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{{Infobox ET}}
{{Infobox ET}}
 
{{ED intro}}
The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s.


== Theory ==
== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13.  
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]].  


It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]].  
It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]].  


In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]].  
It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]].  


It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&80 temperament, and [[buzzard]], the 53&58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|111|columns=11}}
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}
 
=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit.
 
=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.
 
== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
<nowiki/>* As a 23-limit temperament, inconsistently mapped intervals in ''italic''
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}


== 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]]
Line 27: Line 333:
|-
|-
| 2.3
| 2.3
| {{monzo| 176 -111 }}
| {{Monzo| 176 -111 }}
| [{{val| 111 176 }}]
| {{Mapping| 111 176 }}
| -0.236
| −0.236
| 0.236
| 0.236
| 2.18
| 2.18
Line 35: Line 341:
| 2.3.5
| 2.3.5
| 78732/78125, 67108864/66430125
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| {{Mapping| 111 176 258 }}
| -0.570
| −0.570
| 0.510
| 0.510
| 4.72
| 4.72
Line 42: Line 348:
| 2.3.5.7
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| {{Mapping| 111 176 258 312 }}
| -0.797
| −0.797
| 0.591
| 0.591
| 5.47
| 5.47
Line 49: Line 355:
| 2.3.5.7.11
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| {{Mapping| 111 176 258 312 384 }}
| -0.639
| −0.639
| 0.615
| 0.615
| 5.69
| 5.69
Line 56: Line 362:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| {{Mapping| 111 176 258 312 384 411 }}
| -0.655
| −0.655
| 0.562
| 0.562
| 5.21
| 5.21
Line 63: Line 369:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| {{Mapping| 111 176 258 312 384 411 454 }}
| -0.672
| −0.672
| 0.523
| 0.523
| 4.84
| 4.84
Line 70: Line 376:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| −0.740
| 0.521
| 0.521
| 4.83
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed.  
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.  


{| 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 ratio<br>(reduced)
! Cents*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
Line 138: Line 453:
| 41\111
| 41\111
| 443.24
| 443.24
| 162/125
| 22/17
| [[Sensipent]] / [[warrior]]
| [[Warrior]]
|-
|-
| 1
| 1
Line 151: Line 466:
| 475.68
| 475.68
| 21/16
| 21/16
| [[Vulture]] / [[buzzard]]
| [[Buzzard]]
|-
|-
| 1
| 1
Line 198: Line 513:
| 23\111<br>(14\111)
| 23\111<br>(14\111)
| 248.65<br>(151.35)
| 248.65<br>(151.35)
| 231/200<br>(12/11)
| 15/13<br>(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
Line 206: Line 521:
| 4/3<br>(18/17~19/18)
| 4/3<br>(18/17~19/18)
| [[Misty]]
| [[Misty]]
|-
| 37
| 46\111<br>(1\111)
| 497.30<br>(10.81)
| 4/3<br>(169/168)
| [[Rubidium]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
{| class="wikitable"
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)
|+Circulating temperaments in 111EDO
 
! Tones
== Instruments ==
! Pattern
* [[Lumatone mapping for 111edo]]
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}


== Music ==
== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Semisept]]
Line 495: Line 547:
[[Category:Guanyin]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Valinorsmic]]
[[Category:Listen]]

Latest revision as of 13:30, 13 March 2026

← 110edo 111edo 112edo →
Prime factorization 3 × 37
Step size 10.8108 ¢ 
Fifth 65\111 (702.703 ¢)
Semitones (A1:m2) 11:8 (118.9 ¢ : 86.49 ¢)
Consistency limit 21
Distinct consistency limit 15

111 equal divisions of the octave (abbreviated 111edo or 111ed2), also called 111-tone equal temperament (111tet) or 111 equal temperament (111et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 111 equal parts of about 10.8 ¢ each. Each step represents a frequency ratio of 21/111, or the 111th root of 2.

Theory

111edo is consistent through to the 21-odd-limit, and is the smallest edo distinctly consistent through the 15-odd-limit, marking it as an important higher limit tuning. It has a sharp tendency, with primes 3 through 19 all tuned sharp. Since 111 = 3 × 37, 111edo shares the mappings for 5, 7, 11, and 13 with 37edo.

It is also significant for lower limits, especially in terms of what it tempers out in its patent val; for example, it tempers out 176/175 and gives an excellent optimal patent val for the corresponding 11-limit rank-4 temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125, and 5120/5103, and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372.

