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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 37
{{ED intro}}
| Step size = 10.81081¢
| Fifth = 65\111 (702.70¢)
| Major 2nd = 19\111 (205¢)
| Minor 2nd = 8\111 (86¢)
| Augmented 1sn = 11\111  (119¢)
}}
 
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 10.8 [[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-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 34: 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 42: 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 49: 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 56: 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 63: 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 70: 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 77: 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 37et 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 105: Line 413:
| 13/12
| 13/12
| [[Quanic]]
| [[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
| 1
Line 115: Line 453:
| 41\111
| 41\111
| 443.24
| 443.24
| 162/125
| 22/17
| [[Sensipent]]
| [[Warrior]]
|-
|-
| 1
| 1
Line 128: Line 466:
| 475.68
| 475.68
| 21/16
| 21/16
| [[Vulture]] / [[buzzard]]
| [[Buzzard]]
|-
|-
| 1
| 1
Line 135: Line 473:
| 4/3
| 4/3
| [[Kwai]]
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
|-
| 3
| 3
Line 141: Line 491:
| 24/23
| 24/23
| [[Terture]]
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
|-
| 3
| 3
Line 157: 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 165: 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 10.81 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]]
|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:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Buzzard]]
Retrieved from "https://en.xen.wiki/w/111edo"