115edo: Difference between revisions

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'''115edo''' is the [[EDO|equal division of the octave]] into 115 parts of 10.4348 cents each. It tempers out 1594323/1562500 (unicorn comma) and 2109375/2097152 (semicomma) in the 5-limit, as well as 43046721/41943040 and 30958682112/30517578125 ([[Trisedodge family|trisedodge comma]]); 225/224, 1728/1715, and 9920232/9765625 in the 7-limit. 115edo supports the [[Semicomma family|newspeak temperament]], tempering out 441/440, 1375/1372, and 1944/1925 in the 11-limit. The wizardharry comma, 4000/3993 is also tempered out in 115edo.
{{Infobox ET}}
{{ED intro}}


== Theory ==
The equal temperament [[tempering out|tempers out]] 1594323/1562500 ([[unicorn comma]]) and 2109375/2097152 ([[semicomma]]) in the 5-limit, as well as 43046721/41943040 ([[python comma]]) and {{monzo| 19 10 -15 }} ([[trisedodge comma]]); [[225/224]], [[1728/1715]], and 9920232/9765625 in the 7-limit. Using the [[patent val]], it [[support]]s the [[semicomma family #Newspeak|newspeak temperament]], tempering out [[441/440]], [[1375/1372]], and 1944/1925 in the 11-limit. The wizardharry comma, [[4000/3993]] is also tempered out.


Since 115edo has a step of 10.4348 cents, it also allows one to use its MOS scales as circulating temperaments.
=== Odd harmonics ===
{| class="wikitable"
{{Harmonics in equal|115}}
|+Circulating temperaments in 115edo
 
!Tones
== Intervals ==
!Pattern
{{Interval table}}
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[1L 5s]]
|20:19
|-
|7
|[[3L 4s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[7L 2s]]
|13:12
|-
|10
|[[5L 5s]]
|12:11
|-
|11
|[[5L 6s]]
|11:10
|-
|12
|[[7L 5s]]
|10:9
|-
|13
|[[11L 2s]]
| rowspan="2" |9:8
|-
|14
|[[3L 11s]]
|-
|15
|[[10L 5s]]
| rowspan="2" |8:7
|-
|16
|[[3L 13s]]
|-
|17
|13L 4s
| rowspan="3" |7:6
|-
|18
|7L 11s
|-
|19
|1L 18s
|-
|20
|15L 5s
| rowspan="3" |6:5
|-
|21
|[[10L 11s]]
|-
|22
|[[5L 17s]]
|-
|23
|[[23edo]]
|equal
|-
|24
|[[19L 5s]]
| rowspan="5" |5:4
|-
|25
|15L 10s
|-
|26
|11L 15s
|-
|27
|7L 20s
|-
|28
|3L 25s
|-
|29
|28L 1s
| rowspan="10" |4:3
|-
|30
|25L 5s
|-
|31
|22L 9s
|-
|32
|19L 13s
|-
|33
|16L 17s
|-
|34
|13L 21s
|-
|35
|10L 25s
|-
|36
|7L 29s
|-
|37
|4L 33s
|-
|38
|1L 37s
|-
|39
|37L 2s
| rowspan="19" |3:2
|-
|40
|35L 5s
|-
|41
|33L 8s
|-
|42
|31L 11s
|-
|43
|29L 14s
|-
|44
|27L 17s
|-
|45
|25L 20s
|-
|46
|23L 23s
|-
|47
|21L 26s
|-
|48
|19L 29s
|-
|49
|17L 32s
|-
|50
|15L 35s
|-
|51
|13L 38s
|-
|52
|11L 41s
|-
|53
|9L 44s
|-
|54
|7L 47s
|-
|55
|5L 50s
|-
|56
|3L 53s
|-
|57
|1L 56s
|-
|58
|57L 1s
| rowspan="34" |2:1
|-
|59
|56L 3s
|-
|60
|55L 5s
|-
|61
|54L 7s
|-
|62
|53L 9s
|-
|63
|52L 11s
|-
|64
|51L 13s
|-
|65
|50L 15s
|-
|66
|49L 17s
|-
|67
|48L 19s
|-
|68
|47L 21s
|-
|69
|46L 23s
|-
|70
|45L 25s
|-
|71
|44L 27s
|-
|72
|43L 29s
|-
|73
|42L 31s
|-
|74
|41L 33s
|-
|75
|40L 35s
|-
|76
|39L 37s
|-
|77
|38L 39s
|-
|78
|37L 41s
|-
|79
|36L 43s
|-
|80
|35L 45s
|-
|81
|34L 47s
|-
|82
|33L 49s
|-
|83
|32L 51s
|-
|84
|31L 53s
|-
|85
|30L 55s
|-
|86
|29L 57s
|-
|87
|28L 59s
|-
|88
|27L 61s
|-
|89
|26L 63s
|-
|90
|25L 65s
|-
|91
|24L 67s
|}
[[Category:Equal divisions of the octave]]

