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 temper out the meantone comma itself, nor the porcupine comma itself, but instead tempers out whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that tempers out neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot tempers out 81/80 × 250/243, and dicot tempers out 81/80 × 243/250. | Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't temper out the meantone comma itself, nor the porcupine comma itself, but instead tempers out whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that tempers out neither meantone nor porcupine, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot tempers out 81/80 × 250/243, and dicot tempers out 81/80 × 243/250. | ||
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 arithmetic on maps is essentially navigation of the scale tree for the rank-2 temperament they share; for more information on this, see [[Douglas Blumeyer's RTT How-To#Scale trees]]. | 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 arithmetic on maps is essentially navigation of the scale tree for the rank-2 temperament they share; for more information on this, see [[Douglas Blumeyer's RTT How-To#Scale trees]]. 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 effects | Ultimately, these two effects are the primary applications of temperament arithmetic.<ref>It has also been asserted that there exists a connection between temperament arithmetic and "Fokker groups" as discussed on this page: [[Fokker block]], but the connection remains unclear to this author.</ref> | ||
=A note on variance= | =A note on variance= | ||