User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
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Add a more dot-producty version of the rank-2 procedure |
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This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament. | This generator might not be the most meaningful musically, but it's enough to build [[MOS]] scales for rank-2 temperaments which along with the POTE tuning is the main thing you need to compose music in the chosen temperament. | ||
=== Geometric algebraic version === | |||
Canonize rank-2 temperament <math>\mathbf{T}</math> to its wedgie form <math>\mathbf{W}</math> and find integral <math>\overrightarrow{g}</math> such that | |||
:<math>\overrightarrow{g} \cdot (e_1 \cdot \mathbf{W}) = d</math> | |||
where <math>d = gcd(...(e_1 \cdot \mathbf{W}))</math>. The period can be expressed (non-integrally) as <math>\overrightarrow{p} = e_1 / d</math> and a valid generator is <math>\overrightarrow{g}</math>. Other valid generators are of the form | |||
:<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math> | |||