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


== 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 distinctly [[consistent]] in the [[19-odd-limit]] and consistent to the [[21-odd-limit]]. It shares the same [[5/1|5th]] and [[7/1|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 [[3/1|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 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 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]].


=== Prime harmonics ===
=== Prime harmonics ===
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== JI approximation ==
== JI approximation ==
=== 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''.  
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''.  
 
{{15-odd-limit|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|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-
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| 497.70
| 497.70
| 4/3
| 4/3
| [[Gary]] / [[cotoneum]]
| [[Cotoneum]]
|-
|-
| 1
| 1
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* [[Cotoneum53]]
* [[Cotoneum53]]


[[Category:217edo| ]]<!-- main article -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Arch]]
[[Category:Arch]]
[[Category:Birds]]
[[Category:Birds]]
[[Category:Cotoneum]]
[[Category:Cotoneum]]
Retrieved from "https://en.xen.wiki/w/217edo"