186edt

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186edt

187edt →

Prime factorization	2 × 3 × 31
Step size	10.2256 ¢
Octave	117\186edt (1196.39 ¢) (→ 39\62edt)
Consistency limit	3
Distinct consistency limit	3

186 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 186edt or 186ed3), is a nonoctave tuning system that divides the interval of 3/1 into 186 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of 3^1/186, or the 186th root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	10.23	6.99
2	20.45	13.98
3	30.68	20.97	55/54
4	40.9	27.96
5	51.13	34.95	34/33
6	61.35	41.94
7	71.58	48.92	25/24
8	81.8	55.91	43/41, 65/62
9	92.03	62.9	39/37, 58/55
10	102.26	69.89	52/49
11	112.48	76.88
12	122.71	83.87	29/27
13	132.93	90.86
14	143.16	97.85	63/58
15	153.38	104.84	47/43
16	163.61	111.83
17	173.83	118.82
18	184.06	125.81
19	194.29	132.8
20	204.51	139.78
21	214.74	146.77	43/38
22	224.96	153.76
23	235.19	160.75	63/55
24	245.41	167.74	38/33
25	255.64	174.73
26	265.86	181.72	7/6
27	276.09	188.71	34/29
28	286.32	195.7	46/39
29	296.54	202.69	51/43
30	306.77	209.68	37/31
31	316.99	216.67
32	327.22	223.66
33	337.44	230.65
34	347.67	237.63	11/9
35	357.89	244.62
36	368.12	251.61	47/38
37	378.35	258.6	51/41
38	388.57	265.59
39	398.8	272.58	34/27
40	409.02	279.57
41	419.25	286.56
42	429.47	293.55
43	439.7	300.54	58/45
44	449.92	307.53
45	460.15	314.52
46	470.38	321.51
47	480.6	328.49
48	490.83	335.48
49	501.05	342.47
50	511.28	349.46
51	521.5	356.45
52	531.73	363.44
53	541.95	370.43
54	552.18	377.42
55	562.41	384.41	18/13
56	572.63	391.4
57	582.86	398.39	7/5
58	593.08	405.38	31/22
59	603.31	412.37
60	613.53	419.35
61	623.76	426.34
62	633.99	433.33
63	644.21	440.32	45/31
64	654.44	447.31	54/37
65	664.66	454.3
66	674.89	461.29	31/21, 65/44
67	685.11	468.28	52/35, 55/37
68	695.34	475.27
69	705.56	482.26
70	715.79	489.25
71	726.02	496.24
72	736.24	503.23
73	746.47	510.22
74	756.69	517.2	65/42
75	766.92	524.19
76	777.14	531.18	58/37
77	787.37	538.17
78	797.59	545.16	46/29
79	807.82	552.15
80	818.05	559.14
81	828.27	566.13
82	838.5	573.12
83	848.72	580.11	49/30
84	858.95	587.1
85	869.17	594.09	38/23
86	879.4	601.08
87	889.62	608.06
88	899.85	615.05	37/22
89	910.08	622.04	22/13
90	920.3	629.03	63/37
91	930.53	636.02
92	940.75	643.01	31/18
93	950.98	650
94	961.2	656.99	54/31
95	971.43	663.98
96	981.65	670.97	37/21
97	991.88	677.96	39/22, 55/31
98	1002.11	684.95	66/37
99	1012.33	691.94
100	1022.56	698.92	65/36
101	1032.78	705.91
102	1043.01	712.9
103	1053.23	719.89
104	1063.46	726.88
105	1073.68	733.87
106	1083.91	740.86	43/23, 58/31
107	1094.14	747.85
108	1104.36	754.84
109	1114.59	761.83
110	1124.81	768.82
111	1135.04	775.81
112	1145.26	782.8
113	1155.49	789.78
114	1165.71	796.77	49/25
115	1175.94	803.76
116	1186.17	810.75
117	1196.39	817.74
118	1206.62	824.73
119	1216.84	831.72
120	1227.07	838.71	63/31
121	1237.29	845.7	47/23
122	1247.52	852.69	37/18
123	1257.74	859.68	31/15
124	1267.97	866.67	52/25
125	1278.2	873.66
126	1288.42	880.65
127	1298.65	887.63
128	1308.87	894.62	66/31
129	1319.1	901.61	15/7
130	1329.32	908.6
131	1339.55	915.59	13/6
132	1349.77	922.58
133	1360	929.57
134	1370.23	936.56
135	1380.45	943.55
136	1390.68	950.54
137	1400.9	957.53
138	1411.13	964.52
139	1421.35	971.51
140	1431.58	978.49
141	1441.8	985.48
142	1452.03	992.47
143	1462.26	999.46
144	1472.48	1006.45
145	1482.71	1013.44
146	1492.93	1020.43
147	1503.16	1027.42
148	1513.38	1034.41
149	1523.61	1041.4	41/17
150	1533.83	1048.39
151	1544.06	1055.38
152	1554.29	1062.37	27/11
153	1564.51	1069.35
154	1574.74	1076.34
155	1584.96	1083.33
156	1595.19	1090.32
157	1605.41	1097.31	43/17
158	1615.64	1104.3
159	1625.86	1111.29
160	1636.09	1118.28	18/7
161	1646.32	1125.27
162	1656.54	1132.26
163	1666.77	1139.25	55/21
164	1676.99	1146.24
165	1687.22	1153.23
166	1697.44	1160.22
167	1707.67	1167.2
168	1717.89	1174.19
169	1728.12	1181.18
170	1738.35	1188.17
171	1748.57	1195.16
172	1758.8	1202.15	58/21
173	1769.02	1209.14
174	1779.25	1216.13
175	1789.47	1223.12
176	1799.7	1230.11
177	1809.92	1237.1	37/13
178	1820.15	1244.09
179	1830.38	1251.08
180	1840.6	1258.06
181	1850.83	1265.05
182	1861.05	1272.04
183	1871.28	1279.03
184	1881.5	1286.02
185	1891.73	1293.01
186	1901.96	1300	3/1

Harmonics

Approximation of harmonics in 186edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-3.61	+0.00	+3.01	-4.96	-3.61	-4.62	-0.60	+0.00	+1.66	+0.26	+3.01
	Relative (%)	-35.3	+0.0	+29.4	-48.5	-35.3	-45.1	-5.9	+0.0	+16.2	+2.6	+29.4
Steps (reduced)		117 (117)	186 (0)	235 (49)	272 (86)	303 (117)	329 (143)	352 (166)	372 (0)	390 (18)	406 (34)	421 (49)

Approximation of harmonics in 186edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.63	+2.00	-4.96	-4.21	+3.32	-3.61	+5.04	-1.95	-4.62	-3.35	+1.50
	Relative (%)	-25.7	+19.6	-48.5	-41.2	+32.4	-35.3	+49.3	-19.1	-45.1	-32.7	+14.7
Steps (reduced)		434 (62)	447 (75)	458 (86)	469 (97)	480 (108)	489 (117)	499 (127)	507 (135)	515 (143)	523 (151)	531 (159)