178edt

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178edt

179edt →

Prime factorization	2 × 89
Step size	10.6851 ¢
Octave	112\178edt (1196.74 ¢) (→ 56\89edt)
Consistency limit	3
Distinct consistency limit	3

178 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 178edt or 178ed3), is a nonoctave tuning system that divides the interval of 3/1 into 178 equal parts of about 10.7 ¢ each. Each step represents a frequency ratio of 3^1/178, or the 178th root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	10.69	7.3
2	21.37	14.61
3	32.06	21.91
4	42.74	29.21
5	53.43	36.52
6	64.11	43.82	27/26
7	74.8	51.12	47/45
8	85.48	58.43	21/20, 41/39
9	96.17	65.73	37/35
10	106.85	73.03	50/47
11	117.54	80.34	46/43
12	128.22	87.64
13	138.91	94.94
14	149.59	102.25
15	160.28	109.55	34/31, 45/41
16	170.96	116.85
17	181.65	124.16	10/9
18	192.33	131.46	19/17
19	203.02	138.76
20	213.7	146.07	43/38
21	224.39	153.37	33/29
22	235.07	160.67
23	245.76	167.98
24	256.44	175.28	29/25, 51/44
25	267.13	182.58	7/6
26	277.81	189.89	27/23
27	288.5	197.19	13/11
28	299.18	204.49	44/37
29	309.87	211.8
30	320.55	219.1
31	331.24	226.4	23/19, 63/52
32	341.92	233.71
33	352.61	241.01	38/31
34	363.29	248.31	37/30, 58/47
35	373.98	255.62
36	384.67	262.92
37	395.35	270.22	44/35, 54/43
38	406.04	277.53	43/34
39	416.72	284.83
40	427.41	292.13
41	438.09	299.44	58/45
42	448.78	306.74	35/27, 57/44
43	459.46	314.04	30/23
44	470.15	321.35
45	480.83	328.65	33/25
46	491.52	335.96
47	502.2	343.26
48	512.89	350.56	39/29
49	523.57	357.87	23/17
50	534.26	365.17	49/36
51	544.94	372.47	37/27, 63/46
52	555.63	379.78	51/37
53	566.31	387.08	43/31
54	577	394.38	60/43
55	587.68	401.69
56	598.37	408.99	41/29
57	609.05	416.29	27/19
58	619.74	423.6
59	630.42	430.9
60	641.11	438.2
61	651.79	445.51	51/35
62	662.48	452.81	22/15, 63/43
63	673.16	460.11	31/21
64	683.85	467.42	46/31
65	694.53	474.72
66	705.22	482.02
67	715.9	489.33
68	726.59	496.63	35/23
69	737.27	503.93
70	747.96	511.24	57/37
71	758.64	518.54	31/20
72	769.33	525.84	39/25
73	780.02	533.15
74	790.7	540.45	30/19
75	801.39	547.75	27/17
76	812.07	555.06
77	822.76	562.36	37/23
78	833.44	569.66	34/21
79	844.13	576.97	57/35
80	854.81	584.27
81	865.5	591.57
82	876.18	598.88	63/38
83	886.87	606.18
84	897.55	613.48
85	908.24	620.79
86	918.92	628.09	17/10
87	929.61	635.39
88	940.29	642.7	31/18
89	950.98	650
90	961.66	657.3	54/31
91	972.35	664.61
92	983.03	671.91	30/17
93	993.72	679.21
94	1004.4	686.52
95	1015.09	693.82
96	1025.77	701.12	38/21, 47/26
97	1036.46	708.43
98	1047.14	715.73
99	1057.83	723.03	35/19
100	1068.51	730.34	63/34
101	1079.2	737.64
102	1089.88	744.94
103	1100.57	752.25	17/9
104	1111.25	759.55	19/10
105	1121.94	766.85	44/23
106	1132.62	774.16	25/13
107	1143.31	781.46	60/31
108	1154	788.76	37/19
109	1164.68	796.07
110	1175.37	803.37
111	1186.05	810.67
112	1196.74	817.98
113	1207.42	825.28
114	1218.11	832.58
115	1228.79	839.89	63/31
116	1239.48	847.19	43/21, 45/22
117	1250.16	854.49	35/17
118	1260.85	861.8
119	1271.53	869.1
120	1282.22	876.4
121	1292.9	883.71	19/9
122	1303.59	891.01
123	1314.27	898.31	47/22
124	1324.96	905.62	43/20, 58/27
125	1335.64	912.92
126	1346.33	920.22	37/17
127	1357.01	927.53	46/21
128	1367.7	934.83
129	1378.38	942.13	51/23
130	1389.07	949.44	29/13
131	1399.75	956.74
132	1410.44	964.04
133	1421.12	971.35	25/11
134	1431.81	978.65
135	1442.49	985.96	23/10
136	1453.18	993.26	44/19
137	1463.86	1000.56
138	1474.55	1007.87
139	1485.23	1015.17
140	1495.92	1022.47
141	1506.6	1029.78	43/18
142	1517.29	1037.08
143	1527.98	1044.38
144	1538.66	1051.69
145	1549.35	1058.99
146	1560.03	1066.29
147	1570.72	1073.6	52/21, 57/23
148	1581.4	1080.9
149	1592.09	1088.2
150	1602.77	1095.51
151	1613.46	1102.81	33/13
152	1624.14	1110.11	23/9
153	1634.83	1117.42	18/7
154	1645.51	1124.72	44/17
155	1656.2	1132.02
156	1666.88	1139.33
157	1677.57	1146.63	29/11
158	1688.25	1153.93
159	1698.94	1161.24
160	1709.62	1168.54	51/19
161	1720.31	1175.84	27/10
162	1730.99	1183.15
163	1741.68	1190.45	41/15
164	1752.36	1197.75
165	1763.05	1205.06
166	1773.73	1212.36
167	1784.42	1219.66
168	1795.1	1226.97
169	1805.79	1234.27
170	1816.47	1241.57	20/7
171	1827.16	1248.88
172	1837.84	1256.18	26/9
173	1848.53	1263.48
174	1859.21	1270.79
175	1869.9	1278.09
176	1880.58	1285.39
177	1891.27	1292.7
178	1901.96	1300	3/1

Harmonics

Approximation of harmonics in 178edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-3.26	+0.00	+4.16	+2.51	-3.26	-3.01	+0.89	+0.00	-0.76	+5.20	+4.16
	Relative (%)	-30.5	+0.0	+38.9	+23.5	-30.5	-28.1	+8.4	+0.0	-7.1	+48.7	+38.9
Steps (reduced)		112 (112)	178 (0)	225 (47)	261 (83)	290 (112)	315 (137)	337 (159)	356 (0)	373 (17)	389 (33)	403 (47)

Approximation of harmonics in 178edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+4.49	+4.41	+2.51	-2.37	-0.48	-3.26	-0.70	-4.02	-3.01	+1.94	-0.22
	Relative (%)	+42.0	+41.3	+23.5	-22.2	-4.5	-30.5	-6.6	-37.6	-28.1	+18.1	-2.1
Steps (reduced)		416 (60)	428 (72)	439 (83)	449 (93)	459 (103)	468 (112)	477 (121)	485 (129)	493 (137)	501 (145)	508 (152)