Tenney–Euclidean temperament measures: Difference between revisions
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== Introduction == | == Introduction == | ||
Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [ | Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [[wikipedia:Exterior_algebra|wedge product]] of weighted vals or monzos, we may define a norm by means of the usual [[wikipedia:Norm_(mathematics)#Euclidean norm|Euclidean norm]] (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[wikipedia:Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||<sub>RMS</sub>. | ||
=== A Preliminary Note on Scaling Factors === | === A Preliminary Note on Scaling Factors === | ||
These metrics are mainly used to rank temperaments relative to one another. In that regard, it doesn't matter much if an RMS or an < | These metrics are mainly used to rank temperaments relative to one another. In that regard, it doesn't matter much if an RMS or an ''L''<sup>2</sup> | ||
norm is used, because these two are equivalent up to a scaling factor, so they will rank temperaments identically. | norm is used, because these two are equivalent up to a scaling factor, so they will rank temperaments identically. | ||
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# Taking an RMS | # Taking an RMS | ||
# Taking an RMS and also normalizing for the temperament rank | # Taking an RMS and also normalizing for the temperament rank | ||
# Taking the simple < | # Taking the simple ''L''<sup>2</sup> norm | ||
# Any of the above and also dividing by the norm of the JIP | # Any of the above and also dividing by the norm of the JIP | ||
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Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [http://en.wikipedia.org/wiki/Gramian_matrix Gramian]. | Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [http://en.wikipedia.org/wiki/Gramian_matrix Gramian]. | ||
Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The L<sup>2</sup> norm is one of the standard measures, | Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures, | ||
<math>\displaystyle | <math>\displaystyle | ||
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is an '''adjusted error''' which makes the error of a rank ''r'' temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. | is an '''adjusted error''' which makes the error of a rank ''r'' temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament. | ||
By Graham Breed's definition, TE error may be accessed directly via [[TE tuning map]]. If T is the tuning map, then the TE error G can be found by | By Graham Breed's definition, TE error may be accessed directly via [[Tenney-Euclidean Tuning|TE tuning map]]. If T is the tuning map, then the TE error G can be found by | ||
<math>\displaystyle | <math>\displaystyle | ||
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! TE simple badness | ! TE simple badness | ||
|- | |- | ||
! Standard | ! Standard ''L''<sup>2</sup> norm<ref>See also [[wikipedia:Norm (mathematics)#Euclidean norm|Norm (mathematics) #Euclidean norm - Wikipedia]]</ref> | ||
| 7.195 : 5.400 | | 7.195 : 5.400 | ||
| 2.149 : 2.763 | | 2.149 : 2.763 | ||
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|} | |} | ||
<references/> | <references/> | ||
[[Category:math]] | [[Category:math]] | ||
[[Category:measure]] | [[Category:measure]] | ||