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Added an algorithmically generated table of intervals, among some other things, need to complete the table (it's got many gaps and badly formatted)
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== Theory ==
== Theory ==
217edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 217 {{=}} 7 × 31 }}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to [[31edo]], its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|19]]—in fact, this edo has an extremely accurate [[13/1|13th harmonic]], as well as the [[19/15]] interval. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. If one desires higher consistency and precision, [[311edo]] offers a much better palette.  
217edo is a strong [[19-limit]] system, the smallest [[consistency|distinctly consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]] as well as the no-23 [[31-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap| 217 {{=}} 7 × 31 }}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to [[31edo]], its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|19]], excelling as a [[2.3.5.13 subgroup]]. It can be used as a decent approximation of the [[31-limit]], ''almost'' being consistent through the [[31-odd-limit]] except for [[23/14]], [[23/21]], [[29/23]] and their [[octave complement]]s, with errors below the melodic [[just-noticeable difference]]. If one desires higher consistency and precision, [[311edo]] offers a much better palette.  


The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]] and [[823543/819200]] in the 7-limit; [[441/440]], [[4000/3993]], [[5632/5625]], and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]] and [[4096/4095]] in the 13-limit; [[595/594]], [[833/832]], [[936/935]], [[1156/1155]], [[1225/1224]], [[1701/1700]] in the 17-limit; [[343/342]], [[476/475]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]] and [[1540/1539]] in the 19-limit. It allows [[minor minthmic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 32 -7 -9 }} in the 5-limit; [[3136/3125]], [[4375/4374]], [[10976/10935]], [[823543/819200]] and the [[garischisma]], [25 -14 0 -1⟩ in the 7-limit; [[441/440]], [[4000/3993]], [[5632/5625]], and [[16384/16335]] in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]], [[2080/2079]], [[4096/4095]] and [[123201/123200]] in the 13-limit; [[595/594]], [[833/832]], [[936/935]], [[1156/1155]], [[1225/1224]], [[1701/1700]] in the 17-limit; [[343/342]], [[476/475]], [[969/968]], [[1216/1215]], [[1445/1444]], [[1521/1520]] and [[1540/1539]] in the 19-limit. It allows [[minor minthmic chords]], [[werckismic chords]], and [[sinbadmic chords]] in the 13-odd-limit, in addition to [[island chords]] and [[nicolic chords]] in the 15-odd-limit. It provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|217}}
{{Harmonics in equal|217|columns=19|intervals=odd}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 217 factors into primes as {{nowrap| 7 × 31 }}, a product of two {{w|Mersenne prime}}s, 217edo contains [[7edo]] and 31edo as subset edos.  
Since 217 factors into primes as {{nowrap| 7 × 31 }}, a product of two {{w|Mersenne prime}}s, 217edo contains [[7edo]] and 31edo as subset edos.  
== Intervals ==
217edo is not a very high-limit system, but it also manages to be ''almost'' consistent through the no-37 [[39-odd-limit]], thanks to its good approximations of primes 3,5,7,13, missing [[33/28]], [[33/29]] and their octave complements.
Here below is an algorithmically generated table of no-37 39-odd-limit intervals of 217edo using [[User:Godtone#My Python 3 code|Godtone's code]], with some manually added intervals outside that limit for completeness.
