Temperament addition: Difference between revisions
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Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish. | Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish. | ||
Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; | Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Scale trees|see here]] for more information on this. So if you understand the effects on individual maps, then you can apply those to changes of maps within a more complex temperament. | ||
Ultimately, these two effects are the primary applications of temperament addition.<ref>It has also been asserted that there exists a connection between temperament addition and "Fokker groups" as discussed on this page: [[Fokker block]], but the connection remains unclear to this author.</ref> | Ultimately, these two effects are the primary applications of temperament addition.<ref>It has also been asserted that there exists a connection between temperament addition and "Fokker groups" as discussed on this page: [[Fokker block]], but the connection remains unclear to this author.</ref> | ||
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And the canonical form of [{{vector|-2 -2 1 1}} {{vector|5 -5 0 1}}] is [{{vector|-7 3 1 0}} {{vector|5 -5 0 1}}], so that's yet another possible temperament resulting from | And the canonical form of [{{vector|-2 -2 1 1}} {{vector|5 -5 0 1}}] is [{{vector|-7 3 1 0}} {{vector|5 -5 0 1}}], so that's yet another possible temperament resulting from attempting to add these non-addable temperaments. | ||
==== Fourth example: other side of duality ==== | ==== Fourth example: other side of duality ==== | ||
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In order to compare this result directly with our other three results, let's take the dual of this {{rket|{{map|6 8 -4 1}} {{map|0 1 5 10}}}}, which is [{{vector|22 -15 3 0}} {{vector|41 -30 2 2}}] (in canonical form), so we can see that's yet a fourth possible result<ref> | In order to compare this result directly with our other three results, let's take the dual of this {{rket|{{map|6 8 -4 1}} {{map|0 1 5 10}}}}, which is [{{vector|22 -15 3 0}} {{vector|41 -30 2 2}}] (in canonical form), so we can see that's yet a fourth possible result.<ref> | ||
It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{rket|{{map|97 152 220 259}} {{map|-30 -47 -68 -80}}}} + {{rket|{{map|-95 -152 -212 -266}} {{map|30 48 67 84}}}}.</ref> | It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{rket|{{map|97 152 220 259}} {{map|-30 -47 -68 -80}}}} + {{rket|{{map|-95 -152 -212 -266}} {{map|30 48 67 84}}}}.</ref> | ||