Temperament addition: Difference between revisions

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Collinearity has been defined for the matrices and multivectors that represent temperaments, but it can also be defined for temperaments themselves. The conditions of temperament arithmetic motivate a definition of collinearity for temperaments whereby temperaments are considered collinear if ''either of their mappings or their comma bases are collinear''<ref>or — equivalently, in EA — either their multimaps or their multicommas are collinear</ref>.

For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{map|7 11 16}}}} may at first seem to be noncollinear, because the vectors visible in their mappings are clearly noncollinear (when comparing two vectors, the only way they could be collinear is if they are multiples of each other, as discussed here ~~###~~). And indeed their ''mappings'' are noncollinear. But these two ''temperaments'' are collinear, because if we consider their corresponding comma bases, we will find that they share the vector of the meantone comma {{vector|4 -4 1}}.

For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{map|7 11 16}}}} may at first seem to be noncollinear, because the vectors visible in their mappings are clearly noncollinear (when comparing two vectors, the only way they could be collinear is if they are multiples of each other, as discussed [[Collinearity#Individual_vector_collinearity|here]]). And indeed their ''mappings'' are noncollinear. But these two ''temperaments'' are collinear, because if we consider their corresponding comma bases, we will find that they share the vector of the meantone comma {{vector|4 -4 1}}.

====3. Temperament noncollinearity====

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 Collinearity has been defined for the matrices and multivectors that represent temperaments, but it can also be defined for temperaments themselves. The conditions of temperament arithmetic motivate a definition of collinearity for temperaments whereby temperaments are considered collinear if ''either of their mappings or their comma bases are collinear''<ref>or — equivalently, in EA — either their multimaps or their multicommas are collinear</ref>.
-For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{map|7 11 16}}}} may at first seem to be noncollinear, because the vectors visible in their mappings are clearly noncollinear (when comparing two vectors, the only way they could be collinear is if they are multiples of each other, as discussed here ###). And indeed their ''mappings'' are noncollinear. But these two ''temperaments'' are collinear, because if we consider their corresponding comma bases, we will find that they share the vector of the meantone comma {{vector|4 -4 1}}.
+For example, 5-limit 5-ET and 5-limit 7-ET, represented by the mappings {{ket|{{map|5 8 12}}}} and {{ket|{{map|7 11 16}}}} may at first seem to be noncollinear, because the vectors visible in their mappings are clearly noncollinear (when comparing two vectors, the only way they could be collinear is if they are multiples of each other, as discussed [[Collinearity#Individual_vector_collinearity|here]]). And indeed their ''mappings'' are noncollinear. But these two ''temperaments'' are collinear, because if we consider their corresponding comma bases, we will find that they share the vector of the meantone comma {{vector|4 -4 1}}.
 ====3. Temperament noncollinearity====