{| class="wikitable mw-collapsible mw-collapsed"
|+Table of 217edo intervals
|'''#'''
|'''Cents'''
|'''Marks'''
|Approximate intervals
|-
|0
|0
|P1
|
|-
|1
|5.53
|
|
|-
|2
|11.06
|
|
|-
|3
|16.59
|
|
|-
|4
|22.12
|
|[[81/80]]
|-
|5
|27.65
|
|[[64/63]], ''[[531441/524288]]''
|-
|6
|33.18
|
|
|-
|7
|38.71
|
|
|-
| 8
| 44.24
|
| [[40/39]], [[39/38]]
|-
| 9
| 49.77
|
| [[36/35]], [[35/34]], [[34/33]]
|-
| 10
| 55.3
|
| [[33/32]], [[32/31]], [[31/30]]
|-
| 11
| 60.83
|
| [[30/29]], [[29/28]], [[28/27]]
|-
| 12
| 66.36
|
| [[27/26]], [[26/25]]
|-
| 13
| 71.89
|
| [[25/24]], [[24/23]]
|-
| 14
| 77.42
|
| [[23/22]]
|-
| 15
| 82.95
|
| [[22/21]], [[21/20]]
|-
| 16
| 88.48
|m2
| [[20/19]], [[256/243]]
|-
| 17
| 94.01
|
| [[19/18]]
|-
| 18
| 99.54
|
| [[18/17]], [[35/33]]
|-
| 19
| 105.07
|
| [[17/16]]
|-
| 20
| 110.6
|
| [[33/31]], [[16/15]]
|-
| 21
| 116.13
|A1
| [[31/29]], ''[[2187/2048]]''
|-
| 22
| 121.66
|
| [[15/14]], [[29/27]]
|-
| 23
| 127.19
|
| [[14/13]]
|-
| 24
| 132.72
|
| [[27/25]]
|-
| 25
| 138.25
|
| [[13/12]]
|-
| 26
| 143.78
|
| [[38/35]], [[25/23]]
|-
| 27
| 149.31
|
| [[12/11]]
|-
| 28
| 154.84
|
| [[35/32]]
|-
| 29
| 160.37
|
| **[[23/21]]**, [[34/31]]
|-
| 30
| 165.9
|
| [[11/10]]
|-
| 31
| 171.43
|
| [[32/29]], [[21/19]]
|-
| 32
| 176.96
|
| [[31/28]]
|-
| 33
| 182.49
|
| [[10/9]]
|-
| 34
| 188.02
|
| [[39/35]], [[29/26]]
|-
| 35
| 193.55
|
| [[19/17]], [[28/25]]
|-
| 37
| 204.61
|M2
| [[9/8]]
|-
| 38
| 210.14
|
| [[44/39]], [[35/31]], [[26/23]]
|-
| 39
| 215.67
|
| [[17/15]]
|-
| 40
| 221.2
|
| [[25/22]]
|-
| 41
| 226.73
|
| **[[33/29]]**
|-
| 42
| 232.26
|
| [[8/7]]
|-
| 43
| 237.79
|
| [[39/34]], [[31/27]]
|-
| 44
| 243.32
|
| [[23/20]], [[38/33]]
|-
| 45
| 248.85
|
| [[15/13]]
|-
| 46
| 254.38
|
| [[22/19]], [[29/25]]
|-
| 47
| 259.91
|
| [[36/31]]
|-
| 48
| 265.44
|
| [[7/6]]
|-
| 50
| 276.5
|
| [[34/29]], [[27/23]]
|-
| 51
| 282.03
|
| [[20/17]]
|-
| 52
| 287.56
|
| **[[33/28]]**, [[46/39]], [[13/11]]
|-
| 53
| 293.09
|m3
| [[32/27]]
|-
| 54
| 298.62
|
| [[19/16]]
|-
| 55
| 304.15
|
| [[25/21]], [[31/26]]
|-
| 57
| 315.21
|
| [[6/5]]
|-
| 59
| 326.27
|
| [[35/29]], [[29/24]]
|-
| 60
| 331.8
|
| [[23/19]], [[40/33]]
|-
| 61
| 337.33
|
| [[17/14]], **[[28/23]]**
|-
| 62
| 342.86
|
| [[39/32]]
|-
| 63
| 348.39
|
| [[11/9]]
|-
| 64
| 353.92
|
| [[38/31]], [[27/22]]
|-
| 65
| 359.45
|
| [[16/13]]
|-
| 66
| 364.98
|
| [[21/17]]
|-
| 67
| 370.51
|
| [[26/21]], [[31/25]]
|-
| 68
