Mintaka: Difference between revisions

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| Edo join 1 = 5 | Edo join 2 = 17

| Generator = 11/7 | Generator tuning = 778.703 | Optimization method = CWE

| MOS scales = {{sl|2L 3s|3/1}}, {{sl|5L 2s|3/1}}, {{sl|5L 7s|3/1}}, {{sl|5L 12s|3/1}}

| Mapping = 1; -3 -2

| 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 [[5L 2s (3/1~~-equivalent)|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 [[5L 7s (3/1~~-equivalent)|5L 7s]]~~ 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 [[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).

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|BPS~~]] 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]].

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]].

[[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s).

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| Edo join 1 = 17 | Edo join 2 = 22

| Generator = 11/7 | Generator tuning = 780.428 | Optimization method = CWE

| MOS scales = {{sl|2L 3s|3/1}}, {{sl|5L 2s|3/1}}, {{sl|5L 7s|3/1}}, {{sl|5L 12s|3/1}}, {{sl|17L 5s|3/1}}

| Mapping = 1; 6 -3 -2 13

| Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = 22

| 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 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 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.

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== Interval chains ==

One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to 99/49 and placed four generators up. In flatter tunings of the generator, this is closer to a true octave. This interval is meriting of special treatment in terms of consonance and dissonance.

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|+ style="font-size: 105%;" | Mintaka

|-

! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ~~Ratios~~

! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios

|-

! 3.7.11 subgroup !! 3.7.11.20.23/4 extension

@@ Line 4: / Line 4: @@
 | Edo join 1 = 5 | Edo join 2 = 17
 | Generator = 11/7 | Generator tuning = 778.703 | Optimization method = CWE
-| MOS scales = [[2L 3s (3/1-equivalent)|2L&nbsp;3s]], [[5L 2s (3/1-equivalent)|5L&nbsp;2s]], [[5L 7s (3/1-equivalent)|5L&nbsp;7s]], [[5L 12s (3/1-equivalent)|5L&nbsp;12s]]
+| MOS scales = {{sl|2L 3s|3/1}}, {{sl|5L 2s|3/1}}, {{sl|5L 7s|3/1}}, {{sl|5L 12s|3/1}}
 | Mapping = 1; -3 -2
 | 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 [[5L 2s (3/1-equivalent)|5L&nbsp;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 [[5L 7s (3/1-equivalent)|5L&nbsp;7s]] 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 [[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).
-As perhaps the simplest temperament of this subgroup delivering decent accuracy&mdash;and, in particular, the simplest supported by tunings such as 17edt and 22edt&ndash;Mintaka can be considered the 3.7.11 analog of 3.5.7 [[Bohlen-Pierce-Stearns|BPS]] 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]].
+As perhaps the simplest temperament of this subgroup delivering decent accuracy&mdash;and, in particular, the simplest supported by tunings such as 17edt and 22edt&ndash;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]].
 [[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L&nbsp;3s), 7 (5L&nbsp;2s), 12 (5L&nbsp;7s), or 17 (5L&nbsp;12s).
@@ Line 36: / Line 36: @@
 | Edo join 1 = 17 | Edo join 2 = 22
 | Generator = 11/7 | Generator tuning = 780.428 | Optimization method = CWE
-| MOS scales = [[2L 3s (3/1-equivalent)|2L&nbsp;3s]], [[5L 2s (3/1-equivalent)|5L&nbsp;2s]], [[5L 7s (3/1-equivalent)|5L&nbsp;7s]], [[5L 12s (3/1-equivalent)|5L&nbsp;12s]], [[17L 5s (3/1-equivalent)|17L&nbsp;5s]]
+| MOS scales = {{sl|2L 3s|3/1}}, {{sl|5L 2s|3/1}}, {{sl|5L 7s|3/1}}, {{sl|5L 12s|3/1}}, {{sl|17L 5s|3/1}}
 | Mapping = 1; 6 -3 -2 13
 | Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = 22
 | 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 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 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.
 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 55: / Line 56: @@
 == Interval chains ==
 One important feature of subgroups involving 3 and 11 is the quasi-octave at the interval designated 243/121; in this temperament, it is equated to 99/49 and placed four generators up. In flatter tunings of the generator, this is closer to a true octave. This interval is meriting of special treatment in terms of consonance and dissonance.
@@ Line 64: / Line 64: @@
 |+ style="font-size: 105%;" | Mintaka
 |-
-! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! colspan="2" | Approximate Ratios
+! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! colspan="2" | Approximate ratios
 |-
 ! 3.7.11 subgroup !! 3.7.11.20.23/4 extension

Mintaka: Difference between revisions

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