181edo

← 180edo 181edo 182edo →
Prime factorization	181 (prime)
Step size	6.62983 ¢
Fifth	106\181 (702.762 ¢)
Semitones (A1:m2)	18:13 (119.3 ¢ : 86.19 ¢)
Consistency limit	7
Distinct consistency limit	7

181 equal divisions of the octave (abbreviated 181edo or 181ed2), also called 181-tone equal temperament (181tet) or 181 equal temperament (181et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 181 equal parts of about 6.63 ¢ each. Each step represents a frequency ratio of 2^1/181, or the 181st root of 2.

Theory

181edo is only consistent to the 7-odd-limit, though except for 9/5, 23/20 and their octave complements, it is consistent to the 23-odd-limit. Beyond that, it does well on primes 37 and 43, and has unambiguous though not accurate approximations to 29, 31, and 41. However, the composite harmonics 25, 27, 35, and 39 cause inconsistencies, with harmonic 25 itself being inconsistent.

As an equal temperament, 181et tempers out 2109375/2097152 (semicomma) and [14 -22 9⟩ in the 5-limit; 2401/2400, 5120/5103, and 390625/387072 in the 7-limit (supporting the hemififths and the cotritone). Using the patent val, it tempers out 385/384, 1375/1372, 2200/2187, and 4000/3993 in the 11-limit; and 325/324, 352/351, 847/845, and 1575/1573 in the 13-limit. It tempers out 375/374, 595/594, and 1275/1274 in the 17-limit, 400/399 in the 19-limit, and 300/299 in the 23-limit.

Because its harmonic 5 causes some inconsistencies, and is less accurate than the other harmonics, 181edo can reasonably be treated as a no-5 system, where it is purely consistent^{[idiosyncratic term]} (meaning all harmonics have under 25% relative error) up to the 23-odd-limit. It tempers out [15 -13 2⟩ and [-31 -7 15⟩ in the 2.3.7 subgroup; 26411/26244, 43923/43904, and 131072/130977 in the 2.3.7.11 subgroup; and 352/351, 20449/20412, 31213/31104, and 53361/53248 in the 2.3.7.11.13 subgroup. It tempers out 833/832 and 1089/1088 in the no-5 17-limit, 343/342, 1729/1728, and 2432/2431 in the no-5 19-limit, and 392/391 in the no-5 23-limit.

Prime harmonics

Approximation of prime harmonics in 181edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.81	-1.78	-0.87	-1.04	+1.46	+1.12	+0.83	+1.56	-1.95	+1.93
Error	Relative (%)	+0.0	+12.2	-26.9	-13.1	-15.7	+22.0	+16.9	+12.5	+23.5	-29.5	+29.0
Steps (reduced)		181 (0)	287 (106)	420 (58)	508 (146)	626 (83)	670 (127)	740 (16)	769 (45)	819 (95)	879 (155)	897 (173)

Approximation of prime harmonics in 181edo (continued)
Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.59	+1.88	-1.02	-2.52	+1.63	+1.60	-3.07	+0.25	-0.69	-2.38	+0.10
Error	Relative (%)	+8.9	+28.3	-15.4	-38.1	+24.6	+24.2	-46.3	+3.8	-10.4	-35.8	+1.6
Steps (reduced)		943 (38)	970 (65)	982 (77)	1005 (100)	1037 (132)	1065 (160)	1073 (168)	1098 (12)	1113 (27)	1120 (34)	1141 (55)

Subsets and supersets

181edo is the 42nd prime edo.

