Wedgie/Archived version: Difference between revisions

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<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>

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 <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 {{nowrap|E ≤ {{frac|1|4''q''5''q''7}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\val{1 & q_3 & q_5 & q_7}</math> 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 <math>\wedgie{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math> and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' − ''be'' + ''cd'' {{=}} 0}}). 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 <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 {{nowrap|E ≤ {{frac|1|4''q''5''q''7}}}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\val{1 & q_3 & q_5 & q_7}</math> 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 <math>\wedgie{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math> and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' − ''be'' + ''cd'' {{=}} 0}}). 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 {{nowrap|E ≤ {{frac|1|4''q''5''q''7}}, {{nowrap|B ≤ {{frac|20|4''q''5''q''7 {{=}} 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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 <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>
-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 <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 {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\val{1 & q_3 & q_5 & q_7}</math> 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 <math>\wedgie{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math> and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' &minus; ''be'' + ''cd'' {{=}} 0}}). 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 <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 {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\val{1 & q_3 & q_5 & q_7}</math> 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 <math>\wedgie{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math> and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' &minus; ''be'' + ''cd'' {{=}} 0}}). 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 {{nowrap|E ≤ {{frac|1|4''q''<sub>5</sub>''q''<sub>7</sub>}}, {{nowrap|B ≤ {{frac|20|4''q''<sub>5</sub>''q''<sub>7</sub> {{=}} 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.