Wedgie/Archived version: Difference between revisions

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<math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\~~twedgie~~{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E19 ∧ E31}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E19 ∧ E31}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: Greatest common divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s.

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== Conditions on being a wedgie ==

If we take any three integers <math>\~~twedgie~~{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined.

If we take any three integers <math>\bitval{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined.

However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that {{nowrap|W ∧ W {{=}} 0}}, where the "0" means the multival of rank 2''r'' obtained by wedging W with W. For prime limits 7 and 11 this condition suffices for rank two, but in general we need to check, for every prime {{nowrap|''q'' ≤ ''p''}} and every basis val ''v'' sending ''q'' to 1 and everything else to 0, that {{nowrap|(W ∨ ''q'') ∧ W {{=}} 0}} and {{nowrap|(W ∧ ''v'')º ∧ Wº {{=}} 0}}, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[Wikipedia:Plücker embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank-''r'' multival, {{nowrap|W ∨ ''q''}} is a rank-({{nowrap|''r'' − 1}}) multival corresponding to tempering out all the commas of W, as well as ''q''.

In the 7-limit case, if we wedge a prospective rank two multival <math>W = \~~twedgie~~{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' − ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \~~twedgie~~{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''6''w''1 − ''w''5''w''2 + ''w''4''w''3}} = {{nowrap|''w''9''w''1 − ''w''8''w''2 + ''w''7''w''3}} = {{nowrap|''w''10''w''1 − ''w''8''w''4 + ''w''7''w''5}} = {{nowrap|''w''10''w''2 − ''w''9''w''4 + ''w''7''w''6}} = {{nowrap|''w''10''w''3 − ''w''9''w''5 + ''w''8''w''6 {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}.

In the 7-limit case, if we wedge a prospective rank two multival <math>W = \bitval{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' − ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \bitval{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''6''w''1 − ''w''5''w''2 + ''w''4''w''3}} = {{nowrap|''w''9''w''1 − ''w''8''w''2 + ''w''7''w''3}} = {{nowrap|''w''10''w''1 − ''w''8''w''4 + ''w''7''w''5}} = {{nowrap|''w''10''w''2 − ''w''9''w''4 + ''w''7''w''6}} = {{nowrap|''w''10''w''3 − ''w''9''w''5 + ''w''8''w''6 {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}.

== Constrained wedgies ==

@@ Line 102: / Line 102: @@
 <math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>
-We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\twedgie{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.
+We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E<sub>19</sub> ∧ E<sub>31</sub>}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.
 This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: Greatest common divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s.
@@ Line 139: / Line 139: @@
 == Conditions on being a wedgie ==
-If we take any three integers <math>\twedgie{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined.
+If we take any three integers <math>\bitval{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined.
 However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that {{nowrap|W ∧ W {{=}} 0}}, where the "0" means the multival of rank 2''r'' obtained by wedging W with W. For prime limits 7 and 11 this condition suffices for rank two, but in general we need to check, for every prime {{nowrap|''q'' ≤ ''p''}} and every basis val ''v'' sending ''q'' to 1 and everything else to 0, that {{nowrap|(W ∨ ''q'') ∧ W {{=}} 0}} and {{nowrap|(W ∧ ''v'')º ∧ Wº {{=}} 0}}, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[Wikipedia:Plücker embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank-''r'' multival, {{nowrap|W ∨ ''q''}} is a rank-({{nowrap|''r'' &minus; 1}}) multival corresponding to tempering out all the commas of W, as well as ''q''.
-In the 7-limit case, if we wedge a prospective rank two multival <math>W = \twedgie{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' &minus; ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \twedgie{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''<sub>6</sub>''w''<sub>1</sub> &minus; ''w''<sub>5</sub>''w''<sub>2</sub> + ''w''<sub>4</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>9</sub>''w''<sub>1</sub> &minus; ''w''<sub>8</sub>''w''<sub>2</sub> + ''w''<sub>7</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>1</sub> &minus; ''w''<sub>8</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>5</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>2</sub> &minus; ''w''<sub>9</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>6</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>3</sub> &minus; ''w''<sub>9</sub>''w''<sub>5</sub> + ''w''<sub>8</sub>''w''<sub>6</sub> {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}.
+In the 7-limit case, if we wedge a prospective rank two multival <math>W = \bitval{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' &minus; ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \bitval{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''<sub>6</sub>''w''<sub>1</sub> &minus; ''w''<sub>5</sub>''w''<sub>2</sub> + ''w''<sub>4</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>9</sub>''w''<sub>1</sub> &minus; ''w''<sub>8</sub>''w''<sub>2</sub> + ''w''<sub>7</sub>''w''<sub>3</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>1</sub> &minus; ''w''<sub>8</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>5</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>2</sub> &minus; ''w''<sub>9</sub>''w''<sub>4</sub> + ''w''<sub>7</sub>''w''<sub>6</sub>}} = {{nowrap|''w''<sub>10</sub>''w''<sub>3</sub> &minus; ''w''<sub>9</sub>''w''<sub>5</sub> + ''w''<sub>8</sub>''w''<sub>6</sub> {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}.
 == Constrained wedgies ==