Diminished (temperament): Difference between revisions
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| en = Diminished (temperament) | | en = Diminished (temperament) | ||
| de = Verminderte Temperaturen | | de = Verminderte Temperaturen | ||
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{{Infobox regtemp | |||
| Title = Diminished | |||
| Subgroups = 2.3.5, 2.3.5.7 | |||
| Comma basis = [[648/625]] (2.3.5); <br>[[36/35]], [[50/49]] (2.3.5.7) | |||
| Edo join 1 = 4 | Edo join 2 = 12 | |||
| Mapping = 4; 1 1 1 | |||
| Generators = 3/2 | |||
| Generators tuning = 696.0 | |||
| Optimization method = CWE | |||
| MOS scales = [[4L 4s]], [[4L 8s]], [[12L 4s]] | |||
| Pergen = (P8/4, P5) | |||
| Color name = Quadguti | |||
| Odd limit 1 = 5 | Mistuning 1 = 15.6 | Complexity 1 = 8 | |||
| Odd limit 2 = 7-limit 21 | Mistuning 2 = 33.1 | Complexity 2 = 12 | |||
}} | }} | ||
'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] that [[tempering out|tempers out]] the diminished comma, [[648/625]], in the [[5-limit]], and [[36/35]] and [[50/49]] in the [[7-limit]]. It has a 1/4-[[octave]] [[period]] and is [[generator|generated]] by a [[~]][[3/2]] perfect fifth. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned. | '''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] that [[tempering out|tempers out]] the diminished comma, [[648/625]], in the [[5-limit]], and [[36/35]] and [[50/49]] in the [[7-limit]]. It has a 1/4-[[octave]] [[period]] and is [[generator|generated]] by a [[~]][[3/2]] perfect fifth. The main interest in this temperament is in its [[mos scale]]s, featuring [[tetrawood]] (4L 4s) when properly tuned. | ||
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It can be extended to the 2.3.5.7.19-[[subgroup]] where the 1/4-octave period stands in for both ~6/5 and ~19/16 since this ~19/16 is more accurate, though its mos structure of 4L 4s is very flexible, so one could use 3\4 minus ~8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example. | It can be extended to the 2.3.5.7.19-[[subgroup]] where the 1/4-octave period stands in for both ~6/5 and ~19/16 since this ~19/16 is more accurate, though its mos structure of 4L 4s is very flexible, so one could use 3\4 minus ~8/7 as a ~670{{cent}} fifth for a 2.7.19 subgroup version of diminished, for example. | ||
[[12edo]] is an obvious tuning. Another possible tuning is [[16edo]] which has the interesting feature of being relatively good in the 2.7.19 subgroup so that the fifth is approximately [[28/19]], or [[28edo]], which uses something like {{nowrap| ~[[95/64]] {{=}} ~([[19/16]])⋅([[5/4]]) }} as its fifth, of which the latter is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[ | [[12edo]] is an obvious tuning. Another possible tuning is [[16edo]] which has the interesting feature of being relatively good in the 2.7.19 subgroup so that the fifth is approximately [[28/19]], or [[28edo]], which uses something like {{nowrap| ~[[95/64]] {{=}} ~([[19/16]])⋅([[5/4]]) }} as its fifth, of which the latter is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate [[5/4]]'s, being a [[consistent circle]] of them. | ||
See [[Diminished family #Diminished]] for technical data. | See [[Diminished family #Diminished]] for technical data. | ||
__TOC__ | |||
{{Clear}} | |||
== Interval chain == | == Interval chain == | ||
In the following table, odd harmonics 1–9 are in '''bold'''. | In the following table, odd harmonics 1–9 are in '''bold'''. | ||
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! Comments | ! Comments | ||
|- | |- | ||
| 2\4 | | [[4edo|2\4]] | ||
| | | | ||
| 600.000 | | 600.000 | ||
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|- | |- | ||
| | | | ||
| 49/48 | | [[49/48]] | ||
| 635.697 | | 635.697 | ||
| | | | ||
|- | |- | ||
| | | | ||
| 7/4 | | [[7/4]] | ||
| 668.826 | | 668.826 | ||
| | | | ||
|- | |- | ||
| | | | ||
| 25/24 | | [[25/24]] | ||
| 670.672 | | 670.672 | ||
| 1/2-comma | | 1/2-comma | ||
|- | |- | ||
| 9\16 | | [[16edo|9\16]] | ||
| | | | ||
| 675.000 | | 675.000 | ||
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|- | |- | ||
| | | | ||
| 21/20 | | [[21/20]] | ||
| 684.467 | | 684.467 | ||
| | | | ||
|- | |- | ||
| | | | ||
| 21/16 | | [[21/16]] | ||
| 685.390 | | 685.390 | ||
| | | | ||
|- | |||
| [[28edo|16\28]] | |||
| | |||
| 685.714 | |||
| | |||
|- | |- | ||
| | | | ||
| 5/4 | | [[5/4]] | ||
| 686.314 | | 686.314 | ||
| 1/4-comma | | 1/4-comma | ||
|- | |||
| [[40edo|23\40]] | |||
| | |||
| 690.000 | |||
| 40d val | |||
|- | |||
| [[52edo|30\52]] | |||
| | |||
| 692.308 | |||
| 52dd val | |||
|- | |||
| [[64edo|37\64]] | |||
| | |||
| 693.750 | |||
| 64dd val | |||
|- | |- | ||
| | | | ||
| 15/8 | | [[15/8]] | ||
| 694.134 | | 694.134 | ||
| 1/8-comma | | 1/8-comma | ||
|- | |- | ||
| 7\12 | | [[12edo|7\12]] | ||
| | | | ||
| 700.000 | | 700.000 | ||
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|- | |- | ||
| | | | ||
| 3/2 | | [[3/2]] | ||
| 701.955 | | 701.955 | ||
| Untempered | | Untempered | ||
|- | |||
| [[32edo|19\32]] | |||
| | |||
| 712.500 | |||
| 32cd val | |||
|- | |- | ||
| | | | ||
| 9/5 | | [[9/5]] | ||
| 717.596 | | 717.596 | ||
| -1/4-comma | | -1/4-comma | ||
|- | |- | ||
| | | | ||
| 15/14 | | [[15/14]] | ||
| 719.443 | | 719.443 | ||
| | | | ||
|- | |||
| [[20edo|12\20]] | |||
| | |||
| 720.000 | |||
| 20cd val | |||
|- | |- | ||
| | | | ||
| 9/7 | | [[9/7]] | ||
| 735.084 | | 735.084 | ||
| | | | ||
|- | |- | ||
| 5\8 | | [[8edo|5\8]] | ||
| | | | ||
| 750.000 | | 750.000 | ||