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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 a very accurate 13th harmonic, as well as the [[19/15]] interval. It can also be used in the 23-limit. The only inconsistently mapped intervals in the [[23-odd-limit]] are [[23/14]], [[23/21]], and their [[octave complement]]s.  
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|2.3.5.13-subgroup]] temperament. 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]]. [[224edo]], only a bit bigger, offers a much more accurate [[13-limit]], at the cost of worse higher limits. If one desires even 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]], 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]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|217}}
{{Harmonics in equal|217}}
=== 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.
== Intervals ==
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 useful intervals outside that limit. Intervals in italics are inconsistently mapped.
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 217edo intervals
! #
! Cents
! Marks
! Approximate intervals
|-
| 0
| 0
| P1
|
|-
| 1
| 5.53
|
|[[352/351]], [[5120/5103]]'', [[32805/32768]]''
|-
| 2
| 11.06
|
|[[144/143]], [[169/168]], ''[[225/224]]''
|-
| 3
| 16.59
|
|[[4131/4096]]
|-
| 4
| 22.12
|
| [[81/80]]
|-
| 5
| 27.65
|
| [[64/63]], ''[[531441/524288]]''
|-
| 6
| 33.18
|
|[[49/48]]
|-
| 7
| 38.71
|
|[[128/125]]
|-
| 8
| 44.24
|
| [[40/39]], [[39/38]]
|-
| 9
| 49.77
|
| [[36/35]], [[35/34]], [[34/33]], [[1053/1024]]
|-
| 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 ==
=== Selected just intervals ===
=== Selected just intervals ===
{{Q-odd-limit intervals|217|23}}
{{Q-odd-limit intervals|217|31}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 20: Line 979:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 27: Line 986:
|-
|-
| 2.3
| 2.3
| {{monzo| 344 -217 }}
| {{Monzo| 344 -217 }}
| {{mapping| 217 344 }}
| {{Mapping| 217 344 }}
| −0.110
| −0.110
| 0.1101
| 0.1101
Line 34: Line 993:
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| {{Monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| {{mapping| 217 344 504 }}
| {{Mapping| 217 344 504 }}
| −0.186
| −0.186
| 0.1398
| 0.1398
Line 42: Line 1,001:
| 2.3.5.7
| 2.3.5.7
| 3136/3125, 4375/4374, 823543/819200
| 3136/3125, 4375/4374, 823543/819200
| {{mapping| 217 344 504 609 }}
| {{Mapping| 217 344 504 609 }}
| −0.043
| −0.043
| 0.2757
| 0.2757
Line 49: Line 1,008:
| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 3136/3125, 4000/3993, 4375/4374
| 441/440, 3136/3125, 4000/3993, 4375/4374
| {{mapping| 217 344 504 609 751 }}
| {{Mapping| 217 344 504 609 751 }}
| −0.131
| −0.131
| 0.3034
| 0.3034
Line 56: Line 1,015:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 364/363, 441/440, 676/675, 3136/3125, 4375/4374
| 364/363, 441/440, 676/675, 3136/3125, 4375/4374
| {{mapping| 217 344 504 609 751 803 }}
| {{Mapping| 217 344 504 609 751 803 }}
| −0.111
| −0.111
| 0.2808
| 0.2808
Line 63: Line 1,022:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125
| 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125
| {{mapping| 217 344 504 609 751 803 887 }}
| {{Mapping| 217 344 504 609 751 803 887 }}
| −0.099
| −0.099
| 0.2616
| 0.2616
Line 70: Line 1,029:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215
| 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215
| {{mapping| 217 344 504 609 751 803 887 922 }}
| {{Mapping| 217 344 504 609 751 803 887 922 }}
| −0.119
| −0.119
| 0.2504
| 0.2504
Line 77: Line 1,036:
| 2.3.5.7.11.13.17.19.23
| 2.3.5.7.11.13.17.19.23
| 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675
| 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675
| {{mapping| 217 344 504 609 751 803 887 922 982 }}
| {{Mapping| 217 344 504 609 751 803 887 922 982 }}
| −0.158
| −0.158
| 0.2610
| 0.2610
Line 83: Line 1,042:
|}
|}
* 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo|72]] in the 19-limit and [[193edo|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo|243e]] for absolute error and [[270edo|270]] for relative error.  
* 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo|72]] in the 19-limit and [[193edo|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo|243e]] for absolute error and [[270edo|270]] for relative error.  
* 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better.  
* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.  
* It is also prominent in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo|183]] and superseded by [[224edo|224]].  
* It is also notable in the 17-limit, with a lower absolute error than any previous equal temperaments, beating [[183edo|183]] and superseded by [[224edo|224]].  


