Tenney–Euclidean temperament measures: Difference between revisions
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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 | == Example in each 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. | 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 | Here is a demonstration from [[7-limit]] [[magic]] and [[meantone]] compared in each definitions. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
|+7-limit magic vs meantone in TE temperament measures | |+7-limit magic (left) vs meantone (right) in TE temperament measures | ||
! | ! | ||
! TE complexity | ! TE complexity | ||
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|- | |- | ||
! Standard L2 norm | ! Standard L2 norm | ||
| 7.195 : 5.400 | | 7.195 : 5.400 | ||
| 2.149 : 2.763 | | 2.149 : 2.763 | ||
| 12.882×10<sup>-3</sup> : 12.435×10<sup>-3</sup> | | 12.882×10<sup>-3</sup> : 12.435×10<sup>-3</sup> | ||
|- | |- | ||
! Breed's RMS norm | ! Breed's RMS norm | ||
| 1.799 : 1.350 | | 1.799 : 1.350 | ||
| 1.074 : 1.382 | | 1.074 : 1.382 | ||
| 1.610×10<sup>-3</sup> : 1.554×10<sup>-3</sup> | | 1.610×10<sup>-3</sup> : 1.554×10<sup>-3</sup> | ||
|- | |- | ||
! Smith's RMS norm | ! Smith's RMS norm | ||
| 2.937 : 2.204 | | 2.937 : 2.204 | ||
| 2.631 : 3.384 | | 2.631 : 3.384 | ||
| 6.441×10<sup>-3</sup> : 6.218×10<sup>-3</sup> | | 6.441×10<sup>-3</sup> : 6.218×10<sup>-3</sup> | ||
|} | |} | ||
[[Category:math]] | [[Category:math]] | ||
[[Category:measure]] | [[Category:measure]] | ||