Mintaka: Difference between revisions

Line 4:

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

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

| 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 [[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).

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]].

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

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

Line 36:

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

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

| Mapping = 1; 6 -3 -2 13

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

@@ 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 3s]], [[5L 2s (3/1-equivalent)|5L 2s]], [[5L 7s (3/1-equivalent)|5L 7s]], [[5L 12s (3/1-equivalent)|5L 12s]]
+| 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]]
 | 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 [[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).
 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]].
-[[Mos scale]]s of reasonable tunings have cardinalities of 5 (2L 3s), 7 (5L 2s), 12 (5L 7s), or 17 (5L 12s).
+[[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).
 {{tdlink|No-twos subgroup temperaments#Mintaka}}
@@ 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 3s]], [[5L 2s (3/1-equivalent)|5L 2s]], [[5L 7s (3/1-equivalent)|5L 7s]], [[5L 12s (3/1-equivalent)|5L 12s]], [[17L 5s (3/1-equivalent)|17L 5s]]
+| 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]]
 | Mapping = 1; 6 -3 -2 13
 | Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = 22