ni” should be mapped to “B flat”, not to “B”. Furthermore, the per­fect fifth “Ni-Ga” implies that “Ga” is also “E flat” rather than “E”. The Sa-grama and Ma-grama scales are there­fore “D modes”. This is why “Ga” and “Ni” are under­lined in the diagrams.

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

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

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

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

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

The des­ig­na­tion of the small­est inter­val as “pramāņa ṣru­ti” is of great epis­temic impor­tance and deserves a brief expla­na­tion. The seman­tics of “slack­ness or ten­sion” clear­ly belong to “pratyakṣa pramāṇa”, the means of acquir­ing knowl­edge through per­cep­tu­al expe­ri­ence. More pre­cise­ly, “pramāṇa” (प्रमाण) refers to “valid per­cep­tion, mea­sure and struc­ture” (Wisdom Library), a notion of evi­dence shared by all tra­di­tion­al Indian schools of phi­los­o­phy (Raju 2007 page 63). We will return to this notion in the conclusion.

An equiv­a­lent way of con­nect­ing the two cycles of fifths would be to define a 7-shru­ti inter­val, for exam­ple “Ni-Re”. If the pramāņa ṣru­ti were a syn­ton­ic com­ma then this inter­val would be a har­mon­ic major third with a ratioof 5/4. As men­tioned in Just into­na­tion, a gen­er­al frame­work, the inven­tion of the major third as a con­so­nant inter­val dates back to the ear­ly 16th cen­tu­ry in Europe. In Natya Shastra this 7-shru­ti inter­val was clas­si­fied as “asso­nant” (anu­va­di).

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

The pic­ture above shows the two vina-s tuned iden­ti­cal­ly on Sa-grama. Matching notes are marked with yel­low dots. The inner part of the blue cir­cle will be the mov­ing vina in the fol­low­ing trans­po­si­tions, and the out­er part the fixed vina.

First lowering

Bharata writes:

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

In short, this is a pro­ce­dure for low­er­ing all the notes of the mov­able vina by one pramāņa ṣru­ti. First low­er its “Pa” — e.g. make it con­so­nant with the “Re” of the fixed vina — to obtain Ma-grama on the mov­able vina. Then read­just its entire scale to obtain Sa-grama. Note that low­er­ing “Re” and “Dha” means revalu­ing the size of a pramāņa ṣru­ti while main­tain­ing the ‘Re-Dha’ con­so­nant inter­val. The result is as follows:

The two vinas after a low­er­ing of pramāņa ṣru­ti

The pic­ture illus­trates the fact that there is no longer a match between the two vina-s.

Interpreting shruti-s as vari­ables in some metrics

This sit­u­a­tion can be trans­lat­ed into alge­bra. Let “a”, “b”, “c” … “v” be the unknown sizes of the shru­ti-s in the scale (see pic­ture on the side). A met­ric that “trans­lates” Bharata’s mod­el will be nec­es­sary to test it on sound struc­tures pro­duced by an elec­tron­ic instru­ment — the com­put­er. The scope of this trans­la­tion remains valid as long as no addi­tion­al asser­tion is made that is not root­ed in the orig­i­nal model.

Using the sym­bol “#>” to indi­cate that two notes do not match, this first low­er­ing can be sum­marised by the fol­low­ing set of inequalities:

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

Second lowering

The next step is anoth­er low­er­ing by one shru­ti using a dif­fer­ent procedure.

