Generator complexity: Difference between revisions

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If B = &lt;0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the //STD complexity// (a term due to Graham Breed) is P STD(B), where "STD" means the standard deviation. If μ(V) is the mean of the components of the vector V, and J is the [[JIP]] &lt;1 1 1 ... 1|, then  ₱(V) = V - μ(V)J is the projection of V onto the subspace of vectors with zero mean value. We have STD(V) = √( ₱(V) ∙ ₱(V) / dim(V)), where dim(V) is the dimension of V and the " ∙ " denotes the dot product. If M = [M0, M1] is the [[Temperament Mapping Matrices (M-maps)|mapping matrix]] in weighted coordinates in the standard [[Normal lists#x-Normal%20val%20lists|normal val list]] form, then we may express STD complexity as STDcom(M) = M0[1] STD(M1).

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If B = &amp;lt;0 B₃ B₅ B₇ ... Bp| is the generator mapping val in weighted coordinates, and P is the period, then the &lt;em&gt;STD complexity&lt;/em&gt; (a term due to Graham Breed) is P STD(B), where &amp;quot;STD&amp;quot; means the standard deviation. If μ(V) is the mean of the components of the vector V, and J is the &lt;a class="wiki_link" href="/JIP"&gt;JIP&lt;/a&gt; &amp;lt;1 1 1 ... 1|, then  ₱(V) = V - μ(V)J is the projection of V onto the subspace of vectors with zero mean value. We have STD(V) = √( ₱(V) ∙ ₱(V) / dim(V)), where dim(V) is the dimension of V and the &amp;quot; ∙ &amp;quot; denotes the dot product. If M = [M0, M1] is the &lt;a class="wiki_link" href="/Temperament%20Mapping%20Matrices%20%28M-maps%29"&gt;mapping matrix&lt;/a&gt; in weighted coordinates in the standard &lt;a class="wiki_link" href="/Normal%20lists#x-Normal%20val%20lists"&gt;normal val list&lt;/a&gt; form, then we may express STD complexity as STDcom(M) = M0[1] STD(M1).&lt;br /&gt;

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