Diminished (temperament)
Diminished is a rank-2 temperament with a 1/4-octave period generated by a ~3/2 perfect fifth or more generally by anything that melodically qualifies as a fifth. As a 5-limit temperament it tempers out the diminished comma, 648/625, and 36/35 and 50/49 in the 7-limit. A far more accurate interpretation is as a 2.3.19 temperament that makes ~19/16 = 1\4 = 300 ¢, 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. The main interest in this temperament is in its mos scales, featuring tetrawood (4L 4s) when properly tuned. 12edo is the simplest nontrivial tuning. Other possible tunings include 16edo and 28edo, both of which having the interesting feature of being good in the 2.7.19 subgroup, so that the fifth is approximately 28/19. 28edo is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate 5/4's, being a strongly consistent circle of them.
See Dimipent family #Diminished for technical data on the 5-limit temperament.
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 | 92.0 | 15/14, 21/20, 25/24, 49/48 | 392.0 | 5/4, 9/7 | 692.0 | 3/2 | 992.0 | 7/4, 9/5 |
| 2 | 183.9 | 9/8 | 483.9 | 21/16 | 783.9 | 45/28, 63/40 | 1083.9 | 15/8 |
* in 7-limit CTE tuning
Scales
- Diminished12 – in 44edo tuning
Tunings
Prime-optimized tunings
- 5-limit
- 7-limit
Others
- 5-limit DKW: ~6/5 = 1\4, ~3/2 = 690.289
Tuning spectrum
| Edo Generator |
Eigenmonzo (Unchanged-interval)* |
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 | ||
| 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)