| 376.04
|
| [[36/29]]
|-
| 70
| 387.1
|
| [[5/4]]
|-
| 72
| 398.16
|
| [[44/35]], [[39/31]], [[34/27]], **[[29/23]]**
|-
| 73
| 403.69
|
| [[24/19]]
|-
| 74
| 409.22
|M3
| [[19/15]], [[81/64]]
|-
| 75
| 414.75
|
| [[33/26]], [[14/11]]
|-
| 77
| 425.81
|
| [[23/18]], [[32/25]]
|-
| 78
| 431.34
|
| [[50/39]]
|-
| 79
| 436.87
|
| [[9/7]]
|-
| 80
| 442.4
|
| [[40/31]], [[31/24]]
|-
| 81
| 447.93
|
| [[22/17]], [[35/27]]
|-
| 82
| 453.46
|
| [[13/10]]
|-
| 83
| 458.99
|
| [[30/23]]
|-
| 84
| 464.52
|
| [[17/13]]
|-
| 85
| 470.05
|
| [[38/29]], [[21/16]]
|-
| 86
| 475.58
|
| [[46/35]], [[25/19]], [[29/22]]
|-
| 87
| 481.11
|
| [[33/25]]
|-
| 90
| 497.7
|P4
| [[4/3]]
|-
| 93
| 514.29
|
| [[39/29]], [[35/26]], [[31/23]]
|-
| 94
| 519.82
|
| [[27/20]]
|-
| 95
| 525.35
|
| [[23/17]], [[42/31]]
|-
| 96
| 530.88
|
| [[19/14]], [[34/25]]
|-
| 97
| 536.41
|
| [[15/11]]
|-
| 98
| 541.94
|
| [[26/19]]
|-
| 99
| 547.47
|
| [[48/35]]
|-
| 100
| 553.0
|
| [[11/8]]
|-
| 101
| 558.53
|
| [[40/29]], [[29/21]]
|-
| 102
| 564.06
|
| [[18/13]]
|-
| 103
| 569.59
|
| [[25/18]], [[32/23]]
|-
| 104
| 575.12
|
| [[39/28]], [[46/33]]
|-
| 105
| 580.65
|
| [[7/5]]
|-
|106
|586.18
|d5
|[[1024/729]]
|-
| 107
| 591.71
|
| [[38/27]], [[31/22]]
|-
| 108
| 597.24
|
| [[24/17]]
|-
| 109
| 602.76
|
| [[17/12]]
|-
| 110
| 608.29
|
| [[44/31]], [[27/19]]
|-
|111
|613.82
|A4
|[[729/512]]
|-
| 112
| 619.35
|
| [[10/7]]
|-
| 113
| 624.88
|
| [[33/23]], [[56/39]]
|-
| 114
| 630.41
|
| [[23/16]], [[36/25]]
|-
| 115
| 635.94
|
| [[13/9]]
|-
| 116
| 641.47
|
| [[42/29]], [[29/20]]
|-
| 117
| 647.0
|
| [[16/11]]
|-
| 118
| 652.53
|
| [[35/24]]
|-
| 119
| 658.06
|
| [[19/13]]
|-
| 120
| 663.59
|
| [[22/15]]
|-
| 121
| 669.12
|
| [[25/17]], [[28/19]]
|-
| 122
| 674.65
|
| [[31/21]], [[34/23]]
|-
| 123
| 680.18
|
| [[40/27]]
|-
| 124
| 685.71
|
| [[46/31]], [[52/35]], [[58/39]]
|-
| 127
| 702.3
|P5
| [[3/2]]
|-
| 130
| 718.89
|
| [[50/33]]
|-
| 131
| 724.42
|
| [[44/29]], [[38/25]], [[35/23]]
|-
| 132
| 729.95
|
| [[32/21]], [[29/19]]
|-
| 133
| 735.48
|
| [[26/17]]
|-
| 134
| 741.01
|
| [[23/15]]
|-
| 135
| 746.54
|
| [[20/13]]
|-
| 136
| 752.07
|
| [[54/35]], [[17/11]]
|-
| 137
| 757.6
|
| [[48/31]], [[31/20]]
|-
| 138
| 763.13
|
| [[14/9]]
|-
| 139
| 768.66
|
| [[39/25]]
|-
| 140
| 774.19
|
| [[25/16]], [[36/23]]
|-
| 142
| 785.25
|
| [[11/7]], [[52/33]]
|-
| 143
| 790.78
|m6
| [[30/19]], [[128/81]]
|-
| 144
| 796.31
|
| [[19/12]]
|-
| 145
