Diminished (temperament)

Diminished

Subgroups	2.3.5, 2.3.5.7
Comma basis	648/625 (2.3.5); 36/35, 50/49 (2.3.5.7)
Reduced mapping	⟨4; 1 1 1]
ET join	4 & 12
Generators (CWE)	~3/2 = 696.0 ¢
MOS scales	4L 4s, 4L 8s, 12L 4s
Ploidacot	tetraploid monocot
Pergen	(P8/4, P5)
Color name	Quadguti
Minimax error	5-odd-limit: 15.6 ¢; 7-limit 21-odd-limit: 33.1 ¢
Target scale size	5-odd-limit: 8 notes; 7-limit 21-odd-limit: 12 notes

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. Another possible tuning is 16edo which has the interesting feature of being relatively good in the 2.7.19 subgroup so that the fifth is approximately 28/19, or 28edo, which uses something like ~95/64 = ~(19/16)⋅(5/4) as its fifth, of which the latter is notable as a tuning for the 5-limit temperament, dimipent, as it has very accurate 5/4's, being a consistent circle of them.

See Diminished 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	96.0	15/14, 21/20, 25/24, 49/48	396.0	5/4, 9/7	696.0	3/2	996.0	7/4, 9/5
2	191.9	9/8	491.9	21/16	791.9	45/28, 63/40	1091.9	15/8

* In 7-limit CWE tuning, octave reduced

Scales

• Diminished12 – in 44edo tuning

Tunings

Norm-based tunings

5-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 696.9833 ¢	CWE: ~3/2 = 698.2661 ¢	POTE: ~3/2 = 699.5072 ¢

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~3/2 = 691.9545 ¢	CWE: ~3/2 = 695.9618 ¢	POTE: ~3/2 = 699.5235 ¢

Others

• 5-limit DKW: ~6/5 = 300.000 ¢, ~3/2 = 690.289 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	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
16\28		685.714
	5/4	686.314	1/4-comma
23\40		690.000
30\52		692.308
37\64		693.750
	15/8	694.134	1/8-comma
7\12		700.000	9-odd-limit diamond monotone (singleton)
	3/2	701.955	Untempered
19\32		712.500	32c val
	9/5	717.596	-1/4-comma
	15/14	719.443
12\20		720.000	20c val
	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)