Keemun extensions

< Keemun

Keemun has various competing extensions to the 11- and 13-limit. The following temperaments are discussed in this article:

• Tridecimal keemun (15 & 19) – tempering out 49/48, 56/55, 91/90, 100/99;
• Kimah (15f & 19) – tempering out 49/48, 56/55, 65/64, 100/99;
• Qeema (15e & 19) – tempering out 45/44, 49/48, 78/77, 126/125;
• Kumbaya (15 & 19f) – tempering out 40/39, 49/48, 56/55, 66/65;
• Darjeeling (15 & 19e) – tempering out 49/48, 55/54, 66/65, 77/75.

See Kleismic family #Keemun, #Kimah, #Kumbaya, #Qeema, and #Darjeeling for technical data.

Interval chain

#	Approximate ratios
	Cents*	7-limit	13-limit extensions
			Keemun	Kimah	Qeema	Kumbaya	Darjeeling
0	0.0	1/1
1	316.8	6/5				13/11, 16/13	11/9, 13/11
2	633.7	10/7	13/9, 16/11	16/11	13/9	16/11	13/9, 22/15
3	950.5	7/4, 12/7	26/15	22/13	22/13, 26/15		26/15
4	67.3	21/20, 25/24, 36/35	27/26, 33/32	33/32	27/26	14/13, 33/32	27/26, 88/45
5	384.1	5/4	14/11	14/11, 16/13	11/9	14/11
6	701.0	3/2			22/15	20/13
7	1017.8	9/5	20/11	20/11		20/11, 24/13	11/6
8	134.6	15/14	12/11, 13/12	12/11, 14/13	13/12	12/11	11/10, 13/12
9	451.5	9/7, 21/16	13/10		13/10		13/10
10	768.3	25/16		20/13			11/7
11	1085.1	15/8	13/7	24/13	11/6, 13/7		13/7
12	202.0	9/8			11/10	15/13
13	518.8	27/20	15/11	15/11		15/11, 18/13	11/8
14	835.6	45/28	13/8, 18/11	18/11, 21/13	11/7, 13/8	18/11	13/8
15	1152.4	27/14		25/13
16	269.3	75/64, 81/70	13/11	15/13
17	586.1	45/32		18/13	11/8
18	902.9	27/16
19	19.8	81/80	45/44	45/44		45/44	33/32

* In 7-limit CWE tuning, octave reduced

Tunings

11-limit norm-based tunings (keemun)

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~6/5 = 317.5063 ¢	CWE: ~6/5 = 317.5546 ¢	POTE: ~6/5 = 317.5756 ¢

Tuning spectra

Tridecimal keemun

Edo generator	Eigenmonzo (unchanged interval)*	Generator (¢)	Comments
1\4		300.000	4f val, lower bound of 5- and 7-odd-limit diamond monotone
	7/5	308.744
	7/6	311.043
6\23		313.043	23dff val
	15/14	314.930
	9/7	315.009
	13/7	315.609
	5/3	315.641
5\19		315.789	Lower bound of 9- and 11-odd-limit diamond monotone
	9/5	316.799	1/7-kleisma
14\53		316.981	53de val
	3/2	316.993	5-, 7- and 9-odd-limit minimax, 1/6-kleisma
	15/8	317.115	2/11-kleisma
	13/10	317.134
	13/8	317.181
	5/4	317.263	1/5-kleisma
	13/12	317.322
	15/13	317.420
9\34		317.647
	11/9	318.042	11-, 13- and 15-odd-limit minimax
	13/10	318.076
	15/11	318.227
	11/10	319.285
	13/9	318.309
13\49		318.367	49df val
	11/6	318.830
4\15		320.000	Upper bound of 9- and 11-odd-limit diamond monotone
	7/4	322.942
	11/7	323.502
	11/8	324.341
3\11		327.273	11bff val, upper bound of 5- and 7-odd-limit diamond monotone

* Besides the octave