121edo: Difference between revisions

Line 1:

The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 ~~cents~~ each and being the square closest to division of the octave by the Germanic [[Wikipedia:~~Long_hundred~~|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]].

The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]].

== Theory ==

121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[~~island_tetrad~~|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===

== 13-limit detempering of 121et ==

~~: ''~~See also~~: [[~~Detempering~~]].''~~

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

@@ Line 1: / Line 1: @@
-The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [[Wikipedia:Long_hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]].
+The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]].
+== Theory ==
 edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].
-Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
+Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.
+=== Prime harmonics ===
 {{Primes in edo|121|columns=10}}
 == 13-limit detempering of 121et ==
-: ''See also: [[Detempering]].''
+{{See also| Detempering }}
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