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Tenney–Euclidean temperament measures: Difference between revisions

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Interpretation of Graham's RMS
Start documenting Sintel's design
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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.


It is also possible to normalize the metrics to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023 is the first attempt at this goal.  
It is also possible to normalize the metrics to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023, called ''Dirichlet coefficients'', is the first attempt at this goal<ref name="sintel">Sintel. [https://github.com/Sin-tel/temper/blob/c0d5c36e3c189f64860f4aea288ff3ff3bc34982/lib_temper/temper.py "Collection of functions for dealing with regular temperaments"], Temperament Calculator.</ref>.  


== TE complexity ==
== TE complexity ==
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where {{!}}·{{!}} denotes the determinant, and {{t}} denotes the transpose.  
where {{!}}·{{!}} denotes the determinant, and {{t}} denotes the transpose.  


Graham Breed and [[Gene Ward Smith]] have proposed different RMS norms. Let us denote the RMS norm of ''M'' as ‖''M''‖<sub>RMS</sub>. In Graham's paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as
Graham Breed and [[Gene Ward Smith]] have proposed different RMS norms. Let us denote the RMS norm of ''M'' as ‖''M''‖<sub>RMS</sub>. In Graham's paper<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref>, an RMS norm is proposed as


$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$
$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$
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: '''Note''': that is the definition used by Graham Breed's temperament finder.  
: '''Note''': that is the definition used by Graham Breed's temperament finder.  


Gene Ward Smith has defined another RMS norm,
Gene Ward Smith's RMS norm is given as


$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M_W}_2}{\sqrt {C(n, r)}} $$
$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M_W}_2}{\sqrt {C(n, r)}} $$
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where {{nowrap|C(''n'', ''r'')}} is the number of combinations of ''n'' things taken ''r'' at a time without repetition, which equals the number of entries of the wedgie in the usual, compressed form.  
where {{nowrap|C(''n'', ''r'')}} is the number of combinations of ''n'' things taken ''r'' at a time without repetition, which equals the number of entries of the wedgie in the usual, compressed form.  


: '''Note''': that is the definition currently used throughout the wiki, unless stated otherwise.  
We may also note {{nowrap|{{!}}''V''<sub>''W''</sub>''V''<sub>''W''</sub>{{t}}{{!}} {{=}} {{!}}''VW''<sup>2</sup>''V''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P''<sub>''W''</sub>, which is ''V''<sub>''W''</sub>{{t}}(''V''<sub>''W''</sub>''V''<sub>''W''</sub>{{t}}){{inv}}''V''<sub>''W''</sub>, and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''V''{{t}}(''VW''<sup>2</sup>''V''{{t}}){{inv}}''V''}}.


We may also note {{nowrap|{{!}}''V''<sub>''W''</sub>''V''<sub>''W''</sub>{{t}}{{!}} {{=}} {{!}}''VW''<sup>2</sup>''V''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P''<sub>''W''</sub>, which is ''V''<sub>''W''</sub>{{t}}(''V''<sub>''W''</sub>''V''<sub>''W''</sub>{{t}}){{inv}}''V''<sub>''W''</sub>, and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''V''{{t}}(''VW''<sup>2</sup>''V''{{t}}){{inv}}''V''}}.
Sintel has defined a complexity measure that serves as an intermediate step for his badness metric<ref name="sintel"/>, which we will get to later. To obtain this complexity, we normalize the Tenney-weighting matrix ''W'' to ''U'' such that {{nowrap| det(''U'') {{=}} 1 }}, and then take the ''L''<sup>2</sup> norm of ''M''<sub>''U''</sub>. It can be shown that
 
$$ U = W / \det(W)^{1/n} $$
 
and so the complexity is
 
$$ \norm{M_U}_2 = \norm{M_W}_2 / \det(W)^{r/n} $$


== TE error ==
== TE error ==
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