TERNARY RHYTHM NOTATION: A NOVEL GRAMMATICAL CONSTRUCTION OF MUSICAL RHYTHM OVER AN ALGEBRAIC LATTICE

摘要

We present a novel musical notation, generated by a formal grammar, which symbolizes musical rhythms asudstrings over a ternary alphabet. This notative system is shown to be an improvement over the conventionaludbinary rhythm notation, because it uniquely encodes any rhythm given in classical, sheet music notation.udWe leverage this grammatical system to compute the number of distinct rhythms up to a specified timeunit.udWe then define a binary operation on the strings generated by the rhythm grammar to coincide withudthe musical notion of ”additive rhythm.” We show that the set of musical rhythms with a given time-unitudbecomes a finite commutative monoid under this operation, and moreover that this operation imbues the setudwith an algebraic lattice structure. Finally, we use the algebraic properties of this representation of musicaludrhythms to define a number of similarity measures between rhythms, which we suggest to have applicationsudin automated music transcription.