Xenharmonic Wiki

156edo: Difference between revisions

← Older edit
VisualWikitext
merge from https://xen.miraheze.org/wiki/156edo
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,289 edits
m →Prime harmonics: good 41-limit system
 
(14 intermediate revisions by 10 users not shown)
Line 1: Line 1:
'''156edo''' is the [[EDO|equal division of the octave]] into 156 parts of 7.6923 cents each. It tempers out 531441/524288 (pythagorean comma) and 1220703125/1207959552 (ditonic comma) in the 5-limit, as well as 1224440064/1220703125 (parakleisma); 225/224, 250047/250000, and 589824/588245 in the 7-limit. Using the patent val, it tempers out 441/440, 1375/1372, 4375/4356, and 65536/65219 in the 11-limit; 351/350, 364/363, 625/624, 1625/1617, and 13122/13013 in the 13-limit. Using the 156e val, it tempers out 385/384, 540/539, 1331/1323, and 78408/78125 in the 11-limit; 351/350, 625/624, 847/845, and 1001/1000 in the 13-limit.
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
It [[tempering out|tempers out]] 531441/524288 ([[Pythagorean comma]]) and {{monzo| -27 -2 13 }} (ditonmic comma) in the 5-limit, as well as {{monzo| 8 14 -13 }} ([[parakleisma]]); [[225/224]], [[250047/250000]], and [[589824/588245]] in the 7-limit. Using the patent val, it tempers out [[441/440]], 1375/1372, 4375/4356, and 65536/65219 in the 11-limit; [[351/350]], [[364/363]], [[625/624]], 1625/1617, and 13122/13013 in the 13-limit. Using the 156e val, it tempers out [[385/384]], [[540/539]], 1331/1323, and 78408/78125 in the 11-limit; 351/350, 625/624, [[847/845]], and [[1001/1000]] in the 13-limit. It [[support]]s [[compton]] and gives a good tuning for the 5- and 7-limit version thereof.


={{PAGENAME}}=
=== Prime harmonics ===
{{Harmonics in equal|156|intervals=prime|columns=13}}


Welcome to {{PAGENAME}}
=== Subsets and supersets ===
Sinece 156 factors into {{factorization|156}}, 156edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 13, 26, 39, 52, and 78 }}. It is the smallest edo to contain both [[12edo]] and [[13edo]] as subsets.


=={{PAGENAME}} = [[12edo]] × [[13edo]]==
== Intervals ==
 
{{Interval table}}
Distinctively interesting choice for an edo, due to its divisibility by both 12 and 13. This means it can be used to play [[12edo]] music and [[13edo]] music at the same time, using the [[K/N|156/13 and 156/12]] vals respectively.
 
==7.692==
 
{{PAGENAME}} has a step size of {{#expr: 1200.0/156.0}} cents. Microtonal enthusiasts might notice it seems close to the [[225/224]] ratio, but it isn't in the patent val, because {{PAGENAME}} tempers out [[225/224]]. The edo step size closest to [[225/224]], taking into account the primes that compose it, is in [[171edo]]. You have to use the [[K*N|2*156]] val, sacrificing 3 for a new 9, to make this approximation come true. It is up to the microtonalist to decide whether this tradeoff is worth it.
 
==Selected just intervals by error==
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 156edo (ordered by absolute error).
 
