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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 7 × 31
{{ED intro}}
| Step size = 5.52995¢
| Fifth = 127\217 (702.30¢)
| Semitones = 21:16 (116.13¢ : 88.48¢)
| Consistency = 21
}}
The '''217 equal divisions of the octave''' ('''217edo'''), or the '''217(-tone) equal temperament''' ('''217tet''', '''217et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 217 parts of about 5.53 [[cent]]s each.


== Theory ==
== Theory ==
217edo is a strong [[19-limit]] system, the smallest uniquely [[consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]]. It shares the same 5th and 7th [[Harmonic series|harmonics]] with [[31edo]] (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, 11, 13, 17 and 19 – in fact, this edo has a very accurate 13th harmonic, as well as the [[19/15]] interval.  
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.  


It 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 [[gentle 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|columns=11}}
{{Harmonics in equal|217}}


== JI approximation ==
== Approximation to JI ==
=== Selected just intervals ===
=== Selected just intervals ===
The following table shows how [[23-odd-limit|23-odd-limit intervals]] are represented in 217EDO. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{{Q-odd-limit intervals|217|23}}
 
{| class="wikitable center-all"
|+Direct mapping (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[16/13]], [[13/8]]'''
| '''0.025'''
|-
| [[19/15]], [[30/19]]
| 0.028
|-
| [[10/9]], [[9/5]]
| 0.085
|-
| [[17/13]], [[26/17]]
| 0.088
|-
| '''[[17/16]], [[32/17]]'''
| '''0.114'''
|-
| [[24/17]], [[17/12]]
| 0.235
|-
| [[20/19]], [[19/10]]
| 0.321
|-
| [[13/12]], [[24/13]]
| 0.324
|-
| '''[[4/3]], [[3/2]]'''
| '''0.349'''
|-
| [[19/18]], [[36/19]]
| 0.406
|-
| [[6/5]], [[5/3]]
| 0.434
|-
| [[23/22]], [[44/23]]
| 0.463
|-
| [[15/11]], [[22/15]]
| 0.545
|-
| [[22/19]], [[19/11]]
| 0.573
|-
| [[18/17]], [[17/9]]
| 0.585
|-
| [[20/17]], [[17/10]]
| 0.669
|-
| [[18/13]], [[13/9]]
| 0.673
|-
| [[9/8]], [[16/9]]
| 0.698
|-
| [[21/16]], [[32/21]]
| 0.735
|-
| [[24/19]], [[19/12]]
| 0.755
|-
| [[26/21]], [[21/13]]
| 0.760
|-
| [[13/10]], [[20/13]]
| 0.758
|-
| '''[[5/4]], [[8/5]]'''
| '''0.783'''
|-
| [[21/17]], [[34/21]]
| 0.849
|-
| [[11/10]], [[20/11]]
| 0.894
|-
| [[11/9]], [[18/11]]
| 0.979
|-
| [[19/17]], [[34/19]]
| 0.991
|-
| [[30/23]], [[23/15]]
| 1.008
|-
| [[17/15]], [[30/17]]
| 1.018
|-
| [[23/19]], [[38/23]]
| 1.036
|-
| [[26/19]], [[19/13]]
| 1.079
|-
| '''[[8/7]], [[7/4]]'''
| '''1.084'''
|-
| '''[[19/16]], [[32/19]]'''
| '''1.104'''
|-
| [[15/13]], [[26/15]]
| 1.107
|-
| [[14/13]], [[13/7]]
| 1.109
|-
| [[16/15]], [[15/8]]
| 1.132
|-
| [[17/14]], [[28/17]]
| 1.198
|-
| [[12/11]], [[11/6]]
| 1.328
|-
| [[23/20]], [[40/23]]
| 1.357
|-
| [[7/6]], [[12/7]]
| 1.433
|-
| [[23/18]], [[36/23]]
| 1.442
|-
| [[21/20]], [[40/21]]
| 1.518
|-
| [[22/17]], [[17/11]]
| 1.564
|-
| [[13/11]], [[22/13]]
| 1.652
|-
| '''[[11/8]], [[16/11]]'''
| '''1.677'''
|-
| [[9/7]], [[14/9]]
| 1.782
|-
| [[24/23]], [[23/12]]
| 1.791
|-
| [[21/19]], [[38/21]]
| 1.839
|-
| [[7/5]], [[10/7]]
| 1.867
|-
| [[23/17]], [[34/23]]
| 2.027
|-
| [[26/23]], [[23/13]]
| 2.115
|-
| '''[[32/23]], [[23/16]]'''
| '''2.140'''
|-
| [[19/14]], [[28/19]]
| 2.188
|-
| [[15/14]], [[28/15]]
| 2.216
|-
| ''[[28/23]], [[23/14]]''
| ''2.306''
|-
| [[22/21]], [[21/11]]
| 2.412
|-
| ''[[23/21]], [[42/23]]''
| ''2.655''
|-
| [[14/11]], [[11/7]]
| 2.761
|}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 214: Line 28:
| 2.3
| 2.3
| {{monzo| 344 -217 }}
| {{monzo| 344 -217 }}
| [{{val| 217 344 }}]
| {{mapping| 217 344 }}
| -0.110
| −0.110
| 0.1101
| 0.1101
| 1.99
| 1.99
Line 221: Line 35:
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| [{{val| 217 344 504 }}]
| {{mapping| 217 344 504 }}
| -0.186
| −0.186
| 0.1398
| 0.1398
| 2.53
| 2.53
|-
|-
| 2.3.5.7
| 2.3.5.7
| 3136/3125, 4375/4374, <br>823543/819200
