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. | ||
Ultimately, this effect is the primary application of temperament arithmetic. | Ultimately, this effect is the primary application 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> | ||
Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. | Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. | ||
=A note on variance= | =A note on variance= | ||