User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
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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'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}</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 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. | ||
== Higher dimensions == | |||
The same math works in higher [[Harmonic limit|prime limits]], but now the wedge product of two vals is not a pseudovector. For example in the 7-limit which is 4-dimensional <math>\overleftarrow{19} \wedge \overleftarrow{12}</math> is a rank-2 temperament while <math>\overrightarrow{126/125}i</math> is a rank-3 temperament. To combine these objects (which we might call pseudovals) into lower-ranked temperaments we use the vee product. | |||
:<math>v \vee u := \overline{ \overline{v} \wedge \overline{u} }</math> | |||
where the overline represents the Hodge dual. | |||
To give an example we have Septimal Meantone <math>= \overleftarrow{19} \wedge \overleftarrow{12} = \overrightarrow{81/80}i \vee \overrightarrow{126/125}i</math> with the numerical value: | |||
:<math>e_{12} + 4 e_{13} + 10 e_{14} + 4 e_{23} + 13 e_{24} + 12 e_{34}</math> | |||
I do not yet know the significance of these numbers, but it is of great practical use that rank-2 temperaments in higher dimensions get unique integral representations (up to scalar multiplication). Because these values can be made canonical they can be used as keys in a database for querying information about temperaments. | |||