Latest revision as of 13:25, 6 March 2025

← 114edo 115edo 116edo →
Prime factorization 5 × 23
Step size 10.4348 ¢ 
Fifth 67\115 (699.13 ¢)
Semitones (A1:m2) 9:10 (93.91 ¢ : 104.3 ¢)
Consistency limit 7
Distinct consistency limit 7

115 equal divisions of the octave (abbreviated 115edo or 115ed2), also called 115-tone equal temperament (115tet) or 115 equal temperament (115et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 115 equal parts of about 10.4 ¢ each. Each step represents a frequency ratio of 21/115, or the 115th root of 2.

Theory

The equal temperament tempers out 1594323/1562500 (unicorn comma) and 2109375/2097152 (semicomma) in the 5-limit, as well as 43046721/41943040 (python comma) and [19 10 -15⟩ (trisedodge comma); 225/224, 1728/1715, and 9920232/9765625 in the 7-limit. Using the patent val, it supports the newspeak temperament, tempering out 441/440, 1375/1372, and 1944/1925 in the 11-limit. The wizardharry comma, 4000/3993 is also tempered out.

Odd harmonics

Approximation of odd harmonics in 115edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -2.82 -0.23 +1.61 +4.79 +1.73 +4.69 -3.05 -0.61 +5.10 -1.22 -2.19
Relative (%) -27.1 -2.2 +15.4 +45.9 +16.5 +44.9 -29.2 -5.8 +48.8 -11.7 -21.0
Steps
(reduced) 		182
(67) 		267
(37) 		323
(93) 		365
(20) 		398
(53) 		426
(81) 		449
(104) 		470
(10) 		489
(29) 		505
(45) 		520
(60)

