Mintaka: Difference between revisions
mNo edit summary |
accidental self edit conflict |
||
| (8 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox | {{Infobox regtemp | ||
| Subgroups = 3.7.11 | | Subgroups = 3.7.11 | ||
| Comma basis = [[1331/1323]] | | Comma basis = [[1331/1323]] | ||
| Edo join 1 = | | Edo join 1 = b5 | Edo join 2 = b17 | ||
| Mapping = 1; -3 -2 | | Mapping = 1; -3 -2 | ||
| Odd limit 1 = | | Generators = 11/7 | Generators tuning = 778.7 | 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]] | |||
| Ploidacot = alpha-trigem | |||
| Odd limit 1 = 3.7.11 11 | Mistuning 1 = 3.48 | Complexity 1 = 7 | |||
}} | }} | ||
'''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 | '''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 [[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]] 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]]. | ||
| Line 14: | Line 15: | ||
[[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). | ||
{{ | {{Tdlink|No-twos subgroup temperaments #Mintaka}} | ||
| Line 38: | Line 39: | ||
| Title = Mintra | | Title = Mintra | ||
| Subgroups = 3.5.7.11, 3.5.7.11.13 | | Subgroups = 3.5.7.11, 3.5.7.11.13 | ||
| Comma basis = [[245/243]], [[1331/1323]] (3.5.7.11);<br | | Comma basis = [[245/243]], [[1331/1323]] (3.5.7.11);<br>[[245/243]], [[275/273]], [[1331/1323]] (3.5.7.11.13) | ||
| Edo join 1 = | | Edo join 1 = b17 | Edo join 2 = b22 | ||
| Mapping = 1; 6 -3 -2 13 | | Mapping = 1; 6 -3 -2 13 | ||
| Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = | | Generators = 11/7 | Generators tuning = 780.4 | Optimization method = CWE | ||
| Odd limit 2 = | | 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]] | ||
| Odd limit 1 = 11 | Mistuning 1 = 6.16 | Complexity 1 = 12 | |||
| Odd limit 2 = 3.5.7.11.13 25 | Mistuning 2 = 8.77 | Complexity 2 = 17 | |||
}} | }} | ||
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. | ||
| Line 379: | Line 380: | ||
* [https://archive.org/details/TuneIn22Edt Tune in 22edt] – [[Peter Kosmorsky]] (2011) – uses the LssLssLsssLssLsss MOS (Mintaka[17]) | * [https://archive.org/details/TuneIn22Edt Tune in 22edt] – [[Peter Kosmorsky]] (2011) – uses the LssLssLsssLssLsss MOS (Mintaka[17]) | ||
[[Category: | [[Category:Mintaka| ]] <!-- main article --> | ||
[[Category:Rank-2 temperaments]] | [[Category:Rank-2 temperaments]] | ||
[[Category: | [[Category:Non-octave temperaments]] | ||