User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions

@@ Line 72: / Line 72: @@
 === Geometric algebraic version ===
-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
+Given a rank-2 temperament <math>\mathbf{T}</math> 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{T}) = 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{T}))</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>
+If the temperament is not expressed in lowest terms the number of divisions will miscompensate and you will end up with the wrong equave.