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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|217}}
{{ED intro}}


== 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]] (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]]—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.  


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]].
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== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 344 -217 }}
| {{monzo| 344 -217 }}
| {{mapping| 217 344 }}
| {{mapping| 217 344 }}
| &minus;0.110
| −0.110
| 0.1101
| 0.1101
| 1.99
| 1.99
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| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
| {{mapping| 217 344 504 }}
| {{mapping| 217 344 504 }}
| &minus;0.186
| −0.186
| 0.1398
| 0.1398
| 2.53
| 2.53
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| 3136/3125, 4375/4374, 823543/819200
| 3136/3125, 4375/4374, 823543/819200
| {{mapping| 217 344 504 609 }}
| {{mapping| 217 344 504 609 }}
| &minus;0.043
| −0.043
| 0.2757
| 0.2757
| 4.99
| 4.99
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| 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 }}
| &minus;0.131
| −0.131
| 0.3034
| 0.3034
| 5.49
| 5.49
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| 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 }}
| &minus;0.111
| −0.111
| 0.2808
| 0.2808
| 5.08
| 5.08
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| 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 }}
| &minus;0.099
| −0.099
| 0.2616
| 0.2616
| 4.73
| 4.73
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| 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 }}
| &minus;0.119
| −0.119
| 0.2504
| 0.2504
| 4.53
| 4.53
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| 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 }}
| &minus;0.158
| −0.158
| 0.2610
| 0.2610
| 4.72
| 4.72
{{comma basis end}}
|}
* 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 the best, with the no-23 29- and 31-limit approximated even better.  
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperament
|-
|-
| 1
| 1
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| 4/3<br />(243/242)
| 4/3<br />(243/242)
| [[Birds]]
| [[Birds]]
{{rank-2 end}}
|}
{{orf}}
<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>>>
|}


== Scales ==
== Scales ==
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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
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>
</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]]:
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>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217=toneji(r217text) # r217
>>> r217=toneji(r217text) # r217
>>> r31s = [[r217[7 * i + j] for i in range(31)] + [r217[j] * 2] for j in range(7)]
>>> 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]
>>> r31s2 = [ toneji(':'.join([ str(h) for h in r31 ]),True) for r31 in r31s ]
>>> for i in range(7):
>>> for i in range(7):
   print(str(i) + "th: ", ":".join([str(h) for h in r31s2[i]]))
   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
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
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
Line 192: Line 409:
>>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val
>>> # 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)
>>> for i in range(7): # (output omitted to avoid spam)
   print(str(i) + "th:\n")
   print(str(i)+'th:\n')
   worstneji(r31s2[i], 9)
   worstneji(r31s2[i],9)
   print("\n" * 2)
   print('\n'*2)
</syntaxhighlight>
</syntaxhighlight>


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