111ed12

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111ed12

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Prime factorization	3 × 37
Step size	38.7564 ¢
Octave	31\111ed12 (1201.45 ¢)
Twelfth	49\111ed12 (1899.06 ¢)
Consistency limit	12
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. 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.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	38.8	43/42, 44/43, 45/44, 46/45, 47/46
2	77.5	23/22, 45/43
3	116.3	31/29, 46/43, 47/44
4	155	35/32, 47/43
5	193.8	19/17, 47/42
6	232.5	8/7
7	271.3
8	310.1
9	348.8	11/9
10	387.6	5/4
11	426.3	23/18, 32/25
12	465.1	17/13
13	503.8
14	542.6	26/19, 41/30
15	581.3	7/5
16	620.1
17	658.9	19/13, 41/28
18	697.6
19	736.4	26/17
20	775.1	36/23, 47/30
21	813.9	8/5
22	852.6	18/11
23	891.4
24	930.2
25	968.9	7/4
26	1007.7	34/19, 43/24
27	1046.4
28	1085.2	43/23
29	1123.9	44/23
30	1162.7	45/23, 47/24
31	1201.4	2/1
32	1240.2	43/21, 45/22
33	1279	23/11, 44/21
34	1317.7	15/7
35	1356.5	35/16, 46/21
36	1395.2	47/21
37	1434
38	1472.7
39	1511.5
40	1550.3
41	1589
42	1627.8	41/16
43	1666.5	34/13
44	1705.3
45	1744
46	1782.8	14/5
47	1821.5	43/15
48	1860.3	41/14
49	1899.1
50	1937.8	46/15
51	1976.6	47/15
52	2015.3	16/5
53	2054.1	36/11
54	2092.8
55	2131.6	24/7
56	2170.4	7/2
57	2209.1	43/12
58	2247.9	11/3
59	2286.6	15/4
60	2325.4	23/6
61	2364.1	47/12
62	2402.9
63	2441.7	41/10
64	2480.4
65	2519.2	30/7
66	2557.9
67	2596.7
68	2635.4
69	2674.2
70	2712.9
71	2751.7
72	2790.5
73	2829.2	41/8
74	2868
75	2906.7
76	2945.5
77	2984.2	28/5
78	3023
79	3061.8	41/7
80	3100.5	6/1
81	3139.3
82	3178
83	3216.8
84	3255.5
85	3294.3
86	3333
87	3371.8
88	3410.6	43/6
89	3449.3	22/3
90	3488.1	15/2
91	3526.8	23/3
92	3565.6	47/6
93	3604.3
94	3643.1	41/5
95	3681.9
96	3720.6
97	3759.4
98	3798.1
99	3836.9
100	3875.6
101	3914.4
102	3953.1
103	3991.9
104	4030.7	41/4
105	4069.4	21/2
106	4108.2
107	4146.9
108	4185.7
109	4224.4
110	4263.2
111	4302	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)