Mintaka: Difference between revisions

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=== 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. In the ''flatter'' generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)2, placing it 16 generators down; this leads to the 3.5.7.11.13 subgroup version of ''Nekkar'' temperament.

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.

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.

==== Eshurizel ====

==== Nekkar and Eshurizel ====

In Nekkar, as soon as harmonic 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 add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).

In the ''flatter'' generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)2, placing it 16 generators down; this leads to the 3.5.7.11.13 subgroup version of ''Nekkar'' temperament.

Nekkar, as soon as harmonic 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 add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).

Even without the mappings for other primes, this method can be used to introduce octaves into Mintaka in a manner alike to [[sensi]] and [[hedgehog]] being produced as extensions of BPS. Equating the false octave (243/121~99/49) to 2/1 provides 2.3.7.11 [[skwares]] temperament, to which the aforementioned Eshurizel is but an extension.

@@ Line 17: / Line 17: @@
 === 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. In the ''flatter'' generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)<sup>2</sup>, placing it 16 generators down; this leads to the 3.5.7.11.13 subgroup version of ''Nekkar'' temperament.
+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.
 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.
-==== Eshurizel ====
+==== Nekkar and Eshurizel ====
-In Nekkar, as soon as harmonic 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 add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).
+In the ''flatter'' generator range (supported by the Minalzidar extension), the optimal representation of 5 is instead that obtained by tempering out [[120285/117649]], which equates 5 with (529/243)<sup>2</sup>, placing it 16 generators down; this leads to the 3.5.7.11.13 subgroup version of ''Nekkar'' temperament.
+Nekkar, as soon as harmonic 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 add-19 add-23 extension of 11-limit [[squares]] (with commas 81/80, [[99/98]], and [[243/242]]).
 Even without the mappings for other primes, this method can be used to introduce octaves into Mintaka in a manner alike to [[sensi]] and [[hedgehog]] being produced as extensions of BPS. Equating the false octave (243/121~99/49) to 2/1 provides 2.3.7.11 [[skwares]] temperament, to which the aforementioned Eshurizel is but an extension.