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121 isn't consistent in the 21-odd-limit, hence the table shows odd harmonics as per the template doc
 
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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]].
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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]].
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[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_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 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 ===
=== Odd harmonics ===
{{Primes in edo|121|columns=10}}
{{Harmonics in equal|121}}
 
=== Subsets and supersets ===
Since 121 factors into 11<sup>2</sup>, 121edo contains [[11edo]] as its only nontrivial subset.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 22: Line 27:
| 2.3
| 2.3
| {{monzo| 192 -121 }}
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| {{mapping| 121 192 }}
| -0.687
| −0.687
| 0.687
| 0.687
| 6.93
| 6.93
Line 29: Line 34:
| 2.3.5
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| {{mapping| 121 192 281 }}
| -0.524
| −0.524
| 0.606
| 0.606
| 6.11
| 6.11
Line 36: Line 41:
| 2.3.5.7
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| {{mapping| 121 192 281 340 }}
| -0.667
| −0.667
| 0.580
| 0.580
| 5.85
| 5.85
Line 43: Line 48:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| {{mapping| 121 192 281 340 419 }}
| -0.768
| −0.768
| 0.556
| 0.556
| 5.61
| 5.61
Line 50: Line 55:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| {{mapping| 121 192 281 340 419 448 }}
| -0.750
| −0.750
| 0.510
| 0.510
| 5.14
| 5.14
|-
|-
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 }}]
| {{mapping| 121 192 281 340 419 448 495 }}
| -0.787
| −0.787
| 0.480
| 0.480
| 4.85
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 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 [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br />ratio*
! Temperament
|-
|-
| 1
| 1
Line 77: Line 91:
| 21/20
| 21/20
| [[Slithy]]
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
|-
| 1
| 1
Line 83: Line 103:
| 15/14
| 15/14
| [[Subsedia]]
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
|-
| 1
| 1
Line 100: Line 126:
| 317.36
| 317.36
| 6/5
| 6/5
| [[Hanson]] / [[metakleismic]]
| [[Metakleismic]]
|-
|-
| 1
| 1
Line 121: Line 147:
|-
|-
| 1
| 1
| 46/121
| 46\121
| 456.20
| 456.20
| 125/96
| 125/96
Line 127: Line 153:
|-
|-
| 1
| 1
| 47/121
| 47\121
| 466.12
| 466.12
| 55/42
| 55/42
Line 133: Line 159:
|-
|-
| 1
| 1
| 48/121
| 48\121
| 476.03
| 476.03
| 21/16
| 21/16
Line 163: Line 189:
|-
|-
| 11
| 11
| 50\121<br>(5\121)
| 50\121<br />(5\121)
| 495.87<br>(49.59)
| 495.87<br />(49.59)
| 4/3<br>(36/35)
| 4/3<br />(36/35)
| [[Hendecatonic]]
| [[Hendecatonic]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== 13-limit detempering of 121et ==
== 13-limit detempering of 121et ==
Line 174: Line 201:
[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]
[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]


[[Category:Equal divisions of the octave]]
== Miscellany ==
[[Category:Gentle]]
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.
 
[[Category:Grendel]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Quintupole]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]

Latest revision as of 15:38, 24 June 2025

← 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. It 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.

Odd 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 distinct

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.

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