Mintaka: Difference between revisions

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=== Add 13 and 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 [[~~1045~~/~~1029~~]]). 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.

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 260/243 in quality, and therefore can be identified with 20/19 by tempering out [[20020/19683]], equating 27/13 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 [[539539/531441]]). The alternative extension to include prime 13, known as ''Minalzidar'', works better for tunings ''flatter'' than 9\22edt, where it is the most accurate to find [[13/9]] at 3(9/7)<sup>-3</sup>, 9 generators up, tempering out the comma [[351/343]]. The two representations meet at 22edt.

Once either representation of prime 19 is added to the system, it is reasonable to temper out [[247/243]], which equates [[13/9]] to [[27/19]].

Once either representation of prime 13 is added to the system, it is reasonable to temper out [[247/243]], which equates [[19/9]] to [[27/13]].

=== Add 4 and 5 ===

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 === Add 13 and 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 [[1045/1029]]). 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.
+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 260/243 in quality, and therefore can be identified with 20/19 by tempering out [[20020/19683]], equating 27/13 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 [[539539/531441]]). The alternative extension to include prime 13, known as ''Minalzidar'', works better for tunings ''flatter'' than 9\22edt, where it is the most accurate to find [[13/9]] at 3(9/7)<sup>-3</sup>, 9 generators up, tempering out the comma [[351/343]]. The two representations meet at 22edt.
-Once either representation of prime 19 is added to the system, it is reasonable to temper out [[247/243]], which equates [[13/9]] to [[27/19]].
+Once either representation of prime 13 is added to the system, it is reasonable to temper out [[247/243]], which equates [[19/9]] to [[27/13]].
 === Add 4 and 5 ===