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. 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 an interval to produce another interval.