121edo: Difference between revisions
→Regular temperament properties: 13- to 23-limit notability |
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 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 34: | Line 26: | ||
| 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 41: | Line 33: | ||
| 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 48: | Line 40: | ||
| 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 55: | Line 47: | ||
| 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 62: | Line 54: | ||
| 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 69: | Line 61: | ||
| 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 | ||
{{comma basis end}} | |||
* 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errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]].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 === | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 187: | Line 173: | ||
|- | |- | ||
| 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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== 13-limit detempering of 121et == | == 13-limit detempering of 121et == | ||
| Line 200: | Line 186: | ||
== Miscellany == | == 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. | 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:Quintupole]] | [[Category:Quintupole]] | ||
Revision as of 06:08, 16 November 2024
| ← 120edo | 121edo | 122edo → |
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
| 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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
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.