111edo

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← 110edo111edo112edo →
Prime factorization 3 × 37
Step size 10.8108¢ 
Fifth 65\111 (702.703¢)
Semitones (A1:m2) 11:8 (118.9¢ : 86.49¢)
Consistency limit 21
Distinct consistency limit 15

111 equal divisions of the octave (abbreviated 111edo or 111ed2), also called 111-tone equal temperament (111tet) or 111 equal temperament (111et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 111 equal parts of about 10.8 ¢ each. Each step represents a frequency ratio of 21/111, or the 111th root of 2.

Theory

111edo is consistent through to the 21-odd-limit, and is the smallest edo uniquely consistent through the 15-odd-limit, marking it as an important higher limit tuning. It has a sharp tendency, with harmonics 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for 5, 7, 11, and 13 with 37edo.

It is also significant for lower limits, especially in terms of what it tempers out in its patent val; for example, it tempers out 176/175 and gives an excellent optimal patent val for the corresponding 11-limit rank-4 temperament.

In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the quartisma.

It is a particularly good tuning for the 11- or 13-limit versions of semisept, the 31 & 80 temperament, and buzzard, the 53 & 58 temperament. Gene Ward Smith's trio in #Music section is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

Prime harmonics

Approximation of prime harmonics in 111edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.75 +2.88 +4.15 +0.03 +2.72 +3.15 +5.19 -1.25 -2.55 +0.91
Relative (%) +0.0 +6.9 +26.6 +38.4 +0.3 +25.1 +29.2 +48.0 -11.5 -23.6 +8.4
Steps
(reduced)
111
(0)
176
(65)
258
(36)
312
(90)
384
(51)
411
(78)
454
(10)
472
(28)
502
(58)
539
(95)
550
(106)

Subsets and supersets

333edo, which slices the step of 111edo in three, is a significant tuning.

