Tenney–Euclidean metrics: Difference between revisions
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If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW<sup>2</sup>R<sup>T</sup>)<sup>-1</sup>. In the case of marvel, we obtain S = [[(p3)<sup>2</sup>(4(p5)<sup>2</sup>+(p7)<sup>2</sup>) -4(p3)<sup>2</sup>(p5)<sup>2</sup>], [-4(p3)<sup>2</sup>(p5)<sup>2</sup> (p5)<sup>2</sup>(4(p3)<sup>2</sup>+(p7)<sup>2</sup>)]]/H. If k = {{monzo| ''k''<sub>1</sub> ''k''<sub>2</sub> }} is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt (k<sup>T</sup>Sk) gives the OE complexity of the note class. | If we start from a normal val list and remove the first val, the remaining vals map to the octave classes of the notes of the temperament. If we call this reduced list of vals R, then the inner product on note classes in this basis is defined by the symmetric matrix S = (RW<sup>2</sup>R<sup>T</sup>)<sup>-1</sup>. In the case of marvel, we obtain S = [[(p3)<sup>2</sup>(4(p5)<sup>2</sup>+(p7)<sup>2</sup>) -4(p3)<sup>2</sup>(p5)<sup>2</sup>], [-4(p3)<sup>2</sup>(p5)<sup>2</sup> (p5)<sup>2</sup>(4(p3)<sup>2</sup>+(p7)<sup>2</sup>)]]/H. If k = {{monzo| ''k''<sub>1</sub> ''k''<sub>2</sub> }} is a note class of marvel in the coordinates defined by the truncated val list R, which in this case has a basis corresponding to tempered 3 and 5, then sqrt (k<sup>T</sup>Sk) gives the OE complexity of the note class. | ||
[[Category:Regular temperament theory]] | |||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Metric]] | [[Category:Metric]] | ||
{{todo|reduce mathslang}} | {{todo|reduce mathslang}} | ||