User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
Add notes about the projection formula. |
→Geometric interpretation of temperaments: Add POTE mapping for meantone. Move (pseudo)scalar multiples to the right side of the vector. |
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:<math>\overleftarrow{12} \cdot \overrightarrow{15/8} = 11</math> | :<math>\overleftarrow{12} \cdot \overrightarrow{15/8} = 11</math> | ||
Indeed 15/8 (the major seventh) is worth 11 steps of 12-TET. | Indeed 15/8 (the major seventh) is worth 11 steps of 12-TET. | ||
However the commas defining temperaments ''do'' belong to the algebra. It makes no difference if you temper out 81/80 or its square 6561/6400 <math> | However the commas defining temperaments ''do'' belong to the algebra. It makes no difference if you temper out 81/80 or its square 6561/6400 <math> \mapsto \overrightarrow{81/80} \cdot 2</math>. You still get the Meantone temperament. As suggested by the observations above we will define [[Rank-2 temperament|rank-2 temperaments]] in 3 dimensions to be comma monzos interpreted as pseudovectors. Thus Meantone <math>= \overrightarrow{81/80} \cdot i</math>. | ||
The geometric interpretation of a temperament is therefore a plane spanned by two vals, or a line that passes through two points thinking projectively. We can project the [[JIP|just intonation point]] to such a line to find the mapping that is as close to [[Just intonation|just intonation]] as possible. In the case of Meantone: | The geometric interpretation of a temperament is therefore a plane spanned by two vals, or a line that passes through two points thinking projectively. We can project the [[JIP|just intonation point]] to such a line to find the mapping that is as close to [[Just intonation|just intonation]] as possible. In the case of Meantone: | ||
:<math>(<1200.0, 1901.9, 2786.3] \cdot \overrightarrow{81/80}i) (\overrightarrow{81/80}i)^{-1} \approx <1202.6, 1899.3, 2787.0]</math> | :<math>(<1200.0, 1901.9, 2786.3] \cdot \overrightarrow{81/80}i) (\overrightarrow{81/80}i)^{-1} \approx <1202.6, 1899.3, 2787.0]</math> | ||
We can then normalize the first component for pure octaves to get <math><1200.0, 1895, 2780.9]</math>. | We can then normalize the first component for pure octaves to get <math><1200.0, 1895, 2780.9]</math>. | ||
We're again abusing <math>\mathcal G(3,0)</math>. [[Tuning map|Tuning maps]] such as the JIP do not belong in the algebra. Their size matters. Another factor we're missing is weights. When we projected the JIP to Meantone we measured distances in cents which is not always perceptually optimal. To get a [[POTE]] tuning we would need to Tenney-weigh the JIP and inverse-weigh the coordinates of <math>\overrightarrow{81/80}i</math> before calculating the projection and inverse-weight the result. For the same result instead of inverse-weighing the pseudovector we could calculate Meantone <math>= \overleftarrow{5} \wedge \overleftarrow{7}</math> in weighted coordinates. | We're again abusing <math>\mathcal G(3,0)</math>. [[Tuning map|Tuning maps]] such as the JIP do not belong in the algebra. Their size matters. Another factor we're missing is weights. When we projected the JIP to Meantone we measured distances in cents which is not always perceptually optimal. To get a [[POTE]] tuning we would need to Tenney-weigh the JIP and inverse-weigh the coordinates of <math>\overrightarrow{81/80}i</math> before calculating the projection and inverse-weight the result. For the same result instead of inverse-weighing the pseudovector we could calculate Meantone <math>= \overleftarrow{5} \wedge \overleftarrow{7}</math> in weighted coordinates. For reference the POTE mapping vector for Meantone is <math><1200.0, 1896.2, 2785.0]</math> | ||
== Higher dimensions == | == Higher dimensions == | ||