| 801.84
|
| **[[46/29]]**, [[27/17]], [[62/39]], [[35/22]]
|-
| 147
| 812.9
|
| [[8/5]]
|-
| 149
| 823.96
|
| [[29/18]]
|-
| 150
| 829.49
|
| [[50/31]], [[21/13]]
|-
| 151
| 835.02
|
| [[34/21]]
|-
| 152
| 840.55
|
| [[13/8]]
|-
| 153
| 846.08
|
| [[44/27]], [[31/19]]
|-
| 154
| 851.61
|
| [[18/11]]
|-
| 155
| 857.14
|
| [[64/39]]
|-
| 156
| 862.67
|
| **[[23/14]]**, [[28/17]]
|-
| 157
| 868.2
|
| [[33/20]], [[38/23]]
|-
| 158
| 873.73
|
| [[48/29]], [[58/35]]
|-
| 160
| 884.79
|
| [[5/3]]
|-
| 162
| 895.85
|
| [[52/31]], [[42/25]]
|-
| 163
| 901.38
|
| [[32/19]]
|-
| 164
| 906.91
|M6
| [[27/16]]
|-
| 165
| 912.44
|
| [[22/13]], [[39/23]], **[[56/33]]**
|-
| 166
| 917.97
|
| [[17/10]]
|-
| 167
| 923.5
|
| [[46/27]], [[29/17]]
|-
| 169
| 934.56
|
| [[12/7]]
|-
| 170
| 940.09
|
| [[31/18]]
|-
| 171
| 945.62
|
| [[50/29]], [[19/11]]
|-
| 172
| 951.15
|
| [[26/15]]
|-
| 173
| 956.68
|
| [[33/19]], [[40/23]]
|-
| 174
| 962.21
|
| [[54/31]], [[68/39]]
|-
| 175
| 967.74
|
| [[7/4]]
|-
| 176
| 973.27
|
| **[[58/33]]**
|-
| 177
| 978.8
|
| [[44/25]]
|-
| 178
| 984.33
|
| [[30/17]]
|-
| 179
| 989.86
|
| [[23/13]], [[62/35]], [[39/22]]
|-
| 180
| 995.39
|m7
| [[16/9]]
|-
| 182
| 1006.45
|
| [[25/14]], [[34/19]]
|-
| 183
| 1011.98
|
| [[52/29]], [[70/39]]
|-
| 184
| 1017.51
|
| [[9/5]]
|-
| 185
| 1023.04
|
| [[56/31]]
|-
| 186
| 1028.57
|
| [[38/21]], [[29/16]]
|-
| 187
| 1034.1
|
| [[20/11]]
|-
| 188
| 1039.63
|
| [[31/17]], **[[42/23]]**
|-
| 189
| 1045.16
|
| [[64/35]]
|-
| 190
| 1050.69
|
| [[11/6]]
|-
| 191
| 1056.22
|
| [[46/25]], [[35/19]]
|-
| 192
| 1061.75
|
| [[24/13]]
|-
| 193
| 1067.28
|
| [[50/27]]
|-
| 194
| 1072.81
|
| [[13/7]]
|-
| 195
| 1078.34
|
| [[54/29]], [[28/15]]
|-
| 196
| 1083.87
|
| [[58/31]]
|-
| 197
| 1089.4
|
| [[15/8]], [[62/33]]
|-
| 198
| 1094.93
|
| [[32/17]]
|-
| 199
| 1100.46
|
| [[66/35]], [[17/9]]
|-
| 200
| 1105.99
|
| [[36/19]]
|-
| 201
| 1111.52
|M7
| [[19/10]], [[243/128]]
|-
| 202
| 1117.05
|
| [[40/21]], [[21/11]]
|-
| 203
| 1122.58
|
| [[44/23]]
|-
| 204
| 1128.11
|
| [[23/12]], [[48/25]]
|-
| 205
| 1133.64
|
| [[25/13]], [[52/27]]
|-
| 206
| 1139.17
|
| [[27/14]], [[56/29]], [[29/15]]
|-
| 207
| 1144.7
|
| [[60/31]], [[31/16]], [[64/33]]
|-
| 208
| 1150.23
|
| [[33/17]], [[68/35]], [[35/18]]
|-
| 209
| 1155.76
|
| [[76/39]], [[39/20]]
|-
|217
|1200.
|P8
|[[2/1]]
|}


== Approximation to JI ==
== Approximation to JI ==
Retrieved from "https://en.xen.wiki/w/217edo"