Diminished (temperament): Difference between revisions
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| de = Verminderte Temperaturen | | de = Verminderte Temperaturen | ||
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{{Infobox | {{Infobox regtemp | ||
| Title = Diminished | | Title = Diminished | ||
| Subgroups = 2.3.5, 2.3.5.7 | | Subgroups = 2.3.5, 2.3.5.7 | ||
| Comma basis = [[648/625]] (2.3.5) | | Comma basis = [[648/625]] (2.3.5); <br>[[36/35]], [[50/49]] (2.3.5.7) | ||
| Edo join 1 = 4 | Edo join 2 = 12 | | Edo join 1 = 4 | Edo join 2 = 12 | ||
| Mapping = 4; 1 1 1 | | Mapping = 4; 1 1 1 | ||
| | | Generators = 3/2 | ||
| | | Generators tuning = 696.0 | ||
| Optimization method = CWE | | Optimization method = CWE | ||
| MOS scales = [[4L 4s]], [[4L 8s]], [[12L 4s]] | | MOS scales = [[4L 4s]], [[4L 8s]], [[12L 4s]] | ||
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| Color name = Quadguti | | Color name = Quadguti | ||
| Odd limit 1 = 5 | Mistuning 1 = 15.6 | Complexity 1 = 8 | | Odd limit 1 = 5 | Mistuning 1 = 15.6 | Complexity 1 = 8 | ||
| Odd limit 2 = | | 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. | ||
Revision as of 10:02, 9 February 2026
| Diminished |
36/35, 50/49 (2.3.5.7)
7-limit 21-odd-limit: 33.1 ¢
7-limit 21-odd-limit: 12 notes
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)