Wedgie: Difference between revisions

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A '''wedgie''' is an object that uniquely characterizes a temperament regardless of choice of period vs. generator or of equave. A wedgie takes the form <code>⟨⟨x y z]]</code>, with a number of entries (not necessarily 3; in fact that is the simplest possible case) listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, etc). Wedgies can be thought of as a generalization of vals, called ''multivals,'' so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping]] matrix, which specifies a particular equave. A wedgie takes the form <code>⟨⟨x y z]]</code>, with a number of entries (not necessarily 3; in fact that is the simplest possible case) listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, etc). Wedgies can be thought of as a generalization of vals, called ''multivals,'' so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

== How to read a wedgie ==

Following intuitions from [[ploidacot]], one way to characterize a temperament is how many equal parts it splits the perfect fifth (3/2) into. For an example, meantone doesn't split it at all, so we say it is ''monocot.'' We also say it is ''haploid'', since it doesn't split the octave (2/1) at all.

A wedgie is ~~essentially~~ a ~~way to generalize~~ ploidacot information to all possible combinations of primes within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament.

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 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.

For 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.

For the final entry, which is for the 3.5 subgroup, we have another <code>4</code>. But this time, we're thinking tritave-equivalently now, so we'll be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2s.

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 {{Beginner|Plücker coordinates}}
-A '''wedgie''' is an object that uniquely characterizes a temperament regardless of choice of period vs. generator or of equave. A wedgie takes the form <code>⟨⟨x y z]]</code>, with a number of entries (not necessarily 3; in fact that is the simplest possible case) listed in between multiple val brackets (double brackets for rank-2, triple brackets for rank-3, etc). Wedgies can be thought of as a generalization of vals, called ''multivals,'' so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
+A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping]] matrix, which specifies a particular equave. A wedgie takes the form <code>⟨⟨x y z]]</code>, with a number of entries (not necessarily 3; in fact that is the simplest possible case) listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, etc). Wedgies can be thought of as a generalization of vals, called ''multivals,'' so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.
 == How to read a wedgie ==
 Following intuitions from [[ploidacot]], one way to characterize a temperament is how many equal parts it splits the perfect fifth (3/2) into. For an example, meantone doesn't split it at all, so we say it is ''monocot.'' We also say it is ''haploid'', since it doesn't split the octave (2/1) at all.
-A wedgie is essentially a way to generalize ploidacot information to all possible combinations of primes within a temperament, and format that information in a concise manner; as it turns out, this is enough to uniquely characterize the temperament.
+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 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.
+For 2.5, the procedure generalizes, with the entry, <code>4</code>, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.
 For the final entry, which is for the 3.5 subgroup, we have another <code>4</code>. But this time, we're thinking tritave-equivalently now, so we'll be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2s.