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| {{Infobox ET}} | | {{Infobox ET}} |
| '''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]].
| | {{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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| Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.
| | === Odd harmonics === |
| | {{Harmonics in equal|112|intervals=odd}} |
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| {| class="wikitable"
| | === Subsets and supersets === |
| |- | | 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 | |
| | [[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|###]] <!-- 3-digit number --> | | == See also == |
| | [[Skip fretting system 112 9 11]] |
| | [[Category:Listen]] |