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, 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.

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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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, {{nowrap| 5/4 ⋅ 1 = 5/4 }}. This corresponds to the number 1 being the multiplicative identity.

* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, {{nowrap| 5/4 ⋅ 1 {{=}} 5/4 }}. This corresponds to the number 1 being the multiplicative identity.

* Stacking intervals is associative; for example {{nowrap| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) = (5/4 ⋅ 6/5) ⋅ 4/3 = [[2/1|2]] }}. This corresponds to the fact that multiplication is associative.

* Stacking intervals is associative; for example {{nowrap| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) {{=}} (5/4 ⋅ 6/5) ⋅ 4/3 {{=}} [[2/1|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, {{nowrap| 5/4 ⋅ 4/5 = 1 }}. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.

* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, {{nowrap| 5/4 ⋅ 4/5 {{=}} 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, {{nowrap| 5/4 ⋅ 6/5 = 6/5 ⋅ 5/4 = 3/2 }}. This corresponds to the fact that multiplication of real numbers is commutative.

* For example, {{nowrap| 5/4 ⋅ 6/5 {{=}} 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").

@@ Line 1: / Line 1: @@
 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, 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.
 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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 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, {{nowrap| 5/4 ⋅ 1 = 5/4 }}. This corresponds to the number 1 being the multiplicative identity.
+* Stacking a [[unison]] on another interval produces no change; in other words, the unison is the identity element. For example, {{nowrap| 5/4 ⋅ 1 {{=}} 5/4 }}. This corresponds to the number 1 being the multiplicative identity.
-* Stacking intervals is associative; for example {{nowrap| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) = (5/4 ⋅ 6/5) ⋅ 4/3 = [[2/1|2]] }}. This corresponds to the fact that multiplication is associative.
+* Stacking intervals is associative; for example {{nowrap| [[5/4]] ⋅ ([[6/5]] ⋅ [[4/3]]) {{=}} (5/4 ⋅ 6/5) ⋅ 4/3 {{=}} [[2/1|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, {{nowrap| 5/4 ⋅ 4/5 = 1 }}. This corresponds to the descending interval being the multiplicative inverse of the ascending interval.
+* There are ascending and descending versions of each interval. Stacking the ascending and descending versions of an interval produces the unison. For example, {{nowrap| 5/4 ⋅ 4/5 {{=}} 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, {{nowrap| 5/4 ⋅ 6/5 = 6/5 ⋅ 5/4 = 3/2 }}. This corresponds to the fact that multiplication of real numbers is commutative.
+* For example, {{nowrap| 5/4 ⋅ 6/5 {{=}} 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").