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 [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 ~~intra-subgroup~~ comparisons, this can be more complicated.

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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Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [[wikipedia: Gramian matrix|Gramian]].

Let's denote a weighted mapping matrix, whose rows are the weighted vals ''vi'', as V. The ''L''2 norm is one of the standard measures,

Let's denote a weighted mapping matrix, whose rows are the weighted vals ''vi'', as V. The ''L''2 norm is one of the standard complexity measures:

<math>\displaystyle

@@ Line 25: / Line 25: @@
 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 intra-subgroup comparisons, this can be more complicated.
+Note that the above is mainly for comparing temperaments within the same subgroup; when making intersubgroup comparisons, this can be more complicated.
 == TE complexity ==
@@ Line 32: / Line 32: @@
 Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [[wikipedia: Gramian matrix|Gramian]].
-Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures,
+Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard complexity measures:
 <math>\displaystyle