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'''217EDO''' is the [[EDO|equal division of the octave]] into 217 parts of 5.529954 [[cent]]s each. It is a strong 19-limit system, the smallest uniquely [[consistent]] in the 19-limit and consistent to the 21-limit. It shares the same 5th and 7th [[Overtone series|harmonics]] with [[31edo|31EDO]] (217 = 7 × 31), as well as the [[11/9]] interval (supporting the [[31-comma temperaments|birds temperament]]). However, compared to 31EDO, its patent vals 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. 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 and 5632/5625 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 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 provides the [[optimal patent val]] for the 11- and 13-limit [[Hemimean clan #Arch|arch]] and the 11- and 13-limit [[Hemimage temperaments #Cotoneum|cotoneum]].
'''217EDO''' is the [[EDO|equal division of the octave]] into 217 parts of 5.529954 [[cent]]s each.  


== Just approximation ==
== Theory ==
{| class="wikitable"
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.
|+Approximation of primary intervals in 217 EDO
! colspan="2" |Prime number
!2
!3
!5
!7
!11
!13
!17
!19
!23
!29
!31
|-
! rowspan="2" |Error
! absolute ([[Cent|¢]])
| 0.0
| +0.349
| +0.783
| -1.084
| +1.677
| +0.025
| +0.114
| +1.104
| +2.140
| -1.006
| -0.335
|-
! [[Relative error|relative]] (%)
| 0.0
| +6.31
| +14.16
| -19.60
| +30.33
| +0.46
| +2.06
| +19.97
| +38.71
| -18.19
| -6.06
|-
! colspan="2" |Degree ([[Octave reduction|reduced]])
|217 (0)
|344 (127)
|504 (70)
|609 (175)
|751 (100)
|803 (152)
|887 (19)
|922 (54)
|982 (114)
|1054 (186)
|1075 (207)
|}


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]] and 5632/5625 in the 11-limit; [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] 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 provides the [[optimal patent val]] for the 11- and 13-limit [[Hemimean clan #Arch|arch]] and the 11- and 13-limit [[Hemimage temperaments #Cotoneum|cotoneum]].
=== Prime harmonics ===
{{Primes in edo|217}}
== 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''.  


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