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111edo: Difference between revisions

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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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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-08-07 02:23:03 UTC</tt>.<br>
 
: The original revision id was <tt>155516399</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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 revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
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]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 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 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 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;111 temperament, and vulture, the 58&amp;111 temperament.</pre></div>
 
<h4>Original HTML content:</h4>
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]].  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;111edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;111 equal temperament&lt;/em&gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 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 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 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and vulture, the 58&amp;amp;111 temperament.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 ===
{{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 ==
{| 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
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{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
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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 temperament|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<br>(14\111)
| 248.65<br>(151.35)
| 15/13<br>(12/11)
| [[Hemimist]]
|-
| 3
| 46\111<br>(9\111)
| 497.30<br>(97.30)
| 4/3<br>(18/17~19/18)
| [[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 ==
* 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 ==
; [[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:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
Retrieved from "https://en.xen.wiki/w/111edo"