Mintaka: Difference between revisions

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Several extensions of this temperament are possible to incorporate additional harmonics.

=== Add ~~20 and~~ 23/4 ===

=== Add 23/4 & 20 ===

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 S222 * 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 13 ~~and 19~~ ===

=== Add 13 ===

There are two reasonable ways to incorporate prime 13 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 260/243 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 [[218491/216513]]). 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)-3, 9 generators up, tempering out the comma [[351/343]]. The two representations meet at 22edt.

There are two reasonable ways to incorporate prime 13 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 260/243 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 [[218491/216513]]); this is the extension listed as "tridecimal Mintaka". 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)-3, 9 generators up, tempering out the comma [[351/343]]. The two representations meet at 22edt.

~~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 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]] (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. In this range, the "canonical" extension to prime 13 makes sense, though it is worth noting that the Minalzidar extension corresponds directly to the 3.5.7.13 extension of BPS.

~~=== Add 4 and 5 ===~~

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; though the more interesting case with regard to harmonic 20 is documented below.

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. In this range, the "canonical" extension to primes 13 and 19 makes sense, though it is worth noting that the Minalzidar extension corresponds directly to the 3.5.7.13.19 extension of BPS. 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, 13, 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 10: / Line 10: @@
 Several extensions of this temperament are possible to incorporate additional harmonics.
-=== Add 20 and 23/4 ===
+=== Add 23/4 & 20 ===
 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 13 and 19 ===
+=== Add 13 ===
-There are two reasonable ways to incorporate prime 13 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 260/243 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 [[218491/216513]]). 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.
+There are two reasonable ways to incorporate prime 13 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 260/243 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 [[218491/216513]]); this is the extension listed as "tridecimal Mintaka". 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 13 is added to the system, it is reasonable to temper out [[247/243]], which equates [[19/9]] to [[27/13]].
+=== Add 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]] (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. In this range, the "canonical" extension to prime 13 makes sense, though it is worth noting that the Minalzidar extension corresponds directly to the 3.5.7.13 extension of BPS.
-=== Add 4 and 5 ===
+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; though the more interesting case with regard to harmonic 20 is documented below.
-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. In this range, the "canonical" extension to primes 13 and 19 makes sense, though it is worth noting that the Minalzidar extension corresponds directly to the 3.5.7.13.19 extension of BPS. 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, 13, 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 ====