{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent]]s)'''
|-
| style="text-align:center;" | [[13/12]], [[24/13]]
| style="text-align:center;" | 0.111
|-
| style="text-align:center;" | [[6/5]], [[5/3]]
| style="text-align:center;" | 0.257
|-
| style="text-align:center;" | [[13/10]], [[20/13]]
| style="text-align:center;" | 0.368
|-
| style="text-align:center;" | [[8/7]], [[7/4]]
| style="text-align:center;" | 0.405
|-
| style="text-align:center;" | [[15/13]], [[26/15]]
| style="text-align:center;" | 1.587
|-
| style="text-align:center;" | [[5/4]], [[8/5]]
| style="text-align:center;" | 1.698
|-
| style="text-align:center;" | [[18/13]], [[13/9]]
| style="text-align:center;" | 1.844
|-
| style="text-align:center;" | [[4/3]], [[3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[16/13]], [[13/8]]
| style="text-align:center;" | 2.066
|-
| style="text-align:center;" | [[7/5]], [[10/7]]
| style="text-align:center;" | 2.103
|-
| style="text-align:center;" | [[14/11]], [[11/7]]
| style="text-align:center;" | 2.123
|-
| style="text-align:center;" | [[10/9]], [[9/5]]
| style="text-align:center;" | 2.212
|-
| style="text-align:center;" | [[7/6]], [[12/7]]
| style="text-align:center;" | 2.36
|-
| style="text-align:center;" | [[14/13]], [[13/7]]
| style="text-align:center;" | 2.471
|-
| style="text-align:center;" | [[11/8]], [[16/11]]
| style="text-align:center;" | 2.528
|-
| style="text-align:center;" | [[16/15]], [[15/8]]
| style="text-align:center;" | 3.653
|-
| style="text-align:center;" | [[9/8]], [[16/9]]
| style="text-align:center;" | 3.91
|-
| style="text-align:center;" | [[15/14]], [[28/15]]
| style="text-align:center;" | 4.058
|-
| style="text-align:center;" | [[11/10]], [[20/11]]
| style="text-align:center;" | 4.227
|-
| style="text-align:center;" | [[9/7]], [[14/9]]
| style="text-align:center;" | 4.315
|-
| style="text-align:center;" | [[12/11]], [[11/6]]
| style="text-align:center;" | 4.483
|-
| style="text-align:center;" | [[13/11]], [[22/13]]
| style="text-align:center;" | 4.594
|-
| style="text-align:center;" | [[15/11]], [[22/15]]
| style="text-align:center;" | 6.182
|-
| style="text-align:center;" | [[11/9]], [[18/11]]
| style="text-align:center;" | 6.438
|}
 
