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: 4.6 ¢; 9-odd-limit: 6.3 ¢; 17-odd-limit: 8.9 ¢; 23-odd-limit: 9.0 ¢
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  ¢

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	182.404
	13/10	186.447
5 ⧵ 32		187.500	32cddefgh val Lower bound of 7-odd-limit diamond monotone
	23/12	187.720
	13/9	187.794
	23/13	188.208
8 ⧵ 51		188.235	51cdh val Lower bound of 9-odd-limit diamond monotone
	23/18	188.291
	17/11	188.409
	13/12	188.452
	15/14	188.492
	13/8	188.546
11 ⧵ 70		188.571	Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
	7/5	188.593
	17/13	188.605
	21/20	188.621
	13/11	188.623	13-odd-limit minimax
	23/14	188.648
	17/16	188.652
	23/21	188.654
	17/12	188.657
	17/9	188.660
	3/2	188.672	9, 15 and 17-odd-limit minimax
	11/9	188.685	11-odd-limit minimax
	19/13	188.687	19, 21 and 23-odd-limit minimax
	11/6	188.689
	23/15	188.6959
	11/8	188.6963
	23/20	188.711
	21/17	188.738
	19/18	188.747
	17/14	188.748
	21/11	188.758
14 ⧵ 89		188.764	19, 21 and 23-odd-limit diamond monotone (singleton)
	19/12	188.766
	11/7	188.773
	17/15	188.782
	21/16	188.791
	21/19	188.793
	19/16	188.797
	17/10	188.806
	19/14	188.811
	15/11	188.814
	7/4	188.823
	11/10	188.846
	23/17	188.851
	19/15	188.854
	7/6	188.880	7-odd-limit minimax
	19/10	188.891
	15/8	188.913
	23/22	188.922
	23/19	188.975
	9/7	189.006
	21/13	189.036
	5/4	189.040	5-odd-limit minimax
	19/11	189.239
	13/7	189.308
	5/3	189.455
3 ⧵ 19		189.473	Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
	15/13	190.452
	23/16	190.575
	19/17	192.558