A multicultural model of consonance

A frame­work for tun­ing just-intonation scales via two series of fifths

For more than twen­ty cen­turies, musi­cians, instru­ment mak­ers and musi­col­o­gists fig­ured out scale mod­els and tun­ing pro­ce­dures for the cre­ation of music embody­ing the con­cept of “con­so­nance”. There was a shared notion of the octave and the major fifth (inter­val “C” to “G”) being the build­ing blocks of these mod­els, and the har­mon­ic major third (inter­val “C” to “E”) late­ly played a sig­nif­i­cant role in European baroque and clas­si­cal music.

Computer-controlled elec­tron­ic instru­ments open new avenues for the imple­men­ta­tion of micro­tonal­i­ty includ­ing just-intonation frame­works divid­ing the octave in more than 12 grades (https://bolprocessor.org/microtonality/). Throughout cen­turies, Indian art music claimed its adher­ence to a divi­sion of 22 inter­vals (the ṣruti-swara sys­tem) the­o­rized in Nāṭyaśāstra, a Sanskrit trea­tise dat­ing back between 400 BCE to 200 CE. Since con­so­nance (saṃvādī) is the basis of both ancient Indian and European tonal sys­tems, we felt the urge for a the­o­ret­i­cal frame­work encom­pass­ing all models.

Unfortunately, the top­ic of “just into­na­tion” is exposed in an alto­geth­er con­fus­ing and reduc­tive man­ner (read Wikipedia) due to musi­col­o­gists’ focus on inte­ger ratios aimed at reflect­ing the dis­tri­b­u­tion of high­er par­tials in peri­od­i­cal sounds. While these spec­u­la­tive mod­els of into­na­tion may respond to beliefs in the mys­ti­cal prop­er­ties of nat­ur­al num­bers — as claimed by Pythagoreanists — they were rarely checked against non-directed musi­cal prac­tice. Instrument tuners most­ly rely on their own audi­to­ry per­cep­tion of inter­vals rather than resort­ing to num­bers, despite the avail­abil­i­ty of “elec­tron­ic tuners”…

Interestingly, the ancient Indian the­o­ry of nat­ur­al scales does not rely on arith­metics. This should be sur­pris­ing giv­en that in Vedic times mathematicians/philosophers had laid out the foun­da­tions of cal­cu­lus and infin­i­tes­i­mals which much lat­er were 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 decrypt the mod­el depict­ed by the author(s) of Nāṭyaśāstra via a thought exper­i­ment: the two-vina exper­i­ment (https://bolprocessor.org/two-vina-experiment).

Earlier inter­pre­ta­tions of this mod­el, mim­ic­k­ing the Western habit of deal­ing with inter­vals as fre­quen­cy ratios, failed to explain the inter­val­ic struc­ture of ragas in Hindustani clas­si­cal music. In fact, the implic­it mod­el of Nāṭyaśāstra is a “flex­i­ble” one because the size of the major third (or equiv­a­lent­ly the pramāņa ṣru­ti) is not stat­ed in advance. Read Raga into­na­tion (https://bolprocessor.org/raga-intonation) and lis­ten to exam­ples to grasp the artic­u­la­tion between the the­o­ry and prac­tice of into­na­tion in this context.

In Europe, the har­mon­ic major third was final­ly accept­ed as a “celes­tial inter­val” after the Council of Trent (1545-1563) putting an end to the ban­ish­ment of poly­phon­ic singing in reli­gious gath­er­ings. Along with the devel­op­ment of fixed-pitch key­board instru­ments, this gave way to the elab­o­ra­tion of the­o­ret­i­cal mod­els and tun­ing pro­ce­dures attempt­ing to include this inter­val in “pure into­na­tion”. In the­o­ry, this is not fea­si­ble on a chro­mat­ic (12-grade scale) but it can be fig­ured out and applied to Western har­mo­ny if more grades (29 to 41) are per­mit­ted. However, choos­ing enhar­mon­ic posi­tions suit­able for a har­mon­ic con­text remains an uncer­tain venture.

Once again, the Indian mod­el came to the res­cue because it can be extend­ed to pro­duce a con­sis­tent series of twelve “opti­mal­ly con­so­nant” chro­mat­ic scales in com­pli­ance with chord inter­vals in Western har­mo­ny. Each scale con­tains 12 grades, which is more than the notes of chords it is applic­a­ble for. Sound exam­ples are pro­vid­ed to illus­trate this process (https://bolprocessor.org/just-intonation-framework/).

Tuning mechan­i­cal key­board instru­ments (church organ, harp­si­chord, pianoforte) for 12-grade scales made it nec­es­sary to dis­trib­ute unwant­ed dis­so­nance (the syn­ton­ic com­ma) in an accept­able man­ner over series of fifths and fourths. Many tem­pered tun­ing pro­ce­dures were designed, dur­ing the 16th to 19th cen­turies, with empha­sis on either “per­fect fifths” or “pure major thirds”, in response to the con­straints of spe­cif­ic musi­cal repertoires.

These tech­niques have been doc­u­ment­ed in detail by organ play­er and instru­ment design­er Pierre-Yves Asselin, along with meth­ods for achiev­ing the tun­ing on a mechan­i­cal instru­ment such as the harp­si­chord. His book Musique et tem­péra­ment (soon avail­able in English) was a guide­line for imple­ment­ing a sim­i­lar approach in the Bol Processor (https://bolprocessor.org/microtonality/). This frame­work will make it pos­si­ble to lis­ten to baroque and clas­si­cal works, using Csound instru­ments, with the very tun­ings their com­posers had been favor­ing — accord­ing to his­tor­i­cal sources.

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