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. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to 37edo, with which it shares the mappings for 5, 7, 11, and 13.

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 +7 +27 +38 +0 +25 +29 +48 -12 -24 +8
Steps
(reduced) 		111
(0) 		176
(65) 		258
(36) 		312
(90) 		384
(51) 		411
(78) 		454
(10) 		472
(28) 		502
(58) 		539
(95) 		550
(106)

Intervals

Steps Cents Ups and downs notation Approximate ratios
0 0 D 1/1
1 10.8108 ↑D, ↓7E♭
2 21.6216 ↑↑D, ↓6E♭ 78/77, 81/80
3 32.4324 ↑3D, ↓5E♭ 50/49, 55/54, 56/55
4 43.2432 ↑4D, ↓4E♭ 40/39, 77/75
5 54.0541 ↑5D, ↓3E♭ 33/32, 65/63
6 64.8649 ↑6D, ↓↓E♭ 26/25, 27/26, 28/27, 80/77
7 75.6757 ↑7D, ↓E♭
8 86.4865 ↑8D, E♭ 21/20, 81/77
9 97.2973 ↑9D, ↓10E 55/52
10 108.108 ↑10D, ↓9E 16/15
11 118.919 D♯, ↓8E 15/14, 77/72
12 129.73 ↑D♯, ↓7E 14/13, 27/25
13 140.541 ↑↑D♯, ↓6E 13/12
14 151.351 ↑3D♯, ↓5E 12/11, 49/45
15 162.162 ↑4D♯, ↓4E 11/10
16 172.973 ↑5D♯, ↓3E 72/65
17 183.784 ↑6D♯, ↓↓E 10/9, 39/35
18 194.595 ↑7D♯, ↓E 28/25
19 205.405 E 9/8, 44/39
20 216.216 ↑E, ↓7F
21 227.027 ↑↑E, ↓6F 8/7
22 237.838 ↑3E, ↓5F 55/48, 63/55
23 248.649 ↑4E, ↓4F 15/13, 52/45, 81/70
24 259.459 ↑5E, ↓3F 64/55, 65/56
25 270.27 ↑6E, ↓↓F 7/6
26 281.081 ↑7E, ↓F 33/28
27 291.892 F 13/11, 32/27, 77/65
28 302.703 ↑F, ↓7G♭ 25/21
29 313.514 ↑↑F, ↓6G♭ 6/5
30 324.324 ↑3F, ↓5G♭ 65/54, 77/64
31 335.135 ↑4F, ↓4G♭ 40/33, 63/52
32 345.946 ↑5F, ↓3G♭ 11/9, 39/32
33 356.757 ↑6F, ↓↓G♭ 16/13, 27/22
34 367.568 ↑7F, ↓G♭ 26/21
35 378.378 ↑8F, G♭ 56/45, 81/65
36 389.189 ↑9F, ↓10G 5/4
37 400 ↑10F, ↓9G 63/50
38 410.811 F♯, ↓8G 33/26, 80/63, 81/64
39 421.622 ↑F♯, ↓7G 14/11
40 432.432 ↑↑F♯, ↓6G 9/7, 50/39, 77/60
41 443.243 ↑3F♯, ↓5G
42 454.054 ↑4F♯, ↓4G 13/10
43 464.865 ↑5F♯, ↓3G 55/42, 72/55
44 475.676 ↑6F♯, ↓↓G
45 486.486 ↑7F♯, ↓G 65/49
46 497.297 G 4/3
47 508.108 ↑G, ↓7A♭ 75/56
48 518.919 ↑↑G, ↓6A♭ 27/20, 35/26
49 529.73 ↑3G, ↓5A♭
50 540.541 ↑4G, ↓4A♭ 15/11
51 551.351 ↑5G, ↓3A♭ 11/8
52 562.162 ↑6G, ↓↓A♭ 18/13
53 572.973 ↑7G, ↓A♭ 25/18, 39/28
54 583.784 ↑8G, A♭ 7/5
55 594.595 ↑9G, ↓10A 55/39
56 605.405 ↑10G, ↓9A 78/55
57 616.216 G♯, ↓8A 10/7, 77/54
