121edo: Difference between revisions
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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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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. | 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 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. | 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 === | ||
{{Harmonics in equal|121}} | {{Harmonics in equal|121}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 121 factors into | 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" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{monzo| 192 -121 }} | | {{monzo| 192 -121 }} | ||
| {{mapping| 121 192 }} | | {{mapping| 121 192 }} | ||
| | | −0.687 | ||
| 0.687 | | 0.687 | ||
| 6.93 | | 6.93 | ||
| Line 26: | Line 35: | ||
| 15625/15552, {{monzo| 31 -21 1 }} | | 15625/15552, {{monzo| 31 -21 1 }} | ||
| {{mapping| 121 192 281 }} | | {{mapping| 121 192 281 }} | ||
| | | −0.524 | ||
| 0.606 | | 0.606 | ||
| 6.11 | | 6.11 | ||
| Line 33: | Line 42: | ||
| 4000/3969, 6144/6125, 10976/10935 | | 4000/3969, 6144/6125, 10976/10935 | ||
| {{mapping| 121 192 281 340 }} | | {{mapping| 121 192 281 340 }} | ||
| | | −0.667 | ||
| 0.580 | | 0.580 | ||
| 5.85 | | 5.85 | ||
| Line 40: | Line 49: | ||
| 540/539, 896/891, 1375/1372, 4375/4356 | | 540/539, 896/891, 1375/1372, 4375/4356 | ||
| {{mapping| 121 192 281 340 419 }} | | {{mapping| 121 192 281 340 419 }} | ||
| | | −0.768 | ||
| 0.556 | | 0.556 | ||
| 5.61 | | 5.61 | ||
| Line 47: | Line 56: | ||
| 325/324, 352/351, 364/363, 540/539, 625/624 | | 325/324, 352/351, 364/363, 540/539, 625/624 | ||
| {{mapping| 121 192 281 340 419 448 }} | | {{mapping| 121 192 281 340 419 448 }} | ||
| | | −0.750 | ||
| 0.510 | | 0.510 | ||
| 5.14 | | 5.14 | ||
| Line 54: | Line 63: | ||
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441 | | 256/255, 325/324, 352/351, 364/363, 375/374, 442/441 | ||
| {{mapping| 121 192 281 340 419 448 495 }} | | {{mapping| 121 192 281 340 419 448 495 }} | ||
| | | −0.787 | ||
| 0.480 | | 0.480 | ||
| 4.85 | | 4.85 | ||
| Line 61: | Line 70: | ||
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374 | | 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374 | ||
| {{mapping| 121 192 281 340 419 448 495 514 }} | | {{mapping| 121 192 281 340 419 448 495 514 }} | ||
| | | −0.689 | ||
| 0.519 | | 0.519 | ||
| 5.23 | | 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]]. | * 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" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperament | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 177: | Line 193: | ||
| 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 == | ||
Latest revision as of 15:38, 24 June 2025
| ← 120edo | 121edo | 122edo → |
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
| 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
| 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
[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.