| 3136/3125, 4375/4374, 823543/819200
| [{{val| 217 344 504 609 }}]
| {{mapping| 217 344 504 609 }}
| -0.043
| −0.043
| 0.2757
| 0.2757
| 4.99
| 4.99
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 3136/3125, <br>4000/3993, 4375/4374
| 441/440, 3136/3125, 4000/3993, 4375/4374
| [{{val| 217 344 504 609 751 }}]
| {{mapping| 217 344 504 609 751 }}
| -0.131
| −0.131
| 0.3034
| 0.3034
| 5.49
| 5.49
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 364/363, 441/440, 676/675, <br>3136/3125, 4375/4374
| 364/363, 441/440, 676/675, 3136/3125, 4375/4374
| [{{val| 217 344 504 609 751 803 }}]
| {{mapping| 217 344 504 609 751 803 }}
| -0.111
| −0.111
| 0.2808
| 0.2808
| 5.08
| 5.08
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 364/363, 441/440, 595/594, <br>676/675, 1156/1155, 3136/3125
| 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125
| [{{val| 217 344 504 609 751 803 887 }}]
| {{mapping| 217 344 504 609 751 803 887 }}
| -0.099
| −0.099
| 0.2616
| 0.2616
| 4.73
| 4.73
|-
|-
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 343/342, 364/363, 441/440, <br>476/475, 595/594, 676/675, <br>1216/1215
| 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215
| [{{val| 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
| 4.53
| 4.53
|-
| 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
| {{mapping| 217 344 504 609 751 803 887 922 982 }}
| −0.158
| 0.2610
| 4.72
|}
|}
* 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.
* 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]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br />ratio*
! Temperament
|-
|-
| 1
| 1
Line 329: Line 154:
| 497.70
| 497.70
| 4/3
| 4/3
| [[Gary]] / [[cotoneum]]
| [[Cotoneum]]
|-
|-
| 1
| 1
Line 338: Line 163:
|-
|-
| 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
== Notation ==
=== Sagittal ===
217edo can be written in Sagittal using almost the entire Athenian extension (except for {{sagittal| |\ }} {{sagittal| !/ }} {{sagittal| /|| }} {{sagittal| \!! }} since it tempers out [[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. 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"/>
{| class="wikitable center-all"
|+Sagittal notation
! colspan="2" | Steps
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
|-
! rowspan="2" | Symbol
! Evo
| 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|#}}{{sagittal| !) }}
| {{Sagittal|#}}{{sagittal| \! }}
| {{Sagittal|#}}{{sagittal| ~!( }}
| {{Sagittal|#}}{{sagittal| )!( }}
| {{Sagittal|#}}{{sagittal| !( }}
| {{Sagittal|#}}
|-
! Revo
| {{Sagittal| (|) }}
| {{Sagittal| (|\ }}
| {{Sagittal| )||( }}
| {{Sagittal| ~||( }}
| {{Sagittal| )||~ }}
| {{Sagittal| ||) }}
| {{Sagittal| ||\ }}
| {{Sagittal| (||( }}
| {{Sagittal| //|| }}
| {{Sagittal| /||) }}
| {{Sagittal| /||\ }}
|}
=== Ups-and-downs notation ===
The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.
{| class="wikitable center-all"
|+Ups-and-downs notation
! Steps
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
|-
! rowspan="2" | Symbol
| ^
| ^^
| ^^^
| v>
| >
| ^>
| ^^>
| ^^^>
| v>>
| >>
|-
| <<<<#
| ^<<<<#
| vvv<<<#
| vv<<<#
| v<<<#
| <<<#
| ^<<<#
| vvv<<#
| vv<<#
| v<<#
|-
! Steps
| 11
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
|-
! rowspan="2" | Symbol
| ^>>
| ^^>>
| ^^^>>
| v>>>
| >>>
| ^>>>
| ^^>>>
| ^^^>>>
| v>>>>
| >>>>
| rowspan="2" | #
|-
| <<#
| ^<<#
| vvv<#
| vv<#
| v<#
| <#
| ^<#
| vvv#
| vv#
| v#
|}
=== 31edo-based notation ===
Since {{nowrap| 217 {{=}} 31 × 7 }}, one ''could'' base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.
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.
{| class="wikitable center-all"
|+Alternative 31edo-based notation
|-
! Steps
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| 11
| 12
| 13
| 14
|-
! rowspan="2" | Symbol
| rowspan="2" | ^
| rowspan="2" | ^^
| rowspan="2" | ^^^
| vvvt
| vvt
| vt
| t
| ^t
| ^^t
| ^^^t
| v#
| vv#
| vvv#
| #
|-
| v>
| >
| ^>
| ^^>
| ^^^>
| v>>
| >>
| ^>>
| ^^>>
| ^^^>>
| v>>>
|}
|}


