Mystery: Difference between revisions

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'''Mystery''' is the [[regular temperament]] which takes [[29edo|1\29]] [[period]] and adds a single [[generator]] to correct the harmonic 5, 7, 11, and 13 to almost just versions. 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''' 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 also| Hemifamity temperaments #Mystery }}
See [[Hemifamity temperaments #Mystery]] for technical details.


== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–21 are in '''bold'''.
{| class="wikitable center-1 right-2 right-4"
{| class="wikitable center-1 right-2 right-4"
! rowspan="2" |Period
! rowspan="2" | Period
! colspan="2" |Generator 0
! colspan="2" | Generator 0
! colspan="2" |Generator 1
! colspan="2" | Generator 1
|-
|-
! Cents
! Cents
! Approx. Ratios
! Approx. ratios
! Cents
! Cents
! Approx. Ratios
! Approx. ratios
|-
|-
| 0
| 0
| 0.00
| 0.00
| 1/1
| '''1/1'''
| 15.94
| 15.49
| 91/90, 100/99, 105/104, 121/120
| 91/90, 100/99, 105/104, 121/120
|-
|-
Line 23: Line 25:
| 41.38
| 41.38
| 40/39, 45/44, 50/49
| 40/39, 45/44, 50/49
| 57.32
| 56.87
| 28/27, 33/32
| 28/27, 33/32
|-
|-
| 2
| 2
| 82.76
| 82.76
| 21/20
| 21/20, 22/21
| 98.70
| 98.25
| 35/33
| 35/33
|-
|-
Line 35: Line 37:
| 124.14
| 124.14
| 14/13, 15/14
| 14/13, 15/14
| 140.08
| 139.63
| 13/12
| 13/12
|-
|-
Line 41: Line 43:
| 165.52
| 165.52
| 11/10
| 11/10
| 181.46
| 181.01
| 10/9
| 10/9
|-
|-
| 5
| 5
| 206.90
| 206.90
| 9/8
| '''9/8'''
| 222.84
| 222.38
| 25/22
| 25/22
|-
|-
Line 53: Line 55:
| 248.28
| 248.28
| 15/13
| 15/13
| 264.22
| 263.76
| 7/6
| 7/6
|-
|-
Line 59: Line 61:
| 289.66
| 289.66
| 13/11, 32/27, 33/28
| 13/11, 32/27, 33/28
| 305.60
| 305.14
| 25/21
| 25/21
|-
|-
Line 65: Line 67:
| 331.03
| 331.03
| 40/33
| 40/33
| 346.97
| 346.52
| 11/9, 39/32
| 11/9, 39/32
|-
|-
Line 71: Line 73:
| 372.41
| 372.41
| 26/21
| 26/21
| 388.35
| 387.90
| '''5/4'''
| '''5/4'''
|-
|-
Line 77: Line 79:
| 413.79
| 413.79
| 14/11, 33/26, 80/63
| 14/11, 33/26, 80/63
| 429.73
| 429.28
|  
| 77/60
|-
|-
| 11
| 11
| 455.17
| 455.17
| 13/10
| 13/10
| 471.11
| 470.66
| 21/16
| '''21/16'''
|-
|-
| 12
| 12
| 496.55
| 496.55
| '''4/3'''
| 4/3
| 512.49
| 512.04
| 35/26, 75/56
| 35/26, 75/56
|-
|-
Line 95: Line 97:
| 537.93
| 537.93
| 15/11
| 15/11
| 553.87
| 553.42
| '''11/8'''
| '''11/8'''
|-
|-
Line 101: Line 103:
| 579.31
| 579.31
| 7/5
| 7/5
| 595.25
| 594.80
| 45/32
| 45/32
|-
|-
Line 107: Line 109:
| 620.69
| 620.69
| 10/7
| 10/7
| 636.63
| 636.18
| 13/9
| 13/9
|-
|-
Line 113: Line 115:
| 662.07
| 662.07
| 22/15
| 22/15
| 678.01
| 677.56
| 40/27
| 40/27
|-
|-
Line 119: Line 121:
| 703.45
| 703.45
| '''3/2'''
| '''3/2'''
| 719.39
| 718.94
| 50/33
| 50/33
|-
|-
Line 125: Line 127:
| 744.83
| 744.83
| 20/13
| 20/13
| 760.77
| 760.32
| 14/9
| 14/9
|-
|-
Line 131: Line 133:
| 786.21
| 786.21
| 11/7, 52/33, 63/40
| 11/7, 52/33, 63/40
| 802.15
| 801.69
| 35/22
| 35/22
|-
|-
Line 137: Line 139:
| 827.59
| 827.59
| 21/13
| 21/13
| 843.53
| 843.07
| '''13/8''', 44/27
| '''13/8''', 44/27
|-
|-
Line 143: Line 145:
| 868.97
| 868.97
| 33/20
| 33/20
| 884.91
| 884.45
| 5/3
| 5/3
|-
|-
Line 149: Line 151:
| 910.34
| 910.34
| 22/13, 27/16, 56/33
| 22/13, 27/16, 56/33
| 926.29
| 925.83
|  
| 77/45
|-
|-
| 23
| 23
| 951.72
| 951.72
| 26/15
| 26/15
| 967.66
| 967.21
| '''7/4'''
| '''7/4'''
|-
|-
Line 161: Line 163:
| 993.10
| 993.10
| 16/9
| 16/9
| 1009.04
| 1008.59
| 25/14, 70/39
| 25/14, 70/39
|-
|-
Line 167: Line 169:
| 1034.48
| 1034.48
| 20/11
| 20/11
| 1050.42
| 1049.97
| 11/6
| 11/6
|-
|-
Line 173: Line 175:
| 1075.86
| 1075.86
| 13/7, 28/15
| 13/7, 28/15
| 1090.80
| 1091.35
| 15/8
| '''15/8'''
|-
|-
| 27
| 27
| 1117.24
| 1117.24
| 40/21
| 21/11, 40/21
| 1133.18
| 1133.73
| 25/13, 52/27
| 25/13, 52/27
|-
|-
Line 185: Line 187:
| 1158.62
| 1158.62
| 39/20, 49/25, 88/45
| 39/20, 49/25, 88/45
| 1174.56
| 1174.11
| 63/32, 65/33, 77/39, 160/81
| 63/32, 65/33, 77/39, 160/81
|-
|-
| 29
| 29
| 1200.00
| 1200.00
| 2/1
| '''2/1'''
|
|
|}
<nowiki>*</nowiki> in 13-limit CWE tuning
 
== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 9\29
|  
|  
| 372.414
| Lower bound of 7- to 15-odd-limit, <br>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, <br>and 13-limit 21-odd-limit diamond monotone
|-
| 10\29
|  
|  
| 413.793
| 29cdef val, upper bound of 7- to 11-odd-limit diamond monotone
|}
|}
<nowiki>*</nowiki> in 13-limit POTE tuning
<nowiki/>* Besides the octave


[[Category:Temperaments]]
[[Category:Mystery| ]] <!-- main article -->
[[Category:Mystery| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Hemifamity temperaments]]
[[Category:Hemifamity temperaments]]

Latest revision as of 06:56, 21 June 2025

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

Tunings

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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