Stacking

In the context of tuning theory, stacking is the group operation of a free abelian group of intervals. It corresponds to multiplying or dividing the pitch ratios corresponding to the intervals, or adding or subtracting their cent values. Depending on context, octave reduction (or the analogue for other equaves) is sometimes assumed. In a rank-n temperament, there are n generators which can be stacked to produce any interval in the group.

A simple example of stacking can be seen in Pythagorean tuning, or 3-limit JI, a rank-2 temperament that tempers out no commas, which is generated by stacking (multiplying or dividing by) the primes 3 and 2. For example, the interval 531441/524288 can be produced by multiplying by 3 twelve times, and then dividing by 2 nineteen times. Its descending counterpart, 524288/531441, can be produced by multiplying by 2 nineteen times, then dividing by 3 twelve times. In terms of cent values, this corresponds to adding or subtracting steps of 1200 cents or log2(3)*1200 ≈ 1901.955 cents.

Stacking is used to explain regular temperaments, which are often described in terms of stacking multiple instances of a single interval to produce another interval, and commas, which are describable as the difference between a stack of one interval and a stack of a different interval.