User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
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== Higher dimensions == | == 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. | 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 (where <math>\mathcal{G}(4,0)</math> is the relevant algebra) <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> | :<math>v \vee u := \overline{ \overline{v} \wedge \overline{u} }</math> | ||
where the overline represents the Hodge dual. | where the overline represents the Hodge dual. | ||
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:<math>e_{12} + 4 e_{13} + 10 e_{14} + 4 e_{23} + 13 e_{24} + 12 e_{34}</math> | :<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. | 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. | ||
The projection formula for calculating the optimal tuning for a temperament | |||
:<math>\overleftarrow{JIP} \cdot \mathbf{T} / \mathbf{T}</math> | |||
works for temperament <math>\mathbf{T}</math> of any rank in just intonation subgroups of any size and most notably without using a single matrix operation! | |||