Wedgie/Archived version: Difference between revisions
Undo revision 180720 by Domin (talk). Wedgies are useful for uniquely identifying temps and computing complexity more straightly than mappings, which help ppl to search/choose a temp so that they can make music in it Tag: Undo |
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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''. | 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 = \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''<sub>6</sub>''w''<sub>1</sub> − ''w''<sub>5</sub>''w''<sub>2</sub> + ''w''<sub>4</sub>''w''<sub>3</sub>}} | 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''<sub>6</sub>''w''<sub>1</sub> − ''w''<sub>5</sub>''w''<sub>2</sub> + ''w''<sub>4</sub>''w''<sub>3</sub> {{=}} ''w''<sub>9</sub>''w''<sub>1</sub> − ''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> − ''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> − ''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> − ''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 == | == Constrained wedgies == | ||