Mintaka: Difference between revisions
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In the flatter generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)<sup>2</sup>, placing it 16 generators down. ''However'', as soon as harmonic 20 is inserted, this also equates 5 with (20/9)<sup>2</sup>, tempering [[81/80]] in the 3.4.5 subgroup. ''Furthermore'', this then equates 4/3 to 27/20, 8 generators up, therefore creating a square root of 4 at 4 generators up and making this an [[insane]] restriction of [[meantone]] that must be fixed by including a mapping for 2, which turns out to equate it to the false octave of 243/121 or 99/49. Therefore, as soon as prime 5 is incorporated, this temperament folds into ''Eshurizel'', an elaborate add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]). | In the flatter generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)<sup>2</sup>, placing it 16 generators down. ''However'', as soon as harmonic 20 is inserted, this also equates 5 with (20/9)<sup>2</sup>, tempering [[81/80]] in the 3.4.5 subgroup. ''Furthermore'', this then equates 4/3 to 27/20, 8 generators up, therefore creating a square root of 4 at 4 generators up and making this an [[insane]] restriction of [[meantone]] that must be fixed by including a mapping for 2, which turns out to equate it to the false octave of 243/121 or 99/49. Therefore, as soon as prime 5 is incorporated, this temperament folds into ''Eshurizel'', an elaborate add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]). | ||
Even without the mappings for other primes, this method can be used to introduce octaves into Mintaka in a manner alike to [[sensi]] and [[hedgehog]] being produced as extensions of BPS. Equating the false octave (243/121~99/49) to 2/1 provides [[skwares]] temperament, to which the aforementioned Eshurizel is but an extension. | Even without the mappings for other primes, this method can be used to introduce octaves into Mintaka in a manner alike to [[sensi]] and [[hedgehog]] being produced as extensions of BPS. Equating the false octave (243/121~99/49) to 2/1 provides 2.3.7.11 [[skwares]] temperament, to which the aforementioned Eshurizel is but an extension. | ||
== Interval chains == | == Interval chains == | ||