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 90: Line 1,049:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 117: Line 1,076:
| 10\217
| 10\217
| 55.30
| 55.30
| 16875/16384
| 33/32
| [[Escapade]]
| [[Escapade]]
|-
|-
Line 147: Line 1,106:
| 86\217
| 86\217
| 475.58
| 475.58
| 320/243
| 25/19
| [[Vulture]]
| [[Vulture]]
|-
|-
Line 163: Line 1,122:
|-
|-
| 7
| 7
| 94\217<br />(1\217)
| 94\217<br>(1\217)
| 519.82<br />(5.53)
| 519.82<br>(5.53)
| 27/20<br />(325/324)
| 27/20<br>(325/324)
| [[Brahmagupta]]
| [[Brahmagupta]]
|-
|-
| 31
| 31
| 90\217<br />(1\217)
| 90\217<br>(1\217)
| 497.70<br />(5.53)
| 497.70<br>(5.53)
| 4/3<br />(243/242)
| 4/3<br>(243/242)
| [[Birds]]
| [[Birds]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Notation ==
== Notation ==
=== Sagittal ===
217edo can be written in Sagittal using ''almost'' the entire Athenian extension by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">[[George Secor|George D. Secor]] and [[David Keenan|David C. Keenan]], [https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''], p. 11.</ref>
It shares the same exact symbol system as the Athenian notation for just intonation or ''Medium-precision JI notation.''<ref name=":1"/>


