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| '''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].
| | {{Infobox ET}} |
| | {{ED intro}} |
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| | == Theory == |
| | 112edo has two great [[3/2|perfect fifth]]s, the lower of which approximates [[quarter-comma meantone]] (just a tad lower), and the upper of which—the [[patent fifth]]—is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27]] while −8 fifths gives a near-just [[39/32]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17]]. |
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| One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure. | | One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure. |
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| | === Odd harmonics === |
| | {{Harmonics in equal|112|intervals=odd}} |
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| Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments. | | === Subsets and supersets === |
| {| class="wikitable" | | Since 112 factors into {{factorization|112}}, 112edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 16, 28, and 56 }}. [[224edo]], which doubles it, is a strong 13-limit system. |
| |+Circulating temperaments in 112edo
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| !Tones
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| !Pattern
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| !L:s
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| |-
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| |5
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| |[[2L 3s]]
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| |23:22
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| |-
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| |6
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| |[[4L 2s]]
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| |19:18
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| |-
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| |7
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| |[[7edo]]
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| | rowspan="2" |equal
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| |-
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| |8
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| |[[8edo]]
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| |-
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| |9
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| |[[4L 5s]]
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| |13:12
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| |-
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| |10
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| |[[2L 8s]]
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| |12:11
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| |-
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| |11
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| |[[2L 9s]]
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| |11:10
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| |-
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| |12
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| |[[4L 8s]]
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| |10:9
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| |-
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| |13
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| |[[8L 5s]]
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| |9:8
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| |-
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| |14
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| |[[14edo]]
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| |equal
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| |-
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| |15
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| |[[6L 9s]]
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| |8:7
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| |-
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| |16
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| |[[16edo]]
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| |equal
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| |-
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| |17
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| |[[10L 7s]]
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| | rowspan="2" |7:6
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| |-
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| |18
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| |4L 14s
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| |-
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| |19
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| |[[17L 2s]]
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| | rowspan="4" |6:5
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| |-
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| |20
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| |12L 8s
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| |-
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| |21
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| |7L 14s
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| |-
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| |22
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| |2L 20s
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| |-
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| |23
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| |20L 3s
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| | rowspan="5" |5:4
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| |-
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| |24
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| |16L 8s
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| |-
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| |25
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| |12L 13s
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| |-
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| |26
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| |8L 18s
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| |-
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| |27
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| |4L 23s
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| |-
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| |28
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| |[[28edo]]
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| |equal
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| |-
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| |29
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| |25L 4s
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| | rowspan="9" |4:3
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| |-
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| |30
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| |22L 8s
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| |-
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| |31
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| |19L 12s
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| |-
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| |32
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| |16L 16s
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| |-
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| |33
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| |13L 20s
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| |-
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| |34
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| |10L 24s
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| |-
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| |35
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| |7L 28s
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| |-
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| |36
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| |4L 32s
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| |-
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| |37
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| |1L 36s
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| |-
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| |38
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| |36L 2s
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| | rowspan="18" |3:2
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| |-
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| |39
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| |34L 5s
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| |-
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| |40
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| |32L 8s
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| |-
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| |41
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| |30L 11s
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| |-
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| |42
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| |28L 14s
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| |-
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| |43
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| |26L 17s
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| |-
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| |44
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| |24L 20s
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| |-
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| |45
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| |22L 23s
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| |-
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| |46
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| |20L 26s
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| |-
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| |47
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| |18L 29s
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| |-
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| |48
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| |16L 32s
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| |-
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| |49
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| |14L 35s
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| |-
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| |50
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| |12L 38s
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| |-
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| |51
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| |10L 41s
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| |-
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| |52
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| |8L 44s
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| |-
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| |53
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| |6L 47s
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| |-
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| |54
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| |4L 50s
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| |-
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| |55
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| |2L 53s
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| |-
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| |56
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| |[[56edo]]
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| |equal
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| |-
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| |57
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| |55L 2s
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| | rowspan="33" |2:1
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| |-
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| |58
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| |54L 4s
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| |-
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| |59
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| |53L 6s
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| |-
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| |60
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| |52L 8s
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| |-
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| |61
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| |51L 10s
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| |-
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| |62
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| |50L 12s
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| |-
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| |63
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| |49L 14s
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| |-
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| |64
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| |48L 16s
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| |-
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| |65
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| |47L 18s
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| |-
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| |66
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| |46L 20s
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| |-
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| |67
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| |45L 22s
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| |-
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| |68
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| |44L 24s
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| |-
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| |69
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| |43L 26s
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| |-
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| |70
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| |42L 28s
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| |-
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| |71
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| |41L 30s
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| |-
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| |72
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| |40L 32s
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| |-
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| |73
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| |39L 34s
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| |-
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| |74
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| |38L 36s
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| |-
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| |75
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| |37L 38s
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| |-
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| |76
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| |36L 40s
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| |-
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| |77
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| |35L 42s
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| |-
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| |78
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| |34L 44s
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| |-
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| |79
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| |33L 46s
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| |-
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| |80
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| |32L 48s
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| |-
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| |81
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| |31L 50s
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| |-
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| |82
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| |30L 52s
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| |-
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| |83
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| |29L 54s
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| |-
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| |84
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| |28L 56s
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| |-
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| |85
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| |27L 58s
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| |-
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| |86
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| |26L 60s
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| |-
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| |87
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| |25L 62s
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| |-
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| |88
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| |24L 64s
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| |-
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| |89
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| |23L 66s
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| |}
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| ==Music in 112EDO== | | == Intervals == |
| | {{Interval table}} |
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| *[https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]] | | == Music == |
| | ; [[Cam Taylor]] |
| | * [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv''] |
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| [[Category:Equal divisions of the octave]]
| | == See also == |
| [[Category:what is]] | | [[Skip fretting system 112 9 11]] |
| [[Category:wiki]] | | [[Category:Listen]] |