Mintaka: Difference between revisions

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| Odd limit 1 = (3.7.11) 11 | Mistuning 1 = 3.48 | Complexity 1 = 7

}}

'''Mintaka''' is a [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a {{sl|5L 2s~~|3/1~~}} (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) and that all intervals are extremely stretched, though the {{sl|5L ~~7s|3/1~~}} macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. [[22edt|9\22]]edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]]edt and [[39edt|16\39]]edt are also useful, especially for extensions involving primes 5 and 13 (see below).

'''Mintaka''' is a [[non-octave]] [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a {{sl|5L 2s}} (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) and that all intervals are extremely stretched, though the {{sl|5L 2s}} macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. [[22edt|9\22]]edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]]edt and [[39edt|16\39]]edt are also useful, especially for extensions involving primes 5 and 13 (see below).

As perhaps the simplest temperament of this subgroup delivering decent accuracy—and, in particular, the simplest supported by tunings such as 17edt and 22edt—Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen–Pierce–Stearns]] or 2.3.5 [[meantone]], using [[7:9:11]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]].

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=== 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]]); 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 down, 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 does not include 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 down, tempering out the comma [[351/343]]. The two representations meet at 22edt.

=== Add 5 ===

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| Odd limit 2 = (13-limited) 25 | Mistuning 2 = 8.77 | Complexity 2 = 39

}}

For tunings of the generator that possess a sharp 9/7 (sharper than {{frac|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.

For tunings of the generator that possess a sharp 9/7 (sharper than {{frac|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 {{nowrap|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.

@@ Line 8: / Line 8: @@
 | Odd limit 1 = (3.7.11) 11 | Mistuning 1 = 3.48 | Complexity 1 = 7
 }}
-'''Mintaka''' is a [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a {{sl|5L 2s|3/1}} (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) and that all intervals are extremely stretched, though the {{sl|5L 7s|3/1}} macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. [[22edt|9\22]]edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]]edt and [[39edt|16\39]]edt are also useful, especially for extensions involving primes 5 and 13 (see below).
+'''Mintaka''' is a [[non-octave]] [[temperament]] in the 3.7.11 [[subgroup]] where [[~]][[11/7]] is a [[generator]], and the comma [[1331/1323]] is [[tempering out|tempered out]], so a stack of two generators represents [[27/11]] in addition to 121/49, and a stack of three generators, [[3/1|tritave]]-reduced, represents [[9/7]]. As 11/7 as a generator against the tritave produces a {{sl|5L 2s}} (macrodiatonic) scale, with the generator here occupying the role of a [[4/3|perfect fourth]], it is possible to use an analogue of the [[chain-of-fifths notation]] that is standardly used for [[diatonic]] scales, with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) and that all intervals are extremely stretched, though the {{sl|5L 2s}} macrochromatic scale is suggested for musical use due to the hardness of the macrodiatonic and the increased breadth of the tritave. [[22edt|9\22]]edt is a very good tuning for the generator, and 22edt overall excels in the 3.7.11 subgroup, but other tunings such as [[17edt|7\17]]edt and [[39edt|16\39]]edt are also useful, especially for extensions involving primes 5 and 13 (see below).
 As perhaps the simplest temperament of this subgroup delivering decent accuracy—and, in particular, the simplest supported by tunings such as 17edt and 22edt—Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen–Pierce–Stearns]] or 2.3.5 [[meantone]], using [[7:9:11]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]].
@@ Line 26: / Line 26: @@
 === 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]]); 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 down, 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 does not include 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 down, tempering out the comma [[351/343]]. The two representations meet at 22edt.
 === Add 5 ===
@@ Line 41: / Line 41: @@
 | Odd limit 2 = (13-limited) 25 | Mistuning 2 = 8.77 | Complexity 2 = 39
 }}
-For tunings of the generator that possess a sharp 9/7 (sharper than {{frac|1|3}}&nbsp;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.
+For tunings of the generator that possess a sharp 9/7 (sharper than {{frac|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 {{nowrap|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.