164edt

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164 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 164edt or 164ed3), is a nonoctave tuning system that divides the interval of 3/1 into 164 equal parts of about 11.6 ¢ each. Each step represents a frequency ratio of 3^1/164, or the 164th root of 3.

← 163edt

164edt

165edt →

Prime factorization	22 × 41
Step size	11.5973 ¢
Octave	103\164edt (1194.52 ¢)
Consistency limit	3
Distinct consistency limit	3

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	11.6	7.93
2	23.19	15.85
3	34.79	23.78	50/49
4	46.39	31.71	38/37
5	57.99	39.63
6	69.58	47.56
7	81.18	55.49	22/21, 43/41
8	92.78	63.41	58/55
9	104.38	71.34
10	115.97	79.27	31/29, 46/43
11	127.57	87.2
12	139.17	95.12
13	150.76	103.05
14	162.36	110.98	45/41
15	173.96	118.9
16	185.56	126.83	49/44
17	197.15	134.76	37/33
18	208.75	142.68
19	220.35	150.61	25/22
20	231.95	158.54
21	243.54	166.46	38/33
22	255.14	174.39
23	266.74	182.32	7/6
24	278.33	190.24	27/23
25	289.93	198.17	13/11
26	301.53	206.1	25/21
27	313.13	214.02
28	324.72	221.95	41/34
29	336.32	229.88
30	347.92	237.8	11/9
31	359.52	245.73
32	371.11	253.66	26/21
33	382.71	261.59
34	394.31	269.51	54/43
35	405.91	277.44	43/34
36	417.5	285.37
37	429.1	293.29
38	440.7	301.22	58/45
39	452.29	309.15
40	463.89	317.07	17/13
41	475.49	325	54/41
42	487.09	332.93
43	498.68	340.85
44	510.28	348.78	51/38
45	521.88	356.71
46	533.48	364.63	34/25, 49/36
47	545.07	372.56	37/27
48	556.67	380.49	51/37
49	568.27	388.41	25/18
50	579.86	396.34
51	591.46	404.27	38/27
52	603.06	412.2
53	614.66	420.12
54	626.25	428.05	33/23
55	637.85	435.98	13/9
56	649.45	443.9
57	661.05	451.83
58	672.64	459.76
59	684.24	467.68	52/35
60	695.84	475.61
61	707.43	483.54
62	719.03	491.46
63	730.63	499.39
64	742.23	507.32
65	753.82	515.24	17/11
66	765.42	523.17
67	777.02	531.1	58/37
68	788.62	539.02	41/26
69	800.21	546.95	27/17
70	811.81	554.88
71	823.41	562.8	37/23
72	835	570.73	34/21, 47/29
73	846.6	578.66	31/19
74	858.2	586.59
75	869.8	594.51	38/23, 43/26
76	881.39	602.44
77	892.99	610.37
78	904.59	618.29
79	916.19	626.22
80	927.78	634.15
81	939.38	642.07	43/25
82	950.98	650	26/15, 45/26
83	962.57	657.93
84	974.17	665.85
85	985.77	673.78
86	997.37	681.71
87	1008.96	689.63
88	1020.56	697.56
89	1032.16	705.49
90	1043.76	713.41
91	1055.35	721.34	46/25, 57/31
92	1066.95	729.27
93	1078.55	737.2	41/22
94	1090.14	745.12
95	1101.74	753.05	17/9
96	1113.34	760.98
97	1124.94	768.9
98	1136.53	776.83
99	1148.13	784.76	33/17
100	1159.73	792.68	43/22
101	1171.33	800.61
102	1182.92	808.54
103	1194.52	816.46
104	1206.12	824.39
105	1217.72	832.32
106	1229.31	840.24
107	1240.91	848.17	43/21
108	1252.51	856.1
109	1264.1	864.02	27/13
110	1275.7	871.95	23/11
111	1287.3	879.88
112	1298.9	887.8
113	1310.49	895.73
114	1322.09	903.66
115	1333.69	911.59	54/25
116	1345.29	919.51	37/17
117	1356.88	927.44	46/21
118	1368.48	935.37
119	1380.08	943.29
120	1391.67	951.22	38/17
121	1403.27	959.15
122	1414.87	967.07
123	1426.47	975	41/18
124	1438.06	982.93	39/17
125	1449.66	990.85
126	1461.26	998.78
127	1472.86	1006.71
128	1484.45	1014.63
129	1496.05	1022.56
130	1507.65	1030.49	43/18
131	1519.24	1038.41
132	1530.84	1046.34
133	1542.44	1054.27
134	1554.04	1062.2	27/11
135	1565.63	1070.12
136	1577.23	1078.05
137	1588.83	1085.98
138	1600.43	1093.9	58/23
139	1612.02	1101.83	33/13
140	1623.62	1109.76	23/9
141	1635.22	1117.68	18/7
142	1646.81	1125.61
143	1658.41	1133.54
144	1670.01	1141.46
145	1681.61	1149.39
146	1693.2	1157.32
147	1704.8	1165.24
148	1716.4	1173.17
149	1728	1181.1
150	1739.59	1189.02	41/15
151	1751.19	1196.95
152	1762.79	1204.88
153	1774.38	1212.8
154	1785.98	1220.73
155	1797.58	1228.66
156	1809.18	1236.59
157	1820.77	1244.51
158	1832.37	1252.44
159	1843.97	1260.37
160	1855.57	1268.29
161	1867.16	1276.22
162	1878.76	1284.15
163	1890.36	1292.07
164	1901.96	1300	3/1

Harmonics

Approximation of harmonics in 164edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-5.48	+0.00	+0.64	-2.96	-5.48	-5.61	-4.84	+0.00	+3.15	+0.51	+0.64
	Relative (%)	-47.2	+0.0	+5.5	-25.6	-47.2	-48.4	-41.7	+0.0	+27.2	+4.4	+5.5
Steps (reduced)		103 (103)	164 (0)	207 (43)	240 (76)	267 (103)	290 (126)	310 (146)	328 (0)	344 (16)	358 (30)	371 (43)

Approximation of harmonics in 164edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+1.23	+0.51	-2.96	+1.28	+0.70	-5.48	+5.29	-2.33	-5.61	-4.97	-0.74
	Relative (%)	+10.6	+4.4	-25.6	+11.0	+6.0	-47.2	+45.6	-20.1	-48.4	-42.8	-6.4
Steps (reduced)		383 (55)	394 (66)	404 (76)	414 (86)	423 (95)	431 (103)	440 (112)	447 (119)	454 (126)	461 (133)	468 (140)