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

Note that it is no longer pos­si­ble to rely on a low­ered “Pa” to eval­u­ate a pramāņa ṣru­ti for the low­er­ing. The instruc­tion is to low­er the tun­ing of the mov­able vina until either “Re” and “Ga” or “Dha” and “Ni” merge, which is claimed to be the same because of the final low­er­ing of two shru­ti-s (from the ini­tial state):

The two vina-s after the sec­ond low­er­ing (2 shru­ti-s)

Now we have an equa­tion which tells us that the two-shru­ti inter­vals are equal in size:

q + r = d + e

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

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

We should bear in mind that the author is describ­ing a phys­i­cal process, not an abstract “move­ment” by which the mov­ing wheel (or vina) would “jump” in space from its ini­tial to final posi­tion. Therefore, we pay atten­tion to what hap­pens and what does not hap­pen dur­ing the tun­ing of the vina or the rota­tion of the wheel by look­ing at the tra­jec­to­ries of the dots rep­re­sent­ing the note posi­tions (along the blue cir­cle). Things that do not hap­pen (mis­matched notes) give rise to inequa­tions that are nec­es­sary to make sense of the alge­bra­ic model.

This step of the exper­i­ment con­firms that it is wrong to place Sa in the posi­tion of Ni in order to iden­ti­fy Sa-grama with the Western scale. In this case the cor­re­spond­ing notes would not be Re-Ga and Dha-Ni, but Ga-Ma and Ni-Sa.

Third lowering

Bharata writes:

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

The two vinas after the third low­er­ing (3 shruti-s)

This leads to equation

n + o + p = a + b + c

and inequa­tions:

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

Fourth lowering

The pro­ce­dure:

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

The two vinas after the fourth low­er­ing (4 shruti-s)

This yields 2 equations:

j + k + l + m = f + g + h + i
s + t + u + v = f + g + h + i

Algebraic interpretation

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

(S1) d + e > m
(S2) a + b + c > d + e
(S3) f + g + h + i > a + b + c
(S4) j + k + l + m = f + g + h + i
(S5) s + t + u + v = f + g + h + i
(S6) n + o + p = a + b + c
(S7) q + r = d + e

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

We still have 22 vari­ables and only 4 equa­tions. These vari­ables can be “packed” into a set of 8 vari­ables rep­re­sent­ing the “macro-intervals”, i.e. the steps of the gra­ma-s. In this approach the shru­ti-s are a kind of “sub­atom­ic” par­ti­cles of which these “macro-intervals” are made… Now we need only 4 aux­il­iary equa­tions to deter­mine the scale. These can be pro­vid­ed by acoustic infor­ma­tion where the inter­vals are count­ed in cents. First we express that the sum of the vari­ables, the octave, is equal to 1200 cents. (A larg­er val­ue, e.g. 1204, could be used to devise extend­ed octaves).

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

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

(S9) (a + b + c) + (d + e) + (f + g + h + i) + (j + k + l) + m = 701.9 (Sa-Pa)
(S10) (j + k + l) + m + (n + o + p) + (q + r) + (s + t + u + v) = 701.9 (Ma-Sa)
(S11) (d + e) + (f + g + h + i) + (j + k + l) + m + (n + o + p) = 701.9 (Re-Dha)
(S12) (f + g + h + i) + (j + k + l) + m + (n + o + p) + (q + r) = 701.9 (Ga-Ni)

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

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

S10, S11 and S12 can all be derived from S9. So these equa­tions can be dis­card­ed. We still need one more equa­tion to solve the sys­tem. At this stage there are many options in terms of tun­ing pro­ce­dures. As sug­gest­ed above, set­ting the har­mon­ic major third to the ratio 5/4 (386.3 cents) would pro­vide the miss­ing equa­tion. This is equiv­a­lent to set­ting the vari­able “m” to 21.4 cents (syn­ton­ic com­ma). However, this major third can be any size up to the Pythagorean third (81/64 = 407.8 cents), for which we would get m = 0.

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

Extensions of the model

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

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

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

An equal-tempered scale with octave stretched at 1204 cents. (Image cre­at­ed by Bol Processor BP3)

Another option is to get as close as pos­si­ble to “just into­na­tion” with­out chang­ing per­fect fifths and octaves. This is pos­si­ble by allow­ing the com­ma (vari­able “m”) to take any val­ue between 0 and 56.8 cents. Limits are imposed by the inequa­tions derived from the two-vina experiment.

