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 Approximate Ratios Ups and Downs Notation
0 0 1/1 D
1 10.811 ^D, v7E♭
2 21.622 78/77, 81/80 ^^D, v6E♭
3 32.432 50/49, 55/54, 56/55 ^3D, v5E♭
4 43.243 40/39, 77/75 ^4D, v4E♭
5 54.054 33/32, 65/63 ^5D, v3E♭
6 64.865 26/25, 27/26, 28/27, 80/77 ^6D, vvE♭
7 75.676 ^7D, vE♭
8 86.486 21/20, 81/77 ^8D, E♭
9 97.297 55/52 ^9D, v10E
10 108.108 16/15 ^10D, v9E
11 118.919 15/14, 77/72 D♯, v8E
12 129.73 14/13, 27/25 ^D♯, v7E
13 140.541 13/12 ^^D♯, v6E
14 151.351 12/11, 49/45 ^3D♯, v5E
15 162.162 11/10 ^4D♯, v4E
16 172.973 72/65 ^5D♯, v3E
17 183.784 10/9, 39/35 ^6D♯, vvE
18 194.595 28/25 ^7D♯, vE
19 205.405 9/8, 44/39 E
20 216.216 ^E, v7F
21 227.027 8/7 ^^E, v6F
22 237.838 55/48, 63/55 ^3E, v5F
23 248.649 15/13, 52/45, 81/70 ^4E, v4F
24 259.459 64/55, 65/56 ^5E, v3F
25 270.27 7/6 ^6E, vvF
26 281.081 33/28 ^7E, vF
27 291.892 13/11, 32/27, 77/65 F
28 302.703 25/21 ^F, v7G♭
29 313.514 6/5 ^^F, v6G♭
30 324.324 65/54, 77/64 ^3F, v5G♭
31 335.135 40/33, 63/52 ^4F, v4G♭
32 345.946 11/9, 39/32 ^5F, v3G♭
33 356.757 16/13, 27/22 ^6F, vvG♭
34 367.568 26/21 ^7F, vG♭
35 378.378 56/45, 81/65 ^8F, G♭
36 389.189 5/4 ^9F, v10G
37 400 63/50 ^10F, v9G
38 410.811 33/26, 80/63, 81/64 F♯, v8G
39 421.622 14/11 ^F♯, v7G
40 432.432 9/7, 50/39, 77/60 ^^F♯, v6G
41 443.243 ^3F♯, v5G
42 454.054 13/10 ^4F♯, v4G
43 464.865 55/42, 72/55 ^5F♯, v3G
44 475.676 ^6F♯, vvG
45 486.486 65/49 ^7F♯, vG
46 497.297 4/3 G
47 508.108 75/56 ^G, v7A♭
48 518.919 27/20, 35/26 ^^G, v6A♭
49 529.73 ^3G, v5A♭
50 540.541 15/11 ^4G, v4A♭
51 551.351 11/8 ^5G, v3A♭
52 562.162 18/13 ^6G, vvA♭
53 572.973 25/18, 39/28 ^7G, vA♭
54 583.784 7/5 ^8G, A♭
55 594.595 55/39 ^9G, v10A
56 605.405 78/55 ^10G, v9A
57 616.216 10/7, 77/54 G♯, v8A
58 627.027 36/25, 56/39 ^G♯, v7A
59 637.838 13/9, 75/52, 81/56 ^^G♯, v6A
60 648.649 16/11 ^3G♯, v5A
61 659.459 22/15 ^4G♯, v4A
62 670.27 81/55 ^5G♯, v3A
63 681.081 40/27, 52/35, 77/52 ^6G♯, vvA
64 691.892 ^7G♯, vA
65 702.703 3/2 A
66 713.514 ^A, v7B♭
67 724.324 ^^A, v6B♭
68 735.135 55/36, 75/49 ^3A, v5B♭
69 745.946 20/13, 77/50 ^4A, v4B♭
70 756.757 65/42 ^5A, v3B♭
71 767.568 14/9, 39/25, 81/52 ^6A, vvB♭
72 778.378 11/7 ^7A, vB♭
73 789.189 52/33, 63/40 ^8A, B♭
74 800 ^9A, v10B
75 810.811 8/5 ^10A, v9B
76 821.622 45/28, 77/48 A♯, v8B
77 832.432 21/13, 81/50 ^A♯, v7B
78 843.243 13/8, 44/27 ^^A♯, v6B
79 854.054 18/11, 64/39 ^3A♯, v5B
80 864.865 33/20 ^4A♯, v4B
81 875.676 ^5A♯, v3B
82 886.486 5/3 ^6A♯, vvB
83 897.297 42/25 ^7A♯, vB
84 908.108 22/13, 27/16 B
85 918.919 56/33 ^B, v7C
86 929.73 12/7, 77/45 ^^B, v6C
87 940.541 55/32 ^3B, v5C
88 951.351 26/15, 45/26 ^4B, v4C
89 962.162 ^5B, v3C
90 972.973 7/4 ^6B, vvC
91 983.784 ^7B, vC
92 994.595 16/9, 39/22 C
93 1005.405 25/14 ^C, v7D♭
94 1016.216 9/5, 70/39 ^^C, v6D♭
95 1027.027 65/36 ^3C, v5D♭
96 1037.838 20/11 ^4C, v4D♭
97 1048.649 11/6 ^5C, v3D♭
98 1059.459 24/13, 81/44 ^6C, vvD♭
99 1070.27 13/7, 50/27 ^7C, vD♭
100 1081.081 28/15 ^8C, D♭
101 1091.892 15/8 ^9C, v10D
102 1102.703 ^10C, v9D
103 1113.514 40/21 C♯, v8D
104 1124.324 ^C♯, v7D
105 1135.135 25/13, 27/14, 52/27, 77/40 ^^C♯, v6D
106 1145.946 64/33 ^3C♯, v5D
107 1156.757 39/20 ^4C♯, v4D
108 1167.568 49/25, 55/28 ^5C♯, v3D
109 1178.378 77/39 ^6C♯, vvD
110 1189.189 ^7C♯, vD
111 1200 2/1 D

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
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