It further tempers out among others 351/350, 352/351, 640/637, 676/675, 847/845, 1001/1000, 1188/1183, 1573/1568 in the 13-limit; 256/255, 325/324, 442/441 in the 17-limit; 286/285, 400/399, 476/475 in the 19-limit. It excels as a full 23-limit temperament, tempering out 253/252 and 276/275. The 23 is tuned a little flat, unlike the lower primes. 23/19, 23/21 and their octave complements are the only inconsistently mapped intervals in the 23-odd-limit.

It is a particularly good tuning for the 11- or 13-limit versions of semisept, the 31 & 80 temperament, and buzzard, the 53 & 58 temperament. Gene Ward Smith's trio in #Music section is in guanyin temperament, the rank-3 temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

Prime harmonics

Approximation of prime harmonics in 111edo
Harmonic 2 3 5 7 11 13 17 19 23
Error Absolute (¢) +0.00 +0.75 +2.88 +4.15 +0.03 +2.72 +3.15 +5.19 -1.25
Relative (%) +0.0 +6.9 +26.6 +38.4 +0.3 +25.1 +29.2 +48.0 -11.5
Steps
(reduced) 		111
(0) 		176
(65) 		258
(36) 		312
(90) 		384
(51) 		411
(78) 		454
(10) 		472
(28) 		502
(58)
Approximation of prime harmonics in 111edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) -2.55 +0.91 -2.70 +3.37 -3.41 +4.76 +2.17 +0.29 -3.37
Relative (%) -23.6 +8.4 -24.9 +31.2 -31.5 +44.1 +20.1 +2.7 -31.2
Steps
(reduced) 		539
(95) 		550
(106) 		578
(23) 		595
(40) 		602
(47) 		617
(62) 		636
(81) 		653
(98) 		658
(103)

Octave stretch

111edo can benefit from slightly compressing the octave if that is acceptable, using tunings such as 176edt or 287ed6. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit.

Subsets and supersets

Since 111 factors into primes as 3 × 37, 111edo contains 3edo and 37edo as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's 13-odd-limit. 333edo, which slices the step of 111edo in three, is a significant tuning.

Intervals

# Cents Approximated ratios* Ups and downs notation
0 0.0 1/1 D
1 10.8 121/120, 126/125, 144/143, 161/160, 169/168, 196/195, 225/224 ^D, ^4E♭♭
2 21.6 64/63, 81/80, 91/90, 100/99, 105/104 ^^D, ^5E♭♭
3 32.4 46/45, 50/49, 55/54, 56/55, 57/56, 65/64 ^3D, v5E♭
4 43.2 36/35, 39/38, 40/39, 45/44, 49/48 ^4D, v4E♭
5 54.1 33/32, 34/33, 35/34 ^5D, v3E♭
6 64.9 26/25, 27/26, 28/27 v5D♯, vvE♭
7 75.7 22/21, 23/22, 24/23, 25/24 v4D♯, vE♭
8 86.5 20/19, 21/20 v3D♯, E♭
9 97.3 18/17, 19/18 vvD♯, ^E♭
10 108.1 16/15, 17/16 vD♯, ^^E♭
11 118.9 15/14 D♯, ^3E♭
12 129.7 14/13 ^D♯, ^4E♭
13 140.5 13/12 ^^D♯, ^5E♭
14 151.4 12/11 ^3D♯, v5E
15 162.2 11/10 ^4D♯, v4E
16 173.0 21/19 ^5D♯, v3E
17 183.8 10/9 v5D𝄪, vvE
18 194.6 19/17, 28/25 v4D𝄪, vE
19 205.4 9/8 E
20 216.2 17/15, 26/23 ^E, ^4F♭
21 227.0 8/7 ^^E, ^5F♭
22 237.8 23/20 ^3E, v5F
23 248.6 15/13, 22/19 ^4E, v4F
24 259.5 ^5E, v3F
25 270.3 7/6 v5E♯, vvF
26 281.1 20/17 v4E♯, vF
27 291.9 13/11 F
28 302.7 19/16, 25/21 ^F, ^4G♭♭
29 313.5 6/5 ^^F, ^5G♭♭
30 324.3 23/19, 77/64 ^3F, v5G♭
31 335.1 17/14, 40/33 ^4F, v4G♭
32 345.9 11/9, 28/23, 39/32 ^5F, v3G♭
33 356.8 16/13, 27/22 v5F♯, vvG♭
34 367.6 21/17, 26/21 v4F♯, vG♭
35 378.4 56/45 v3F♯, G♭
36 389.2 5/4 vvF♯, ^G♭
37 400.0 24/19, 34/27 vF♯, ^^G♭
38 410.8 19/15 F♯, ^3G♭
39 421.6 14/11, 23/18 ^F♯, ^4G♭
40 432.4 9/7 ^^F♯, ^5G♭
41 443.2 22/17 ^3F♯, v5G
42 454.1 13/10 ^4F♯, v4G
43 464.9 17/13 ^5F♯, v3G
44 475.7 21/16, 25/19 v5F𝄪, vvG
45 486.5 45/34, 65/49 v4F𝄪, vG
46 497.3 4/3 G
47 508.1 51/38 ^G, ^4A♭♭
48 518.9 23/17, 27/20 ^^G, ^5A♭♭
49 529.7 19/14 ^3G, v5A♭
50 540.5 15/11, 26/19 ^4G, v4A♭
51 551.4 11/8 ^5G, v3A♭
52 562.2 18/13 v5G♯, vvA♭
53 573.0 32/23 v4G♯, vA♭
54 583.8 7/5 v3G♯, A♭
55 594.6 24/17 vvG♯, ^A♭
… … … …

