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.

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

Tunings

Prime-optimized tunings

  • 5-limit
    • CTE: ~6/5 = 1\4, ~3/2 = 696.9833
    • CWE: ~6/5 = 1\4, ~3/2 = 698.2661
  • 7-limit
    • CTE: ~6/5 = 1\4, ~3/2 = 691.9545
    • CWE: ~6/5 = 1\4, ~3/2 = 695.9618

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