Tenney–Euclidean temperament measures: Difference between revisions
Some basic improvements: stating the purpose of these things. The introduction section is really all about scaling factors. |
Resolve contradictions in the note: the normalization was meant to make comparison across ranks and subgroups possible but it wasn't successful. |
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== Note on scaling factors == | == Note on scaling factors == | ||
Given a [[Wedgies and multivals|multival]] or multimonzo which is a {{w|Exterior algebra|wedge product}} of weighted vals or monzos (where the weighting factors are 1/log<sub>2</sub>(''p'') for the entry corresponding to ''p''), we may define a norm by means of the usual {{w|Norm (mathematics) #Euclidean norm|Euclidean norm}} (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). | Given a [[Wedgies and multivals|multival]] or multimonzo which is a {{w|Exterior algebra|wedge product}} of weighted vals or monzos (where the weighting factors are 1/log<sub>2</sub>(''p'') for the entry corresponding to ''p''), we may define a norm by means of the usual {{w|Norm (mathematics) #Euclidean norm|Euclidean norm}} (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). We can rescale this several ways, for example by taking a {{w|Root mean square|root mean square}} (RMS) average of the entries of the multivector. These metrics are mainly used to rank temperaments relative to one another. In that regard, it does not 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. As a result, it is somewhat common to equivocate between the various choices of scaling factor, and treat the entire thing as "the" Tenney-Euclidean norm, so that we are really only concerned with the results of these metrics up to that equivalence. | ||
We can rescale this | |||
These metrics are mainly used to rank temperaments relative to one another. In that regard, it does not 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. As a result, it is somewhat common to equivocate between the various choices of scaling factor, and treat the entire thing as "the" Tenney-Euclidean norm, so that we are really only concerned with the results of these metrics up to that equivalence. | |||
Because of this, there are different "standards" for scaling that are commonly in use: | Because of this, there are different "standards" for scaling that are commonly in use: | ||
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Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | ||
An important point of this normalization is to allow us to meaningfully compare measures of corresponding temperaments in different [[just intonation subgroup]]s. However, none of them has been quite successful at this goal until [[Sintel]] developed a scheme in 2023. | |||
== TE complexity == | == TE complexity == | ||
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where det () denotes the determinant, and V<sup>T</sup> denotes the transpose of V. | where det () denotes the determinant, and V<sup>T</sup> denotes the transpose of V. | ||
In Graham Breed's paper, an RMS norm is proposed as | We denote the RMS norm as ‖M‖<sub>RMS</sub>. In Graham Breed's paper, an RMS norm is proposed as | ||
<math>\displaystyle | <math>\displaystyle | ||