==Table of {{PAGENAME}} intervals==
 
{| class="wikitable"
|-
| | Step
| | Five limit
| | Seven limit
| | Eleven limit
| | Thirteen limit
|-
| | 1
| | 15625/15552
| | 1029/1024
| | 242/243
| | 144/143
|-
| | 2
| | 81/80
| | 81/80
| | 81/80
| | 78/77
|-
| | 3
| | 3125/3072
| | 245/243
| | 121/120
| | 65/64
|-
| | 4
| | 2048/2025
| | 50/49
| | 45/44
| | 45/44
|-
| | 5
| | 16875/16384
| | 49/48
| | 49/48
| | 49/48
|-
| | 6
| | 128/125
| | 36/35
| | 36/35
| | 36/35
|-
| | 7
| | 250/243
| | 250/243
| | 33/32
| | 33/32
|-
| | 8
| | 648/625
| | 648/625
| | 80/77
| | 27/26
|-
| | 9
| | 25/24
| | 25/24
| | 25/24
| | 25/24
|-
| | 10
| | 6561/6250
| | 256/245
| | 81/77
| | 81/77
|-
| | 11
| | 135/128
| | 21/20
| | 21/20
| | 21/20
|-
| | 12
| | 16384/15625
| | 729/686
| | 128/121
| | 96/91
|-
| | 13
| | 256/243
| | 200/189
| | 35/33
| | 35/33
|-
| | 14
| | 62500/59049
| | 343/324
| | 343/324
| | 143/135
|-
| | 15
| | 16/15
| | 15/14
| | 15/14
| | 15/14
|-
| | 16
| | 3125/2916
| | 343/320
| | 77/72
| | 77/72
|-
| | 17
| | 27/25
| | 27/25
| | 27/25
| | 14/13
|-
| | 18
| | 625/576
| | 175/162
| | 121/112
| | 13/12
|-
| | 19
| | 2187/2000
| | 160/147
| | 12/11
| | 12/11
|-
| | 20
| | 1125/1024
| | 35/32
| | 35/32
| | 35/32
|-
| | 21
| | 2048/1875
| | 54/49
| | 54/49
| | 54/49
|-
| | 22
| | 800/729
| | 441/400
| | 11/10
| | 11/10
|-
| | 23
| | 3456/3125
| | 972/875
| | 135/121
| | 72/65
|-
| | 24
| | 10/9
| | 10/9
| | 10/9
| | 10/9
|-
| | 25
| | 17496/15625
| | 384/343
| | 384/343
| | 143/128
|-
| | 26
| | 9/8
| | 9/8
| | 9/8
| | 9/8
|-
| | 27
| | 15625/13824
| | 1944/1715
| | 121/108
| | 91/81
|-
| | 28
| | 729/640
| | 500/441
| | 25/22
| | 25/22
|-
| | 29
| | 9375/8192
| | 245/216
| | 245/216
| | 91/80
|-
| | 30
| | 256/225
| | 8/7
| | 8/7
| | 8/7
|-
| | 31
| | 2500/2187
| | 147/128
| | 55/48
| | 55/48
|-
| | 32
| | 144/125
| | 81/70
| | 81/70
| | 15/13
|-
| | 33
| | 125/108
| | 125/108
| | 121/105
| | 52/45
|-
| | 34
| | 729/625
| | 400/343
| | 64/55
| | 64/55
|-
| | 35
| | 75/64
| | 7/6
| | 7/6
| | 7/6
|-
| | 36
| | 32768/28125
| | 288/245
| | 288/245
| | 168/143
|-
| | 37
| | 1215/1024
| | 147/125
| | 33/28
| | 13/11
|-
| | 38
| | 18432/15625
| | 1458/1225
| | 144/121
| | 108/91
|-
| | 39
| | 32/27
| | 25/21
| | 25/21
| | 25/21
|-
| | 40
| | 15625/13122
| | 343/288
| | 343/288
| | 143/120
|-
| | 41
| | 6/5
| | 6/5
| | 6/5
| | 6/5
|-
| | 42
| | 3125/2592
| | 875/729
| | 77/64
| | 65/54
|-
| | 43
| | 243/200
| | 243/200
| | 40/33
| | 40/33
|-
| | 44
| | 625/512
| | 98/81
| | 98/81
| | 39/32
|-
| | 45
| | 4096/3375
| | 60/49
| | 27/22
| | 27/22
|-
| | 46
| | 8000/6561
| | 49/40
| | 11/9
| | 11/9
|-
| | 47
| | 768/625
| | 216/175
| | 150/121
| | 16/13
|-
| | 48
| | 100/81
| | 100/81
| | 99/80
| | 26/21
|-
| | 49
| | 3888/3125
| | 1280/1029
| | 96/77
| | 81/65
|-
| | 50
| | 5/4
| | 5/4
| | 5/4
| | 5/4
|-
| | 51
| | 19683/15625
| | 432/343
| | 432/343
| | 169/135
