User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions
Add a more dot-producty version of the rank-2 procedure |
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=== Geometric algebraic version === | === Geometric algebraic version === | ||
Canonize rank-2 temperament <math>\mathbf{T}</math> to its wedgie form <math>\mathbf{W}</math> and find | Canonize rank-2 temperament <math>\mathbf{T}</math> to its wedgie form <math>\mathbf{W}</math> and find <math>\overrightarrow{g}</math> (with integer coefficients) such that | ||
:<math>\overrightarrow{g} \cdot (e_1 \cdot \mathbf{W}) = d</math> | :<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 | 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> | :<math>n \overrightarrow{p} ± \overrightarrow{g}, n \in \mathbb{Z}</math> | ||