Wedgie: Difference between revisions

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The elements of a wedgie each represent the number of parts into which a particular [[subgroup]] is split by the temperament, or the number of distinct sets of notes within the temperament linked by motions within that subgroup. They can be thought of as a generalization of the ploidacot information to all possible combinations of [[prime harmonic|primes]] within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament.

For example, take the wedgie for meantone: <code>⟨⟨1 4 4]]</code>. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is <code>1</code>, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).

For an intuition as to why multiplication is performed, consider hemipyth (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use [[Radical interval|radical]] steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes.

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''See also: [[Linear algebra formalism#Wedge product]]''

Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone. Note that if you take a wedgie and flip all the signs, it still results in the same wedgie, so ⟨⟨-1 -4 -4]] is the same as ⟨⟨1 4 4]] from before.

More than two vals can be combined into a higher-rank wedgie by an analogous method.

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 The elements of a wedgie each represent the number of parts into which a particular [[subgroup]] is split by the temperament, or the number of distinct sets of notes within the temperament linked by motions within that subgroup. They can be thought of as a generalization of the ploidacot information to all possible combinations of [[prime harmonic|primes]] within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament.
-For example, take the wedgie for meantone: <code>⟨⟨1 4 4]]</code>. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is <code>1</code>, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).
+For example, take the wedgie for meantone: <code>⟨⟨1 4 4]]</code>.  Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is <code>1</code>, telling us that the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).
 For an intuition as to why multiplication is performed, consider hemipyth (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use [[Radical interval|radical]] steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes.
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 ''See also: [[Linear algebra formalism#Wedge product]]''
-Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.
+Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone. Note that if you take a wedgie and flip all the signs, it still results in the same wedgie, so ⟨⟨-1 -4 -4]] is the same as ⟨⟨1 4 4]] from before.
 More than two vals can be combined into a higher-rank wedgie by an analogous method.