Plücker coordinates: Difference between revisions

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[[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]

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More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.

These two representations are related via the [[Hodge dual]].

~~== How to read a wedgie ==~~

Following intuitions from [[ploidacot]], one way to characterize a temperament is how many generators 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.

== Definition ==

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-{{Expert}}
+{{Expert|Wedgie}}
 [[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]]
 {{Wikipedia|Plücker embedding}}
@@ Line 9: / Line 9: @@
 More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.
 These two representations are related via the [[Hodge dual]].
-== How to read a wedgie ==
-Following intuitions from [[ploidacot]], one way to characterize a temperament is how many generators 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.
 == Definition ==