111ed12

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111ed12

112ed12 →

Prime factorization	3 × 37
Step size	38.7564¢
Octave	31\111ed12 (1201.45¢)
Twelfth	49\111ed12 (1899.06¢)
Consistency limit	11
Distinct consistency limit	9

111 equal divisions of the 12th harmonic (abbreviated 111ed12) is a nonoctave tuning system that divides the interval of 12/1 into 111 equal parts of about 38.8 ¢ each. Each step represents a frequency ratio of 12^1/111, or the 111th root of 12.

111ed12 is nearly identical to 31edo but with the 12/1 rather than the 2/1 being just. The octave is about 1.45 cents stretched compared to 31edo.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	38.756	43/42, 44/43, 45/44, 46/45, 47/46
2	77.513	23/22, 45/43
3	116.269	31/29, 46/43, 47/44
4	155.025	35/32, 47/43
5	193.782	19/17, 47/42
6	232.538	8/7
7	271.294
8	310.051
9	348.807	11/9
10	387.564	5/4
11	426.32	23/18, 32/25
12	465.076	17/13
13	503.833
14	542.589	26/19, 41/30
15	581.345	7/5
16	620.102
17	658.858	19/13, 41/28
18	697.614
19	736.371	26/17
20	775.127	36/23, 47/30
21	813.883	8/5
22	852.64	18/11
23	891.396
24	930.152
25	968.909	7/4
26	1007.665	34/19, 43/24
27	1046.421
28	1085.178	43/23
29	1123.934	44/23
30	1162.691	45/23, 47/24
31	1201.447	2/1
32	1240.203	43/21, 45/22
33	1278.96	23/11, 44/21
34	1317.716	15/7
35	1356.472	35/16, 46/21
36	1395.229	47/21
37	1433.985
38	1472.741
39	1511.498
40	1550.254
41	1589.01
42	1627.767	41/16
43	1666.523	34/13
44	1705.279
45	1744.036
46	1782.792	14/5
47	1821.549	43/15
48	1860.305	41/14
49	1899.061
50	1937.818	46/15
51	1976.574	47/15
52	2015.33	16/5
53	2054.087	36/11
54	2092.843
55	2131.599	24/7
56	2170.356	7/2
57	2209.112	43/12
58	2247.868	11/3
59	2286.625	15/4
60	2325.381	23/6
61	2364.137	47/12
62	2402.894
63	2441.65	41/10
64	2480.406
65	2519.163	30/7
66	2557.919
67	2596.676
68	2635.432
69	2674.188
70	2712.945
71	2751.701
72	2790.457
73	2829.214	41/8
74	2867.97
75	2906.726
76	2945.483
77	2984.239	28/5
78	3022.995
79	3061.752	41/7
80	3100.508	6/1
81	3139.264
82	3178.021
83	3216.777
84	3255.534
85	3294.29
86	3333.046
87	3371.803
88	3410.559	43/6
89	3449.315	22/3
90	3488.072	15/2
91	3526.828	23/3
92	3565.584	47/6
93	3604.341
94	3643.097	41/5
95	3681.853
96	3720.61
97	3759.366
98	3798.122
99	3836.879
100	3875.635
101	3914.391
102	3953.148
103	3991.904
104	4030.661	41/4
105	4069.417	21/2
106	4108.173
107	4146.93
108	4185.686
109	4224.442
110	4263.199
111	4301.955	12/1

Harmonics

Approximation of harmonics in 111ed12

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+1.4	-2.9	+2.9	+4.1	-1.4	+3.0	+4.3	-5.8	+5.6	-4.4	+0.0
	Relative (%)	+3.7	-7.5	+7.5	+10.7	-3.7	+7.7	+11.2	-14.9	+14.4	-11.3	+0.0
Steps (reduced)		31 (31)	49 (49)	62 (62)	72 (72)	80 (80)	87 (87)	93 (93)	98 (98)	103 (103)	107 (107)	111 (0)

Approximation of harmonics in 111ed12

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+16.5	+4.4	+1.2	+5.8	+17.1	-4.3	+18.3	+7.0	+0.1	-2.9	-2.4
	Relative (%)	+42.5	+11.4	+3.2	+14.9	+44.1	-11.2	+47.3	+18.2	+0.2	-7.6	-6.2
Steps (reduced)		115 (4)	118 (7)	121 (10)	124 (13)	127 (16)	129 (18)	132 (21)	134 (23)	136 (25)	138 (27)	140 (29)