121edo

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 9.9174 cents each, and being the square closest to division of the octave by the Germanic 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 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 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 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.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[192 -121⟩	[⟨121 192]]	-0.687	0.687	6.93
2.3.5	15625/15552, [31 -21 1⟩	[⟨121 192 281]]	-0.524	0.606	6.11
2.3.5.7	4000/3969, 6144/6125, 10976/10935	[⟨121 192 281 340]]	-0.667	0.580	5.85
2.3.5.7.11	540/539, 896/891, 1375/1372, 4375/4356	[⟨121 192 281 340 419]]	-0.768	0.556	5.61
2.3.5.7.11.13	325/324, 352/351, 364/363, 540/539, 625/624	[⟨121 192 281 340 419 448]]	-0.750	0.510	5.14
2.3.5.7.11.13.17	256/255, 325/324, 352/351, 364/363, 375/374, 442/441	[⟨121 192 281 340 419 448 495]]	-0.787	0.480	4.85
2.3.5.7.11.13.17.19	190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374	[⟨121 192 281 340 419 448 495 514]]	-0.689	0.519	5.23

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	9\121	89.26	21/20	Slithy
1	10\121	99.17	18/17	Quintupole
1	12\121	119.01	15/14	Subsedia
1	13\121	128.93	14/13	Tertiathirds
1	16\121	158.68	35/32	Hemikleismic
1	27\121	267.77	7/6	Hemimaquila
1	32\121	317.36	6/5	Hanson / metakleismic
1	39\121	386.78	5/4	Grendel
1	40\121	396.69	44/35	Squarschmidt
1	42\121	416.53	14/11	Sqrtphi
1	46\121	456.20	125/96	Qak
1	47\121	466.12	55/42	Hemiseptisix
1	48\121	476.03	21/16	Subfourth
1	50\121	495.87	4/3	Leapday / polypyth
1	51\121	505.79	75/56	Marfifths / marf / diatessic
1	54\121	535.54	512/375	Maquila
1	59\121	585.12	7/5	Pluto
11	50\121 (5\121)	495.87 (49.59)	4/3 (36/35)	Hendecatonic

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]