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.

There have been several ~~ways to define~~ TE temperament measures, ~~and~~ each will be discussed below. Nonetheless, the following relationship always holds:

[[Cangwu_badness|Cangwu badness]] is an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error.

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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== Introduction ==

Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [http://en.wikipedia.org/wiki/Exterior_algebra wedge product] of weighted vals or monzos, we may define a norm by means of the usual Euclidean 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 ([http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Exterior_algebra wedge product] of weighted vals or monzos, we may define a norm by means of the usual Euclidean (<math>\ell_2</math>) 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 ([http://en.wikipedia.org/wiki/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>.

=== A Preliminary Note on Scaling Factors ===

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 <math>\ell_2</math>

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:

# Taking an RMS

# Taking an RMS and also normalizing for the temperament rank

# Taking the simple <math>\ell_2</math> norm

Graham Breed's original definitions from his "Primerr.pdf" paper tend to use the second definition, as do *parts* of his [http://x31eq.com/temper/ the 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 intra-subgroup 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.
-There have been several ways to define TE temperament measures, and each will be discussed below. Nonetheless, the following relationship always holds:
+[[Cangwu_badness|Cangwu badness]] is an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error.
+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 10: @@
 == Introduction ==
-Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [http://en.wikipedia.org/wiki/Exterior_algebra wedge product] of weighted vals or monzos, we may define a norm by means of the usual Euclidean 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 ([http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Exterior_algebra wedge product] of weighted vals or monzos, we may define a norm by means of the usual Euclidean (<math>\ell_2</math>) 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 ([http://en.wikipedia.org/wiki/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>.
+=== A Preliminary Note on Scaling Factors ===
+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 <math>\ell_2</math>
+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:
+# Taking an RMS
+# Taking an RMS and also normalizing for the temperament rank
+# Taking the simple <math>\ell_2</math> norm
+Graham Breed's original definitions from his "Primerr.pdf" paper tend to use the second definition, as do *parts* of his [http://x31eq.com/temper/ the 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 intra-subgroup comparisons, this can be more complicated.
 == TE Complexity ==