Diminished (temperament): Difference between revisions
rework to note more accurate structures |
Diminished is specifically {648/625} and {36/35, 50/49} and has relevance in common-practice music, a consensus dating back to 2002. If you're interested in something else, create a different article. Rework to integrate the extension with prime 19. |
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'''Diminished''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] | '''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. | ||
See [[Dimipent family #Diminished]] for technical data | 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. 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. | |||
== Interval chain == | == Interval chain == | ||
Revision as of 07:38, 22 September 2024
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. 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.
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)