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

Abstract

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