Intervals

Todo: complete table

Steps	Cents	Approximate ratios*
0	0	1/1
1	6.63
2	13.26
3	19.89
4	26.52	64/63, 65/64, 66/65
5	33.15	49/48, 50/49, 52/51
6	39.78	45/44, 51/50
7	46.41	35/34
8	53.04	33/32, 34/33, 36/35
9	59.67
10	66.3	25/24, 27/26
11	72.93	24/23, 26/25
12	79.56	22/21, 23/22
13	86.19	20/19, 21/20
14	92.82	19/18
15	99.45	18/17
16	106.08	17/16
17	112.71	16/15
18	119.34	15/14
19	125.97	14/13
20	132.6
21	139.23	13/12, 25/23, 27/25
22	145.86
23	152.49	12/11
24	159.12	23/21
25	165.75	11/10
26	172.38	21/19
27	179.01	10/9, 51/46
28	185.64	49/44
29	192.27	19/17
30	198.9	55/49
31	205.52	9/8
32	212.15	26/23
33	218.78	17/15
34	225.41
35	232.04	8/7
36	238.67	39/34
37	245.3	15/13, 23/20, 38/33
38	251.93
39	258.56	65/56
40	265.19
41	271.82
42	278.45	27/23
43	285.08	33/28, 46/39
44	291.71	45/38
45	298.34	19/16
46	304.97
47	311.6
48	318.23	6/5
49	324.86
50	331.49	23/19, 63/52
51	338.12	62/51
52	344.75
53	351.38	49/40, 60/49
54	358.01
55	364.64	21/17
56	371.27	57/46
57	377.9	56/45
58	384.53	5/4
59	391.16
60	397.79
61	404.42	24/19
62	411.05
63	417.68	14/11
64	424.31	23/18
65	430.94
66	437.57
67	444.2
68	450.83
69	457.46
70	464.09	17/13
71	470.72	21/16
72	477.35
73	483.98
74	490.61
75	497.24	4/3
76	503.87
77	510.5	51/38
78	517.13
79	523.76	23/17, 65/48
80	530.39
81	537.02	15/11
82	543.65	26/19, 63/46
83	550.28	11/8
84	556.91
85	563.54	18/13
86	570.17
87	576.8
88	583.43	7/5
89	590.06	45/32
90	596.69	24/17
91	603.31	17/12
92	609.94	64/45
93	616.57	10/7
94	623.2
95	629.83
96	636.46	13/9
97	643.09
98	649.72	16/11
99	656.35	19/13
100	662.98	22/15
101	669.61
102	676.24	34/23, 65/44
103	682.87
104	689.5
105	696.13
106	702.76	3/2
107	709.39
108	716.02
109	722.65
110	729.28	32/21
111	735.91	26/17
112	742.54
113	749.17
114	755.8	65/42
115	762.43
116	769.06
117	775.69	36/23
118	782.32	11/7
119	788.95
120	795.58	19/12
121	802.21	62/39
122	808.84
123	815.47
124	822.1	45/28
125	828.73
126	835.36	34/21
127	841.99
128	848.62	49/30
129	855.25
130	861.88
131	868.51	38/23
132	875.14	63/38
133	881.77
134	888.4
135	895.03	57/34
136	901.66	32/19
137	908.29
138	914.92	39/23, 56/33
139	921.55	46/27
140	928.18	65/38
141	934.81
142	941.44
143	948.07
144	954.7	33/19
145	961.33
146	967.96	7/4
147	974.59
148	981.22
149	987.85	23/13
150	994.48
151	1001.1
152	1007.73	34/19
153	1014.36
154	1020.99
155	1027.62	38/21
156	1034.25	20/11
157	1040.88
158	1047.51
159	1054.14
160	1060.77	24/13
161	1067.4	63/34
162	1074.03
163	1080.66	28/15
164	1087.29	15/8
165	1093.92	32/17
166	1100.55	17/9
167	1107.18	36/19
168	1113.81	19/10, 40/21
169	1120.44	21/11
170	1127.07	23/12
171	1133.7	52/27
172	1140.33
173	1146.96	64/33
174	1153.59
175	1160.22
176	1166.85	51/26
177	1173.48	63/32, 65/33
178	1180.11
179	1186.74
180	1193.37
181	1200	2/1

*As a 23-limit temperament

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[287 -181⟩	[⟨181 287]]	−0.255	0.255	3.84
2.3.5	2109375/2097152, [14 -22 9⟩	[⟨181 287 420]]	+0.086	0.525	7.92
2.3.5.7	2401/2400, 5120/5103, 390625/387072	[⟨181 287 420 508]]	+0.142	0.465	7.01
2.3.5.7.11	385/384, 1375/1372, 2200/2187, 4000/3993	[⟨181 287 420 508 626]]	+0.174	0.421	6.35
2.3.5.7.11.13	325/324, 352/351, 385/384, 1375/1372, 1575/1573	[⟨181 287 420 508 626 670]]	+0.079	0.439	6.62
2.3.5.7.11.13.17	325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274	[⟨181 287 420 508 626 670 740]]	+0.028	0.425	6.40
2.3.5.7.11.13.17.19	325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274	[⟨181 287 420 508 626 670 740 769]]	+0.000	0.404	6.09

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	18\181	119.34	15/14	Septidiasemi
1	35\181	232.04	8/7	Quadrawell
1	39\181	258.56	[-32 13 5⟩	Lafa
1	41\181	271.82	75/64	Orson
1	53\181	351.38	49/40	Hemififths (7-limit)
1	78\181	517.13	66/49	Cutefourths
1	88\181	583.43	7/5	Cotritone (11-limit)

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium

"Today Or Tomorrow?" from Questions (2024) – Spotify | Bandcamp | YouTube – slurpee in 181edo tuning