Mathematical Models for Binarization and Ternarization of Musical Rhythm

Musical cyclic rhythms with a cycle length (timespan) of 8 or 16 pulses are called binary; those with 6 or 12 pulses are called ternary. The process of mapping a ternary rhythm of, say 12 pulses, to a rhythm of 16 pulses, such that musicologically salient properties are preserved is termed binarization. By analogy, the converse process of mapping a binary rhythm to a ternary rhythm is referred to as ternarization. New algorithms are proposed and investigated for the binarization and ternarization of musical rhythms with the goal of understanding the historical evolution of traditional rhythms through inter-cultural contacts. The algorithms also have applications to automated rhythmic pattern generation, and may be incorporated in composition software tools.

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