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111edo: Difference between revisions

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== 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|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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== 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 10.811 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
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| 5
| 5
| [[1L 4s]]
| [[1L 4s]]
|23:22
| 23:22
|-
|-
| 6
| 6
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| equal
| equal
|-
|-
|38
| 38
|35L 3s
| 35L 3s
| rowspan="18" |3:2
| rowspan="18" |3:2
|-
|-
|39
| 39
|33L 6s
| 33L 6s
|-
|-
| 40
| 40
|31L 9s
| 31L 9s
|-
|-
|41
| 41
|29L 12s
| 29L 12s
|-
|-
|42
| 42
|27L 15s
| 27L 15s
|-
|-
|43
| 43
|25L 18s
| 25L 18s
|-
|-
|44
| 44
|23L 21s
| 23L 21s
|-
|-
|45
| 45
|21L 24s
| 21L 24s
|-
|-
|46
| 46
|19L 27s
| 19L 27s
|-
|-
|47
| 47
|17L 30s
| 17L 30s
|-
|-
|48
| 48
|15L 33s
| 15L 33s
|-
|-
|49
| 49
|13L 36s
| 13L 36s
|-
|-
|50
| 50
|11L 39s
| 11L 39s
|-
|-
|51
| 51
|9L 42s
| 9L 42s
|-
|-
|52
| 52
|7L 45s
| 7L 45s
|-
|-
|53
| 53
|5L 48s
| 5L 48s
|-
|-
|54
| 54
|3L 51s
| 3L 51s
|-
|-
|55
| 55
|1L 54s
| 1L 54s
|-
|-
|56
| 56
|55L 1s
| 55L 1s
| rowspan="33" |2:1
| rowspan="33" |2:1
|-
|-
|57
| 57
|54L 3s
| 54L 3s
|-
|-
|58
| 58
|53L 5s
| 53L 5s
|-
|-
|59
| 59
|52L 7s
| 52L 7s
|-
|-
|60
| 60
|51L 9s
| 51L 9s
|-
|-
|61
| 61
|50L 11s
| 50L 11s
|-
|-
|62
| 62
|49L 13s
| 49L 13s
|-
|-
|63
| 63
|48L 15s
| 48L 15s
|-
|-
|64
| 64
|47L 17s
| 47L 17s
|-
|-
|65
| 65
|46L 19s
| 46L 19s
|-
|-
|66
| 66
|45L 21s
| 45L 21s
|-
|-
|67
| 67
|44L 23s
| 44L 23s
|-
|-
|68
| 68
|43L 25s
| 43L 25s
|-
|-
|69
| 69
|42L 27s
| 42L 27s
|-
|-
|70
| 70
|41L 29s
| 41L 29s
|-
|-
|71
| 71
|40L 31s
| 40L 31s
|-
|-
|72
| 72
|39L 33s
| 39L 33s
|-
|-
|73
| 73
|38L 35s
| 38L 35s
|-
|-
|74
| 74
|37L 37s
| 37L 37s
|-
|-
|75
| 75
|36L 39s
| 36L 39s
|-
|-
|76
| 76
|35L 41s
| 35L 41s
|-
|-
|77
| 77
|34L 43s
| 34L 43s
|-
|-
|78
| 78
|33L 45s
| 33L 45s
|-
|-
|79
| 79
|32L 47s
| 32L 47s
|-
|-
|80
| 80
|31L 49s
| 31L 49s
|-
|-
|81
| 81
|30L 51s
| 30L 51s
|-
|-
|82
| 82
|29L 53s
| 29L 53s
|-
|-
|83
| 83
|28L 55s
| 28L 55s
|-
|-
|84
| 84
|27L 57s
| 27L 57s
|-
|-
|85
| 85
|26L 59s
| 26L 59s
|-
|-
|86
| 86
|25L 61s
| 25L 61s
|-
|-
|87
| 87
|24L 63s
| 24L 63s
|-
|-
|88
| 88
|23L 65s
| 23L 65s
|}
|}


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