Wedgie/Archived version: Difference between revisions

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== Reconstituting wedgies in general ==

Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1~~/(C(~~n, 3~~)lb~~(q)lb(p)) then wedging K = &~~lt;1~~ lb(3) lb(5) ~~...~~ lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie~~. Here~~ p and q are the largest and second largest primes in the prime limit~~, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6~~.

Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if <math>E \leqslant \frac{1}{\binom{n}{3}\log_2\left(q\right)\log_2\left(p\right)}</math> then wedging <math>K = \val{1 & \operatorname{lb}\left(3\right) & \operatorname{lb}\left(5\right) & \ldots & \operatorname{lb}\left(p\right)}</math> with the val consisting of 0 followed by the first {{nowrap|''n'' − 1}} coefficients of the wedgie and rounding will give the wedgie, where ''p'' and ''q'' are the largest and second largest primes in the prime limit.

More generally, we can reconstitute W by rounding Y = (~~W∨2~~)∧K to the nearest integer coefficients, where K is the JI point &~~lt;1 lb~~(3) lb(5) ~~... lb~~(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|~~| ≤ |~~|W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)~~∧K||~~=||(W-Y)~~∧K||~~, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].

More generally, we can reconstitute W by rounding {{nowrap|Y {{=}} (W ∨ 2) ∧ K}} to the nearest integer coefficients, where K is the JI point <math>\tval{1 & log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> in unweighted coordinates. Then we have <math>\left\|\left(W - Y\right) + Y\right\| \leqslant \left\|W-Y\right\| + \left\|Y\right\|</math> by the triangle inequality, and since {{nowrap|{{!!}}W − Y{{!!}}}} is bounded by the fact that W has been obtained by rounding, complexity, which is {{nowrap|{{!!}}(W − Y) + Y{{!!}} {{=}} {{!!}}W{{!!}}}}, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is {{nowrap|{{!!}}((W − Y) + Y) ∧ K{{!!}} {{=}} {{!!}}(W − Y) ∧ K{{!!}}}}, hence relative error is also bounded by the fact that {{nowrap|{{!!}}W − Y{{!!}}}} is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].

In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes E ≤ 1/(~~2√5 q5q7q11~~), then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.

In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes {{nowrap|E ≤ {{frac|1|2√(5)''q''5''q''7''q''11}}}}, then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.

== See also ==

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 == Reconstituting wedgies in general ==
-Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if E ≤ 1/(C(n, 3)lb(q)lb(p)) then wedging K = &lt;1 lb(3) lb(5) ... lb(p)| with the val consisting of 0 followed by the first n-1 coefficients of the wedgie and rounding will give the wedgie. Here p and q are the largest and second largest primes in the prime limit, lb(x) is log base two, and C(n, 3) is n choose three, n(n-1)(n-2)/6.
+Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if <math>E \leqslant \frac{1}{\binom{n}{3}\log_2\left(q\right)\log_2\left(p\right)}</math> then wedging <math>K = \val{1 & \operatorname{lb}\left(3\right) & \operatorname{lb}\left(5\right) & \ldots & \operatorname{lb}\left(p\right)}</math> with the val consisting of 0 followed by the first {{nowrap|''n'' &minus; 1}} coefficients of the wedgie and rounding will give the wedgie, where ''p'' and ''q'' are the largest and second largest primes in the prime limit.
-More generally, we can reconstitute W by rounding Y = (W∨2)∧K to the nearest integer coefficients, where K is the JI point &lt;1 lb(3) lb(5) ... lb(p)| in unweighted coordinates. Then we have ||(W-Y)+Y|| ≤ ||W-Y|| + ||Y|| by the triangle inequality, and since ||W-Y|| is bounded by the fact that W has been obtained by rounding, complexity, which is ||(W-Y)+Y||=||W||, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is ||((W-Y) + Y)∧K||=||(W-Y)∧K||, hence relative error is also bounded by the fact that ||W-Y|| is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].
+More generally, we can reconstitute W by rounding {{nowrap|Y {{=}} (W ∨ 2) ∧ K}} to the nearest integer coefficients, where K is the JI point <math>\tval{1 & log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> in unweighted coordinates. Then we have <math>\left\|\left(W - Y\right) + Y\right\| \leqslant \left\|W-Y\right\| + \left\|Y\right\|</math> by the triangle inequality, and since {{nowrap|{{!!}}W &minus; Y{{!!}}}} is bounded by the fact that W has been obtained by rounding, complexity, which is {{nowrap|{{!!}}(W &minus; Y) + Y{{!!}} {{=}} {{!!}}W{{!!}}}}, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from Y∧K = ((W∨2)∧K)∧K = 0 that relative error, which is ||W∧K||, is {{nowrap|{{!!}}((W &minus; Y) + Y) ∧ K{{!!}} {{=}} {{!!}}(W &minus; Y) ∧ K{{!!}}}}, hence relative error is also bounded by the fact that {{nowrap|{{!!}}W &minus; Y{{!!}}}} is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].
-In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes E ≤ 1/(2√5 q5q7q11), then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.
+In the particular case of the 11-limit in rank three, we have that (W∨2)∧K gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes {{nowrap|E ≤ {{frac|1|2√(5)''q''<sub>5</sub>''q''<sub>7</sub>''q''<sub>11</sub>}}}}, then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.
 == See also ==