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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
Line 353: Line 173:
* [[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]]

Revision as of 08:33, 13 May 2023

← 216edo 217edo 218edo →
Prime factorization 7 × 31
Step size 5.52995 ¢ 
Fifth 127\217 (702.304 ¢)
Semitones (A1:m2) 21:16 (116.1 ¢ : 88.48 ¢)
Consistency limit 21
Distinct consistency limit 19

Template:EDO intro

Theory

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 5th and 7th harmonics with 31edo (217 = 7 × 31), as well as the 11/9 interval (supporting the 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.

The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the escapade comma, [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

Approximation of prime harmonics in 217edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.35 +0.78 -1.08 +1.68 +0.03 +0.11 +1.10 +2.14 -1.01 -0.34
Relative (%) +0.0 +6.3 +14.2 -19.6 +30.3 +0.5 +2.1 +20.0 +38.7 -18.2 -6.1
Steps
(reduced) 		217
(0) 		344
(127) 		504
(70) 		609
(175) 		751
(100) 		803
(152) 		887
(19) 		922
(54) 		982
(114) 		1054
(186) 		1075
(207)

JI approximation

Selected just intervals

The following table shows how 23-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italic. The following tables show how 23-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italics.

23-odd-limit intervals in 217edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.025 0.5
19/15, 30/19 0.028 0.5
9/5, 10/9 0.085 1.5
17/13, 26/17 0.088 1.6
17/16, 32/17 0.114 2.1
17/12, 24/17 0.235 4.3
19/10, 20/19 0.321 5.8
13/12, 24/13 0.324 5.9
3/2, 4/3 0.349 6.3
19/18, 36/19 0.406 7.3
5/3, 6/5 0.434 7.8
23/22, 44/23 0.463 8.4
15/11, 22/15 0.545 9.9
19/11, 22/19 0.573 10.4
17/9, 18/17 0.585 10.6
17/10, 20/17 0.669 12.1
13/9, 18/13 0.673 12.2
9/8, 16/9 0.698 12.6
21/16, 32/21 0.735 13.3
19/12, 24/19 0.755 13.7
13/10, 20/13 0.758 13.7
21/13, 26/21 0.760 13.7
5/4, 8/5 0.783 14.2
21/17, 34/21 0.849 15.3
11/10, 20/11 0.894 16.2
11/9, 18/11 0.979 17.7
19/17, 34/19 0.991 17.9
23/15, 30/23 1.008 18.2
17/15, 30/17 1.018 18.4
23/19, 38/23 1.036 18.7
19/13, 26/19 1.079 19.5
7/4, 8/7 1.084 19.6
19/16, 32/19 1.104 20.0
15/13, 26/15 1.107 20.0
13/7, 14/13 1.109 20.1
15/8, 16/15 1.132 20.5
17/14, 28/17 1.198 21.7
11/6, 12/11 1.328 24.0
23/20, 40/23 1.357 24.5
7/6, 12/7 1.433 25.9
23/18, 36/23 1.442 26.1
21/20, 40/21 1.518 27.4
17/11, 22/17 1.564 28.3
13/11, 22/13 1.652 29.9
11/8, 16/11 1.677 30.3
9/7, 14/9 1.782 32.2
23/12, 24/23 1.791 32.4
21/19, 38/21 1.839 33.3
7/5, 10/7 1.867 33.8
23/17, 34/23 2.027 36.6
23/13, 26/23 2.115 38.2
23/16, 32/23 2.140 38.7
19/14, 28/19 2.188 39.6
15/14, 28/15 2.216 40.1
23/14, 28/23 2.306 41.7
21/11, 22/21 2.412 43.6
23/21, 42/23 2.655 48.0
11/7, 14/11 2.761 49.9
23-odd-limit intervals in 217edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.025 0.5
19/15, 30/19 0.028 0.5
9/5, 10/9 0.085 1.5
17/13, 26/17 0.088 1.6
17/16, 32/17 0.114 2.1
17/12, 24/17 0.235 4.3
19/10, 20/19 0.321 5.8
13/12, 24/13 0.324 5.9
3/2, 4/3 0.349 6.3
19/18, 36/19 0.406 7.3
5/3, 6/5 0.434 7.8
23/22, 44/23 0.463 8.4
15/11, 22/15 0.545 9.9
19/11, 22/19 0.573 10.4
17/9, 18/17 0.585 10.6
17/10, 20/17 0.669 12.1
13/9, 18/13 0.673 12.2
9/8, 16/9 0.698 12.6
21/16, 32/21 0.735 13.3
19/12, 24/19 0.755 13.7
13/10, 20/13 0.758 13.7
21/13, 26/21 0.760 13.7
5/4, 8/5 0.783 14.2
21/17, 34/21 0.849 15.3
11/10, 20/11 0.894 16.2
11/9, 18/11 0.979 17.7
19/17, 34/19 0.991 17.9
23/15, 30/23 1.008 18.2
17/15, 30/17 1.018 18.4
23/19, 38/23 1.036 18.7
19/13, 26/19 1.079 19.5
7/4, 8/7 1.084 19.6
19/16, 32/19 1.104 20.0
15/13, 26/15 1.107 20.0
13/7, 14/13 1.109 20.1
15/8, 16/15 1.132 20.5
17/14, 28/17 1.198 21.7
11/6, 12/11 1.328 24.0
23/20, 40/23 1.357 24.5
7/6, 12/7 1.433 25.9
23/18, 36/23 1.442 26.1
21/20, 40/21 1.518 27.4
17/11, 22/17 1.564 28.3
13/11, 22/13 1.652 29.9
11/8, 16/11 1.677 30.3
9/7, 14/9 1.782 32.2
23/12, 24/23 1.791 32.4
21/19, 38/21 1.839 33.3
7/5, 10/7 1.867 33.8
23/17, 34/23 2.027 36.6
23/13, 26/23 2.115 38.2
23/16, 32/23 2.140 38.7
19/14, 28/19 2.188 39.6
15/14, 28/15 2.216 40.1
21/11, 22/21 2.412 43.6
11/7, 14/11 2.761 49.9
23/21, 42/23 2.875 52.0
23/14, 28/23 3.224 58.3

Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [344 -217⟩ [⟨217 344]] -0.110 0.1101 1.99
2.3.5 [8 14 -13⟩, [32 -7 -9⟩ [⟨217 344 504]] -0.186 0.1398 2.53
2.3.5.7 3136/3125, 4375/4374,
823543/819200 		[⟨217 344 504 609]] -0.043 0.2757 4.99
2.3.5.7.11 441/440, 3136/3125,
4000/3993, 4375/4374 		[⟨217 344 504 609 751]] -0.131 0.3034 5.49
2.3.5.7.11.13 364/363, 441/440, 676/675,
3136/3125, 4375/4374 		[⟨217 344 504 609 751 803]] -0.111 0.2808 5.08
2.3.5.7.11.13.17 364/363, 441/440, 595/594,
676/675, 1156/1155, 3136/3125 		[⟨217 344 504 609 751 803 887]] -0.099 0.2616 4.73
2.3.5.7.11.13.17.19 343/342, 364/363, 441/440,
476/475, 595/594, 676/675,
1216/1215 		[⟨217 344 504 609 751 803 887 922]] -0.119 0.2504 4.53

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 3\217 16.59 100/99 Quincy
1 5\217 27.65 64/63 Arch
1 9\217 49.77 36/35 Hemiquindromeda
1 10\217 55.30 16875/16384 Escapade
1 18\217 99.54 18/17 Quintagar / quintoneum / quinsandra
1 30\217 165.90 11/10 Satin
1 33\217 182.49 10/9 Mitonic / mineral
1 57\217 315.21 6/5 Parakleismic / paralytic
1 86\217 475.58 320/243 Vulture
1 90\217 497.70 4/3 Cotoneum
1 101\217 558.53 112/81 Condor
7 94\217
(1\217) 		519.82
(5.53) 		27/20
(325/324) 		Brahmagupta
31 90\217
(1\217) 		497.70
(5.53) 		4/3
(243/242) 		Birds

Scales

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