Two algorithms

Two algo­rithms for the instan­ci­a­tion of struc­tures of musi­cal objects

This is an extend­ed and revised ver­sion of the paper: Symbolic and Sonic Representations of Sound-Object Structures pub­lished in M. Balaban, K. Ebcioglu & O. Laske (Eds.) “Understanding Music with AI: Perspectives on Music Cognition”, AAAI Press (1992, pp.64-109).

A rep­re­sen­ta­tion­al mod­el of dis­crete struc­tures of musi­cal objects at the sym­bol­ic and sono­log­i­cal lev­els is intro­duced. This mod­el is being used for the design of com­put­er tools for rule-based musi­cal com­po­si­tion in which the low-level musi­cal objects are not notes, but “sound-objects”, there­by mean­ing arbi­trary sequences of mes­sages dis­patched to a real-time dig­i­tal sound processor.

“Polymetric expres­sions” are string rep­re­sen­ta­tions of con­cur­rent process­es that can eas­i­ly be han­dled by for­mal gram­mars. These expres­sions may not con­tain all the infor­ma­tion need­ed for syn­chro­niz­ing the whole struc­ture of sound-objects, i.e. deter­min­ing their strict order­ing on (sym­bol­ic) time. In response to this, the notion of “sym­bol­ic tem­po” is intro­duced: order­ing all objects in a struc­ture is pos­si­ble once their sym­bol­ic tem­pos are known. Rules for assign­ing sym­bol­ic tem­pos to objects are there­fore pro­posed. These form the basis of an algo­rithm inter­pret­ing incom­plete poly­met­ric expres­sions. The rel­e­vant fea­tures of this inter­pre­ta­tion are commented.

An exam­ple is giv­en to illus­trate the advan­tage of using (incom­plete) poly­met­ric rep­re­sen­ta­tions instead of con­ven­tion­al music nota­tion or event tables when the com­plete descrip­tion of the musi­cal piece and/or its vari­ants calls for dif­fi­cult com­pu­ta­tions of durations.

Given a strict order­ing of sound-objects sum­ma­rized in a “phase table” rep­re­sent­ing the com­plete poly­met­ric expres­sion, the next step is to cal­cu­late the dates at which mes­sages should be dis­patched. This requires a descrip­tion of “sound-object pro­to­types” along with their metric/topological prop­er­ties, and var­i­ous para­me­ters relat­ed to the musi­cal per­for­mance (e.g. “smooth” or “stri­at­ed” time, tem­po, etc.). These prop­er­ties are dis­cussed in detail and a time-polynomial constraint-satisfaction algo­rithm for the time-setting of sound-objects in a poly­met­ric struc­ture is intro­duced. Typical exam­ples com­put­ed by this algo­rithm are shown and discussed.

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