Temperament addition: Difference between revisions
Cmloegcmluin (talk | contribs) →g_{\text{min}}>1: reorganize as I had already wanted to anyway, but now also to make room for this new section for adding non-addable temps |
Cmloegcmluin (talk | contribs) →Getting to the side of duality with g_{\text{min}}=1: minor correction |
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==Getting to the side of duality with <math>g_{\text{min}}=1</math>== | ==Getting to the side of duality with <math>g_{\text{min}}=1</math>== | ||
We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and the porcupine mapping {{ket|{{map|1 2 3}} {{map|0 3 5}}}} each consist of two vectors. So these representations | We may be looking at a temperament representation which itself does not consist of a single vector, but its dual does. For example, the meantone mapping {{ket|{{map|1 0 -4}} {{map|0 1 4}}}} and the porcupine mapping {{ket|{{map|1 2 3}} {{map|0 3 5}}}} each consist of two vectors. So these representations require additional labor to compute. But their duals are easy! If we simply find a comma basis for each of these mappings, we get {{bra|{{vector|4 -4 1}}}} and {{bra|{{vector|1 -5 3}}}}. In this form, the temperaments can be entry-wise added, to {{bra|{{vector|5 -9 4}}}} as we saw earlier. And if in the end we're still after a mapping, since we started with mappings, we can take the dual of this comma basis, to find the mapping {{ket|{{map|1 1 1}} {{map|0 4 9}}}}. | ||
==Negation== | ==Negation== | ||