These “just sys­tems” are cal­cu­lat­ed as follows:

a + b + c = j + k + l = n + o + p = Maj - C
d + e = h + i = q + r = u + v = L + C
f + g = s + t = Maj - L - C
m = C

where L = 90.25 cents (lim­ma = 256/243), Maj = 203.9 cents (major who­le­tone = 9/8)
and 0 < C < 56.8 (pramāņa ṣru­ti or syn­ton­ic comma)

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

The “gra­ma” scale used for just into­na­tion, with a syn­ton­ic com­ma of 81/80. Pythagorean cycle of fifths in red, har­mon­ic cycle of fifths in green.

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

The relevance of circular representations

Circular rep­re­sen­ta­tion of tāl Pañjābi, catuśra­jāti
[16 counts] from a Gujarati text in Devanagari script
(J. Kippen, pers. communication)

It is safe to clas­si­fy the two-vina exper­i­ment as a thought exper­i­ment, since it is unlike­ly that it could be car­ried out with mechan­i­cal instru­ments. Representing it on a cir­cu­lar graph (a mov­able wheel inside a fixed crown) achieves the same goal with­out resort­ing to imag­i­nary devices.

Circular rep­re­sen­ta­tions belong to Indian tra­di­tions of var­i­ous schools, includ­ing the descrip­tion of rhyth­mic cycles (tāl-s) used by drum­mers. These dia­grams are meant to out­line the rich inter­nal struc­ture of musi­cal con­struc­tions that can­not be reduced to “beat count­ing” (Kippen 2020).

For exam­ple, the image on the side was used to describe the ţhekkā (cycle of quasi-onomatopoeic syl­la­bles rep­re­sent­ing the beats of the drum) of tāl Pañjābi which reads as follows:

Unfortunately, ear­ly print­ing tech­nol­o­gy may have made the pub­li­ca­tion and trans­mis­sion of these learn­ing aids difficult.

If Bharata’s con­tem­po­raries ever used sim­i­lar cir­cu­lar rep­re­sen­ta­tions to reflect on musi­cal scales, we sus­pect that archae­o­log­i­cal traces might not be prop­er­ly iden­ti­fied, as their draw­ings might be mis­tak­en for yantra-s, astro­log­i­cal charts and the like!

Return to epistemology

Bharata’s exper­i­ment is a typ­i­cal exam­ple of the pref­er­ence for facts derived from empir­i­cal obser­va­tion over a pro­claimed uni­ver­sal log­ic aimed at estab­lish­ing “irrefutable proofs”.

Empirical proofs are uni­ver­sal, not meta­phys­i­cal proofs; elim­i­nat­ing empir­i­cal proofs is con­trary to all sys­tems of Indian phi­los­o­phy. Thus ele­vat­ing meta­phys­i­cal proofs above empir­i­cal proofs, as for­mal math­e­mat­ics does, is a demand to reject all Indian phi­los­o­phy as infe­ri­or. Curiously, like Indian phi­los­o­phy, present-day sci­ence too uses empir­i­cal means of proof, so this is also a demand to reject sci­ence as infe­ri­or (to Christian metaphysics).

Logic is not uni­ver­sal either as Western philoso­phers have fool­ish­ly main­tained: Buddhist [qua­si truth-functional] and Jain [three-valued] log­ics are dif­fer­ent from those cur­rent­ly used in for­mal math­e­mat­i­cal proof. The the­o­rems of math­e­mat­ics would change if those log­ics were used. So, impos­ing a par­tic­u­lar log­ic is a means of cul­tur­al hege­mo­ny. If log­ic is decid­ed empir­i­cal­ly, that would, of course, kill the phi­los­o­phy of meta­phys­i­cal proof. Further, it may result in quan­tum log­ic, sim­i­lar to Buddhist logic […].