|-
| | 52
| | 81/64
| | 63/50
| | 44/35
| | 44/35
|-
| | 53
| | 15625/12288
| | 1225/972
| | 121/96
| | 91/72
|-
| | 54
| | 512/405
| | 80/63
| | 14/11
| | 14/11
|-
| | 55
| | 25000/19683
| | 245/192
| | 245/192
| | 143/112
|-
| | 56
| | 32/25
| | 9/7
| | 9/7
| | 9/7
|-
| | 57
| | 625/486
| | 625/486
| | 77/60
| | 77/60
|-
| | 58
| | 162/125
| | 162/125
| | 100/77
| | 84/65
|-
| | 59
| | 125/96
| | 35/27
| | 35/27
| | 13/10
|-
| | 60
| | 6561/5000
| | 64/49
| | 64/49
| | 64/49
|-
| | 61
| | 675/512
| | 21/16
| | 21/16
| | 21/16
|-
| | 62
| | 4096/3125
| | 324/245
| | 160/121
| | 120/91
|-
| | 63
| | 320/243
| | 250/189
| | 33/25
| | 33/25
|-
| | 64
| | 20736/15625
| | 1715/1296
| | 162/121
| | 143/108
|-
| | 65
| | 4/3
| | 4/3
| | 4/3
| | 4/3
|-
| | 66
| | 15625/11664
| | 343/256
| | 343/256
| | 169/126
|-
| | 67
| | 27/20
| | 27/20
| | 27/20
| | 27/20
|-
| | 68
| | 3125/2304
| | 875/648
| | 121/90
| | 65/48
|-
| | 69
| | 2187/1600
| | 200/147
| | 15/11
| | 15/11
|-
| | 70
| | 5625/4096
| | 49/36
| | 49/36
| | 49/36
|-
| | 71
| | 512/375
| | 48/35
| | 48/35
| | 48/35
|-
| | 72
| | 1000/729
| | 343/250
| | 11/8
| | 11/8
|-
| | 73
| | 864/625
| | 243/175
| | 168/121
| | 18/13
|-
| | 74
| | 25/18
| | 25/18
| | 25/18
| | 25/18
|-
| | 75
| | 4374/3125
| | 480/343
| | 108/77
| | 108/77
|-
| | 76
| | 45/32
| | 7/5
| | 7/5
| | 7/5
|-
| | 77
| | 65536/46875
| | 486/343
| | 486/343
| | 128/91
|-
| | 78
| | 729/512
| | 567/400
| | 99/70
| | 55/39
|-
| | 79
| | 46875/32768
| | 343/243
| | 343/243
| | 91/64
|-
| | 80
| | 64/45
| | 10/7
| | 10/7
| | 10/7
|-
| | 81
| | 3125/2187
| | 343/240
| | 77/54
| | 77/54
|-
| | 82
| | 36/25
| | 36/25
| | 36/25
| | 36/25
|-
| | 83
| | 625/432
| | 350/243
| | 121/84
| | 13/9
|-
| | 84
| | 729/500
| | 500/343
| | 16/11
| | 16/11
|-
| | 85
| | 375/256
| | 35/24
| | 35/24
| | 35/24
|-
| | 86
| | 8192/5625
| | 72/49
| | 72/49
| | 72/49
|-
| | 87
| | 3200/2187
| | 147/100
| | 22/15
| | 22/15
|-
| | 88
| | 4608/3125
| | 729/490
| | 180/121
| | 96/65
|-
| | 89
| | 40/27
| | 40/27
| | 40/27
| | 40/27
|-
| | 90
| | 23328/15625
| | 512/343
| | 512/343
| | 143/96
|-
| | 91
| | 3/2
| | 3/2
| | 3/2
| | 3/2
|-
| | 92
| | 15625/10368
| | 2401/1600
| | 121/81
| | 121/81
|-
| | 93
| | 243/160
| | 189/125
| | 50/33
| | 50/33
|-
| | 94
| | 3125/2048
| | 245/162
| | 121/80
| | 91/60
|-
| | 95
| | 1024/675
| | 32/21
| | 32/21
| | 32/21
|-
| | 96
| | 10000/6561
| | 49/32
| | 49/32
| | 49/32
|-
| | 97
| | 192/125
| | 54/35
| | 54/35
| | 20/13
|-
| | 98
| | 125/81
| | 125/81
| | 77/50
| | 65/42
|-
| | 99
| | 972/625
| | 972/625
| | 120/77
| | 81/52
|-
| | 100
| | 25/16
| | 14/9
| | 14/9
| | 14/9
|-
| | 101
| | 19683/12500
| | 384/245
| | 243/154
| | 169/108
|-
| | 102
| | 405/256
| | 63/40
| | 11/7
| | 11/7
|-
| | 103
| | 24576/15625
| | 1944/1225
| | 192/121
| | 144/91
|-
| | 104
| | 128/81
| | 100/63
| | 35/22
| | 35/22
|-
| | 105
