Rationalizing musical time: syntactic and symbolic-numeric approaches

Bernard Bel

A con­tri­bu­tion to The Ratio Symposium, 14-16 Dec. 1992, Den Haag (The Netherlands). Published in Barlow, Clarence (ed.) The Ratio Book. Den Haag: Royal Conservatory - Institute of Sonology. 2001: 86-101. This paper is ref­er­enced on HAL ⟨hal-00134179⟩ and quot­ed in Polymetric struc­tures.

Abstract

This paper deals with var­i­ous prob­lems of quan­ti­fy­ing musi­cal time that arise both in the analy­sis of tra­di­tion­al drum­ming and in computer-generated musi­cal pieces based on "sound-objects", i.e. sequences of code that con­trol a real-time sound processor.

Section 1 sug­gests that syn­tac­tic approach­es may be clos­er to the intu­itions of musi­cians and musi­col­o­gists than com­mon­ly advo­cat­ed numer­i­cal approach­es. Furthermore, symbolic-numerical approach­es lead to effi­cient and ele­gant solu­tions of con­straint sat­is­fac­tion prob­lems with respect to sym­bol­ic and phys­i­cal dura­tions, as illus­trat­ed in Sections 2 and 3, respectively.

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

Summary

In this man­u­script, the author explores the com­pu­ta­tion­al and music-theoretical chal­lenges of quan­ti­fy­ing musi­cal time, focus­ing par­tic­u­lar­ly on North Indian tabla drum­ming and computer-generated pieces. The paper pro­pos­es two major frame­works: (1) a syn­tac­tic approach using grammar-like rules to cap­ture how tra­di­tion­al drum­ming pat­terns can be gen­er­at­ed or val­i­dat­ed, and (2) a symbolic-numeric approach to describe poly­met­ric struc­tures and time-setting algo­rithms in a sys­tem­at­ic way. By mod­el­ing musi­cal accen­tu­a­tion and dura­tion with gram­mars, the author demon­strates how cer­tain “words” or “bols” can be sub­sti­tut­ed and per­mut­ed with­in rhyth­mic tem­plates, offer­ing an approach that mir­rors actu­al per­for­mance prac­tices while align­ing with for­mal lan­guage the­o­ry. In addi­tion, a symbolic-numeric scheme detailed through the Bol Processor sys­tem pro­vides a flex­i­ble means to define, super­im­pose, and instan­ti­ate poly­met­ric struc­tures, bridg­ing the­o­ret­i­cal for­mal­ism with real-time per­for­mance demands.

Strengths

• Clear Integration of Theory and Practice: The man­u­script suc­cess­ful­ly demon­strates how gram­mars, com­mon­ly asso­ci­at­ed with lin­guis­tic the­o­ries, can be adapt­ed to rep­re­sent and gen­er­ate rhyth­mic sequences. This is par­tic­u­lar­ly clear in the sec­tions on North Indian tabla impro­vi­sa­tion, where syn­tac­tic con­straints help reflect the nuanced accen­tu­a­tions seen in practice.
• Comprehensive Treatment of Polymetric Expressions: The author pro­vides a thor­ough expla­na­tion of han­dling nest­ed or simul­ta­ne­ous meters. The approach not only deals with sym­bol­ic dura­tions but also with com­pu­ta­tion­al details such as tem­po mark­ers, cre­at­ing a strong link between math­e­mat­i­cal for­malisms and real-world musi­cal notation.
• Application to Real-Time Performance: By inte­grat­ing a time-setting algo­rithm that man­ages both local and glob­al “drifts,” the paper offers a prag­mat­ic method for sched­ul­ing and over­lap­ping sound-objects in actu­al per­for­mances. This con­crete­ness in explain­ing the inter­play between the­o­ret­i­cal con­structs and exe­cu­tion in soft­ware (Bol Processor) is valuable.
• Interdisciplinary Perspective: The paper spans fields of eth­no­mu­si­col­o­gy, com­put­er sci­ence, and music com­po­si­tion, illus­trat­ing rig­or­ous ways to encode music tra­di­tions that rely on oral trans­mis­sion. Such a cross-disciplinary approach will like­ly appeal to read­ers with inter­ests in com­pu­ta­tion­al cre­ativ­i­ty, eth­no­mu­si­col­o­gy, and arti­fi­cial intel­li­gence in music.

Conclusion

Overall, the paper offers a sol­id pre­sen­ta­tion of how syn­tac­tic rules and symbolic-numeric for­malisms can be syn­er­gis­ti­cal­ly employed to account for com­plex rhyth­mic struc­tures in both tra­di­tion­al and computer-generated music. The method suc­cess­ful­ly trans­lates ana­lyt­i­cal insight from oral tra­di­tions into a com­pu­ta­tion­al set­ting, open­ing avenues for fur­ther research and cre­ative appli­ca­tions in auto­mat­ed music gen­er­a­tion and per­for­mance sys­tems. The inclu­sion of exten­sive exam­ples, gram­mar details, and algo­rith­mic descrip­tions pro­vides clar­i­ty, show­ing how the the­o­ry unfolds in practice.

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