=== Sagittal ===
{| class="wikitable center-all"
217edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal||\}} {{sagittal|!/}} {{sagittal|/||}} {{sagittal|\!!}} since it tempers [[1240029/1239040]]), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian<ref name=":0">[[Ragismic microtemperaments#Brahmagupta]]</ref>. It is identical to [[224edo]]'s Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.<ref name=":1">https://sagittal.org/sagittal.pdf p. 11</ref>
It shares the same exact symbol system as the Athenian notation for Just Intonation or ''Medium-precision JI notation.''<ref name=":1" />
{| class="wikitable"
|+Sagittal notation
|+Sagittal notation
!217edosteps
! colspan="2" | Steps
!-21
! 1
!-20
! 2
!-19
! 3
!-18
! 4
!-17
! 5
!-16
! 6
!-15
! 7
!-14
! 8
!-13
! 9
!-12
! 10
!-11
! 11
!-10
! 12
!-9
! 13
!-8
! 14
!-7
! 15
!-6
! 16
!-5
! 17
!-4
! 18
!-3
! 19
!-2
! 20
!-1
! 21
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
!21
|-
|-
|Revo
! rowspan="2" | Symbol
|{{sagittal|\!!/}}
! Evo
|{{sagittal|\!!)}}
| rowspan="2" | <big>{{Sagittal| |( }}</big>
|{{sagittal|\\!!}}
| rowspan="2" | <big>{{Sagittal| )|( }}</big>
|{{sagittal|(!!(}}
| rowspan="2" | <big>{{Sagittal| ~|( }}</big>
|{{sagittal|!!/}}
| rowspan="2" | <big>{{Sagittal| /| }}</big>
|{{sagittal|!!)}}
| rowspan="2" | <big>{{Sagittal| |) }}</big>
|{{sagittal|)!!~}}
| rowspan="2" | <big>{{Sagittal| (| }}</big>
|{{sagittal|~!!(}}
| rowspan="2" | <big>{{Sagittal| (|( }}</big>
|{{sagittal|)!!(}}
| rowspan="2" | <big>{{Sagittal| //| }}</big>
|{{sagittal|(!/}}
| rowspan="2" | <big>{{Sagittal| /|) }}</big>
|{{sagittal|(!)}}
| rowspan="2" | <big>{{Sagittal| /|\ }}</big>
| rowspan="2" |{{sagittal|\!/}}
| <small>{{Sagittal|#}}{{sagittal| \!/ }}</small>
| rowspan="2" |{{sagittal|\!)}}
| <small>{{Sagittal|#}}{{sagittal| \!) }}</small>
| rowspan="2" |{{sagittal|\\!}}
| <small>{{Sagittal|#}}{{sagittal| \\! }}</small>
| rowspan="2" |{{sagittal|(!(}}
| <small>{{Sagittal|#}}{{sagittal| (!( }}</small>
| rowspan="2" |{{sagittal|(!}}
| <small>{{Sagittal|#}}{{sagittal| (! }}</small>
| rowspan="2" |{{sagittal|!)}}
| <small>{{Sagittal|#}}{{sagittal| !) }}</small>
| rowspan="2" |{{sagittal|\!}}
| <small>{{Sagittal|#}}{{sagittal| \! }}</small>
| rowspan="2" |{{sagittal|~!(}}
| <small>{{Sagittal|#}}{{sagittal| ~!( }}</small>
| rowspan="2" |{{sagittal|)!(}}
| <small>{{Sagittal|#}}{{sagittal| )!( }}</small>
| rowspan="2" |{{sagittal|!(}}
| <small>{{Sagittal|#}}{{sagittal| !( }}</small>
| rowspan="2" |{{sagittal||//|}}
| <small>{{Sagittal|#}}</small>
| rowspan="2" |{{sagittal||(}}
| rowspan="2" |{{sagittal|)|(}}
| rowspan="2" |{{sagittal|~|(}}
| rowspan="2" |{{sagittal|/|}}
| rowspan="2" |{{sagittal||)}}
| rowspan="2" |{{sagittal|(|}}
| rowspan="2" |{{sagittal|(|(}}
| rowspan="2" |{{sagittal|//|}}
| rowspan="2" |{{sagittal|/|)}}
| rowspan="2" |{{sagittal|/|\}}
|{{sagittal|(|)}}
|{{sagittal|(|\}}
|{{sagittal|)||(}}
|{{sagittal|~||(}}
|{{sagittal|)||~}}
|{{sagittal|||)}}
|{{sagittal|||\}}
|{{sagittal|(||(}}
|{{sagittal|//||}}
|{{sagittal|/||)}}
|{{sagittal|/||\}}
|-
|-
|Evo
! Revo
|{{sagittal|b}}
| <big>{{Sagittal| (|) }}</big>
|{{sagittal|b}}{{sagittal||(}}
| <big>{{Sagittal| (|\ }}</big>
|{{sagittal|b}}{{sagittal|)|(}}
| <big>{{Sagittal| )||( }}</big>
|{{sagittal|b}}{{sagittal|~|(}}
| <big>{{Sagittal| ~||( }}</big>
|{{sagittal|b}}{{sagittal|/|}}
| <big>{{Sagittal| )||~ }}</big>
|{{sagittal|b}}{{sagittal||)}}
| <big>{{Sagittal| ||) }}</big>
|{{sagittal|b}}{{sagittal|(|}}
| <big>{{Sagittal| ||\ }}</big>
|{{sagittal|b}}{{sagittal|(|(}}
| <big>{{Sagittal| (||( }}</big>
|{{sagittal|b}}{{sagittal|//|}}
| <big>{{Sagittal| //|| }}</big>
|{{sagittal|b}}{{sagittal|/|)}}
| <big>{{Sagittal| /||) }}</big>
|{{sagittal|b}}{{sagittal|/|\}}
| <big>{{Sagittal| /||\ }}</big>
|{{sagittal|#}}{{sagittal|\!/}}
|{{sagittal|#}}{{sagittal|\!)}}
|{{sagittal|#}}{{sagittal|\\!}}
|{{sagittal|#}}{{sagittal|(!(}}
|{{sagittal|#}}{{sagittal|(!}}
|{{sagittal|#}}{{sagittal|!)}}
|{{sagittal|#}}{{sagittal|\!}}
|{{sagittal|#}}{{sagittal|~!(}}
|{{sagittal|#}}{{sagittal|)!(}}
|{{sagittal|#}}{{sagittal|!(}}
|{{sagittal|#}}
|}
|}
Because it uses the entire Athenian system (except for {{sagittal| |\ }} {{sagittal| !/ }} {{sagittal| /|| }} {{sagittal| \!! }} since it tempers out [[1240029/1239040]]), it allows no accidental enharmonic respellings.


=== Ups-and-downs notation ===
=== Ups-and-downs notation ===
The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.
The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.{{Ups and downs sharpness|217|false}}
{| class="wikitable" style="text-align:center;"
 
|+Ups-and-downs notation
=== 31edo-based meantone notation ===
! rowspan="3" |217edosteps
Since {{nowrap| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.
!-21
 
!-20
This could be useful when [[31edo]] is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches. It also can be written with SZ half-sharps and up/down. The main drawback of this notation is that 3/2 is no longer P5 but '''^P5''' , so that ~4:5:6 is from C written as C-E-^G, compared to C-^<E-G to the patent val chain-of-fifths ups-and-downs notation.
!-19
 