Line 357: Line 383:
* [[Cotoneum29]]
* [[Cotoneum29]]
* [[Cotoneum41]]
* [[Cotoneum41]]
* [[Cotoneum53]]


[[Category:217edo| ]]<!-- main article -->
== Detemperaments ==
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== 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.
<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 ====
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">
>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217=toneji(r217text) # r217
>>> r31s = [ [r217[7*i+j] for i in range(31)]+[r217[j]*2] for j in range(7) ]
>>> r31s2 = [ toneji(':'.join([ str(h) for h in r31 ]),True) for r31 in r31s ]
>>> for i in range(7):
  print(str(i)+'th: ',':'.join([ str(h) for h in r31s2[i] ]))
0th:  274:280:286:293:299:306:313:320:327:334:342:350:358:366:374:383:392:400:409:418:428:438:448:458:468:479:490:500:512:524:535:548
1th:  351:359:367:375:384:393:401:410:420:429:439:449:459:469:480:491:502:514:526:536:549:562:574:588:600:614:628:642:656:672:687:702
2th:  301:308:315:322:329:337:344:352:360:368:376:385:394:402:412:421:430:440:450:460:470:482:492:504:515:527:538:550:563:576:589:602
3th:  258:264:270:276:282:289:295:302:309:316:323:330:338:346:353:361:369:378:386:395:404:413:422:432:442:452:462:472:483:494:505:516
4th:  227:232:237:242:248:253:259:265:271:277:283:290:296:303:310:317:324:331:339:347:354:362:370:379:388:396:406:414:424:434:443:454
5th:  416:425:435:444:455:465:476:486:497:508:520:532:544:556:568:582:594:608:622:636:650:664:680:696:712:728:744:760:778:796:813:832
6th:  213:218:223:228:233:239:244:249:255:261:267:273:279:285:292:298:305:312:319:326:333:341:349:357:365:373:382:390:399:408:417:426
>>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val
>>> for i in range(7): # (output omitted to avoid spam)
  print(str(i)+'th:\n')
  worstneji(r31s2[i],9)
  print('\n'*2)
</syntaxhighlight>
 
== References ==
<ref name=":0" /> [[Ragismic microtemperaments#Brahmagupta]]
[[Category:Arch]]
[[Category:Arch]]
[[Category:Birds]]
[[Category:Birds]]
[[Category:Cotoneum]]
[[Category:Cotoneum]]
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