Mintaka: Difference between revisions

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For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma, or effectively between [[17edt]] and [[22edt]] tuning), 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. A good tuning for this temperament is [[39edt]], the triple BP equalized scale, though others such as [[95edt]] are possible.

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. This extension then is equivalent to tempering out [[275/273]] and equating [[13/11]] to [[25/21]]. Furthermore, 13/11 appears 15 generators up, and has a cube root in the temperament: 35/33. Therefore, as 13/11 = ([[35/33]])([[37/35]])([[39/37]]), it is "free" to equate 35/33 additionally to 37/35 and 39/37, placing the 37th harmonic 8 generators up.

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. This extension then is equivalent to tempering out [[275/273]] and equating [[13/11]] to [[25/21]]. Furthermore, 13/11 appears 15 generators up, and has a cube root in the temperament: 35/33. Therefore, as 13/11 = ([[35/33]])([[37/35]])([[39/37]]), it is "free" to equate 35/33 additionally to 37/35 and 39/37 (which amounts to tempering out [[407/405]]), placing the 37th harmonic 8 generators up.

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.

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 For tunings of the generator that possess a sharp 9/7 (sharper than 1/3 comma, or effectively between [[17edt]] and [[22edt]] tuning), 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. A good tuning for this temperament is [[39edt]], the triple BP equalized scale, though others such as [[95edt]] are possible.
-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. This extension then is equivalent to tempering out [[275/273]] and equating [[13/11]] to [[25/21]]. Furthermore, 13/11 appears 15 generators up, and has a cube root in the temperament: 35/33. Therefore, as 13/11 = ([[35/33]])([[37/35]])([[39/37]]), it is "free" to equate 35/33 additionally to 37/35 and 39/37, placing the 37th harmonic 8 generators up.
+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. This extension then is equivalent to tempering out [[275/273]] and equating [[13/11]] to [[25/21]]. Furthermore, 13/11 appears 15 generators up, and has a cube root in the temperament: 35/33. Therefore, as 13/11 = ([[35/33]])([[37/35]])([[39/37]]), it is "free" to equate 35/33 additionally to 37/35 and 39/37 (which amounts to tempering out [[407/405]]), placing the 37th harmonic 8 generators up.
 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.