C. K. Raju (2013 p. 182-183)
Yuktibhāşā’s proof of the “Pythagorean” the­o­rem.
Source: C. K. Raju (2007 p. 67)

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

Obviously, the proof of the “Pythagorean Theorem” is very easy if you are either (a) allowed to take mea­sure­ments or, equiv­a­lent­ly, (b) allowed to move fig­ures around in space.

C. K. Raju (2013 p. 167)

This process takes place in sev­er­al stages of mov­ing fig­ures, sim­i­lar to the mov­ing scales (or fig­ures rep­re­sent­ing scales) in the two-vina exper­i­ment. The 3 single-shru­ti tone inter­vals can be com­pared to the areas of the 3 squares in Yuktibhāşā. The fol­low­ing com­ment would there­fore apply to Bharata’s procedure:

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

C. K. Raju (2007 p. 67)

Bernard Bel — Dec. 2020


Arnold, E. J. A Mathematical mod­el of the Shruti-Swara-Grama-Murcchana-Jati System. New Delhi, 1982: Journal of the Sangit Natak Akademi.

Arnold, E.J.; Bel, B. A Scientific Study of North Indian Music. Bombay, 1983: NCPA Quarterly Journal, vol. XII Nos. 2 3.

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

Bel, B.; Bor, J. Intonation of North Indian Classical Music: work­ing with the MMA. Video on Dailymotion. Bombay, 1984: National Center for the Performing Arts.

Bel, B.; Bor, J. NCPA/ISTAR Research Collaboration. Bombay, 1985: NCPA Quarterly Journal, vol. XIV, No. 1, p. 45-53.

Bel, B. A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra. Note interne. Marseille, 1988a : Groupe Représentation et Traitement des Connaissances (CNRS).

Bel, B. Raga : approches con­ceptuelles et expéri­men­tales. Actes du col­loque “Structures Musicales et Assistance Informatique”. Marseille, 1988b.

Bharata. Natya Shastra. There is no cur­rent­ly avail­able English trans­la­tion of the first six chap­ters of Bharata’s Natya Shastra. However, most of the infor­ma­tion required for this inter­pre­ta­tion has been repro­duced and com­ment­ed by Śārṅgadeva in his Sangita Ratnakara (13th cen­tu­ry CE), trans­lat­ed by Dr R. K. Shringy, vol.I. Banaras 1978: Motilal Banarsidass.

Bose, N. D. Melodic Types of Hindustan. Bombay, 1960: Jaico.

Burkert, W. Lore and Science in Ancient Pythagoreanism. Cambridge MA, 1972: Harvard University Press.

Casey, J. The First Six Books of the Elements of Euclid, and Propositions I.-XXI. of Book VI. London, 1885: Longmans. Free e-book, Project Gutenberg.

Iyengar, R. N. Concept of Probability in Sanskrit Texts on Classical Music. Bangalore, 2017. Invited Talk at ICPR Seminar on “Science & Technology in the Indic Tradition: Critical Perspectives and Current Relevance”, I. I. Sc.

Kippen, J. Rhythmic Thought and Practice in the Indian Subcontinent. In R. Hartenberger & R. McClelland (Eds.), The Cambridge Companion to Rhythm (Cambridge Companions to Music, p. 241-260). Cambridge, 2020: Cambridge University Press. doi:10.1017/9781108631730.020

Levy, M. Intonation in North Indian Music. New Delhi, 1982: Biblia Impex.

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

Raju, C. K. Euclid and Jesus: How and why the church changed math­e­mat­ics and Christianity across two reli­gious wars. Penang (Malaysia), 2013: Multiversity, Citizens International.

Rao, S.; Van der Meer, W. The Construction, Reconstruction, and Deconstruction of Shruti. Hindustani music: thir­teenth to twen­ti­eth cen­turies (J. Bor), Manohar, New Delhi 2010.

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.

Leave a Reply

Your email address will not be published. Required fields are marked *