Mystery

Mystery is a regular temperament which takes 1\29 period and adds a single generator to correct the harmonics 5, 7, 11, and 13 to almost just qualities. It tempers out the 29-comma in the 5-limit, 5120/5103 in the 7-limit, 441/440, 896/891, 3388/3375, and 4000/3993 in the 11-limit, 196/195, 352/351, 364/363, 676/675, and 847/845 in the 13-limit.

Mystery

Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	5120/5103, 50421/50000 (7-limit); 441/440, 896/891, 3388/3375 (11-limit); 196/195, 352/351, 364/363, 676/675 (13-limit)
Reduced mapping	⟨29; 0 1 1 1 1]
ET join	29 & 58
Generators (CWE)	~5/4 = 387.9 ¢
MOS scales	29L 29s, 58L 29s
Ploidacot	29-ploid acot
Pergen	(P8/29, ^1)
Color name	Quadsawati
Minimax error	9-odd-limit: 3.20 ¢; 13-limit 21-odd-limit: 4.69 ¢
Target scale size	9-odd-limit: 58 notes; 13-limit 21-odd-limit: 58 notes

See Hemifamity temperaments #Mystery for technical details.

Interval chain

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

Period	Generator 0		Generator 1
	Cents	Approx. ratios	Cents	Approx. ratios
0	0.00	1/1	15.49	91/90, 100/99, 105/104, 121/120
1	41.38	40/39, 45/44, 50/49	56.87	28/27, 33/32
2	82.76	21/20, 22/21	98.25	35/33
3	124.14	14/13, 15/14	139.63	13/12
4	165.52	11/10	181.01	10/9
5	206.90	9/8	222.38	25/22
6	248.28	15/13	263.76	7/6
7	289.66	13/11, 32/27, 33/28	305.14	25/21
8	331.03	40/33	346.52	11/9, 39/32
9	372.41	26/21	387.90	5/4
10	413.79	14/11, 33/26, 80/63	429.28	77/60
11	455.17	13/10	470.66	21/16
12	496.55	4/3	512.04	35/26, 75/56
13	537.93	15/11	553.42	11/8
14	579.31	7/5	594.80	45/32
15	620.69	10/7	636.18	13/9
16	662.07	22/15	677.56	40/27
17	703.45	3/2	718.94	50/33
18	744.83	20/13	760.32	14/9
19	786.21	11/7, 52/33, 63/40	801.69	35/22
20	827.59	21/13	843.07	13/8, 44/27
21	868.97	33/20	884.45	5/3
22	910.34	22/13, 27/16, 56/33	925.83	77/45
23	951.72	26/15	967.21	7/4
24	993.10	16/9	1008.59	25/14, 70/39
25	1034.48	20/11	1049.97	11/6
26	1075.86	13/7, 28/15	1091.35	15/8
27	1117.24	21/11, 40/21	1133.73	25/13, 52/27
28	1158.62	39/20, 49/25, 88/45	1174.11	63/32, 65/33, 77/39, 160/81
29	1200.00	2/1

* in 13-limit CWE tuning

Scales

• Mystery58

Tunings

Norm-based tunings

7-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~5/4 = 387.6143 ¢	CWE: ~5/4 = 388.3030 ¢	POTE: ~5/4 = 388.6457 ¢

13-limit norm-based tunings

	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~5/4 = 386.9394 ¢	CWE: ~5/4 = 387.9017 ¢	POTE: ~5/4 = 388.3541 ¢

Tuning spectrum

Edo generator	Unchanged interval (eigenmonzo)*	Generator (¢)	Comments
9\29		372.414	Lower bound of 7- to 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
	15/8	384.820
	13/8	385.355
	11/8	385.801
28\87		386.207
	5/4	386.314
	13/12	386.849
	11/6	387.294
	5/3	387.807
	13/9	388.342
	21/16	388.022
	11/9	388.787
47\145		388.966
	9/5	389.300
	7/4	389.516
	7/6	391.009
	9/7	392.502
19\58		393.103	Upper bound of 13- and 15-odd-limit, and 13-limit 21-odd-limit diamond monotone
10\29		413.793	29cdef val, upper bound of 7- to 11-odd-limit diamond monotone

* Besides the octave