User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
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Because vals can be multiplied by an arbitrary scalar they represent lines that pass through the origin. Because one endpoint of the line is fixed the resulting space of interest is 2-dimensional in the case of three component vals. This is known as the val space and is isomorphic to the [[Projective tuning space|projective tuning space]]. | Because vals can be multiplied by an arbitrary scalar they represent lines that pass through the origin. Because one endpoint of the line is fixed the resulting space of interest is 2-dimensional in the case of three component vals. This is known as the val space and is isomorphic to the [[Projective tuning space|projective tuning space]]. | ||
3D lines that pass through the origin find representation in the Geometric Algebra <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space. | 3D lines that pass through the origin find representation in the [https://en.wikipedia.org/wiki/Geometric_algebra Geometric Algebra] <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space. | ||
:<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math> | :<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math> | ||
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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}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 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 == | == Higher dimensions == | ||