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== Theory ==
== 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 [[harmonic]]s 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]].  
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 [[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]].  
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 {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[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 ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}
{{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 ===
=== Subsets and supersets ===
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| 1
| 1
| 10.8
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|-
|-
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| 23
| 23
| 248.6
| 248.6
| [[15/13]]
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 259.5
| 259.5
| [[22/19]]
|
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|-
|-
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| 32
| 32
| 345.9
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|-
|-
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| 45
| 45
| 486.5
| 486.5
| [[65/49]]
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|-
|-
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| …
| …
|}
|}
<nowiki/>* As a 23-limit temperament
<nowiki/>* As a 23-limit temperament, inconsistently mapped intervals in ''italic''


== Approximation to JI ==
== Approximation to JI ==
=== Zeta peak index ===
=== Interval mappings ===
{| class="wikitable center-all"
{{Q-odd-limit intervals}}
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}


== Regular temperament properties ==
== Regular temperament properties ==
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|-
|-
| 2.3
| 2.3
| {{monzo| 176 -111 }}
| {{Monzo| 176 -111 }}
| {{mapping| 111 176 }}
| {{Mapping| 111 176 }}
| −0.236
| −0.236
| 0.236
| 0.236
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| 2.3.5
| 2.3.5
| 78732/78125, 67108864/66430125
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| {{Mapping| 111 176 258 }}
| −0.570
| −0.570
| 0.510
| 0.510
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| 2.3.5.7
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| {{Mapping| 111 176 258 312 }}
| −0.797
| −0.797
| 0.591
| 0.591
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| 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
| {{mapping| 111 176 258 312 384 }}
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| −0.639
| 0.615
| 0.615
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| 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
| {{mapping| 111 176 258 312 384 411 }}
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| −0.655
| 0.562
| 0.562
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| 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
| {{mapping| 111 176 258 312 384 411 454 }}
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| −0.672
| 0.523
| 0.523
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| 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
| {{mapping| 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]].  
* 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]].  
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
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|-
|-
| 3
| 3
| 23\111<br />(14\111)
| 23\111<br>(14\111)
| 248.65<br />(151.35)
| 248.65<br>(151.35)
| 15/13<br />(12/11)
| 15/13<br>(12/11)
| [[Hemimist]]
| [[Hemimist]]
|-
|-
| 3
| 3
| 46\111<br />(9\111)
| 46\111<br>(9\111)
| 497.30<br />(97.30)
| 497.30<br>(97.30)
| 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 lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* 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)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)
== Instruments ==
* [[Lumatone mapping for 111edo]]


== Music ==
== Music ==
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