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 10.4 ^D, E♭♭
2 20.9 ^^D, ^E♭♭
3 31.3 ^3D, ^^E♭♭
4 41.7 40/39, 41/40, 42/41, 43/42, 44/43 ^4D, ^3E♭♭
5 52.2 33/32, 34/33, 35/34 v4D♯, ^4E♭♭
6 62.6 29/28 v3D♯, v4E♭
7 73 24/23 vvD♯, v3E♭
8 83.5 21/20, 43/41 vD♯, vvE♭
9 93.9 D♯, vE♭
10 104.3 17/16 ^D♯, E♭
11 114.8 31/29, 46/43, 47/44 ^^D♯, ^E♭
12 125.2 43/40 ^3D♯, ^^E♭
13 135.7 40/37 ^4D♯, ^3E♭
14 146.1 25/23, 37/34 v4D𝄪, ^4E♭
15 156.5 23/21, 35/32 v3D𝄪, v4E
16 167 11/10, 43/39 vvD𝄪, v3E
17 177.4 31/28, 41/37 vD𝄪, vvE
18 187.8 29/26, 39/35 D𝄪, vE
19 198.3 28/25, 37/33, 46/41 E
20 208.7 35/31, 44/39 ^E, F♭
21 219.1 42/37 ^^E, ^F♭
22 229.6 8/7 ^3E, ^^F♭
23 240 23/20 ^4E, ^3F♭
24 250.4 37/32 v4E♯, ^4F♭
25 260.9 43/37 v3E♯, v4F
26 271.3 48/41 vvE♯, v3F
27 281.7 20/17 vE♯, vvF
28 292.2 E♯, vF
29 302.6 25/21, 31/26 F
30 313 ^F, G♭♭
31 323.5 35/29, 41/34, 47/39 ^^F, ^G♭♭
32 333.9 40/33 ^3F, ^^G♭♭
33 344.3 39/32 ^4F, ^3G♭♭
34 354.8 43/35 v4F♯, ^4G♭♭
35 365.2 21/17 v3F♯, v4G♭
36 375.7 41/33, 46/37 vvF♯, v3G♭
37 386.1 5/4 vF♯, vvG♭
38 396.5 39/31, 44/35 F♯, vG♭
39 407 43/34 ^F♯, G♭
40 417.4 14/11 ^^F♯, ^G♭
41 427.8 32/25, 41/32 ^3F♯, ^^G♭
42 438.3 ^4F♯, ^3G♭
43 448.7 48/37 v4F𝄪, ^4G♭
44 459.1 30/23, 43/33 v3F𝄪, v4G
45 469.6 21/16, 38/29 vvF𝄪, v3G
46 480 29/22, 33/25 vF𝄪, vvG
47 490.4 F𝄪, vG
48 500.9 G
49 511.3 39/29, 43/32, 47/35 ^G, A♭♭
50 521.7 23/17 ^^G, ^A♭♭
51 532.2 34/25 ^3G, ^^A♭♭
52 542.6 26/19, 41/30 ^4G, ^3A♭♭
53 553 11/8 v4G♯, ^4A♭♭
54 563.5 v3G♯, v4A♭
55 573.9 39/28, 46/33 vvG♯, v3A♭
56 584.3 7/5 vG♯, vvA♭
57 594.8 31/22 G♯, vA♭
58 605.2 44/31 ^G♯, A♭
59 615.7 10/7 ^^G♯, ^A♭
60 626.1 33/23 ^3G♯, ^^A♭
61 636.5 ^4G♯, ^3A♭
62 647 16/11 v4G𝄪, ^4A♭
63 657.4 19/13 v3G𝄪, v4A
64 667.8 25/17 vvG𝄪, v3A
65 678.3 34/23, 37/25 vG𝄪, vvA
66 688.7 G𝄪, vA
67 699.1 A
68 709.6 ^A, B♭♭
69 720 44/29, 47/31 ^^A, ^B♭♭
70 730.4 29/19, 32/21 ^3A, ^^B♭♭
71 740.9 23/15, 43/28 ^4A, ^3B♭♭
72 751.3 37/24 v4A♯, ^4B♭♭
73 761.7 v3A♯, v4B♭
74 772.2 25/16 vvA♯, v3B♭
75 782.6 11/7 vA♯, vvB♭
76 793 A♯, vB♭
77 803.5 35/22 ^A♯, B♭
78 813.9 8/5 ^^A♯, ^B♭
79 824.3 37/23 ^3A♯, ^^B♭
80 834.8 34/21, 47/29 ^4A♯, ^3B♭
81 845.2 v4A𝄪, ^4B♭
82 855.7 41/25 v3A𝄪, v4B
83 866.1 33/20 vvA𝄪, v3B
84 876.5 vA𝄪, vvB
85 887 A𝄪, vB
86 897.4 42/25, 47/28 B
87 907.8 ^B, C♭
88 918.3 17/10 ^^B, ^C♭
89 928.7 41/24 ^3B, ^^C♭
90 939.1 43/25 ^4B, ^3C♭
91 949.6 v4B♯, ^4C♭
92 960 40/23 v3B♯, v4C
93 970.4 7/4 vvB♯, v3C
94 980.9 37/21 vB♯, vvC
95 991.3 39/22 B♯, vC
96 1001.7 25/14, 41/23 C
97 1012.2 ^C, D♭♭
98 1022.6 ^^C, ^D♭♭
99 1033 20/11 ^3C, ^^D♭♭
100 1043.5 42/23 ^4C, ^3D♭♭
101 1053.9 46/25 v4C♯, ^4D♭♭
102 1064.3 37/20 v3C♯, v4D♭
103 1074.8 vvC♯, v3D♭
104 1085.2 43/23 vC♯, vvD♭
105 1095.7 32/17 C♯, vD♭
106 1106.1 ^C♯, D♭
107 1116.5 40/21 ^^C♯, ^D♭
108 1127 23/12 ^3C♯, ^^D♭
109 1137.4 ^4C♯, ^3D♭
110 1147.8 33/17 v4C𝄪, ^4D♭
111 1158.3 39/20, 41/21, 43/22 v3C𝄪, v4D
112 1168.7 vvC𝄪, v3D
113 1179.1 vC𝄪, vvD
114 1189.6 C𝄪, vD
115 1200 2/1 D
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