Tenney–Euclidean temperament measures: Difference between revisions
Clarification (step 2) |
Clarification (step 3). Remove todo cuz it's very applicable now |
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where C(''n'', ''r'') is the number of combinations of ''n'' things taken ''r'' at a time, which equals the number of entries of the wedgie. Note: this is the definition currently used throughout the wiki, unless stated otherwise. | where C(''n'', ''r'') is the number of combinations of ''n'' things taken ''r'' at a time, which equals the number of entries of the wedgie. Note: this is the definition currently used throughout the wiki, unless stated otherwise. | ||
If W is a [http://en.wikipedia.org/wiki/Diagonal_matrix diagonal matrix] with 1, 1/log<sub>2</sub>3, …, 1/log<sub>2</sub>''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV<sup>T</sup>) = det(AW<sup>2</sup>A<sup>T</sup>). This may be related to the [[Tenney-Euclidean_metrics|TE tuning projection matrix]] P, which is V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V, and the corresponding matrix for unweighted monzos '''P''' = A<sup>T</sup>(AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>A. | If W is a [http://en.wikipedia.org/wiki/Diagonal_matrix diagonal matrix] with 1, 1/log<sub>2</sub>3, …, 1/log<sub>2</sub>''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then V = AW and det(VV<sup>T</sup>) = det(AW<sup>2</sup>A<sup>T</sup>). This may be related to the [[Tenney-Euclidean_metrics|TE tuning projection matrix]] P, which is V<sup>T</sup>(VV<sup>T</sup>)<sup>-1</sup>V, and the corresponding matrix for unweighted monzos '''P''' = A<sup>T</sup>(AW<sup>2</sup>A<sup>T</sup>)<sup>-1</sup>A. | ||
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G and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. | G and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case, G = sin θ, where θ is the angle between J and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin θ, TE error as it appears on the temperament finder pages. | ||
== Example in different definitions == | |||
The different definitions yield different results, but they are related from each other by a factor of rank and limit. Meaningful comparison of temperaments in the same rank and limit will be provided by picking any one of them. | |||
Here is a demonstration from [[7-limit]] [[magic]] and [[meantone]] compared in different definitions. | |||
{| class="wikitable center-all" | |||
|+7-limit magic vs meantone in TE temperament measures | |||
! | |||
! TE complexity | |||
! TE error (¢) | |||
! TE simple badness | |||
|- | |||
! Standard L2 norm | |||
| 7.195 : 5.400 = 1.332 | |||
| 2.149 : 2.763 = 0.777 | |||
| 12.882×10<sup>-3</sup> : 12.435×10<sup>-3</sup> = 1.036 | |||
|- | |||
! Breed's RMS norm | |||
| 1.799 : 1.350 = 1.332 | |||
| 1.074 : 1.382 = 0.777 | |||
| 1.610×10<sup>-3</sup> : 1.554×10<sup>-3</sup> = 1.036 | |||
|- | |||
! Smith's RMS norm | |||
| 2.937 : 2.204 = 1.332 | |||
| 2.631 : 3.384 = 0.777 | |||
| 6.441×10<sup>-3</sup> : 6.218×10<sup>-3</sup> = 1.036 | |||
|} | |||
[[Category:math]] | [[Category:math]] | ||
[[Category:measure]] | [[Category:measure]] | ||