* As a 23-limit temperament, inconsistently mapped intervals in italic

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 111edo. Prime harmonics are in bold.

As 111edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 111edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/8, 16/11 0.033 0.3
13/10, 20/13 0.160 1.5
15/14, 28/15 0.524 4.8
11/6, 12/11 0.714 6.6
3/2, 4/3 0.748 6.9
15/13, 26/15 0.908 8.4
13/9, 18/13 1.220 11.3
7/5, 10/7 1.272 11.8
9/5, 10/9 1.380 12.8
13/7, 14/13 1.431 13.2
11/9, 18/11 1.462 13.5
9/8, 16/9 1.495 13.8
13/12, 24/13 1.968 18.2
5/3, 6/5 2.128 19.7
9/7, 14/9 2.652 24.5
13/11, 22/13 2.682 24.8
13/8, 16/13 2.716 25.1
11/10, 20/11 2.842 26.3
5/4, 8/5 2.875 26.6
7/6, 12/7 3.399 31.4
15/11, 22/15 3.590 33.2
15/8, 16/15 3.623 33.5
11/7, 14/11 4.114 38.1
7/4, 8/7 4.147 38.4

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [176 -111⟩ [⟨111 176]] −0.236 0.236 2.18
2.3.5 78732/78125, 67108864/66430125 [⟨111 176 258]] −0.570 0.510 4.72
2.3.5.7 1728/1715, 3136/3125, 5120/5103 [⟨111 176 258 312]] −0.797 0.591 5.47
2.3.5.7.11 176/175, 540/539, 1331/1323, 5120/5103 [⟨111 176 258 312 384]] −0.639 0.615 5.69
2.3.5.7.11.13 176/175, 351/350, 540/539, 676/675, 1331/1323 [⟨111 176 258 312 384 411]] −0.655 0.562 5.21
2.3.5.7.11.13.17 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 [⟨111 176 258 312 384 411 454]] −0.672 0.523 4.84
2.3.5.7.11.13.17.19 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 [⟨111 176 258 312 384 411 454 472]] −0.740 0.521 4.83
2.3.5.7.11.13.17.19.23 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399 [⟨111 176 258 312 384 411 454 472 502]] −0.628 0.586 5.43
  • 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating 94 and 103h before being superseded by 121i.

Rank-2 temperaments

Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 11\111 118.92 15/14 Subsedia
1 13\111 140.54 13/12 Quanic
1 14\111 151.35 12/11 Browser
1 16\111 172.97 400/363 Undetrita
1 20\111 216.22 17/15 Tremka
1 23\111 248.65 15/13 Hemikwai
1 31\111 335.14 17/14 Cohemimabila
1 35\111 378.38 56/45 Subpental
1 41\111 443.24 22/17 Warrior
1 43\111 464.86 17/13 Semisept
1 44\111 475.68 21/16 Buzzard
1 46\111 497.30 4/3 Kwai
1 49\111 529.73 19/14 Tuskaloosa
1 55\111 594.59 55/39 Gaster
3 7\111 75.68 24/23 Terture
3 12\111 129.73 14/13 Trimabila
3 13\111 140.54 243/224 Septichrome
3 17\111 183.55 10/9 Mirkat
3 23\111
(14\111) 		248.65
(151.35) 		15/13
(12/11) 		Hemimist
3 46\111
(9\111) 		497.30
(97.30) 		4/3
(18/17~19/18) 		Misty
37 46\111
(1\111) 		497.30
(10.81) 		4/3
(169/168) 		Rubidium

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

Scales

  • Direct sunlight (subset of Sensi[19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
  • Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

Instruments

Music

Gene Ward Smith
  • Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) – SoundCloud | details | play – in Guanyin[22], 111edo tuning
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