Diminished (temperament): Difference between revisions
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'''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. [[12edo]] is an obvious tuning. Other possible tunings include [[16edo]] and [[28edo]]. | ||
See [[Dimipent family #Diminished]] for technical data. | See [[Dimipent family #Diminished]] for technical data. | ||
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== Scales == | == Scales == | ||
* [[Diminished12]] – in [[44edo]] tuning | * [[Diminished12]] – in [[44edo]] tuning | ||
== Tunings == | |||
=== Prime-optimized tunings === | |||
* 7-limit [[CTE]]: ~6/5 = 1\4, ~3/2 = 696.9833 | |||
* 7-limit [[CWE]]: ~6/5 = 1\4, ~3/2 = 698.2661 | |||
=== Others === | |||
* 5-limit [[DKW theory|DKW]]: ~6/5 = 1\4, ~3/2 = 690.289 | |||
== See also == | == See also == | ||
Revision as of 12:57, 2 August 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. 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
Others
- 5-limit DKW: ~6/5 = 1\4, ~3/2 = 690.289
See also
- Diminished, the disambiguation page