Wedgie/Archived version: Difference between revisions

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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 = \bitval{a & b & c & d & e & f}</math>.

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 = \bival{\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

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 = \bival{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q_5} & \frac{e}{q_3 q_7} & \frac{f}{q_5 q_7}}</math>. We now have

<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>

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 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 = \bitval{a & b & c & d & e & f}</math>.
-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 = \bival{\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
+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 = \bival{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q_5} & \frac{e}{q_3 q_7} & \frac{f}{q_5 q_7}}</math>. We now have
 <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>