Tenney–Euclidean temperament measures: Difference between revisions
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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 [https://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 | 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 == | == 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. | 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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== TE logflat badness == | == 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 | ||
$$ L = B \cdot C^{r/(n - r)} $$ | $$ L = B \cdot C^{r/(n - r)} $$ | ||
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$$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$ | $$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$ | ||
In Sintel's | In Sintel's derivation, | ||
$$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$ | $$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$ | ||