A multicultural model of consonance

A frame­work for tun­ing just-intonation scales via two series of fifths
Image cre­at­ed by Bol Processor based on a mod­el by Pierre-Yves Asselin

For more than twen­ty cen­turies, musi­cians, instru­ment mak­ers and musi­col­o­gists have devised scale mod­els and tun­ing pro­ce­dures to cre­ate tonal music that embod­ies the con­cept of “con­so­nance”.

This does not mean that every style of tonal music aims to achieve con­so­nance. This con­cept is most explic­it in the design and per­for­mance of North Indian raga and Western har­mon­ic music.

There was a com­mon idea that the octave and the major fifth (inter­val ‘C’ to ‘G’) were the build­ing blocks of these mod­els, and the har­mon­ic major third (inter­val ‘C’ to ‘E’) has recent­ly played an impor­tant role in European music.

Computer-controlled elec­tron­ic instru­ments are open­ing up new avenues for the imple­men­ta­tion of micro­tonal­i­ty, includ­ing just into­na­tion frame­works that divide the octave into more than 12 degrees - see the Microtonality page. For cen­turies, Indian art music claimed to adhere to a divi­sion of 22 inter­vals (the ṣruti-swara sys­tem) the­o­rised in the Nāṭyaśāstra, a Sanskrit trea­tise dat­ing from between 400 BCE and 200 CE. Since con­so­nance (saṃvādī) is the basis of both ancient Indian and European tonal sys­tems, we felt the need for a the­o­ret­i­cal frame­work that encom­passed both models.

Unfortunately, the sub­ject of “just into­na­tion” is pre­sent­ed in a whol­ly con­fus­ing and reduc­tive man­ner (read Wikipedia), with musi­col­o­gists focus­ing on inte­ger ratios that reflect the dis­tri­b­u­tion of high­er par­tials in peri­od­ic sounds. While these spec­u­la­tive mod­els of into­na­tion may sup­port beliefs in the mag­i­cal prop­er­ties of nat­ur­al num­bers — as claimed by Pythagoreanists — they have rarely been teste against undi­rect­ed musi­cal prac­tice. Instrument tuners rely on their own audi­to­ry per­cep­tion of inter­vals rather than on num­bers, despite the avail­abil­i­ty of “elec­tron­ic tuners”…

Interestingly, the ancient Indian the­o­ry of nat­ur­al scales did not rely on arith­metic. This may be sur­pris­ing giv­en that in Vedic times mathematicians/philosophers had laid out the foun­da­tions of cal­cu­lus and infin­i­tes­i­mals which were much lat­er export­ed from Kerala to Europe and borrowed/appropriated by European schol­ars — read C.K. Raju’s Cultural Foundations of Mathematics: the nature of math­e­mat­i­cal proof and the trans­mis­sion of the cal­cu­lus from India to Europe in the 16th c. CE. This epis­te­mo­log­i­cal para­dox was an incen­tive to deci­pher the mod­el pre­sent­ed by the author(s) of the Nāṭyaśāstra by means of a thought exper­i­ment: the two-vina exper­i­ment.

Earlier inter­pre­ta­tions of this mod­el, mim­ic­k­ing the Western habit of treat­ing inter­vals as fre­quen­cy ratios, failed to explain the inter­val­ic struc­ture of ragas in Hindustani clas­si­cal music. In real­i­ty, the implic­it mod­el of the Nāṭyaśāstra is a ‘flex­i­ble’ one because the size of the major third (or equiv­a­lent­ly the pramāņa ṣru­ti) is not pre­de­ter­mined. Read the page on Raga into­na­tion and lis­ten to the exam­ples to under­stand the con­nec­tion between the the­o­ry and prac­tice of into­na­tion in this context.

In Europe, the har­mon­ic major third was final­ly accept­ed as a “heav­en­ly inter­val” after the Council of Trent (1545-1563), end­ing the ban on poly­phon­ic singing in reli­gious gath­er­ings. Major chords—  such as {C, E, G} — are vital ele­ments of Western har­mo­ny, and play­ing a major chord with­out unwant­ed beats requires the sim­plest fre­quen­cy ratio (5/4) for the har­mon­ic major third {C, E}.

A key­board with 19 keys per octave (from “A” to “a”) key­board designed by Gioseffo Zarlino (1517-1590) (source)

With the devel­op­ment of fixed-pitch key­board instru­ments, the search for con­so­nant inter­vals gave way to the elab­o­ra­tion of the­o­ret­i­cal mod­els (and tun­ing pro­ce­dures) that attempt­ed to per­form this inter­val in “pure into­na­tion”. Theoretically, this is not pos­si­ble on a chro­mat­ic scale (12 degrees), but it can be worked out and applied to Western har­mo­ny if more degrees (29 to 41) are allowed. Nevertheless, the choice of enhar­mon­ic posi­tions suit­able for a har­mon­ic con­text remains an uncer­tain proposition.

Once again, the Indian mod­el comes to the res­cue, because it can be extend­ed to pro­duce a con­sis­tent series of twelve “opti­mal­ly con­so­nant” chro­mat­ic scales, cor­re­spond­ing to chord inter­vals in west­ern har­mo­ny. Each scale con­tains 12 degrees, which is more than the notes of the chords to which it applies. Sound exam­ples are pro­vid­ed to illus­trate this process — see the Just into­na­tion: a gen­er­al frame­work page.

The tun­ing of mechan­i­cal key­board instru­ments (church organ, harp­si­chord, pianoforte) to 12-degree scales made it nec­es­sary to dis­trib­ute unwant­ed dis­so­nances (the syn­ton­ic com­ma) over series of fifths and fourths in an accept­able man­ner. From the 16th to the 19th cen­turies, many tem­pered tun­ing sys­tems were devel­oped in response to the con­straints of par­tic­u­lar musi­cal reper­toires, with an empha­sis on either “per­fect fifths” or “pure major thirds”.

These tech­niques have been exten­sive­ly doc­u­ment­ed by the organ­ist and instru­ment builder Pierre-Yves Asselin, along with meth­ods for achiev­ing into­na­tion on a mechan­i­cal instru­ment such as the harp­si­chord. His book Musique et tem­péra­ment (Paris: Jobert, 2000, to be pub­lished in English) served as a guide for imple­ment­ing a sim­i­lar approach in the Bol Processor — see the pages Microtonality, Creation of just-intonation scales and Comparing tem­pera­ments. This frame­work should make it pos­si­ble to lis­ten to Baroque and clas­si­cal work­son Csound instru­ments in the tun­ings intend­ed by their com­posers, accord­ing to his­tor­i­cal sources.

Sadly, Pierre-Yves Asselin left this world on 4 December 2023. We hope that the English trans­la­tion of his ground­break­ing work will be com­plet­ed soon.

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