| | 31250/19683
| | 343/216
| | 343/216
| | 143/90
|-
| | 106
| | 8/5
| | 8/5
| | 8/5
| | 8/5
|-
| | 107
| | 3125/1944
| | 1029/640
| | 77/48
| | 77/48
|-
| | 108
| | 81/50
| | 81/50
| | 81/50
| | 21/13
|-
| | 109
| | 625/384
| | 175/108
| | 121/75
| | 13/8
|-
| | 110
| | 6561/4000
| | 80/49
| | 18/11
| | 18/11
|-
| | 111
| | 3375/2048
| | 49/30
| | 44/27
| | 44/27
|-
| | 112
| | 1024/625
| | 81/49
| | 81/49
| | 64/39
|-
| | 113
| | 400/243
| | 400/243
| | 33/20
| | 33/20
|-
| | 114
| | 5184/3125
| | 1458/875
| | 128/77
| | 108/65
|-
| | 115
| | 5/3
| | 5/3
| | 5/3
| | 5/3
|-
| | 116
| | 26244/15625
| | 576/343
| | 539/324
| | 240/143
|-
| | 117
| | 27/16
| | 27/16
| | 27/16
| | 27/16
|-
| | 118
| | 15625/9216
| | 1225/729
| | 121/72
| | 91/54
|-
| | 119
| | 2048/1215
| | 250/147
| | 56/33
| | 22/13
|-
| | 120
| | 28125/16384
| | 245/144
| | 245/144
| | 143/84
|-
| | 121
| | 128/75
| | 12/7
| | 12/7
| | 12/7
|-
| | 122
| | 1250/729
| | 343/200
| | 55/32
| | 55/32
|-
| | 123
| | 216/125
| | 216/125
| | 210/121
| | 45/26
|-
| | 124
| | 125/72
| | 125/72
| | 121/70
| | 26/15
|-
| | 125
| | 2187/1250
| | 256/147
| | 96/55
| | 96/55
|-
| | 126
| | 225/128
| | 7/4
| | 7/4
| | 7/4
|-
| | 127
| | 16384/9375
| | 432/245
| | 432/245
| | 160/91
|-
| | 128
| | 1280/729
| | 441/250
| | 44/25
| | 39/22
|-
| | 129
| | 27648/15625
| | 1715/972
| | 216/121
| | 143/81
|-
| | 130
| | 16/9
| | 16/9
| | 16/9
| | 16/9
|-
| | 131
| | 15625/8748
| | 343/192
| | 343/192
| | 143/80
|-
| | 132
| | 9/5
| | 9/5
| | 9/5
| | 9/5
|-
| | 133
| | 3125/1728
| | 875/486
| | 231/128
| | 65/36
|-
| | 134
| | 729/400
| | 625/343
| | 20/11
| | 20/11
|-
| | 135
| | 1875/1024
| | 49/27
| | 49/27
| | 49/27
|-
| | 136
| | 2048/1125
| | 64/35
| | 64/35
| | 64/35
|-
| | 137
| | 4000/2187
| | 147/80
| | 11/6
| | 11/6
|-
| | 138
| | 1152/625
| | 324/175
| | 224/121
| | 24/13
|-
| | 139
| | 50/27
| | 50/27
| | 50/27
| | 13/7
|-
| | 140
| | 5832/3125
| | 640/343
| | 144/77
| | 144/77
|-
| | 141
| | 15/8
| | 15/8
| | 15/8
| | 15/8
|-
| | 142
| | 59049/31250
| | 648/343
| | 539/288
| | 169/90
|-
| | 143
| | 243/128
| | 189/100
| | 66/35
| | 49/26
|-
| | 144
| | 15625/8192
| | 1225/648
| | 121/64
| | 91/48
|-
| | 145
| | 256/135
| | 40/21
| | 21/11
| | 21/11
|-
| | 146
| | 12500/6561
| | 245/128
| | 154/81
| | 143/75
|-
| | 147
| | 48/25
| | 27/14
| | 27/14
| | 25/13
|-
| | 148
| | 625/324
| | 625/324
| | 77/40
| | 52/27
|-
| | 149
| | 243/125
| | 243/125
| | 64/33
| | 64/33
|-
| | 150
| | 125/64
| | 35/18
| | 35/18
| | 35/18
|-
| | 151
| | 19683/10000
| | 96/49
| | 96/49
| | 96/49
|-
| | 152
| | 2025/1024
| | 49/25
| | 49/25
| | 49/25
|-
| | 153
| | 6144/3125
| | 486/245
| | 240/121
| | 128/65
|-
| | 154
| | 160/81
| | 125/63
| | 99/50
| | 77/39
|-
| | 155
| | 31104/15625
| | 1715/864
| | 243/121
| | 143/72
|-
| | 156
| | 2/1
| | 2/1
| | 2/1
| | 2/1
|}
Retrieved from "https://en.xen.wiki/w/156edo"