58 627.027 ↑G♯, ↓7A 36/25, 56/39
59 637.838 ↑↑G♯, ↓6A 13/9, 75/52, 81/56
60 648.649 ↑3G♯, ↓5A 16/11
61 659.459 ↑4G♯, ↓4A 22/15
62 670.27 ↑5G♯, ↓3A 81/55
63 681.081 ↑6G♯, ↓↓A 40/27, 52/35, 77/52
64 691.892 ↑7G♯, ↓A
65 702.703 A 3/2
66 713.514 ↑A, ↓7B♭
67 724.324 ↑↑A, ↓6B♭
68 735.135 ↑3A, ↓5B♭ 55/36, 75/49
69 745.946 ↑4A, ↓4B♭ 20/13, 77/50
70 756.757 ↑5A, ↓3B♭ 65/42
71 767.568 ↑6A, ↓↓B♭ 14/9, 39/25, 81/52
72 778.378 ↑7A, ↓B♭ 11/7
73 789.189 ↑8A, B♭ 52/33, 63/40
74 800 ↑9A, ↓10B
75 810.811 ↑10A, ↓9B 8/5
76 821.622 A♯, ↓8B 45/28, 77/48
77 832.432 ↑A♯, ↓7B 21/13, 81/50
78 843.243 ↑↑A♯, ↓6B 13/8, 44/27
79 854.054 ↑3A♯, ↓5B 18/11, 64/39
80 864.865 ↑4A♯, ↓4B 33/20
81 875.676 ↑5A♯, ↓3B
82 886.486 ↑6A♯, ↓↓B 5/3
83 897.297 ↑7A♯, ↓B 42/25
84 908.108 B 22/13, 27/16
85 918.919 ↑B, ↓7C 56/33
86 929.73 ↑↑B, ↓6C 12/7, 77/45
87 940.541 ↑3B, ↓5C 55/32
88 951.351 ↑4B, ↓4C 26/15, 45/26
89 962.162 ↑5B, ↓3C
90 972.973 ↑6B, ↓↓C 7/4
91 983.784 ↑7B, ↓C
92 994.595 C 16/9, 39/22
93 1005.41 ↑C, ↓7D♭ 25/14
94 1016.22 ↑↑C, ↓6D♭ 9/5, 70/39
95 1027.03 ↑3C, ↓5D♭ 65/36
96 1037.84 ↑4C, ↓4D♭ 20/11
97 1048.65 ↑5C, ↓3D♭ 11/6
98 1059.46 ↑6C, ↓↓D♭ 24/13, 81/44
99 1070.27 ↑7C, ↓D♭ 13/7, 50/27
100 1081.08 ↑8C, D♭ 28/15
101 1091.89 ↑9C, ↓10D 15/8
102 1102.7 ↑10C, ↓9D
103 1113.51 C♯, ↓8D 40/21
104 1124.32 ↑C♯, ↓7D
105 1135.14 ↑↑C♯, ↓6D 25/13, 27/14, 52/27, 77/40
106 1145.95 ↑3C♯, ↓5D 64/33
107 1156.76 ↑4C♯, ↓4D 39/20
108 1167.57 ↑5C♯, ↓3D 49/25, 55/28
109 1178.38 ↑6C♯, ↓↓D 77/39
110 1189.19 ↑7C♯, ↓D
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

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\111 443.24 162/125 Sensipent / warrior
1 43\111 464.86 17/13 Semisept
1 44\111 475.68 21/16 Vulture / 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) 		231/200
(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], approximated from 27edo)

  • 54.054
  • 129.730
  • 497.297
  • 702.703
  • 756.757
  • 1145.946
  • 1200.000

Hypersakura (subset of Sensi[19], approximated from 27edo)

  • 54.054
  • 497.297
  • 702.703
  • 756.757
  • 1200.000

Music

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