111edo

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← 110edo 111edo 112edo →
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 distinctly 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. Since 111 = 3 × 37, 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

Since 111 factors into 3 × 37, 111edo contains 3edo and 37edo as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s 13-odd-limit. 333edo, which slices the step of 111edo in three, is a significant tuning.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 10.8 ^D, ^4E♭♭
2 21.6 ^^D, ^5E♭♭
3 32.4 ^3D, v5E♭
4 43.2 39/38, 40/39, 41/40, 42/41 ^4D, v4E♭
5 54.1 32/31, 33/32 ^5D, v3E♭
6 64.9 27/26, 28/27 v5D♯, vvE♭
7 75.7 23/22, 24/23, 47/45 v4D♯, vE♭
8 86.5 21/20, 41/39 v3D♯, E♭
9 97.3 18/17 vvD♯, ^E♭
10 108.1 33/31 vD♯, ^^E♭
11 118.9 15/14, 46/43 D♯, ^3E♭
12 129.7 14/13, 41/38 ^D♯, ^4E♭
13 140.5 13/12, 38/35 ^^D♯, ^5E♭
14 151.4 12/11 ^3D♯, v5E
15 162.2 45/41 ^4D♯, v4E
16 173 21/19 ^5D♯, v3E
17 183.8 10/9 v5D𝄪, vvE
18 194.6 19/17, 28/25, 47/42 v4D𝄪, vE
19 205.4 9/8 E
20 216.2 17/15 ^E, ^4F♭
21 227 41/36 ^^E, ^5F♭
22 237.8 31/27, 39/34, 47/41 ^3E, v5F
23 248.6 15/13 ^4E, v4F
24 259.5 36/31, 43/37 ^5E, v3F
25 270.3 v5E♯, vvF
26 281.1 20/17, 47/40 v4E♯, vF
27 291.9 45/38 F
28 302.7 25/21, 31/26 ^F, ^4G♭♭
29 313.5 6/5 ^^F, ^5G♭♭
30 324.3 41/34, 47/39 ^3F, v5G♭
31 335.1 17/14, 40/33 ^4F, v4G♭
32 345.9 11/9 ^5F, v3G♭
33 356.8 v5F♯, vvG♭
34 367.6 21/17, 26/21, 47/38 v4F♯, vG♭
35 378.4 46/37 v3F♯, G♭
36 389.2 vvF♯, ^G♭
37 400 29/23, 34/27 vF♯, ^^G♭
38 410.8 19/15, 33/26 F♯, ^3G♭
39 421.6 37/29 ^F♯, ^4G♭
40 432.4 ^^F♯, ^5G♭
41 443.2 31/24, 40/31 ^3F♯, v5G
42 454.1 13/10 ^4F♯, v4G
43 464.9 17/13 ^5F♯, v3G
44 475.7 25/19 v5F𝄪, vvG
45 486.5 45/34 v4F𝄪, vG
46 497.3 4/3 G
47 508.1 ^G, ^4A♭♭
48 518.9 27/20, 31/23 ^^G, ^5A♭♭
49 529.7 19/14 ^3G, v5A♭
50 540.5 41/30 ^4G, v4A♭
51 551.4 11/8 ^5G, v3A♭
52 562.2 18/13, 47/34 v5G♯, vvA♭
53 573 32/23, 39/28, 46/33 v4G♯, vA♭
54 583.8 7/5 v3G♯, A♭
55 594.6 31/22 vvG♯, ^A♭
56 605.4 44/31 vG♯, ^^A♭
57 616.2 10/7 G♯, ^3A♭
58 627 23/16, 33/23 ^G♯, ^4A♭
59 637.8 13/9 ^^G♯, ^5A♭
60 648.6 16/11 ^3G♯, v5A
61 659.5 41/28 ^4G♯, v4A
62 670.3 28/19 ^5G♯, v3A
63 681.1 40/27, 43/29, 46/31 v5G𝄪, vvA
64 691.9 v4G𝄪, vA
65 702.7 3/2 A
66 713.5 ^A, ^4B♭♭
67 724.3 38/25, 41/27 ^^A, ^5B♭♭
68 735.1 26/17 ^3A, v5B♭
69 745.9 20/13 ^4A, v4B♭
70 756.8 31/20 ^5A, v3B♭
71 767.6 v5A♯, vvB♭
72 778.4 47/30 v4A♯, vB♭
73 789.2 30/19, 41/26 v3A♯, B♭
74 800 27/17, 46/29 vvA♯, ^B♭
75 810.8 vA♯, ^^B♭
76 821.6 37/23, 45/28 A♯, ^3B♭
77 832.4 21/13, 34/21 ^A♯, ^4B♭
78 843.2 ^^A♯, ^5B♭
79 854.1 18/11 ^3A♯, v5B
80 864.9 28/17, 33/20 ^4A♯, v4B
81 875.7 ^5A♯, v3B
82 886.5 5/3 v5A𝄪, vvB
83 897.3 42/25, 47/28 v4A𝄪, vB
84 908.1 B
85 918.9 17/10 ^B, ^4C♭
86 929.7 ^^B, ^5C♭
87 940.5 31/18 ^3B, v5C
88 951.4 26/15, 45/26 ^4B, v4C
89 962.2 ^5B, v3C
90 973 v5B♯, vvC
91 983.8 30/17 v4B♯, vC
92 994.6 16/9 C
93 1005.4 25/14, 34/19 ^C, ^4D♭♭
94 1016.2 9/5 ^^C, ^5D♭♭
95 1027 38/21, 47/26 ^3C, v5D♭
96 1037.8 ^4C, v4D♭
97 1048.6 11/6 ^5C, v3D♭
98 1059.5 24/13, 35/19 v5C♯, vvD♭
99 1070.3 13/7 v4C♯, vD♭
100 1081.1 28/15, 43/23 v3C♯, D♭
101 1091.9 47/25 vvC♯, ^D♭
102 1102.7 17/9 vC♯, ^^D♭
103 1113.5 40/21 C♯, ^3D♭
104 1124.3 23/12, 44/23 ^C♯, ^4D♭
105 1135.1 27/14 ^^C♯, ^5D♭
106 1145.9 31/16 ^3C♯, v5D
107 1156.8 39/20, 41/21 ^4C♯, v4D
108 1167.6 ^5C♯, v3D
109 1178.4 v5C𝄪, vvD
110 1189.2 v4C𝄪, vD
111 1200 2/1 D

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
655zpi 111.059577998833 10.8050113427643 9.038544 1.394739 18.041165 111edo 1199.35625904684 22 16

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 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
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