Diminished (temperament)

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
  • 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

See also

• Diminished, the disambiguation page