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111 does not equal 10 times 11... no idea what this means, so removing it
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]].
{{Infobox ET}}
{{ED intro}}


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 temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&111 temperament, and [[buzzard]], the 58&111 temperament. The Trio piece below 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.
== 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 [[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]].  


The prime factorization is
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]].


<math>111 = 3 \cdot 37</math>
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 {{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.


Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments.
=== Prime harmonics ===
{| class="wikitable"
{{Harmonics in equal|111|columns=9}}
|+Circulating temperaments in 111edo
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}
!Tones
 
!Pattern
=== Octave stretch ===
!L:s
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}}
|-
|-
|5
| 2
| [[1L 4s]]
| 21.6
|23:22
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
|-
|6
| 3
|[[3L 3s]]
| 32.4
|19:18
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
|-
|7
| 4
|[[6L 1s]]
| 43.2
|16:15
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
|-
|8
| 5
| [[7L 1s]]
| 54.1
|14:13
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
|-
|9
| 6
|[[3L 6s]]
| 64.9
|13:12
| [[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
| 10
|[[1L 9s]]
| 108.1
|12:11
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
|-
| 11
| 11
|[[1L 10s]]
| 118.9
|11:10
| [[15/14]]
| {{UDnote|step=11}}
|-
|-
| 12
| 12
|[[3L 9s]]
| 129.7
|10:9
| [[14/13]]
| {{UDnote|step=12}}
|-
|-
|13
| 13
| [[6L 7s]]
| 140.5
|9:8
| [[13/12]]
| {{UDnote|step=13}}
|-
|-
|14
| 14
|[[13L 1s]]
| 151.4
| rowspan="2" |8:7
| [[12/11]]
| {{UDnote|step=14}}
|-
|-
|15
| 15
|[[6L 9s]]
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
|-
|16
| 16
|[[15L 1s]]
| 173.0
| rowspan="3" |7:6
| [[21/19]]
| {{UDnote|step=16}}
|-
|-
|17
| 17
|[[9L 8s]]
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
|-
|18
| 18
|3L 15s
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
|-
|19
| 19
|[[16L 3s]]
| 205.4
| rowspan="4" |6:5
| [[9/8]]
| {{UDnote|step=19}}
|-
|-
|20
| 20
|11L 9s
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
|-
|21
| 21
|6L 15s
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
|-
|22
| 22
|1L 21s
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
|-
|23
| 23
|19L 4s
| 248.6
| rowspan="5" | 5:4
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
|-
|24
| 24
|15L 9s
| 259.5
|
| {{UDnote|step=24}}
|-
|-
|25
| 25
| 11L 14s
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
|-
|26
| 26
|7L 19s
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
|-
|27
| 27
| 3L 24s
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
|-
|28
| 28
|27L 1s
| 302.7
| rowspan="9" |4:3
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
|-
|29
| 29
|24L 5s
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
|-
|30
| 30
|21L 9s
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
|-
|31
| 31
|18L 13s
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
|-
| 32
| 32
|15L 17s
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
|-
|33
| 33
|12L 21s
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
|-
|34
| 34
|9L 25s
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
|-
|35
| 35
|6L 29s
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
|-
|36
| 36
|3L 33s
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
|-
|37
| 37
|[[37edo]]
| 400.0
|equal
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
|-
|38
| 38
|35L 3s
| 410.8
| rowspan="18" |3:2
| [[19/15]]
| {{UDnote|step=38}}
|-
|-
|39
| 39
|33L 6s
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
|-
| 40
| 40
|31L 9s
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
|-
|41
| 41
|29L 12s
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
|-
|42
| 42
|27L 15s
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
|-
|43
| 43
|25L 18s
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
|-
|44
| 44
|23L 21s
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
|-
|45
| 45
|21L 24s
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
|-
|46
| 46
|19L 27s
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
|-
|47
| 47
|17L 30s
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
|-
|48
| 48
|15L 33s
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
|-
|49
| 49
|13L 36s
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
|-
|50
| 50
|11L 39s
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
|-
|51
| 51
|9L 42s
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
|-
|52
| 52
|7L 45s
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
|-
|53
| 53
|5L 48s
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
|-
|54
| 54
|3L 51s
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
|-
|55
| 55
|1L 54s
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
|-
|56
| …
|55L 1s
| …
| rowspan="33" |2:1
| …
| …
|}
<nowiki/>* As a 23-limit temperament, inconsistently mapped intervals in ''italic''
 
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|57
! rowspan="2" | [[Subgroup]]
|54L 3s
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|58
! [[TE error|Absolute]] (¢)
|53L 5s
! [[TE simple badness|Relative]] (%)
|-
|-
|59
| 2.3
|52L 7s
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
|-
|60
| 2.3.5
|51L 9s
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
|-
|61
| 2.3.5.7
|50L 11s
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
|-
|62
| 2.3.5.7.11
|49L 13s
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
|-
|63
| 2.3.5.7.11.13
|48L 15s
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
|-
|64
| 2.3.5.7.11.13.17
|47L 17s
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
|-
|65
| 2.3.5.7.11.13.17.19
|46L 19s
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
|-
|66
| 2.3.5.7.11.13.17.19.23
|45L 21s
| 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 ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.
 
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
|67
! Periods<br>per 8ve
|44L 23s
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
|-
|68
| 1
|43L 25s
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
|-
|69
| 1
|42L 27s
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
|-
|70
| 1
|41L 29s
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
|-
|71
| 1
|40L 31s
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
|-
|72
| 1
|39L 33s
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
|-
|73
| 1
|38L 35s
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
|-
|74
| 1
|37L 37s
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
|-
|75
| 1
|36L 39s
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
|-
|76
| 1
|35L 41s
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
|-
|77
| 1
|34L 43s
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
|-
|78
| 1
|33L 45s
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
|-
|79
| 1
|32L 47s
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
|-
|80
| 1
|31L 49s
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
|-
|81
| 1
|30L 51s
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
|-
|82
| 3
|29L 53s
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
|-
|83
| 3
|28L 55s
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
|-
|84
| 3
|27L 57s
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
|-
|85
| 3
|26L 59s
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
|-
|86
| 3
|25L 61s
| 23\111<br>(14\111)
| 248.65<br>(151.35)
| 15/13<br>(12/11)
| [[Hemimist]]
|-
|-
|87
| 3
|24L 63s
| 46\111<br>(9\111)
| 497.30<br>(97.30)
| 4/3<br>(18/17~19/18)
| [[Misty]]
|-
|-
|88
| 37
|23L 65s
| 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 ==
* 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 ==
* [[Lumatone mapping for 111edo]]
== Music ==
== Music ==
; [[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


[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]]
[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Buzzard]]
Line 295: Line 546:
[[Category:Orwellismic]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
Retrieved from "https://en.xen.wiki/w/111edo"