217edo: Difference between revisions
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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 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 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 [[Hemimean clan|arch temperament]] in the 11 and 13 limits. | '''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 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 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 [[Hemimean clan|arch temperament]] in the 11 and 13 limits. | ||
== Just approximation == | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Approximation of primary intervals in 217 EDO | |+Approximation of primary intervals in 217 EDO | ||
| Line 17: | Line 18: | ||
|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
!absolute ([[Cent|¢]]) | ! absolute ([[Cent|¢]]) | ||
| 0.0 | | 0.0 | ||
| +0.349 | | +0.349 | ||
| Line 30: | Line 31: | ||
| -0.335 | | -0.335 | ||
|- | |- | ||
![[Relative error|relative]] (%) | ! [[Relative error|relative]] (%) | ||
| 0.0 | | 0.0 | ||
| +6.31 | | +6.31 | ||
Revision as of 22:23, 19 March 2021
217EDO is the equal division of the octave into 217 parts of 5.529954 cents each. It is a strong 19-limit system, the smallest uniquely consistent in the 19-limit and consistent to the 21-limit. It 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 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 arch temperament in the 11 and 13 limits.
Just approximation
| Prime number | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.0 | +0.349 | +0.783 | -1.084 | +1.677 | +0.025 | +0.114 | +1.104 | +2.140 | -1.006 | -0.335 |
| relative (%) | 0.0 | +6.31 | +14.16 | -19.60 | +30.33 | +0.46 | +2.06 | +19.97 | +38.71 | -18.19 | -6.06 | |
| Degree (reduced) | 217 (0) | 344 (127) | 504 (70) | 609 (175) | 751 (100) | 803 (152) | 887 (19) | 922 (54) | 982 (114) | 1054 (186) | 1075 (207) | |