!-18
{| class="wikitable center-all"
!-17
|+Alternative 31edo-based notation
!-16
|-
!-15
! Steps
!-14
| 1
!-13
| 2
!-12
| 3
!-11
| 4
!-10
| 5
!-9
| 6
!-8
| 7
!-7
| 8
!-6
| 9
!-5
| 10
!-4
| 11
!-3
| 12
!-2
| 13
!-1
| 14
!0
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
!21
|-
|-
| rowspan="2" |b
! rowspan="2" | Symbol
|<<<<
| rowspan="2" | ^
|^<<<<
| rowspan="2" | ^^
|vvv<<<
| rowspan="2" | ^^^
|vv<<<
| vvvt
|v<<<
| vvt
|<<<
| vt
|^<<<
| t
|vvv<<
| ^t
|vv<<
| ^^t
|v<<
| ^^^t
|<<
| v#
|^<<
| vv#
|vvv<
| vvv#
|vv<
| #
|v<
|<
|^<
|vvv
|vv
|v
| rowspan="2" |h
|^
|^^
|^^^
|v>
|>
|^>
|^^>
|^^^>
|v>>
|>>
|^>>
|^^>>
|^^^>>
|v>>>
|>>>
|^>>>
|^^>>>
|^^^>>>
|v>>>>
|>>>>
| rowspan="2" |#
|-
|-
|^b
| v>
|^^b
| >
|^^^b
| ^>
|v>b
| ^^>
|>b
| ^^^>
|^>b
| v>>
|^^>b
| >>
|^^^>b
| ^>>
|v>>b
| ^^>>
|>>b
| ^^^>>
|^>>b
| v>>>
|^^>>b
|^^^>>b
|v>>>b
|>>>b
|^>>>b
|^^>>>b
|^^^>>>b
|v>>>>b
|>>>>b
|<<<<#
|^<<<<#
|vvv<<<#
|vv<<<#
|v<<<#
|<<<#
|^<<<#
|vvv<<#
|vv<<#
|v<<#
|<<#
|^<<#
|vvv<#
|vv<#
|v<#
|<#
|^<#
|vvv#
|vv#
|v#
|}
|}


=== 31edo-based notation ===
=== 7edo-based whitewood notation ===
Since 217edo is 31*7, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.
Since {{nowrap| 217 {{=}}  7 × 31}}, one ''could'' use the inherited [[Whitewood family|whitewood]] fifth 124\217, ditch sharps and flats, and instead use ups and downs to represent pitch deviations from the 7edo nominals. Since 31=15*2+1, all pitches can be notated with ups, downs, quips and quids. ~4:5:6 becomes C-^^^>E-^^^G.
 
{| class="wikitable center-all"
This could be useful when [[31edo]] as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the [[31edo]] pitches, similar to how one might use [[159edo]] to provide subtle corrections of [[53edo]] pitches.
|-
{| class="wikitable"
! Steps
|+Alternative 31edo-based notation
|0
!-14
| 1
!-13
| 2
!-12
| 3
!-11
| 4
!-10
| 5
!-9
| 6
!-8
| 7
!-7
| 8
!-6
| 9
!-5
| 10
!-4
| 11
!-3
| 12
!-2
| 13
!-1
| 14
!0
|15
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
!11
!12
!13
!14
|-
|-
|b
! Sharp
|^b
Symbol
|^^b
|^^^b
|vvvd
|vvd
|vd
|d
|d^
|d^^
|d^^^
| rowspan="2" |vvv
| rowspan="2" |vv
| rowspan="2" |v
| rowspan="2" |h
| rowspan="2" |h
| rowspan="2" |^
| ^
| rowspan="2" |^^
|^^
| rowspan="2" |^^^
|^^^
|vvv‡
vv>
|vv‡
| v>
|v‡
| >
|‡
| ^>
|^‡
| ^^>
|^^‡
| ^^^>
|^^^‡
vv>>
|v#
| v>>
|vv#
| >>
|vvv#
| ^>>
|#
| ^^>>
| vv>>>
^^^>>
| v>>>
|>>>
|-
|-
|^<<<
!Flat
|vvv<<
symbol
|vv<<
|v
|v<<
|vv
|<<
|vvv
|^<<
^^<
|vvv<
| v<
|vv<
| <
|v<
| v<
|<
| vv<
|^<
| vvv<
|v>
vv<<
|>
| ^<<
|^>
| <<
|^^>
| v<<
|^^^>
| vv<<
|v>>
| vvv<<
|>>
^^<<<
|^>>
| ^<<<
|^^>>
|<<<
|^^^>>
|v>>>
|}
|}


