121edo

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121edo

122edo →

Prime factorization	112
Step size	9.91736 ¢
Fifth	71\121 (704.132 ¢)
Semitones (A1:m2)	13:8 (128.9 ¢ : 79.34 ¢)
Consistency limit	19
Distinct consistency limit	15

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]

Miscellany

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.