Unidec

Unidec

Subgroups	2.3.5.7, 2.3.5.7.11
Comma basis	1029/1024, 4375/4374 (7-limit); 385/384, 441/440, 4375/4374 (11-limit)
Reduced mapping	⟨2; 6 11 -2 -3]
ET join	26 & 46
Generators (CWE)	~14/11 = 416.9 ¢
MOS scales	6L 2s, 6L 8s, 6L 14s, 20L 6s
Ploidacot	diploid gamma-hexacot
Minimax error	9-odd-limit: 2.41 ¢; 11-limit 21-odd-limit: 2.95 ¢
Target scale size	9-odd-limit: 46 notes; 11-limit 21-odd-limit: 46 notes

Unidec is a temperament generated by a wide major third representing 80/63 which can be taken as 14/11 from the 11-limit onwards. Two generators minus a semi-octave period make a ~8/7, and three ~8/7's make a perfect fifth of 3/2, tempering out 1029/1024, the gamelisma, and 4375/4374, the ragisma, with the 11-limit extension adding 385/384 or 441/440 to the comma list.

Unidec has two viable 13-limit extensions: hendec and ekadash, discussed in Unidec extensions.

See Gamelismic clan #Unidec for technical data.

Interval chain

In the following table, odd harmonics 1–21 and their inverses are in bold.

#	Period 0		Period 1
	Cents*	Approximate ratios	Cents*	Approximate ratios
0	0.00	1/1	600.00	99/70, 140/99
1	416.85	14/11	1016.80	9/5
2	833.71	81/50, 160/99	233.71	8/7
3	50.56	33/32, 36/35	650.56	16/11
4	467.42	21/16	1067.42	50/27
5	884.27	5/3	284.27	33/28
6	101.13	35/33	701.13	3/2
7	517.98	27/20	1117.98	21/11, 40/21
8	934.83	12/7	334.83	40/33
9	151.69	12/11	751.69	54/35
10	568.54	25/18	1168.54	55/28, 63/32, 96/49, 108/55
11	985.40	99/56	385.40	5/4
12	202.25	9/8	802.25	35/22
13	619.11	10/7	19.11	81/80, 99/98, 100/99

* In 11-limit CWE tuning, octave reduced

Chords and harmony

See also: Chords of unidec

Scales

• Unidec26

Tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~80/63 = 416.9403 ¢	CWE: ~80/63 = 416.8688 ¢	POTE: ~80/63 = 416.8385 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~14/11 = 416.9262 ¢	CWE: ~14/11 = 416.8543 ¢	POTE: ~14/11 = 416.8350 ¢

Tuning spectrum

Eigenmonzo (unchanged interval)	Generator (¢)	Comments
13/12	415.357
7/4	415.587
13/10	416.198
11/8	416.227
13/9	416.338
15/13	416.516
7/6	416.641
7/5	416.730	7-odd-limit minimax
11/6	416.737
11/10	416.785
9/7	416.792	9- and 11-odd-limit minimax
15/14	416.813
11/9	416.839
15/11	416.848
5/3	416.872
5/4	416.938	5-, 13- and 15-odd-limit minimax
15/8	416.957
3/2	416.993
11/7	417.508
9/5	417.596
13/11	417.842
13/7	417.925
13/8	420.264

Music

Andrew Heathwaite

• "Hypnocloudsmack 2" from Newbeams