121edo

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← 120edo 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

121 equal divisions of the octave (abbreviated 121edo or 121ed2), also called 121-tone equal temperament (121tet) or 121 equal temperament (121et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 121 equal parts of about 9.92 ¢ each. Each step represents a frequency ratio of 21/121, or the 121st root of 2.

Theory

121edo has a distinctly sharp tendency, in that the odd harmonics from 3 to 19 all have sharp tunings. The equal temperament tempers out 15625/15552 (kleisma) 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 major minthmic chords, because it tempers out 364/363 it allows minor minthmic chords, because it tempers out 676/675 it allows island chords and because it tempers out 1575/1573 it allows nicolic chords. 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

Approximation of odd harmonics in 121edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.18 +0.46 +3.07 +4.35 +4.05 +2.45 +2.64 +4.14 +0.01 -4.67 -3.48
Relative (%) +22.0 +4.7 +31.0 +43.9 +40.9 +24.7 +26.6 +41.7 +0.1 -47.0 -35.1
Steps
(reduced)
192
(71)
281
(39)
340
(98)
384
(21)
419
(56)
448
(85)
473
(110)
495
(11)
514
(30)
531
(47)
547
(63)

Subsets and supersets

Since 121 factors into 112, 121edo contains 11edo as its only nontrivial subset.

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
  • 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating 111 before being superseded by 130 in all those limits except for the 17-limit, where it is superseded by 140.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
ratio*
Temperament
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 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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

13-limit detempering of 121et

[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.