Mintaka: Difference between revisions

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The alternative extension to include prime 19, known as ''Minalzidar'', works better for tunings ''flatter'' than 9\22edt, where it is the most accurate to find [[19/9]] at (9/7)^3, 9 generators up, tempering out the comma [[6561/6517]]. The two representations meet at 22edt.

In this range, the optimal representation of 5 is that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)2, placing it 16 generators down. ''However'', as soon as prime 20 is inserted, this also equates 5 with (20/9)2, 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]]. Therefore, as soon as prime 5 is incorporated, this temperament folds into ''Eshurizel'', an elaborate extension of 11-limit [[squares]].

In this range, the optimal representation of 5 is that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)2, placing it 16 generators down. ''However'', as soon as prime 20 is inserted, this also equates 5 with (20/9)2, 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 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).

=== Add 4 and 5 ===

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 The alternative extension to include prime 19, known as ''Minalzidar'', works better for tunings ''flatter'' than 9\22edt, where it is the most accurate to find [[19/9]] at (9/7)^3, 9 generators up, tempering out the comma [[6561/6517]]. The two representations meet at 22edt.
-In this range, the optimal representation of 5 is 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 prime 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]]. Therefore, as soon as prime 5 is incorporated, this temperament folds into ''Eshurizel'', an elaborate extension of 11-limit [[squares]].
+In this range, the optimal representation of 5 is 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 prime 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 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).
 === Add 4 and 5 ===

Mintaka: Difference between revisions

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