Stacking: Difference between revisions

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In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]~~<nowiki/>~~s. It corresponds to multiplying or dividing the pitch [[ratio]]~~<nowiki/>~~s corresponding to the intervals, or adding or subtracting their [[cent]] values. In a rank-n [[temperament]], there are n [[~~Generator|generators~~]] which can be stacked to produce any interval in the group.

In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]s. It corresponds to 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. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s 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 [[Tempering out|tempers out]] no c[[~~Comma|ommas~~]], 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.

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 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~~ temperament~~|regular temperaments~~]], which are often described in terms of stacking an interval to produce another interval.

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.

[[Category:Method]]

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-In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]<nowiki/>s. It corresponds to multiplying or dividing the pitch [[ratio]]<nowiki/>s corresponding to the intervals, or adding or subtracting their [[cent]] values. In a rank-n [[temperament]], there are n [[Generator|generators]] which can be stacked to produce any interval in the group.
+In the context of tuning theory, '''stacking''' is the group operation of a [[free abelian group]] of [[interval]]s. It corresponds to 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. In a [[rank]]-''n'' [[temperament]], there are ''n'' [[generator]]s 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 [[Tempering out|tempers out]] no c[[Comma|ommas]], 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.
+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 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 temperament|regular temperaments]], which are often described in terms of stacking an interval to produce another interval.
+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.
+[[Category:Method]]