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

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 Let's return to three dimensions (or two thinking projectively). Meantone can be decomposed into:
 :<math>< 0, 1, 4 ] \wedge <1, 0, -4]</math>
-As tunings these would be a division of 5/1 into 4 equal parts each representing a 3/1, while the other is the octave "divided" into a single unit. Let's call these vals <math>\overleftarrow{v_0}</math> and <math>\overleftarrow{v_1}</math>. I think they represent the generators of Meantone 3/1 and 2/1, but I need to some more research. (to be continued...)
+As tunings these would be a division of 5/1 into 4 equal parts each representing a 3/1, while the other is the octave "divided" into a single unit. Let's call these vals <math>\overleftarrow{v_0}</math> and <math>\overleftarrow{v_1}</math>. Any val <math>\overleftarrow{v}</math> that supports Meantone can be expressed as a linear combination of these two vals and thus the number of scale steps for a given monzo is
+:<math>\begin{align}
+n &= \overleftarrow{v} \cdot \overrightarrow{m} \\
+  &= (p\overleftarrow{v_0} + q\overleftarrow{v_1}) \cdot \overrightarrow{m} \\
+  &= p(\overleftarrow{v_0} \cdot \overrightarrow{m}) + q(\overleftarrow{v_1} \cdot \overrightarrow{m})
+\end{align}
+</math>
+With a lot of hand waving we can say that 3/1 and 2/1 generate the Meantone temperament. Any frequency/pitch from an interval tuned to a specific version of meantone has a counterpart composed of only 3/1 and 2/1 in the same tuning with the same frequency/pitch.
+Another example might be Blackwood <math>= e_3 \wedge \overleftarrow{5}</math> where the generators are 5/1 (based on <math>e_2</math>) and one step of the octave divided into 5 equal parts (denoted in backslash notation as 1\5). It is impossible to find an integral val with a first component larger than zero but smaller than 5.
+A similar thing happens with Augmented <math>= e_2 \wedge \overleftarrow{3}</math>, but now the generators are 3/1 and 1\3.
+Something without a split octave and a 5/1 generator would be Dicot <math>= < 0, 2, 1] \wedge < 1, 1, 2]</math>.
+There are more musically useful ways to express these generators (the one related to the octave is usually called the period), but observations like this should be enough to make software that finds some generators with brute force and uses those to build [[MOS]] scales for rank 2 temperaments which is what the user most likely cares about.