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{{Infobox ET}}
{{Infobox ET}}
== Theory ==
'''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]].
'''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]].


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 ==
== Music ==
 
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 ==


* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Listen]]

Revision as of 21:16, 30 May 2023

← 111edo 112edo 113edo →
Prime factorization 24 × 7
Step size 10.7143 ¢ 
Fifth 66\112 (707.143 ¢) (→ 33\56)
Semitones (A1:m2) 14:6 (150 ¢ : 64.29 ¢)
Dual sharp fifth 66\112 (707.143 ¢) (→ 33\56)
Dual flat fifth 65\112 (696.429 ¢)
Dual major 2nd 19\112 (203.571 ¢)
Consistency limit 3
Distinct consistency limit 3

Theory

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 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of 7edo) and +9 fifths gives a close approximation to 17:21.

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

Approximation of odd harmonics in 112edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +5.19 -0.60 -4.54 -0.34 -4.89 -4.81 +4.59 +2.19 +2.49 +0.65 +3.87
Relative (%) +48.4 -5.6 -42.4 -3.2 -45.6 -44.9 +42.8 +20.4 +23.2 +6.0 +36.1
Steps
(reduced) 		178
(66) 		260
(36) 		314
(90) 		355
(19) 		387
(51) 		414
(78) 		438
(102) 		458
(10) 		476
(28) 		492
(44) 		507
(59)

Music

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