|
|
| (6 intermediate revisions by 4 users not shown) |
| Line 1: |
Line 1: |
| {{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}} |
| | |
| | == 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]]. |
|
| |
|
| 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. |
|
| |
|
| == Odd harmonics == | | === Odd harmonics === |
| {{Harmonics in equal|112|intervals=odd}} | | {{Harmonics in equal|112|intervals=odd}} |
|
| |
|
| == Circulating temperaments == | | === 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. |
| Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as [[circulating temperament]]s. | |
| | |
| {| class="wikitable" | |
| |-
| |
| |+ Circulating temperaments in 112edo
| |
| |-
| |
| ! Tones
| |
| ! Pattern
| |
| ! L:s
| |
| |-
| |
| | 5
| |
| | [[2L 3s]]
| |
| | 23:22
| |
| |-
| |
| | 6
| |
| | [[4L 2s]]
| |
| | 19:18
| |
| |-
| |
| | 7
| |
| | [[7edo]]
| |
| | rowspan="2" | equal
| |
| |-
| |
| | 8
| |
| | [[8edo]]
| |
| |-
| |
| | 9
| |
| | [[4L 5s]]
| |
| | 13:12
| |
| |-
| |
| | 10
| |
| | [[2L 8s]]
| |
| | 12:11
| |
| |-
| |
| | 11
| |
| | [[2L 9s]]
| |
| | 11:10
| |
| |-
| |
| | 12
| |
| | [[4L 8s]]
| |
| | 10:9
| |
| |-
| |
| | 13
| |
| | [[8L 5s]]
| |
| | 9:8
| |
| |-
| |
| | 14
| |
| | [[14edo]]
| |
| | equal
| |
| |-
| |
| | 15
| |
| | [[6L 9s]]
| |
| | 8:7
| |
| |-
| |
| | 16
| |
| | [[16edo]]
| |
| | equal
| |
| |-
| |
| | 17
| |
| | [[10L 7s]]
| |
| | rowspan="2" | 7:6
| |
| |-
| |
| | 18
| |
| | 4L 14s
| |
| |-
| |
| | 19
| |
| | [[17L 2s]]
| |
| | rowspan="4" | 6:5
| |
| |-
| |
| | 20
| |
| | 12L 8s
| |
| |-
| |
| | 21
| |
| | 7L 14s
| |
| |-
| |
| | 22
| |
| | 2L 20s
| |
| |-
| |
| | 23
| |
| | 20L 3s
| |
| | rowspan="5" | 5:4
| |
| |-
| |
| | 24
| |
| | 16L 8s
| |
| |-
| |
| | 25
| |
| | 12L 13s
| |
| |-
| |
| | 26
| |
| | 8L 18s
| |
| |-
| |
| | 27
| |
| | 4L 23s
| |
| |-
| |
| | 28
| |
| | [[28edo]]
| |
| | equal
| |
| |-
| |
| | 29
| |
| | 25L 4s
| |
| | rowspan="9" | 4:3
| |
| |-
| |
| | 30
| |
| | 22L 8s
| |
| |-
| |
| | 31
| |
| | 19L 12s
| |
| |-
| |
| | 32
| |
| | 16L 16s
| |
| |-
| |
| | 33
| |
| | 13L 20s
| |
| |-
| |
| | 34
| |
| | 10L 24s
| |
| |-
| |
| | 35
| |
| | 7L 28s
| |
| |-
| |
| | 36
| |
| | 4L 32s
| |
| |-
| |
| | 37
| |
| | 1L 36s
| |
| |-
| |
| | 38
| |
| | 36L 2s
| |
| | rowspan="18" | 3:2
| |
| |-
| |
| | 39
| |
| | 34L 5s
| |
| |-
| |
| | 40
| |
| | 32L 8s
| |
| |-
| |
| | 41
| |
| | 30L 11s
| |
| |-
| |
| | 42
| |
| | 28L 14s
| |
| |-
| |
| | 43
| |
| | 26L 17s
| |
| |-
| |
| | 44
| |
| | 24L 20s
| |
| |-
| |
| | 45
| |
| | 22L 23s
| |
| |-
| |
| | 46
| |
| | 20L 26s
| |
| |-
| |
| | 47
| |
| | 18L 29s
| |
| |-
| |
| | 48
| |
| | 16L 32s
| |
| |-
| |
| | 49
| |
| | 14L 35s
| |
| |-
| |
| | 50
| |
| | 12L 38s
| |
| |-
| |
| | 51
| |
| | 10L 41s
| |
| |-
| |
| | 52
| |
| | 8L 44s
| |
| |-
| |
| | 53
| |
| | 6L 47s
| |
| |-
| |
| | 54
| |
| | 4L 50s
| |
| |-
| |
| | 55
| |
| | 2L 53s
| |
| |-
| |
| | 56
| |
| | [[56edo]]
| |
| | equal
| |
| |-
| |
| | 57
| |
| | 55L 2s
| |
| | rowspan="33" | 2:1
| |
| |-
| |
| | 58
| |
| | 54L 4s
| |
| |-
| |
| | 59
| |
| | 53L 6s
| |
| |-
| |
| | 60
| |
| | 52L 8s
| |
| |-
| |
| | 61
| |
| | 51L 10s
| |
| |-
| |
| | 62
| |
| | 50L 12s
| |
| |-
| |
| | 63
| |
| | 49L 14s
| |
| |-
| |
| | 64
| |
| | 48L 16s
| |
| |-
| |
| | 65
| |
| | 47L 18s
| |
| |-
| |
| | 66
| |
| | 46L 20s
| |
| |-
| |
| | 67
| |
| | 45L 22s
| |
| |-
| |
| | 68
| |
| | 44L 24s
| |
| |-
| |
| | 69
| |
| | 43L 26s
| |
| |-
| |
| | 70
| |
| | 42L 28s
| |
| |-
| |
| | 71
| |
| | 41L 30s
| |
| |-
| |
| | 72
| |
| | 40L 32s
| |
| |-
| |
| | 73
| |
| | 39L 34s
| |
| |-
| |
| | 74
| |
| | 38L 36s
| |
| |-
| |
| | 75
| |
| | 37L 38s
| |
| |-
| |
| | 76
| |
| | 36L 40s
| |
| |-
| |
| | 77
| |
| | 35L 42s
| |
| |-
| |
| | 78
| |
| | 34L 44s
| |
| |-
| |
| | 79
| |
| | 33L 46s
| |
| |-
| |
| | 80
| |
| | 32L 48s
| |
| |-
| |
| | 81
| |
| | 31L 50s
| |
| |-
| |
| | 82
| |
| | 30L 52s
| |
| |-
| |
| | 83
| |
| | 29L 54s
| |
| |-
| |
| | 84
| |
| | 28L 56s
| |
| |-
| |
| | 85
| |
| | 27L 58s
| |
| |-
| |
| | 86
| |
| | 26L 60s
| |
| |-
| |
| | 87
| |
| | 25L 62s
| |
| |-
| |
| | 88
| |
| | 24L 64s
| |
| |-
| |
| | 89
| |
| | 23L 66s
| |
| |}
| |
|
| |
|
| == Music in 112EDO == | | == Intervals == |
| | {{Interval table}} |
|
| |
|
| * [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''] |
|
| |
|
| [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | | == See also == |
| | [[Skip fretting system 112 9 11]] |
| | [[Category:Listen]] |