Temperament addition: Difference between revisions

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Linear dependence 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 linear dependence for temperaments whereby temperaments are considered linearly dependent if ''either of their mappings or their comma bases are linearly dependent''<ref>or — equivalently, in EA — either their multimaps or their multicommas are linearly dependent</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 linearly independent, because the basis vectors visible in their mappings are clearly linearly independent (when comparing two vectors, the only way they could be linearly dependent is if they are multiples of each other, as discussed [[Linear dependence#Linear dependence between individual vectors|here]]). And indeed their ''mappings'' are linearly independent. But these two ''temperaments'' are linearly ''de''pendent, because if we consider their corresponding comma bases, we will find that they share the basis vector of the meantone comma {{vector|4 -4 1}}.

To make this point visually, we could say that two temperaments are linearly dependent if they intersect in one or the other of tone space and tuning space. So you have to check both views.<ref>You may be wondering — what about two temperaments which are parallel in tone or tuning space, e.g. compton and blackwood in tuning space? Their comma bases are each nullity 1, and they meet to give a nullity 2 comma basis, which corresponds to a rank-1 mapping, which means it should appear as an ET point on the PTS diagram. But how could that be? Well, here's their meet: {{bra|{{vector|1 0 0}} {{vector|0 1 0}}}}, and so that corresponding mapping is {{ket|{{map|0 0 1}}}}. So it's some degenerate ET. I suppose we could say it's the point at infinity away from the center of the diagram.</ref>

====3. Linear independence between temperaments====

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 Linear dependence 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 linear dependence for temperaments whereby temperaments are considered linearly dependent if ''either of their mappings or their comma bases are linearly dependent''<ref>or — equivalently, in EA — either their multimaps or their multicommas are linearly dependent</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 linearly independent, because the basis vectors visible in their mappings are clearly linearly independent (when comparing two vectors, the only way they could be linearly dependent is if they are multiples of each other, as discussed [[Linear dependence#Linear dependence between individual vectors|here]]). And indeed their ''mappings'' are linearly independent. But these two ''temperaments'' are linearly ''de''pendent, because if we consider their corresponding comma bases, we will find that they share the basis vector of the meantone comma {{vector|4 -4 1}}.
+To make this point visually, we could say that two temperaments are linearly dependent if they intersect in one or the other of tone space and tuning space. So you have to check both views.<ref>You may be wondering — what about two temperaments which are parallel in tone or tuning space, e.g. compton and blackwood in tuning space? Their comma bases are each nullity 1, and they meet to give a nullity 2 comma basis, which corresponds to a rank-1 mapping, which means it should appear as an ET point on the PTS diagram. But how could that be? Well, here's their meet: {{bra|{{vector|1 0 0}} {{vector|0 1 0}}}}, and so that corresponding mapping is {{ket|{{map|0 0 1}}}}. So it's some degenerate ET. I suppose we could say it's the point at infinity away from the center of the diagram.</ref>
 ====3. Linear independence between temperaments====