Intervals

Steps Cents Ups and Downs Notation Approximate Ratios
0 0 D 1/1
1 10.8108 ^D, v7E♭
2 21.6216 ^^D, v6E♭ 78/77, 81/80
3 32.4324 ^3D, v5E♭ 50/49, 55/54, 56/55
4 43.2432 ^4D, v4E♭ 40/39, 77/75
5 54.0541 ^5D, v3E♭ 33/32, 65/63
6 64.8649 ^6D, vvE♭ 26/25, 27/26, 28/27, 80/77
7 75.6757 ^7D, vE♭
8 86.4865 ^8D, E♭ 21/20, 81/77
9 97.2973 ^9D, v10E 55/52
10 108.108 ^10D, v9E 16/15
11 118.919 D♯, v8E 15/14, 77/72
12 129.73 ^D♯, v7E 14/13, 27/25
13 140.541 ^^D♯, v6E 13/12
14 151.351 ^3D♯, v5E 12/11, 49/45
15 162.162 ^4D♯, v4E 11/10
16 172.973 ^5D♯, v3E 72/65
17 183.784 ^6D♯, vvE 10/9, 39/35
18 194.595 ^7D♯, vE 28/25
19 205.405 E 9/8, 44/39
20 216.216 ^E, v7F
21 227.027 ^^E, v6F 8/7
22 237.838 ^3E, v5F 55/48, 63/55
23 248.649 ^4E, v4F 15/13, 52/45, 81/70
24 259.459 ^5E, v3F 64/55, 65/56
25 270.27 ^6E, vvF 7/6
26 281.081 ^7E, vF 33/28
27 291.892 F 13/11, 32/27, 77/65
28 302.703 ^F, v7G♭ 25/21
29 313.514 ^^F, v6G♭ 6/5
30 324.324 ^3F, v5G♭ 65/54, 77/64
31 335.135 ^4F, v4G♭ 40/33, 63/52
32 345.946 ^5F, v3G♭ 11/9, 39/32
33 356.757 ^6F, vvG♭ 16/13, 27/22
34 367.568 ^7F, vG♭ 26/21
35 378.378 ^8F, G♭ 56/45, 81/65
36 389.189 ^9F, v10G 5/4
37 400 ^10F, v9G 63/50
38 410.811 F♯, v8G 33/26, 80/63, 81/64
39 421.622 ^F♯, v7G 14/11
40 432.432 ^^F♯, v6G 9/7, 50/39, 77/60
41 443.243 ^3F♯, v5G
42 454.054 ^4F♯, v4G 13/10
43 464.865 ^5F♯, v3G 55/42, 72/55
44 475.676 ^6F♯, vvG
45 486.486 ^7F♯, vG 65/49
46 497.297 G 4/3
47 508.108 ^G, v7A♭ 75/56
48 518.919 ^^G, v6A♭ 27/20, 35/26
49 529.73 ^3G, v5A♭
50 540.541 ^4G, v4A♭ 15/11
51 551.351 ^5G, v3A♭ 11/8
52 562.162 ^6G, vvA♭ 18/13
53 572.973 ^7G, vA♭ 25/18, 39/28
54 583.784 ^8G, A♭ 7/5
55 594.595 ^9G, v10A 55/39
56 605.405 ^10G, v9A 78/55
57 616.216 G♯, v8A 10/7, 77/54
58 627.027 ^G♯, v7A 36/25, 56/39
59 637.838 ^^G♯, v6A 13/9, 75/52, 81/56
60 648.649 ^3G♯, v5A 16/11
61 659.459 ^4G♯, v4A 22/15
62 670.27 ^5G♯, v3A 81/55
63 681.081 ^6G♯, vvA 40/27, 52/35, 77/52
64 691.892 ^7G♯, vA
65 702.703 A 3/2
66 713.514 ^A, v7B♭
67 724.324 ^^A, v6B♭
68 735.135 ^3A, v5B♭ 55/36, 75/49
69 745.946 ^4A, v4B♭ 20/13, 77/50
70 756.757 ^5A, v3B♭ 65/42
71 767.568 ^6A, vvB♭ 14/9, 39/25, 81/52
72 778.378 ^7A, vB♭ 11/7
73 789.189 ^8A, B♭ 52/33, 63/40
74 800 ^9A, v10B
75 810.811 ^10A, v9B 8/5
76 821.622 A♯, v8B 45/28, 77/48
77 832.432 ^A♯, v7B 21/13, 81/50
78 843.243 ^^A♯, v6B 13/8, 44/27
79 854.054 ^3A♯, v5B 18/11, 64/39
80 864.865 ^4A♯, v4B 33/20
81 875.676 ^5A♯, v3B
82 886.486 ^6A♯, vvB 5/3
83 897.297 ^7A♯, vB 42/25
84 908.108 B 22/13, 27/16
85 918.919 ^B, v7C 56/33
86 929.73 ^^B, v6C 12/7, 77/45
87 940.541 ^3B, v5C 55/32
88 951.351 ^4B, v4C 26/15, 45/26
89 962.162 ^5B, v3C
90 972.973 ^6B, vvC 7/4
91 983.784 ^7B, vC
92 994.595 C 16/9, 39/22
93 1005.41 ^C, v7D♭ 25/14
94 1016.22 ^^C, v6D♭ 9/5, 70/39
95 1027.03 ^3C, v5D♭ 65/36
96 1037.84 ^4C, v4D♭ 20/11
97 1048.65 ^5C, v3D♭ 11/6
98 1059.46 ^6C, vvD♭ 24/13, 81/44
99 1070.27 ^7C, vD♭ 13/7, 50/27
100 1081.08 ^8C, D♭ 28/15
101 1091.89 ^9C, v10D 15/8
102 1102.7 ^10C, v9D
103 1113.51 C♯, v8D 40/21
104 1124.32 ^C♯, v7D
105 1135.14 ^^C♯, v6D 25/13, 27/14, 52/27, 77/40
106 1145.95 ^3C♯, v5D 64/33
107 1156.76 ^4C♯, v4D 39/20
108 1167.57 ^5C♯, v3D 49/25, 55/28
109 1178.38 ^6C♯, vvD 77/39
110 1189.19 ^7C♯, vD
111 1200 D 2/1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [176 -111 [111 176]] -0.236 0.236 2.18
2.3.5 78732/78125, 67108864/66430125 [111 176 258]] -0.570 0.510 4.72
2.3.5.7 1728/1715, 3136/3125, 5120/5103 [111 176 258 312]] -0.797 0.591 5.47
2.3.5.7.11 176/175, 540/539, 1331/1323, 5120/5103 [111 176 258 312 384]] -0.639 0.615 5.69
2.3.5.7.11.13 176/175, 351/350, 540/539, 676/675, 1331/1323 [111 176 258 312 384 411]] -0.655 0.562 5.21
2.3.5.7.11.13.17 176/175, 256/255, 351/350, 442/441, 540/539, 715/714 [111 176 258 312 384 411 454]] -0.672 0.523 4.84
2.3.5.7.11.13.17.19 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475 [111 176 258 312 384 411 454 472]] -0.740 0.521 4.83
  • 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating 94 and 103h before being superseded by 121i.

Rank-2 temperaments

Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperament
1 11\111 118.92 15/14 Subsedia
1 13\111 140.54 13/12 Quanic
1 14\111 151.35 12/11 Browser
1 16\111 172.97 400/363 Undetrita
1 20\111 216.22 17/15 Tremka
1 23\111 248.65 15/13 Hemikwai
1 31\111 335.14 17/14 Cohemimabila
1 35\111 378.38 56/45 Subpental
1 41\111 443.24 22/17 Warrior
1 43\111 464.86 17/13 Semisept
1 44\111 475.68 21/16 Buzzard
1 46\111 497.30 4/3 Kwai
1 49\111 529.73 19/14 Tuskaloosa
1 55\111 594.59 55/39 Gaster
3 7\111 75.68 24/23 Terture
3 12\111 129.73 14/13 Trimabila
3 13\111 140.54 243/224 Septichrome
3 17\111 183.55 10/9 Mirkat
3 23\111
(14\111)
248.65
(151.35)
15/13
(12/11)
Hemimist
3 46\111
(9\111)
497.30
(97.30)
4/3
(18/17~19/18)
Misty

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

  • Direct sunlight (subset of Sensi[19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
  • Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

Music

Gene Ward Smith
  • Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) – SoundCloud | details | play – guanyin[22] in 111edo tuning