282edo: Difference between revisions

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The ''282 equal ~~division~~'' ~~divides~~ the octave into 282 ~~equal~~ parts of 4.~~255 cents~~ each. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[Optimal_patent_val|optimal patent val]] for [[Porwell_family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[~~Porwell_temperaments~~#Septisuperfourth|septisuperfourth]] temperament. In the 13-limit it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079. ~~It is the smallest equal temperament uniquely~~ [[~~consistent|consistent~~]] ~~through to the 23 limit, and also the smallest consistent to the~~ 29 ~~limit. 282 has proper divisors 1, 2, 3, 6, 47, 94, and 141. It therefore divides the steps of 94et into three, but is not contorted beyond the 3~~-limit.

The '''282 equal divisions of the octave''' ('''282EDO'''), or the '''282(-tone) equal temperament''' ('''282TET''', '''282ET''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 282 parts of 4.2553 [[cent]]s each.

== Theory ==

282EDO is the smallest equal temperament uniquely [[consistent]] through to the 23 odd-limit, and also the smallest consistent to the 29 odd-limit. It shares the same 3rd, 7th, and 13th harmonics with [[94edo|94EDO]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (supporting the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[Optimal_patent_val|optimal patent val]] for [[Porwell_family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[Porwell temperaments #Septisuperfourth|septisuperfourth]] temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079.

=== Prime harmonics ===

[[Category:Equal divisions of the octave]]

[[Category:29-limit]]

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-The ''282 equal division'' divides the octave into 282 equal parts of 4.255 cents each. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[Optimal_patent_val|optimal patent val]] for [[Porwell_family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[Porwell_temperaments#Septisuperfourth|septisuperfourth]] temperament. In the 13-limit it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079. It is the smallest equal temperament uniquely [[consistent|consistent]] through to the 23 limit, and also the smallest consistent to the 29 limit. 282 has proper divisors 1, 2, 3, 6, 47, 94, and 141. It therefore divides the steps of 94et into three, but is not contorted beyond the 3-limit.
+The '''282 equal divisions of the octave''' ('''282EDO'''), or the '''282(-tone) equal temperament''' ('''282TET''', '''282ET''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 282 parts of 4.2553 [[cent]]s each.
+== Theory ==
+EDO is the smallest equal temperament uniquely [[consistent]] through to the 23 odd-limit, and also the smallest consistent to the 29 odd-limit. It shares the same 3rd, 7th, and 13th harmonics with [[94edo|94EDO]] (282 = 3 × 94), as well as [[11/10]] and [[20/17]] (supporting the [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[Optimal_patent_val|optimal patent val]] for [[Porwell_family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[Porwell temperaments #Septisuperfourth|septisuperfourth]] temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079.
+=== Prime harmonics ===
+{{Primes in edo|edo=282|columns=10}}
+[[Category:Equal divisions of the octave]]
+[[Category:29-limit]]