The two-vina experiment

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

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

Accurate set­tings of the Shruti Harmonium (1980)

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

After sev­er­al years of exper­i­men­tal work, it had become clear that, even though the into­na­tion of Indian clas­si­cal music is far from a ran­dom process, it would be haz­ardous to assess an inter­pre­ta­tion of ancient scale the­o­ry with the aid of today’s musi­cal data. There at least three rea­sons for this:

  1. There are infi­nite­ly valid inter­pre­ta­tions of the ancient the­o­ry, as we will show.
  2. The con­cept of raga, i.e. the basic prin­ci­ple of Indian clas­si­cal music, appeared first in the lit­er­a­ture cir­ca 900 CE in Matanga’s Brihaddeshi, and it under­went a grad­ual devel­op­ment until 13th cen­tu­ry, when Sharangadeva enlist­ed 264 ragas in his Sangitratnakara.
  3. Drones were not in use at the time of Natya Shastra; the influ­ence of the drone on into­na­tion is con­sid­er­able, if not pre­dom­i­nant, in con­tem­po­rary music performance.

The ancient Indian the­o­ry of scales remains use­ful for its insight into ear­ly melod­ic clas­si­fi­ca­tion (the jāti sys­tem) which lat­er might have engen­dered the raga sys­tem. Therefore, it may be best envis­aged as a topo­log­i­cal descrip­tion of tonal struc­tures. Read Raga into­na­tion for a more detailed account of the­o­ret­i­cal and prac­ti­cal issues.

The top­ic 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 ren­der more comprehensive.

The historical context

Bharata Muni, the author(s) of Natya Shastra must have heard about the­o­ries of musi­cal scales attrib­uted to ancient Greeks. Indian schol­ars were in posi­tion to bor­row the mod­el and expand it con­sid­er­ably because of their deep under­stand­ing of mathematics.

Readers of C.K. Raju — notably his out­stand­ing work Cultural Foundations of Mathematics (2007) — are aware 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, lat­er export­ed by Jesuit priests from Kerala to Europe 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.

Therefore, 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 for­mal­ized with alge­bra rather than a finite 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 may be read as a thought exper­i­ment rather than a process involv­ing phys­i­cal objects. There is no cer­ti­tude that these two vina-s ever exist­ed — and even that “Bharata Muni”, the author/experimenter, was a unique per­son. Their approach is an instance of a val­i­da­tion (pramāņa) resort­ing to empir­i­cal proof, in oth­er words dri­ven by the phys­i­cal­ly man­i­fest (pratyakşa) rather than inferred from “axioms” con­sti­tu­tive of a the­o­ret­i­cal mod­el. This may be summed up as “pre­fer­ring physics to metaphysics”.

Constructing and manip­u­lat­ing vina-s in the man­ner indi­cat­ed by the exper­i­menter seems an insur­montable tech­no­log­i­cal chal­lenge. This has been a sub­ject of dis­cus­sion among a num­ber of authors — read Iyengar (2017 pages 7-sq.) Leaving aside the pos­si­bil­i­ty of a prac­ti­cal real­iza­tion is not a denial of phys­i­cal real­i­ty, as for­mal math­e­mat­ics would sys­tem­at­i­cal­ly man­date. Calling it a “thought exper­i­ment” is a way of assert­ing the link with the phys­i­cal mod­el. In the same man­ner, using cir­cu­lar graphs to rep­re­sent tun­ing schemes and alge­bra to describe rela­tions between inter­vals are aids to under­stand­ing which do not reduce the mod­el to spe­cif­ic, ide­al­is­tic inter­pre­ta­tions sim­i­lar to spec­u­la­tions on inte­ger num­bers cher­ished by Western schol­ars. These graphs aim at facil­i­tat­ing a com­pu­ta­tion­al design of instru­ments mod­el­ing these imag­ined instru­ments — read Raga into­na­tion and Just into­na­tion, a gen­er­al frame­work.

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

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-grade scale do, re, mi, fa, sol, la, si (“C”, “D”, “E”, “F”, “A”, “B”) which some schol­ars have done despite the faulty inter­pre­ta­tion of intervals.

Intervals are notat­ed in shru­ti-s which may, for a start, be tak­en as an order­ing sys­tem rather than a unit of mea­sure­ment. The exper­i­ment will con­firm that a four-shru­ti inter­val is larg­er than a three-shru­ti, a three-shru­ti larg­er than two-shru­ti and the lat­ter larg­er than a sin­gle shru­ti. In dif­fer­ent con­texts the word “shru­ti” des­ig­nates note posi­tions instead of 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 enlists 9-shru­ti (con­so­nant) inter­vals: “Sa-Pa”, “Sa-Ma”, “Ma-Ni”, “Ni-Ga” and “Re-Dha”. In addi­tion, he defines anoth­er scale named “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 of 9 shru­ti-s.

