Mintaka: Difference between revisions

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Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.

=== Add 19 ===

=== 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.

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]].

=== Add 4 and 5 ===

For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with [[BPS]], and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. This is ''mintra'' temperament, which splits the BPS generator in three. With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.

For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with [[BPS]] (as well as [[Deneb]] in the 3.5.11 subgroup), and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. This is ''mintra'' temperament, which splits the BPS generator in three. With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.

If we combine all of the above, using the sharper representation of 5 and 19, we find the complete 3.4.5.7.11.19.23 temperament with commas [[100/99]], [[133/132]], 253/252, 484/483, and 540/539.

If we combine all of the above, using the sharper representation of 5 and 19, we find the complete 3.4.5.7.11.13.19.23 temperament with commas [[100/99]], [[133/132]], 247/245, 253/252, 484/483, and 540/539.

==== Eshurizel ====

@@ Line 13: / Line 13: @@
 Off the bat, given that 1331/1323 is a [[Square superparticular#Sk2_.2A_S.28k_.2B_1.29_and_S.28k_-_1.29_.2A_Sk2_.28lopsided_commas.29|lopsided comma]] with S-expression S22<sup>2</sup> * S23, one can reliably choose to temper both S22 = [[484/483]] and S23 = [[529/528]] in the 3.7.11.23/4 subgroup, which equates the 11/7 generator to [[36/23]], and the interval [[11/9]] to [[28/23]]. Furthermore, the tiny comma S161 = [[25921/25920]] can be tempered to add harmonic 20 to the subgroup, finding it 8 generators down. More neatly, this can be expressed as the temperament that tempers out the commas [[253/252]], 484/483, and [[540/539]] in the 3.7.11.20.23/4 subgroup.
-=== Add 19 ===
+=== 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.
+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]].
 === Add 4 and 5 ===
-For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with [[BPS]], and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. This is ''mintra'' temperament, which splits the BPS generator in three. With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.
+For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma), it is reasonable to combine this temperament with [[BPS]] (as well as [[Deneb]] in the 3.5.11 subgroup), and additionally temper out [[245/243]], thereby equating [[5/3]] to 81/49 at 6 generators up. This is ''mintra'' temperament, which splits the BPS generator in three. With the inclusion of 20 in the subgroup above, [[4/3]] would therefore also appear, at the position of (20/9)/(5/3), 14 generators down.
-If we combine all of the above, using the sharper representation of 5 and 19, we find the complete 3.4.5.7.11.19.23 temperament with commas [[100/99]], [[133/132]], 253/252, 484/483, and 540/539.
+If we combine all of the above, using the sharper representation of 5 and 19, we find the complete 3.4.5.7.11.13.19.23 temperament with commas [[100/99]], [[133/132]], 247/245, 253/252, 484/483, and 540/539.
 ==== Eshurizel ====