Wedgie: Difference between revisions

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~~#REDIRECT~~ [[~~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.

== 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.

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 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.

For another example, take father, which has the wedgie <code>⟨⟨1 -1 -4]]</code>.

Here, we again have a <code>1</code> as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.

Thus, going into our second entry, <code>-1</code> for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth.'' Since we're thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth.

Finally, for 3.5, we have the entry <code>-4</code>. Again, we're tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we'll be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we're in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it - 5/3 is split into four parts, which each contain a negative generator.

For our final example, we will consider blackwood <code>⟨⟨0 5 8]]</code>.

Blackwood's first entry is <code>0</code>, which means that it reduces 2.3 to a rank-1 structure. This can be seen as 3 being found 0 generators from some ploid (since 3/2 in blackwood is 3\5), since 0 times anything is 0.

The next entry, <code>5</code>, is simple: in 2.5, 5 (in this case, 5/4) is found by going up one generator, but remember that each entry is where the two primes are found multiplied together. Since 2 is found at 5 ploids, the entry is 1 * 5 = 5. (Technically, there's a hidden 5 in the 2.3 entry that gets multiplied by 0 and vanishes.)

And then the final entry, for 3.5, is <code>8</code>. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, 1 * 8 = 8.

For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit! If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.

For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.

== The form of a wedgie ==

The notation being used previously, <code>⟨⟨x y z]]</code>, is formally a shorthand for a matrix form, written

<math>\begin{bmatrix}

0 & x & y\\

-x & 0 & z\\

-y & -z & 0

\end{bmatrix}</math>. For a wedgie on a 4-prime subgroup, the structure of <code>⟨⟨a b c d e f]]</code> is actually <math>\begin{bmatrix}

0 & a & b & c\\

-a & 0 & d & e\\

-b & -d & 0 & f\\

-c & -e & -f & 0

\end{bmatrix}</math>. This means that a wedgie can easily be "clipped" by removing columns and corresponding rows to produce a restriction of the temperament, and thus wedgies which have a set of entries produced this way in common are strong extensions of some common structure.

== See also ==

* [[Wedgie/Archived version]]

* [[Catalog of temperaments by wedgie]]

* [[Ploidacot]]

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-#REDIRECT [[Plücker coordinates]]
+{{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.
+== 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.
+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 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.
+For another example, take father, which has the wedgie <code>⟨⟨1 -1 -4]]</code>.
+Here, we again have a <code>1</code> as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.
+Thus, going into our second entry, <code>-1</code> for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth.'' Since we're thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth.
+Finally, for 3.5, we have the entry <code>-4</code>. Again, we're tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we'll be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we're in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it - 5/3 is split into four parts, which each contain a negative generator.
+For our final example, we will consider blackwood  <code>⟨⟨0 5 8]]</code>.
+Blackwood's first entry is <code>0</code>, which means that it reduces 2.3 to a rank-1 structure. This can be seen as 3 being found 0 generators from some ploid (since 3/2 in blackwood is 3\5), since 0 times anything is 0.
+The next entry, <code>5</code>, is simple: in 2.5, 5 (in this case, 5/4) is found by going up one generator, but remember that each entry is where the two primes are found multiplied together. Since 2 is found at 5 ploids, the entry is 1 * 5 = 5. (Technically, there's a hidden 5 in the 2.3 entry that gets multiplied by 0 and vanishes.)
+And then the final entry, for 3.5, is <code>8</code>. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, 1 * 8 = 8.
+For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit! If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.
+For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.
+== The form of a wedgie ==
+The notation being used previously, <code>⟨⟨x y z]]</code>, is formally a shorthand for a matrix form, written
+<math>\begin{bmatrix}
+& x & y\\
+-x & 0 & z\\
+-y & -z & 0
+\end{bmatrix}</math>. For a wedgie on a 4-prime subgroup, the structure of <code>⟨⟨a b c d e f]]</code> is actually <math>\begin{bmatrix}
+& a & b & c\\
+-a & 0 & d & e\\
+-b & -d & 0 & f\\
+-c & -e & -f & 0
+\end{bmatrix}</math>. This means that a wedgie can easily be "clipped" by removing columns and corresponding rows to produce a restriction of the temperament, and thus wedgies which have a set of entries produced this way in common are strong extensions of some common structure.
+== See also ==
+* [[Wedgie/Archived version]]
+* [[Catalog of temperaments by wedgie]]
+* [[Ploidacot]]