Xenial

Xenial

Subgroups	2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
Comma basis	126/125, 177147/175616 (7-limit); 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230 (23-limit)
Reduced mapping	⟨1; -9 -17 -33 22 -21 26 27 -3]
ET join	19 & 70
Generators (CWE)	~10/9 = 188.8 ¢
MOS scales	6L 1s, 6L 7s, 13L 6s, 19L 13s, 19L 32s, 19L 51s
Ploidacot	zeta-enneacot
Pergen	(P8, P11/9)
Minimax error	7-odd-limit: 4.60 ¢; 9-odd-limit: 6.27 ¢; 17-odd-limit: 8.90 ¢; 23-odd-limit: 8.96 ¢
Target scale size	7-odd-limit: 51 notes; 9-odd-limit: 51 notes; 17-odd-limit: 70 notes; 23-odd-limit: 70 notes

Xenial is a rank-2 temperament that is generated by a sharpened minor whole tone of ~10/9, so that nine generators reach 4/3, 17 reach 8/5, 21 reach 16/13 and 33 reach 8/7 with octave reduction. It is also generated by dividing 11th harmonic into 22 equal parts, 17th harmonic into 26 equal parts, or 19th harmonic into 27 equal parts.

See Starling temperaments #Xenial for more technical data.

Interval chain

#	Cents*	Approximate ratios
0	0.000	1/1
1	188.775	10/9, 19/17, 28/25
2	377.551	56/45
3	566.326	18/13, 32/23
4	755.102	17/11, 20/13
5	943.877	19/11, 26/15
6	1132.653	23/12, 27/14
7	121.428	15/14
8	310.204	6/5
9	498.979	4/3
10	687.755	40/27
11	876.530
12	1065.306	13/7, 24/13
13	54.081	26/25, 33/32
14	242.857	23/20
15	431.632	9/7, 23/18
16	620.408	10/7
17	809.183	8/5
18	997.959	16/9, 23/13
19	1186.734
20	175.510
21	364.285	16/13, 26/21
22	553.061	11/8
23	741.836	23/15
24	930.612	12/7
25	1119.387	40/21, 44/23, 48/25
26	108.163	16/15, 17/16
27	296.938	19/16
28	485.714
29	674.439	34/23
30	863.265	38/23, 23/14
31	1052.040	11/6, 46/25
32	40.815	36/35, 46/45, 50/49
33	229.591	8/7
34	418.366	32/25

* In 23-limit CWE tuning

Tunings

Norm-based tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.8535 ¢	CWE: ~10/9 = 188.8544 ¢	POTE: ~10/9 = 188.8548 ¢

11-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.8295 ¢	CWE: ~10/9 = 188.8085 ¢	POTE: ~10/9 = 188.8028 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.7987 ¢	CWE: ~10/9 = 188.7898 ¢	POTE: ~10/9 = 188.7875 ¢

17-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.7811 ¢	CWE: ~10/9 = 188.7677 ¢	POTE: ~10/9 = 188.7655 ¢

19-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.7828 ¢	CWE: ~10/9 = 188.7770 ¢	POTE: ~10/9 = 188.7762 ¢

23-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/9 = 188.7849 ¢	CWE: ~10/9 = 188.7755 ¢	POTE: ~10/9 = 188.7744 ¢

Tuning spectrum

Edo generator	Eigenmonzo (unchanged interval)	Generator (¢)	Comments
	9/5	182.4037
	13/10	186.4465
5 ⧵ 32		187.5000	32cddefgh val Lower bound of 7-odd-limit diamond monotone
	23/12	187.7199
	13/9	187.7941
	23/13	188.2081
8 ⧵ 51		188.2353	51cdh val Lower bound of 9-odd-limit diamond monotone
	23/18	188.2910
	17/11	188.4094
	13/12	188.4523
	15/14	188.4918
	13/8	188.5463
11 ⧵ 70		188.5714	Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
	7/5	188.5930
	17/13	188.6048
	21/20	188.6213
	13/11	188.6230	13-odd-limit minimax
	23/14	188.6483
	17/16	188.6521
	23/21	188.6537
	17/12	188.6572
	17/9	188.6601
	3/2	188.6717	9, 15 and 17-odd-limit minimax
	11/9	188.6852	11-odd-limit minimax
	19/13	188.6872	19, 21 and 23-odd-limit minimax
	11/6	188.6891
	23/15	188.6959
	11/8	188.6963
	23/20	188.7115
	21/17	188.7379
	19/18	188.7467
	17/14	188.7480
	21/11	188.7584
14 ⧵ 89		188.7640	19, 21 and 23-odd-limit diamond monotone (singleton)
	19/12	188.7655
	11/7	188.7726
	17/15	188.7824
	21/16	188.7909
	21/19	188.7932
	19/16	188.7968
	17/10	188.8056
	19/14	188.8115
	15/11	188.8135
	7/4	188.8235
	11/10	188.8463
	23/17	188.8511
	19/15	188.8537
	7/6	188.8804	7-odd-limit minimax
	19/10	188.8909
	15/8	188.9127
	23/22	188.9217
	23/19	188.9746
	9/7	189.0056
	21/13	189.0356
	5/4	189.0404	5-odd-limit minimax
	19/11	189.2390
	13/7	189.3085
	5/3	189.4552
3 ⧵ 19		189.4737	Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
	15/13	190.4518
	23/16	190.5752
	19/17	192.5576