L'intonation juste dans la théorie ancienne de l'Inde : ses applications aux musiques modale et harmonique

E. James Arnold

Revue de musi­colo­gie, JSTOR, 71^e (1-2), p.11-38. Traduction : Bernard Bel.

👉  Cited on page Just into­na­tion: a gen­er­al framework

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Résumé

La théorie de l'intonation juste basée sur deux gammes fon­da­men­tales (grama-s), telle que la décrivent le Natya Shastra de Bharata et d'autres traités musi­cologiques anciens en san­scrit, for­malise les rela­tions internes des gammes dia­toniques avec une éton­nante pré­ci­sion. Quelques mod­i­fi­ca­tions min­imes suff­isent à l'adapter aux gammes non-diatoniques famil­ières de la musique indi­enne con­tem­po­raine. Cet arti­cle émet l'hypothèse que la théorie de Bharata pro­pose une meilleure base psy­choa­cous­tique rationnelle que les expli­ca­tions actuelles pour ce qui con­cerne les heures d'interprétation des ragas. Le mod­èle math­é­ma­tique présen­té ici, d'un sys­tème de rela­tions inter­valliques dans les gammes dia­toniques et celles qui en dérivent, est un out­il pra­tique pour étudi­er les rela­tions plus en pro­fondeur. La dis­cus­sion débouche sur une propo­si­tion d'application du sys­tème indi­en à la musique har­monique en into­na­tion juste.

Excerpts of an AI review of this paper (Claude AI, June 2025)

This schol­ar­ly paper by E.J. Arnold presents a fas­ci­nat­ing explo­ration of ancient Indian music the­o­ry and its poten­tial appli­ca­tions to both modal and har­mon­ic musi­cal sys­tems. The work stands as a sig­nif­i­cant con­tri­bu­tion to com­par­a­tive musi­col­o­gy, bridg­ing Eastern and Western the­o­ret­i­cal frame­works through rig­or­ous math­e­mat­i­cal mod­el­ing and exper­i­men­tal validation.

Theoretical Framework and Methodology

Arnold's cen­tral the­sis revolves around the ancient Indian grāma-mūrcchana sys­tem, a sophis­ti­cat­ed the­o­ret­i­cal con­struct involv­ing fun­da­men­tal scales (grāma-s) and their modal trans­for­ma­tions (mūrcchana-s). The author devel­ops an inno­v­a­tive math­e­mat­i­cal mod­el using a cir­cu­lar com­pu­ta­tion­al disc that visu­al­izes the rela­tion­ships between tonal posi­tions with­in the 22-śru­ti micro­ton­al sys­tem described in Sanskrit trea­tis­es like the Nāṭyaśāstra and Dattilam.

The paper's strength lies in its method­i­cal approach, sys­tem­at­i­cal­ly exam­in­ing how this ancient sys­tem can pro­vide coher­ent expla­na­tions for har­mon­ic rela­tion­ships in both Indian clas­si­cal music and Western tonal har­mo­ny. Arnold's use of Jacques Dudon's sym­bol­ic nota­tion sys­tem for rep­re­sent­ing micro­ton­al inter­vals demon­strates schol­ar­ly rig­or and atten­tion to cross-cultural the­o­ret­i­cal precision.

Historical Context and Scholarly Significance

Arnold effec­tive­ly con­tex­tu­al­izes the research with­in broad­er musi­co­log­i­cal schol­ar­ship, acknowl­edg­ing the work of pio­neers like Sir William Jones and more recent con­tri­bu­tions from schol­ars such as Bharata com­men­ta­tors and con­tem­po­rary Indian musi­col­o­gists. The paper address­es a crit­i­cal gap in under­stand­ing how ancient the­o­ret­i­cal sys­tems might inform mod­ern musi­cal prac­tice, par­tic­u­lar­ly giv­en the his­tor­i­cal dis­rup­tion of liv­ing oral traditions.

The dis­cus­sion of the rāga system's rela­tion­ship to time cycles (sand­hiprakāśa) rep­re­sents par­tic­u­lar­ly valu­able schol­ar­ship. Arnold's analy­sis of how spe­cif­ic rāga-s cor­re­spond to par­tic­u­lar hours of the day or night, sup­port­ed by sys­tem­at­ic tab­u­la­tion of 85 rāga-s with their appro­pri­ate per­for­mance times, pro­vides empir­i­cal ground­ing for what has often remained in the realm of cul­tur­al speculation.

