Arithmetic procedures: Least common multiple and canonical forms

Take two different musical fragments, say measures, a and b. If the first lasts three time units and the second four units (for instance if the first is in 3/4 and the second in 4/4), we can repeat a four times and b three times, so that if we start playing them simultaneously, after 12 time units the whole structure will be repeated, as illustrated in the following scheme:

Least Common Multiple diagram

That is, we have ‘vertical’ structures that repeat after the least common multiple of the durations of a and b. We could stack a third structure of length 7, producing a combined structure of length 3 x 4 x 7 = 84.

This simple idea could be easily applied to generate a minimalist polyphonic composition which could be called a ‘minimal canon’. These procedures can also (or instead) be applied to dynamic, tonal, modulatory, or tone-colour schemes.

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