Intervals of 9 or 13 shru­ti-s are declared “con­so­nant” (sam­va­di). Leaving out 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 dif­fer­ing from 3/2 gen­er­ates 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 fre­quen­cy ratios are expressed log­a­rith­mi­cal­ly with 1200 cents rep­re­sent­ing one octave, and fur­ther con­vert­ed to angles with a full octave on a cir­cle, the descrip­tion of Sa-grama and Ma-grama scales may be sum­ma­rized in a cir­cu­lar dia­gram (see picture).

Two cycles of fifths have been high­light­ed in red and green col­ors. Note that both the “Sa-Ma” and “Ma-Ni” inter­vals are per­fect fifths, dis­card­ing the asso­ci­a­tion of Sa-grama with the con­ven­tion­al Western scale: the “Ni” should be mapped to “B flat”, not to “B”. Further, the “Ni-Ga” per­fect fifth implies that “Ga” is also “E flat” rather than “E”. The Sa-grama and Ma-grama scales are there­fore “D modes”. For this rea­son, “Ga” and “Ni” appear under­lined on the diagrams.

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

We must avoid pre­ma­ture con­clu­sions about 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 with the syn­ton­ic com­ma (fre­quen­cy ratio 81/80). Doing so reduces Bharata’s mod­el to “just into­na­tion”, indeed with inter­est­ing prop­er­ties in its appli­ca­tion to Western har­mo­ny (read page), but with ques­tion­able rel­e­vance to the prac­tice of Indian music. As claimed 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 major epis­temic rel­e­vance and deserves a brief expla­na­tion. The seman­tics of “slack­ness or tense­ness” clear­ly belongs to “pratyakṣa pramāṇa”, the means of acquir­ing knowl­edge by per­cep­tu­al expe­ri­ence. More pre­cise­ly, “pramāṇa (प्रमाण) refers to “valid per­cep­tion, mea­sure and struc­ture”” (Wisdom Library), a notion of proof shared by all Indian tra­di­tion­al schools of phi­los­o­phy (Raju 2007 page 63). We will get back 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 instance “Ni-Re”. If the pramāņa ṣru­ti were a syn­ton­ic com­ma then this inter­val would be a har­mon­ic major third with ratio 5/4. As evoked 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 had been rat­ed “asso­nant” (anu­va­di).

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

Picture above rep­re­sents the two vina-s tuned iden­ti­cal­ly on Sa-grama. Matching notes are marked by yel­low spots. The inner part of the blue cir­cle will be the mov­able vina in the fol­low­ing trans­po­si­tion process­es, 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 brief, this is a pro­ce­dure for low­er­ing all notes of the mov­able vina by one pramāņa ṣru­ti. First low­er its “Pa” — e.g. make it con­so­nant with “Re” of the fixed vina — to obtain Ma-grama on the mov­able vina. Then read­just its whole scale to obtain Sa-grama. Note that low­er­ing “Re” and “Dha” implies appre­ci­at­ing again the size of a pramāņa ṣru­ti while pre­serv­ing 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 are no more match­ing notes between the two vina-s.

Interpreting shruti-s as vari­ables in some metrics

This sit­u­a­tion can be trans­lat­ed to alge­bra. Let “a”, “b”, “c” … “v” be the unknown sizes of shru­ti-s in the scale (see pic­ture on the side). A met­rics “trans­lat­ing” Bharata’s mod­el will be nec­es­sary for check­ing it on sound struc­tures pro­duced by an elec­tron­ic instru­ment — the com­put­er. The scope of this trans­la­tion remains valid as long as no extra asser­tion has been stat­ed which is not root­ed in the orig­i­nal model.

Using sym­bol “#>” to indi­cate that two notes are not match­ing, this first low­er­ing may be sum­ma­rized by the fol­low­ing set of inequations:

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 again a low­er­ing by one shru­ti with 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” are merged, 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 get an equa­tion report­ing 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 keep in mind that the author is describ­ing a phys­i­cal process, not an abstract “move­ment” by which the move­able wheel (or vina) would “jump in space” from its ini­tial to final posi­tion. Therefore we pay atten­tion to things hap­pen­ing and not hap­pen­ing dur­ing the tun­ing of the vina, or rota­tion of the wheel, look­ing at the tra­jec­to­ries of dots rep­re­sent­ing note posi­tions (along the blue cir­cle). Things not hap­pen­ing (non-matching notes) yield inequa­tions required for mak­ing sense of the alge­bra­ic model.

