Wedgie/Archived version: Difference between revisions

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== Constrained wedgies ==

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank ~~two~~ wedgie W = &~~lt;~~&~~lt;a b~~ c d e f||.

Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank-w wedgie <math>W = \wedgie{a & b & c & d & e & f}</math>.

By definition, E = ||~~J∧Z||~~, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &~~lt;~~&~~lt;a/q3 b/q5~~ c~~/q7~~ d~~/(q3q5)~~ e~~/(q3q7)~~ f/~~(q5q7)||~~. We now have

By definition, <math>E = \left\|J \wedge Z\right\|</math>, where Z is the weighted version of W; if ''q''3, ''q''5, and ''q''7 are the logarithms base two of 3, 5, and 7, then <math>Z = \wedgie{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q5} & \frac{e}{q_3 q7} & \frac{f}{q_5 q7}}</math>. We now have

<math>\~~displaystyle~~ (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\

<math>\left(\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3}\right)^2+\left(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3}\right)^2+\left(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5}\right)^2+\left(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5}\right)^2 = 4 E^2</math>

+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2</math>

From this we can conclude that d, e and f satisfy |d - ~~q3b~~ + ~~q5a~~| ~~≤ 2Eq3q5~~, |e - ~~q3c~~ + ~~q7a~~| ~~≤ 2Eq3q7~~ and |f - ~~q5c~~ + ~~q7d~~| ~~≤ 2Eq5q7~~. This has an interesting interpretation: since <1 q3 q5 q7|∧<0 a b c| = <<a b c q3b-q5a q3c-q7a q5c-q7b||, if E ≤ 1/(4q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <<a b c ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound.

From this we can conclude that ''d'', ''e'', and ''f'' satisfy <math>\left|d - q_3b + q_5a\right| \leqslant 2Eq_3q_5</math>, <math>\left|e - q_3c + q_7a\right| \leqslant 2Eq_3q_7</math>, and <math>\left|f - q_5c + q_7d\right| \leqslant 2Eq_5q_7</math>. This has an interesting interpretation: since <1 q3 q5 q7|∧<0 a b c| = <<a b c q3b-q5a q3c-q7a q5c-q7b||<math>\val{1 & q_3 & q_5 & q_7} \wedge \val{0 & a & b & c} = \wedgie{a & b & c & q_3b - q_5a & q_3c - q_7a & q_5c - q_7b}</math>, if E ≤ 1/(4q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <<a b c ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound.

If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean_metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4q5q7), B ≤ 20/(4q5q7) = 0.767. This badness figure is easily met. While simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.

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 == Constrained wedgies ==
-Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank two wedgie W = &lt;&lt;a b c d e f||.
+Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank-w wedgie <math>W = \wedgie{a & b & c & d & e & f}</math>.
-By definition, E = ||J∧Z||, where Z is the weighted version of W; if q3, q5 and q7 are the logarithms base two of 3, 5, and 7, then Z = &lt;&lt;a/q3 b/q5 c/q7 d/(q3q5) e/(q3q7) f/(q5q7)||. We now have
+By definition, <math>E = \left\|J \wedge Z\right\|</math>, where Z is the weighted version of W; if ''q''<sub>3</sub>, ''q''<sub>5</sub>, and ''q''<sub>7</sub> are the logarithms base two of 3, 5, and 7, then <math>Z = \wedgie{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q5} & \frac{e}{q_3 q7} & \frac{f}{q_5 q7}}</math>. We now have
-<math>\displaystyle (\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3})^2+(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3})^2+(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5})^2 \\
+<math>\left(\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3}\right)^2+\left(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3}\right)^2+\left(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5}\right)^2+\left(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5}\right)^2 = 4 E^2</math>
-+(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5})^2 = 4 E^2</math>
-From this we can conclude that d, e and f satisfy |d - q3b + q5a| ≤ 2Eq3q5, |e - q3c + q7a| ≤ 2Eq3q7 and |f - q5c + q7d| ≤ 2Eq5q7. This has an interesting interpretation: since &lt;1 q3 q5 q7|∧&lt;0 a b c| = &lt;&lt;a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||, if E ≤ 1/(4q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples &lt;&lt;a b c ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with &lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound.
+From this we can conclude that ''d'', ''e'', and ''f'' satisfy <math>\left|d - q_3b + q_5a\right| \leqslant 2Eq_3q_5</math>, <math>\left|e - q_3c + q_7a\right| \leqslant 2Eq_3q_7</math>, and <math>\left|f - q_5c + q_7d\right| \leqslant 2Eq_5q_7</math>. This has an interesting interpretation: since &lt;1 q3 q5 q7|∧&lt;0 a b c| = &lt;&lt;a  b  c  q3b-q5a  q3c-q7a  q5c-q7b||<math>\val{1 & q_3 & q_5 & q_7} \wedge \val{0 & a & b & c} = \wedgie{a & b & c & q_3b - q_5a & q_3c - q_7a & q_5c - q_7b}</math>, if E ≤ 1/(4q5q7), then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with &lt;1 q3 q5 q7| and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples &lt;&lt;a b c ...|| (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with &lt;1 q3 q5 q7| and rounding, then checking if the GCD is one and the Pfaffian af-be+cd is zero. Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound.
 If C = ||W|| is the TE complexity, then the formula for the [[Tenney-Euclidean_metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: B = CE. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since E ≤ 1/(4q5q7), B ≤ 20/(4q5q7) = 0.767. This badness figure is easily met. While simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.