Tenney–Euclidean temperament measures: Difference between revisions

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As these metrics are mainly used to rank temperaments within the same [[rank]] and [[just intonation subgroup]], it does not matter much which scheme is used, because they 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.

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 [https://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~~, 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>.

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<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 ==

Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the complexity of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave.

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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.

We may also note {{nowrap|~~{{!}}~~''V''''W''''V''''W''{{t}}~~{{!}}~~ {{=}} ~~{{!}}~~''VW''2''V''{{t}}~~{{!}}~~}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P''''W'', which is ''V''''W''{{t}}(''V''''W''''V''''W''{{t}}){{inv}}''V''''W'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''V''{{t}}(''VW''2''V''{{t}}){{inv}}''V''}}.

We may also note {{nowrap| det(''V''''W''''V''''W''{{t}}) {{=}} det(''VW''2''V''{{t}}) }}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P''''W'', which is ''V''''W''{{t}}(''V''''W''''V''''W''{{t}}){{inv}}''V''''W'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''V''{{t}}(''VW''2''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''2 norm of ''M''''U''. It can be shown that

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gives another error, called the ''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 {{nowrap|(1 + ''ε'')''ψ''}} for any positive ''ε'' results in an infinite set of vals supporting the temperament.

''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J''''W'' and the val in question~~. Multiplying by 1200 to obtain a result in cents leads to 1200 sin(''θ''), the TE error in cents~~.

''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''''W'' and the val in question.

== TE simple badness ==

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== TE logflat badness ==

Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula

Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' (called ''Dirichlet coefficients'' in Sintel's scheme), is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula

$$ L = B \cdot C^{r/(n - r)} $$

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$$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$

In Sintel's ~~Dirichlet coefficients, or Dirichlet badness~~,

In Sintel's derivation,

$$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$

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 As these metrics are mainly used to rank temperaments within the same [[rank]] and [[just intonation subgroup]], it does not matter much which scheme is used, because they 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.
-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 [https://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, 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>.
+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<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 ==
+{{Todo|rework|inline=1|text=Explain without wedgies}}
 Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the complexity of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave.
@@ Line 50: / Line 52: @@
 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.
-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| det(''V''<sub>''W''</sub>''V''<sub>''W''</sub>{{t}}) {{=}} det(''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
@@ Line 96: / Line 98: @@
 gives another error, called the ''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 {{nowrap|(1 + ''ε'')''ψ''}} for any positive ''ε'' results in an infinite set of vals supporting the temperament.
-''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J''<sub>''W''</sub> and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200&nbsp;sin(''θ''), the TE error in cents.
+''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''<sub>''W''</sub> and the val in question.
 == TE simple badness ==
@@ Line 125: / Line 127: @@
 == TE logflat badness ==
-Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula
+Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' (called ''Dirichlet coefficients'' in Sintel's scheme), is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula
 $$ L = B \cdot C^{r/(n - r)} $$
@@ Line 135: / Line 137: @@
 $$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$
-In Sintel's Dirichlet coefficients, or Dirichlet badness,
+In Sintel's derivation,
 $$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$