This step of the exper­i­ment con­firms that it is wrong to locate Sa at the posi­tion of Ni for the sake of iden­ti­fy­ing Sa-grama with the Western scale. In this case, match­ing notes would no 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, con­straints are sum­ma­rized 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 num­bers of shru­ti-s denote an order­ing of the sizes of inter­vals between notes.

Still, we have 22 unknown vari­ables and only 4 equa­tions. These vari­ables can be “packed” to a set of 8 unknown vari­ables which rep­re­sent the “macro-intervals”, i.e. the steps of the gra­ma-s. In this approach, shru­ti-s are sort of “sub­atom­ic” par­ti­cles which these “macro-intervals” are made of… Now we need only 4 aux­il­iary equa­tions to deter­mine the scale. These may be pro­vid­ed by acoustic infor­ma­tion, with inter­vals mea­sured in cents. First we express that the sum of the vari­ables, the octave, is equal to 1200 cents: 

(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 rela­tion­ships 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

S1O, S11 and S12 can all be derived from S9. These equa­tions may there­fore be dis­card­ed. We still need one more equa­tion to solve the sys­tem. At this stage there are many options asso­ci­at­ed with tun­ing pro­ce­dures. As sug­gest­ed above, set­ting the har­mon­ic major third to ratio 5/4 (386.3 cents) would pro­vide the miss­ing equa­tion. This amounts to set­ting vari­able “m” to 21.4 cents (syn­ton­ic com­ma). However, this major third can take any val­ue 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 great amount of pos­si­bil­i­ties includ­ing the tem­pera­ment of some inter­vals which musi­cians might achieve spon­ta­neous­ly in par­al­lel melod­ic move­ments. A set of solu­tions is exposed in A Mathematical Discussion of the Ancient Theory of Scales accord­ing to Natyashastra and a few of them have been tried on the Bol Processor to check musi­cal exam­ples for which they might pro­vide ade­quate scales — read Raga into­na­tion.

Extensions of the model

In order to com­plete his sys­tem of scales, Bharata had to intro­duce two new notes in 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 inves­ti­gate the behav­ior of the new scale in all trans­po­si­tions (mur­ccha­na-s), includ­ing those start­ing with “Ga” and “Ni”, and infer equa­tions cor­re­spond­ing to an opti­mal con­so­nance of the scale. We end up with 11 equa­tions for only 10 unknown vari­ables, which means that this per­fec­tion can­not be achieved. One con­straint must be released.

An option is to release con­straints on major thirds, fifths or octaves, lead­ing to a form of tem­pera­ment. For instance, stretch­ing the octave by 3.7 cents gen­er­ates 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 come as close as pos­si­ble to “just into­na­tion” with­out mod­i­fy­ing per­fect fifths and octaves. This is pos­si­ble if the com­ma (vari­able “m”) is allowed an arbi­trary 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-grade scale named “gra­ma” which we use as a frame­work for con­so­nant chro­mat­ic scales eli­gi­ble 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 page Raga into­na­tion.

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 thought exper­i­ment because of the unlike­li­hood that it could be worked with mechan­i­cal instru­ments. Representing it on a cir­cu­lar graph (a move­able wheel inside a fixed crown) achieves the same goal with­out resort­ing to imag­i­nary devices.

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

For instance, the image above was used to describe the ţhekkā (cycle of quasi-onomatopoeic syl­la­bles rep­re­sent­ing the drum strokes) of tāl Pañjābi which reads as follows:

Unfortunately, ear­ly print­ing press tech­nol­o­gy may have ren­dered uneasy the pub­li­ca­tion and trans­mis­sion of these aids to learning.

If con­tem­po­raries of Bharata ever used sim­i­lar cir­cu­lar rep­re­sen­ta­tions for reflect­ing on musi­cal scales, we guess that arche­o­log­i­cal traces might not be iden­ti­fied prop­er­ly as their draw­ings could 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 inferred from empir­i­cal obser­va­tions over a pro­claimed uni­ver­sal log­ic aimed at estab­lish­ing “irrefragable demonstrations”.

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 likened 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 (ca. 1530 CE), a fig­ure of a right-angle tri­an­gle is drawn on a palm leaf with squares on its two sides and its hypothenuse. Then the fig­ure is cut and rotat­ed in a way high­light­ing that the areas are equal.

Clearly, the proof of the “Pythagorean the­o­rem” is very easy if one is either (a) allowed to make mea­sure­ments, or, equiv­a­lent­ly (b) allowed to move fig­ures about in space.

C. K. Raju (2013 p. 167)

This process takes place on sev­er­al steps of mov­ing fig­ures in a way sim­i­lar to mov­ing scales (or fig­ures rep­re­sent­ing scales) in the two-vina exper­i­ment. The 3 single-shru­ti tonal inter­vals may be likened to the areas of the 3 squares in Yuktibhāşā. The fol­low­ing remark 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


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