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 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 [[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 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]].

== Just approximation ==

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[[Category:Arch]]

[[Category:Birds]]

[[Category:Cotoneum]]

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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 [[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 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]].
 == Just approximation ==
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 [[Category:Arch]]
 [[Category:Birds]]
+[[Category:Cotoneum]]