Stacking: Difference between revisions

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In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.

A simple example of stacking can be seen in [[Pythagorean]] tuning~~, or~~ [[3-limit]] [[JI]]~~, a [[rank-2 temperament]] that [[tempering out|tempers out]] no [[comma]]s~~, 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 {{nowrap| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.

A simple example of stacking can be seen in [[Pythagorean]] tuning a.k.a. [[3-limit]] [[JI]], 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 {{nowrap| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.

Stacking is used to explain [[regular temperament]]s, 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'' [[generator]]s which can be stacked to produce any interval in the group.

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 In tuning theory, '''stacking''' is multiplying or dividing the pitch [[ratio]]s corresponding to the intervals, or adding or subtracting their [[cent]] values. Depending on context, [[octave reduction]] (or the analogue for other [[equave]]s) is sometimes assumed.
-A simple example of stacking can be seen in [[Pythagorean]] tuning, or [[3-limit]] [[JI]], a [[rank-2 temperament]] that [[tempering out|tempers out]] no [[comma]]s, 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 {{nowrap| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.
+A simple example of stacking can be seen in [[Pythagorean]] tuning a.k.a. [[3-limit]] [[JI]], 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 {{nowrap| log<sub>2</sub>(3) ⋅ 1200 ≈ 1901.955 }} cents.
 Stacking is used to explain [[regular temperament]]s, 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'' [[generator]]s which can be stacked to produce any interval in the group.