Diminished (temperament)
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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. 12edo is an obvious tuning. Other possible tunings include 16edo and 28edo.
See Dimipent 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 | 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)