User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions

← Older edit Newer edit →

VisualWikitext

@@ Line 77: / Line 77: @@
 :<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math>
 If the temperament is not expressed in lowest terms the number of divisions will miscompensate and you will end up with the wrong equave.
+=== Mapping ===
+An expression such as <math>\mathbf{T} = \overleftarrow{12} \wedge \overleftarrow{19}</math> is nice for getting the feel of a temperament: Meantone feels like what [[12edo]] and [[19edo]] have in common. However it's hard  to figure out how a given monzo should be mapped to periods and generators.
+We can invert <math>\overrightarrow{p}</math> and <math>\overrightarrow{g}</math> ''(TODO: How?)'' to produce a [[mapping]] for the temperament. Let's choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. The mapping vals will be:
+:<math>\begin{align}
+\overleftarrow{p} &= < 1, 1, 0 ] \\
+\overleftarrow{g} &= < 0, 1, 4 ]
+\end{align}</math>
+Now <math>\mathbf{T} = \overleftarrow{p} \wedge \overleftarrow{g}</math> and for any monzo <math>\overrightarrow{m}</math> we can get the number of periods as <math>\overleftarrow{p} \cdot \overrightarrow{m}</math> and the number of generators as <math>\overleftarrow{g} \cdot \overrightarrow{m}</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 because <math>\frac{5}{3}\frac{81}{80} = \frac{27}{16}</math> and the comma <math>\frac{81}{80}</math> vanishes in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave.
 == Optimizing database keys ==