Tenney–Euclidean temperament measures - Revision history

FloraC: Document my implementation of Sintel's error. Request proof for logflat badness. - editorial notes

2026-08-05T16:07:42Z

Document my implementation of Sintel's error. Request proof for logflat badness. - editorial notes

FloraC: /* Preliminaries */ rework

2026-08-05T09:55:57Z

Preliminaries: rework

← Older revision		Revision as of 09:55, 5 August 2026
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	== Preliminaries ==		== Preliminaries ==
	There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. ~~The reason these~~ differences come up is because we are adopting different averaging methods for the entries of a multivector.		There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. These differences come up because we are adopting different averaging methods for the entries of a multivector. To start with, consider a norm defined by means of the usual {{w\|norm (mathematics) #Euclidean norm\|Euclidean norm}}, a.k.a. ''L''<sup>2</sup> or ℓ<sub>2</sub> norm. The result of this is a kind of a sum of all the entries. We can rescale this in several ways, for example by taking a {{w\|root mean square}} (RMS) average of the entries, by normalizing for the temperament's rank, and/or by normalizing up to the norm of the [[just tuning map]].

	~~To start with, we may define a norm by means of the usual {{w\|norm (mathematics) #Euclidean norm\|Euclidean norm}}, a.k.a.~~ ''L''~~<sup>2</sup> norm or ℓ<sub>2</sub> norm~~. ~~The~~ result ~~of this~~ is ~~a kind~~ of ~~a sum of all~~ the ~~entries. We can rescale this in several ways~~, ~~for example by taking a {{w\|root mean square}} (RMS) average~~ of ~~the entries~~.		''These metrics are meant to be used to rank temperaments within the same [[rank]] and [[subgroup]]''. It does not matter much which scheme is used, because 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.

	~~Here are the different standards for scaling that are commonly in use~~:		[[Graham Breed]]'s original definitions<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref> are based on an RMS norm normalized for the temperament's rank, as are parts of his [https://x31eq.com/temper/ Temperament Finder], although other scaling and normalization methods are sometimes used as well.
	~~# Taking the simple~~ ''L''~~<sup>2~~</~~sup~~> ~~norm~~
	~~# Taking~~ an RMS
	~~# Taking an RMS and also normalizing~~ for the temperament rank
	~~# Any~~ of ~~the above and also dividing by the norm of the just intonation points (~~[~~[JIP~~]]).

	~~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~~.		More recently, it is proposed that the metrics should be normalized 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>, though still with notable caveats.

	Graham Breed's original definitions<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref> 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.~~		Each of these will be discussed in detail below.

	~~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 ==

Fredg999: Properly reference another mention of primerr

2026-07-27T02:14:29Z

Properly reference another mention of primerr

← Older revision		Revision as of 02:14, 27 July 2026
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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.		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 [https://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well.		Graham Breed's original definitions<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref> 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 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>.
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	where det(·) denotes the determinant, and {{t}} denotes the transpose.		where det(·) 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"~~>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		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"/>, an RMS norm is proposed as

	$$ \norm{M_W}_\text{RMS} = \sqrt {\det \left( \frac {V_W V_W^\mathsf{T}}{n} \right)} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$		$$ \norm{M_W}_\text{RMS} = \sqrt {\det \left( \frac {V_W V_W^\mathsf{T}}{n} \right)} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$

Xenllium at 02:39, 29 December 2025

2025-12-29T02:39:26Z

← Older revision		Revision as of 02:39, 29 December 2025
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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 defines the TE error as the ratio ‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap\|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap\| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:		Gene Ward Smith defines the TE error as the ratio {{nowrap\|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}}, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap\|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap\| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:

	$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$		$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$
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	''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.		''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.

			Sintel defines the TE error as the ratio {{nowrap\|''G'' {{=}} ‖''M''<sub>''U''</sub> ∧ ''J''<sub>''U''</sub>‖/‖''M''<sub>''U''</sub>‖}}, using ''U''-weighted norm (see the next section), and it results to the same value of Graham's definition.

	== TE simple badness ==		== TE simple badness ==
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	\| 5.400		\| 5.400
	\| 2.763		\| 2.763
	\| 12.~~435×10~~<sup>−3</sup>		\| 1.244×10<sup>−2</sup>
	\|-		\|-
	\| Septimal magic		\| Septimal magic
	\| 7.195		\| 7.195
	\| 2.149		\| 2.149
	\| 12.~~882×10~~<sup>−3</sup>		\| 1.288×10<sup>−2</sup>
	\|}		\|}
	{\| class="wikitable center-all left-1"		{\| class="wikitable center-all left-1"
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	\| 2.631		\| 2.631
	\| 6.441×10<sup>−3</sup>		\| 6.441×10<sup>−3</sup>
			\|}
			{\| class="wikitable center-all left-1"
			\|+ style="font-size: 105%;" \| Sintel's norm
			\|-
			! Temperament
			! Complexity
			! Error (¢)
			! Simple badness
			\|-
			\| Septimal meantone
			\| 17.357
			\| 1.382
			\| 1.999×10<sup>−2</sup>
			\|-
			\| Septimal magic
			\| 23.126
			\| 1.074
			\| 2.070×10<sup>−2</sup>
	\|}		\|}