Mathematical Innovation and Practical Applications

The paper's most com­pelling con­tri­bu­tion lies in its math­e­mat­i­cal mod­el­ing of the śruti-swara-grāma-mūrcchana sys­tem. The cir­cu­lar disc rep­re­sen­ta­tion allows for imme­di­ate visu­al­iza­tion of har­mon­ic rela­tion­ships and demon­strates how ancient Indian the­o­ry antic­i­pat­ed many con­cepts lat­er devel­oped in Western har­mon­ic analy­sis. Arnold's exper­i­men­tal work with elec­tron­ic instru­ments, includ­ing the śru­ti har­mo­ni­um devel­oped by Bernard Bel, pro­vides cru­cial empir­i­cal val­i­da­tion of the­o­ret­i­cal predictions.

The appli­ca­tion to Western clas­si­cal har­mo­ny rep­re­sents ground­break­ing cross-cultural musi­col­o­gy. Arnold demon­strates how major and minor scales can be under­stood with­in the grāma-mūrcchana frame­work, reveal­ing pre­vi­ous­ly unrec­og­nized con­nec­tions between Eastern and Western the­o­ret­i­cal sys­tems. The analy­sis of mod­u­la­tion pro­ce­dures using the ancient Indian sys­tem offers fresh per­spec­tives on famil­iar har­mon­ic progressions.

Limitations and Areas for Development

While Arnold acknowl­edges the study's lim­i­ta­tions with­in "musi­cal geom­e­try," the paper could ben­e­fit from more exten­sive dis­cus­sion of prac­ti­cal imple­men­ta­tion chal­lenges. The the­o­ret­i­cal ele­gance of the grāma-mūrcchana sys­tem con­trasts with the prac­ti­cal dif­fi­cul­ties con­tem­po­rary musi­cians face when attempt­ing to real­ize these micro­ton­al rela­tion­ships on tra­di­tion­al instruments.

Additionally, while the paper excel­lent­ly demon­strates the­o­ret­i­cal cor­re­spon­dences between ancient Indian and Western sys­tems, it could explore more deeply the aes­thet­ic and cul­tur­al impli­ca­tions of these con­nec­tions. The rela­tion­ship between math­e­mat­i­cal pre­ci­sion and musi­cal expres­sion deserves fur­ther investigation.

Contemporary Relevance and Future Directions

Arnold's work antic­i­pates impor­tant devel­op­ments in con­tem­po­rary music the­o­ry, par­tic­u­lar­ly in the grow­ing inter­est in micro­ton­al and cross-cultural approach­es to har­mo­ny. The paper's sys­tem­at­ic approach to under­stand­ing non-equal-tempered sys­tems has impli­ca­tions for elec­tron­ic music com­po­si­tion, world music fusion, and the devel­op­ment of new the­o­ret­i­cal frame­works for glob­al musi­cal understanding.

The research also con­tributes to broad­er dis­cus­sions about the uni­ver­sal­i­ty of musi­cal prin­ci­ples ver­sus cul­tur­al speci­fici­ty, sug­gest­ing that math­e­mat­i­cal rela­tion­ships under­ly­ing musi­cal sys­tems may tran­scend cul­tur­al bound­aries while main­tain­ing dis­tinct aes­thet­ic identities.

Conclusion

This paper rep­re­sents exem­plary schol­ar­ship in com­par­a­tive musi­col­o­gy, suc­cess­ful­ly bridg­ing ancient Indian music the­o­ry and con­tem­po­rary ana­lyt­i­cal meth­ods. Arnold's math­e­mat­i­cal mod­el­ing pro­vides a robust frame­work for under­stand­ing com­plex micro­ton­al rela­tion­ships, while the prac­ti­cal appli­ca­tions demon­strate the con­tin­ued rel­e­vance of ancient the­o­ret­i­cal sys­tems. The work opens impor­tant avenues for future research in cross-cultural music the­o­ry and pro­vides valu­able tools for both schol­ars and prac­ti­tion­ers inter­est­ed in expand­ing their har­mon­ic vocab­u­lary beyond Western equal temperament.

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