Mystery: Difference between revisions

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| Title = Mystery
| Title = Mystery
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[5120/5103]], [[50421/50000]] (7-limit); <br>[[441/440]], [[896/891]], [[3388/3375]] (11-limit); <br>[[196/195]], [[352/351]], [[364/363]], [[676/675]] <br>(13-limit)
| Comma basis = [[5120/5103]], [[50421/50000]] (7-limit); <br>[[441/440]], [[896/891]], [[3388/3375]] (11-limit); <br>[[196/195]], [[352/351]], [[364/363]], [[676/675]] (13-limit)
| Edo join 1 = 58 | Edo join 2 = 87
| Edo join 1 = 29 | Edo join 2 = 58
| Mapping = 29; 0 1 1 1 1
| Mapping = 29; 0 1 1 1 1
| Generator = 5/4
| Generators = 5/4
| Generator tuning = 387.9
| Generators tuning = 387.9
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[29L&nbsp;29s]], [[58L&nbsp;29s]]
| MOS scales = [[29L&nbsp;29s]], [[58L&nbsp;29s]]
| Pergen = (P8/29, M3<sup>5</sup>)
| Pergen = (P8/29, ^1)
| Color name = Quadsawati
| Color name = Quadsawati
| Odd limit 1 = 9 | Mistuning 1 = 3.20 | Complexity 1 = 58
| Odd limit 1 = 9 | Mistuning 1 = 3.20 | Complexity 1 = 58
| Odd limit 2 = (13-limit) 21 | Mistuning 2 = 4.69 | Complexity 2 = 58
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 4.69 | Complexity 2 = 58
}}
}}
'''Mystery''' is a [[regular temperament]] which takes [[29edo|1\29]] [[period]] and adds a single [[generator]] to correct the [[harmonic]]s [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] to almost just qualities. It [[tempering out|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''' is a [[regular temperament]] which takes [[29edo|1\29]] [[period]] and adds a single [[generator]] to correct the [[harmonic]]s [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] to almost just qualities. It [[tempering out|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.  


See [[Hemifamity temperaments #Mystery]] for technical details.  
See [[Hemifamity temperaments #Mystery]] for technical details.  
 
__TOC__
{{Clear}}
== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are in '''bold'''.  
In the following table, odd harmonics 1–21 are in '''bold'''.  
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|}
|}
<nowiki>*</nowiki> in 13-limit CWE tuning
<nowiki>*</nowiki> in 13-limit CWE tuning
== Scales ==
* [[Mystery58]]


== Tunings ==
== Tunings ==

Latest revision as of 09:31, 9 February 2026

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

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.

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

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

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