Mintaka: Difference between revisions

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=== Add 19 ===

There are two reasonable ways to incorporate prime 19 into the subgroup. For tunings of the generator ''sharper'' than 9\22edt, the step 81/77 approaches or exceeds 20/19 in quality, and therefore can be identified with 20/19 by tempering out [[1540/1539]], equating 19/9 to (77/81)(20/9), 13 generators down (or alternatively, if one refuses to admit the even number 20 into the subgroup, by tempering out [[16929/16807]]). 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.

~~==== Eshurizel ====~~

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 ===

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 === Add 19 ===
 There are two reasonable ways to incorporate prime 19 into the subgroup. For tunings of the generator ''sharper'' than 9\22edt, the step 81/77 approaches or exceeds 20/19 in quality, and therefore can be identified with 20/19 by tempering out [[1540/1539]], equating 19/9 to (77/81)(20/9), 13 generators down (or alternatively, if one refuses to admit the even number 20 into the subgroup, by tempering out [[16929/16807]]). 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.
-==== Eshurizel ====
-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 ===