User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
→Mapping: Claim that the mapping can be calculated using matrix inversion. |
m →Mapping: Comments about non-integral period monzos. |
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For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>. | For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>. | ||
If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', I'm pretty sure the non-integral components of the matrix inverse will fall onto the comma mapping vals so they don't matter in the end. I also believe that its always possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically. | |||
== Optimizing database keys == | == Optimizing database keys == | ||