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| Prime factorization = 3 × 37
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Step size = 10.81081¢
| Fifth = 65\111 (702.70¢)
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205¢)
| Major 2nd = 19\111 (205.4¢)
| Minor 2nd = 8\111 (86¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Augmented 1sn = 11\111  (119¢)
| Consistency = 21
}}
}}


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.  
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 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|37EDO]], with which it shares the mappings for 5, 7, 11, and 13.  


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 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]].  
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=== 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"
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| 13/12
| 13/12
| [[Quanic]]
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
|-
| 1
| 1
Line 168: Line 174:


== Scales ==
== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
Since 111EDO has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
{| class="wikitable"
|+Circulating temperaments in 111edo
|+Circulating temperaments in 111EDO
!Tones
!Tones
!Pattern
!Pattern
!L:s
!L:s
|-
|-
|5
| 5
| [[1L 4s]]
| [[1L 4s]]
|23:22
|23:22
|-
|-
|6
| 6
|[[3L 3s]]
| [[3L 3s]]
|19:18
| 19:18
|-
|-
|7
| 7
|[[6L 1s]]
| [[6L 1s]]
|16:15
| 16:15
|-
|-
|8
| 8
| [[7L 1s]]
| [[7L 1s]]
|14:13
| 14:13
|-
|-
|9
| 9
|[[3L 6s]]
| [[3L 6s]]
|13:12
| 13:12
|-
|-
| 10
| 10
|[[1L 9s]]
| [[1L 9s]]
|12:11
| 12:11
|-
|-
| 11
| 11
|[[1L 10s]]
| [[1L 10s]]
|11:10
| 11:10
|-
|-
| 12
| 12
|[[3L 9s]]
| [[3L 9s]]
|10:9
| 10:9
|-
|-
|13
| 13
| [[6L 7s]]
| [[6L 7s]]
|9:8
| 9:8
|-
|-
|14
| 14
|[[13L 1s]]
| [[13L 1s]]
| rowspan="2" |8:7
| rowspan="2" |8:7
|-
|-
|15
| 15
|[[6L 9s]]
| [[6L 9s]]
|-
|-
|16
| 16
|[[15L 1s]]
| [[15L 1s]]
| rowspan="3" |7:6
| rowspan="3" |7:6
|-
|-
|17
| 17
|[[9L 8s]]
| [[9L 8s]]
|-
|-
|18
| 18
|3L 15s
| 3L 15s
|-
|-
|19
| 19
|[[16L 3s]]
| [[16L 3s]]
| rowspan="4" |6:5
| rowspan="4" |6:5
|-
|-
|20
| 20
|11L 9s
| 11L 9s
|-
|-
|21
| 21
|6L 15s
| 6L 15s
|-
|-
|22
| 22
|1L 21s
| 1L 21s
|-
|-
|23
| 23
|19L 4s
| 19L 4s
| rowspan="5" | 5:4
| rowspan="5" | 5:4
|-
|-
|24
| 24
|15L 9s
| 15L 9s
|-
|-
|25
| 25
| 11L 14s
| 11L 14s
|-
|-
|26
| 26
|7L 19s
| 7L 19s
|-
|-
|27
| 27
| 3L 24s
| 3L 24s
|-
|-
|28
| 28
|27L 1s
| 27L 1s
| rowspan="9" |4:3
| rowspan="9" |4:3
|-
|-
|29
| 29
|24L 5s
| 24L 5s
|-
|-
|30
| 30
|21L 9s
| 21L 9s
|-
|-
|31
| 31
|18L 13s
| 18L 13s
|-
|-
| 32
| 32
|15L 17s
| 15L 17s
|-
|-
|33
| 33
|12L 21s
| 12L 21s
|-
|-
|34
| 34
|9L 25s
| 9L 25s
|-
|-
|35
| 35
|6L 29s
| 6L 29s
|-
|-
|36
| 36
|3L 33s
| 3L 33s
|-
|-
|37
| 37
|[[37edo]]
| [[37edo|37EDO]]
|equal
| equal
|-
|-
|38
|38
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