FloraC: /* TE complexity */ explanation via Mike Battaglia (archive #20598)

2025-08-26T12:54:54Z

TE complexity: explanation via Mike Battaglia (archive #20598)

← Older revision		Revision as of 12:54, 26 August 2025
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	== TE complexity ==		== TE complexity ==
	~~{{Todo\|rework\|inline=1\|text=Explain without wedgies}}~~		TE complexity is the average hypervolume of the parallelepipeds formed by any ''n'' linearly independent generators which form and saturate that lattice. Since this is exactly the same thing as the magnitude of the multivector formed by those vectors, TE complexity is the exact same thing as the average of the coefficients of the [[wedgie]] for that temperament.

	~~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~~.

	Let us define the val weighting matrix ''W'' to be the {{w\|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap\|''Q'' {{=}} {{val\| 2 3 5 … ''p'' }} }},		Let us define the val weighting matrix ''W'' to be the {{w\|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap\|''Q'' {{=}} {{val\| 2 3 5 … ''p'' }} }},
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	If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''<sub>''w''</sub>)<sub>''i''</sub>.		If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''<sub>''w''</sub>)<sub>''i''</sub>.

			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.

	Our first complexity measure of a temperament is given by the ''L''<sup>2</sup> norm of the Tenney-weighted wedgie ''M''<sub>''W''</sub>, which can in turn be obtained from the Tenney-weighted mapping matrix ''V''<sub>''W''</sub>. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w\|Gramian matrix\|Gramian}}:		Our first complexity measure of a temperament is given by the ''L''<sup>2</sup> norm of the Tenney-weighted wedgie ''M''<sub>''W''</sub>, which can in turn be obtained from the Tenney-weighted mapping matrix ''V''<sub>''W''</sub>. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w\|Gramian matrix\|Gramian}}:

FloraC: Undo revision 203571 by VectorGraphics (talk). Readers aren't obligated to see your poor-tasted operation name

2025-06-25T14:25:40Z

Undo revision 203571 by VectorGraphics (talk). Readers aren't obligated to see your poor-tasted operation name

← Older revision		Revision as of 14:25, 25 June 2025
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	{{Texops}}~~{{Todo\|rewrite\|inline=1\|text=Remove all references to wedgies, as per [[User:VectorGraphics/Operation_Loosen_Underpants]]}}~~		{{Texops}}

	The '''Tenney–Euclidean temperament measures''' ('''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. These are evaluations of a temperament's [[complexity]], [[error]], and [[badness]], respectively, and they follow the identity		The '''Tenney–Euclidean temperament measures''' ('''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. These are evaluations of a temperament's [[complexity]], [[error]], and [[badness]], respectively, and they follow the identity

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	== 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.

VectorGraphics at 04:30, 25 June 2025

2025-06-25T04:30:52Z

← Older revision		Revision as of 04:30, 25 June 2025
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	{{Texops}}		{{Texops}}{{Todo\|rewrite\|inline=1\|text=Remove all references to wedgies, as per [[User:VectorGraphics/Operation_Loosen_Underpants]]}}

	The '''Tenney–Euclidean temperament measures''' ('''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. These are evaluations of a temperament's [[complexity]], [[error]], and [[badness]], respectively, and they follow the identity		The '''Tenney–Euclidean temperament measures''' ('''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. These are evaluations of a temperament's [[complexity]], [[error]], and [[badness]], respectively, and they follow the identity

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	== 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.

VectorGraphics at 03:58, 12 June 2025

2025-06-12T03:58:09Z

← Older revision		Revision as of 03:58, 12 June 2025
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	== 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.

FloraC: Dirichlet coeffcients was meant to be a generic substitute for logflat badness

2025-06-06T16:36:12Z

*Dirichlet coeffcients* was meant to be a generic substitute for logflat badness

← Older revision		Revision as of 16:36, 6 June 2025
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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~~, 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 ==		== TE complexity ==
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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 ~~Dirichlet coefficients, or Dirichlet badness~~,		In Sintel's derivation,

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

FloraC: Minor cleanup

2025-06-05T03:56:50Z

Minor cleanup

← Older revision		Revision as of 03:56, 5 June 2025
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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.		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, 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>.
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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.

	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		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
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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.		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 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 ==		== TE simple badness ==