Tenney–Euclidean temperament measures: Difference between revisions

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The '''Tenney-Euclidean temperament measures''' (or '''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness.

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. 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. Each of these variations will be discussed below. Nonetheless, the following relationship always holds:

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. Each of these variations will be discussed below. Nonetheless, the following relationship always holds:

<math>\displaystyle

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TE temperament measures have been extensively studied by [[Graham Breed]] (see [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''), who also proposed [[Cangwu badness]], an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error.

== ~~Introduction~~ ==

== 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/log2(''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''2 norm or ℓ2 norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ({{w|Root mean square|root mean square}}) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different ~~prime limits~~ can be meaningfully compared. If M is a multivector, we denote the RMS norm as ‖M‖RMS.

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/log2(''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''2 norm or ℓ2 norm).

We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ({{w|Root mean square|root mean square}}) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different [[just intonation subgroup]]s can be meaningfully compared. If M is a multivector, we denote the RMS norm as ‖M‖RMS.

~~=== Preliminary note on scaling factors ===~~

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''2 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.

These metrics are mainly used to rank temperaments relative to one another. In that regard, it ~~doesn't~~ matter much if an RMS or an ''L''2 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:

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# 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 [[JIP]] ~~(just intonation points~~).

# Any of the above and also dividing by the norm of the just intonation points ([[JIP]]).

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.

Note that the above is mainly for comparing temperaments within the same subgroup; when making intersubgroup comparisons, this can be more complicated.

== TE complexity ==

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-The '''Tenney-Euclidean temperament measures''' (or '''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness.
+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. 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. Each of these variations will be discussed below. Nonetheless, the following relationship always holds:
-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. Each of these variations will be discussed below. Nonetheless, the following relationship always holds:
 <math>\displaystyle
@@ Line 8: / Line 6: @@
 TE temperament measures have been extensively studied by [[Graham Breed]] (see [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''), who also proposed [[Cangwu badness]], an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error.
-== Introduction ==
+== 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). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ({{w|Root mean square|root mean square}}) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ‖M‖<sub>RMS</sub>.
+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 by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ({{w|Root mean square|root mean square}}) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different [[just intonation subgroup]]s can be meaningfully compared. If M is a multivector, we denote the RMS norm as ‖M‖<sub>RMS</sub>.
-=== Preliminary note on scaling factors ===
+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.
-These metrics are mainly used to rank temperaments relative to one another. In that regard, it doesn't 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:
@@ Line 19: / Line 18: @@
 # 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 [[JIP]] (just intonation points).
+# Any of the above and also dividing by the norm of the just intonation points ([[JIP]]).
 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.
 Note that the above is mainly for comparing temperaments within the same subgroup; when making intersubgroup comparisons, this can be more complicated.
 == TE complexity ==