Stacking
In tuning theory, stacking is 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.
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 Pythagorean comma 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. In a rank-n temperament, there are n generators which can be stacked to produce any interval in the group.
Mathematical definition
Stacking is the group operation on a free abelian group of musical intervals.
In the context of group theory, any space of intervals created by multiplicatively stacking arbitrarily many (or negatively many) of a given set of generators is considered a free abelian group under stacking. Where the set of generators is finite, it is called a finitely generated free abelian group.
The more explicit definition for this follows.
Note: Examples are provided with the group of just intervals, corresponding to the infinite set of primes as generators, and to positive rationals under multiplication. However, the following properties can be shown to apply in other cases, such as tunings of regular temperaments, or any specific subgroup of JI, including prime limits. An intuitive way to confirm this is to see that any set of generators can be seen as a tempering of some just intonation subgroup, if a very inaccurate one.
a) It follows the group theory axioms ("group"):
- Stacking a unison on another interval produces no change; in other words, the unison is the identity element. For example, 5/4. This corresponds to the number 1 being the multiplicative identity.
- Stacking intervals is associative; for example (5/4 ⋅ 6/5) ⋅ 4/3 = 2. This corresponds to the fact that multiplication is associative.
- There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, 1. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.
- Stacking intervals in such a group produces another interval in the group. For example, 5/4 and 6/5 are in the group of just intervals, and thus it follows that 3/2 is as well.
b) Stacking intervals is commutative ("abelian"):
- For example, 6/5 ⋅ 5/4 = 3/2. This corresponds to the fact that multiplication of real numbers is commutative.
c) Nontrivial products of stacks of generators do not ever produce the identity ("free").
- In the case of JI, this corresponds to the fundamental theorem of arithmetic, because every interval has a unique prime factorization.
- In the case of tunings such as equal temperaments, there is only one generator.
- The number of generators is the rank of the temperament, so that equal temperaments are rank-1, temperaments with a generator and period are rank-2, and so on.