Mintaka: Difference between revisions

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=== 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]] (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.

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 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.

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 ====

In the flatter generator range (supported by the Minalzidar extension ~~to prime 19~~), 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. ''However'', 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. ''However'', 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]]).

== Interval chains ==

@@ Line 19: / Line 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]] (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.
+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 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.
+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 ====
-In the flatter generator range (supported by the Minalzidar extension to prime 19), 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. ''However'', 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. ''However'', 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]]).
 == Interval chains ==

Mintaka: Difference between revisions

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