Diminished (temperament): Difference between revisions
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+ Infobox Regtemp; fix error (strongly consistent circle means tempered comma is less than 25% of a step, which is not the case) |
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| en = Diminished (temperament) | | en = Diminished (temperament) | ||
| de = Verminderte Temperaturen | | de = Verminderte Temperaturen | ||
}} | |||
{{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 | |||
| Generator = 3/2 | |||
| Generator 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. | ||
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{{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'''. | ||
Revision as of 02:23, 15 December 2025
Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). Diminished is a rank-2 temperament that 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 generated by a ~3/2 perfect fifth. The main interest in this temperament is in its mos scales, featuring tetrawood (4L 4s) when properly tuned.
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 ¢ 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 ~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.
Interval chain
In the following table, odd harmonics 1–9 are in bold.
| # | Period 0 | Period 1 | Period 2 | Period 3 | ||||
|---|---|---|---|---|---|---|---|---|
| Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | Cents* | Approx. ratios | |
| 0 | 0.0 | 1/1 | 300.0 | 6/5, 7/6 | 600.0 | 7/5, 10/7 | 900.0 | 5/3, 12/7 |
| 1 | 96.0 | 15/14, 21/20, 25/24, 49/48 | 396.0 | 5/4, 9/7 | 696.0 | 3/2 | 996.0 | 7/4, 9/5 |
| 2 | 191.9 | 9/8 | 491.9 | 21/16 | 791.9 | 45/28, 63/40 | 1091.9 | 15/8 |
* In 7-limit CWE tuning, octave reduced
Scales
- Diminished12 – in 44edo tuning
Tunings
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 696.9833 ¢ | CWE: ~3/2 = 698.2661 ¢ | POTE: ~3/2 = 699.5072 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 691.9545 ¢ | CWE: ~3/2 = 695.9618 ¢ | POTE: ~3/2 = 699.5235 ¢ |
Others
- 5-limit DKW: ~6/5 = 300.000 ¢, ~3/2 = 690.289 ¢
Tuning spectrum
| Edo generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 2\4 | 600.000 | Lower bound of 7-odd-limit diamond monotone | |
| 49/48 | 635.697 | ||
| 7/4 | 668.826 | ||
| 25/24 | 670.672 | 1/2-comma | |
| 9\16 | 675.000 | ||
| 21/20 | 684.467 | ||
| 21/16 | 685.390 | ||
| 16\28 | 685.714 | ||
| 5/4 | 686.314 | 1/4-comma | |
| 15/8 | 694.134 | 1/8-comma | |
| 7\12 | 700.000 | 9-odd-limit diamond monotone (singleton) | |
| 3/2 | 701.955 | Untempered | |
| 9/5 | 717.596 | -1/4-comma | |
| 15/14 | 719.443 | ||
| 9/7 | 735.084 | ||
| 5\8 | 750.000 | 8d val, upper bound of 7-odd-limit diamond monotone |
* Besides the octave
See also
- Diminished (disambiguation page)