Tenney–Euclidean temperament measures: Difference between revisions

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: '''Note''': that is the definition currently used throughout the wiki, unless stated otherwise.

If W is a [[Wikipedia: Diagonal matrix|diagonal matrix]] with 1, 1/log23, …, 1/log2''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then {{nowrap|''V'' {{=}} ''AW''}} and {{nowrap|{{!}}''VV''{{t}}{{!}} {{=}} {{!}}''AW''2''A''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P'', which is ''V''{{t}}(''VV''{{t}}){{inv}}''V'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''A''{{t}}(''AW''2''A''{{t}}){{inv}}''A''}}.

If ''W'' is a [[Wikipedia: Diagonal matrix|diagonal matrix]] with 1, 1/log23, …, 1/log2''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then {{nowrap|''V'' {{=}} ''AW''}} and {{nowrap|{{!}}''VV''{{t}}{{!}} {{=}} {{!}}''AW'' 2''A''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P'', which is ''V''{{t}}(''VV''{{t}}){{inv}}''V'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''A''{{t}}(''AW'' 2''A''{{t}}){{inv}}''A''}}.

== TE simple badness ==

@@ Line 50: / Line 50: @@
 : '''Note''': that is the definition currently used throughout the wiki, unless stated otherwise.
-If W is a [[Wikipedia: Diagonal matrix|diagonal matrix]] with 1, 1/log<sub>2</sub>3, …, 1/log<sub>2</sub>''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then {{nowrap|''V'' {{=}} ''AW''}} and {{nowrap|{{!}}''VV''{{t}}{{!}} {{=}} {{!}}''AW''<sup>2</sup>''A''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P'', which is ''V''{{t}}(''VV''{{t}}){{inv}}''V'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''A''{{t}}(''AW''<sup>2</sup>''A''{{t}}){{inv}}''A''}}.
+If ''W'' is a [[Wikipedia: Diagonal matrix|diagonal matrix]] with 1, 1/log<sub>2</sub>3, …, 1/log<sub>2</sub>''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then {{nowrap|''V'' {{=}} ''AW''}} and {{nowrap|{{!}}''VV''{{t}}{{!}} {{=}} {{!}}''AW''&#x2009;<sup>2</sup>''A''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P'', which is ''V''{{t}}(''VV''{{t}}){{inv}}''V'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''A''{{t}}(''AW''&#x2009;<sup>2</sup>''A''{{t}}){{inv}}''A''}}.
 == TE simple badness ==