217edo: Difference between revisions

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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 5.~~529954~~ [[cent]]s each.

{{Infobox ET

| Prime factorization = 7 × 31

| Step size = 5.52995¢

| Fifth = 127\217 (702.30¢)

| Semitones = 21:16 (116.13¢ : 88.48¢)

| Consistency = 21

}}

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.

== 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~~|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 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.

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]].

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 provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].

=== Prime harmonics ===

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-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 5.529954 [[cent]]s each.
+{{Infobox ET
+| Prime factorization = 7 × 31
+| Step size = 5.52995¢
+| Fifth = 127\217 (702.30¢)
+| Semitones = 21:16 (116.13¢ : 88.48¢)
+| Consistency = 21
+}}
+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.
 == Theory ==
-EDO 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|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.
+edo 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.
-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]].
+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 provides the [[optimal patent val]] for the 11- and 13-limit [[arch]] and the 11- and 13-limit [[cotoneum]].
 === Prime harmonics ===