Line 533: Line 1,337:
== Detemperaments ==
== Detemperaments ==
=== Ringer 217 ===
=== Ringer 217 ===
217edo is the basis for an exceptional Ringer scale that maps an unusually very large amount of the harmonic series (without having to omit any other harmonics) compared to other edos in this size range. Specifically, it maps the complete mode 167 of the harmonic series, corresponding to the ''entire'' 333-odd-limit. As 217 = 31 * 7, this can be used to derive 7 possible 31nejis.
217edo is the basis for an exceptional Ringer scale that maps an unusually very large amount of the harmonic series (without having to omit any other harmonics) compared to other edos in this size range. Specifically, it maps the complete mode 167 of the harmonic series, corresponding to the ''entire'' 333-odd-limit. As {{nowrap| 217 {{=}} 31 × 7 }}, this can be used to derive 7 possible 31nejis.
<pre>
<pre>167:168:337/2:169:339/2:170:341/2:171:687/4:172:173:347/2:174:349/2:175:351/2:176:353/2:177:178:357/2:179:359/2:180:361/2:181:182:365/2:183:367/2:184:369/2:185:186:373/2:187:375/2:188:189:379/2:190:191:383/2:192:385/2:193:194:389/2:195:196:393/2:197:395/2:198:199:399/2:200:401/2:201:202:203:813/4:204:409/2:205:206:413/2:207:208:417/2:209:210:421/2:211:212:425/2:213:214:429/2:215:216:217:435/2:218:219:439/2:220:221:443/2:222:223:224:449/2:225:226:227:455/2:228:229:459/2:230:231:232:465/2:233:234:469/2:235:236:237:238:239:479/2:240:241:483/2:242:243:244:245:491/2:246:247:248:497/2:249:250:251:252:505/2:253:254:255:256:257:515/2:258:259:260:261:262:263:527/2:264:265:266:267:535/2:268:269:270:271:272:273:274:549/2:275:276:277:278:279:280:281:563/2:282:283:284:285:286:287:288:289:290:291:292:293:294:589/2:295:296:297:298:299:300:301:302:303:304:305:306:307:308:309:310:311:312:313:314:315:316:317:318:319:320:321:322:323:324:325:326:327:328:329:330:331:332:333:334</pre>
167:168:337/2:169:339/2:170:341/2:171:687/4:172:173:347/2:174:349/2:175:351/2:176:353/2:177:178:357/2:179:359/2:180:361/2:181:182:365/2:183:367/2:184:369/2:185:186:373/2:187:375/2:188:189:379/2:190:191:383/2:192:385/2:193:194:389/2:195:196:393/2:197:395/2:198:199:399/2:200:401/2:201:202:203:813/4:204:409/2:205:206:413/2:207:208:417/2:209:210:421/2:211:212:425/2:213:214:429/2:215:216:217:435/2:218:219:439/2:220:221:443/2:222:223:224:449/2:225:226:227:455/2:228:229:459/2:230:231:232:465/2:233:234:469/2:235:236:237:238:239:479/2:240:241:483/2:242:243:244:245:491/2:246:247:248:497/2:249:250:251:252:505/2:253:254:255:256:257:515/2:258:259:260:261:262:263:527/2:264:265:266:267:535/2:268:269:270:271:272:273:274:549/2:275:276:277:278:279:280:281:563/2:282:283:284:285:286:287:288:289:290:291:292:293:294:589/2:295:296:297:298:299:300:301:302:303:304:305:306:307:308:309:310:311:312:313:314:315:316:317:318:319:320:321:322:323:324:325:326:327:328:329:330:331:332:333:334
</pre>


==== Deriving 31nejis ====
==== Deriving 31nejis ====
This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of [[User:Godtone]]'s [[User:Godtone#My_Python_3_code|copyleft Python 3 code]]:
This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of [[User:Godtone]]'s [[User:Godtone #My Python 3 code|copyleft Python 3 code]]:
<syntaxhighlight lang="python">
<syntaxhighlight lang="python">
>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217text = '[paste the above Ringer 217 data here]'
Line 562: Line 1,364:


== References ==
== References ==
<ref name=":0" /> [[Ragismic microtemperaments#Brahmagupta]]
<references/>
 
[[Category:Arch]]
[[Category:Arch]]
[[Category:Birds]]
[[Category:Birds]]
[[Category:Cotoneum]]
[[Category:Cotoneum]]
Retrieved from "https://en.xen.wiki/w/217edo"