138edt

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138edt

139edt →

Prime factorization	2 × 3 × 23
Step size	13.7823¢
Octave	87\138edt (1199.06¢) (→29\46edt)
Consistency limit	16
Distinct consistency limit	13

138 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 138edt or 138ed3), is a nonoctave tuning system that divides the interval of 3/1 into 138 equal parts of about 13.8 ¢ each. Each step represents a frequency ratio of 3^1/138, or the 138th root of 3.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	13.782
2	27.565
3	41.347	41/40, 42/41, 43/42, 44/43
4	55.129	31/30, 32/31
5	68.911	26/25
6	82.694	43/41
7	96.476
8	110.258	16/15
9	124.041	29/27, 43/40
10	137.823	13/12
11	151.605	12/11
12	165.387	11/10
13	179.17	51/46
14	192.952	19/17
15	206.734
16	220.517	25/22
17	234.299
18	248.081	15/13
19	261.863	50/43
20	275.646	34/29
21	289.428	13/11
22	303.21	25/21, 31/26
23	316.993	6/5
24	330.775	23/19
25	344.557	50/41
26	358.339	16/13
27	372.122	31/25
28	385.904	5/4
29	399.686	29/23, 34/27
30	413.468	33/26, 47/37
31	427.251	32/25
32	441.033	40/31
33	454.815	13/10
34	468.598	38/29
35	482.38	41/31
36	496.162
37	509.944	43/32, 51/38
38	523.727	23/17
39	537.509	15/11
40	551.291	11/8
41	565.074	18/13, 43/31
42	578.856
43	592.638	31/22, 38/27
44	606.42	44/31
45	620.203
46	633.985
47	647.767	16/11
48	661.55	22/15, 41/28
49	675.332	31/21, 34/23
50	689.114
51	702.896	3/2
52	716.679
53	730.461	29/19, 32/21
54	744.243	20/13, 43/28
55	758.026	31/20, 48/31
56	771.808	25/16
57	785.59	52/33
58	799.372	27/17, 46/29
59	813.155	8/5
60	826.937	29/18, 50/31
61	840.719	13/8
62	854.502
63	868.284	33/20, 38/23
64	882.066
65	895.848	52/31
66	909.631	22/13
67	923.413	29/17, 46/27
68	937.195	43/25
69	950.978	26/15, 45/26
70	964.76
71	978.542	44/25, 51/29
72	992.324	39/22
73	1006.107	34/19
74	1019.889
75	1033.671	20/11
76	1047.453
77	1061.236	24/13
78	1075.018	54/29
79	1088.8	15/8
80	1102.583	17/9
81	1116.365	40/21
82	1130.147	48/25
83	1143.929	31/16
84	1157.712	39/20, 41/21
85	1171.494
86	1185.276
87	1199.059	2/1
88	1212.841
89	1226.623
90	1240.405	43/21, 45/22
91	1254.188	33/16
92	1267.97	52/25
93	1281.752	44/21
94	1295.535
95	1309.317
96	1323.099
97	1336.881	13/6
98	1350.664	24/11
99	1364.446	11/5
100	1378.228	51/23
101	1392.011	38/17
102	1405.793
103	1419.575
104	1433.357
105	1447.14	30/13
106	1460.922
107	1474.704
108	1488.487	26/11
109	1502.269	50/21
110	1516.051	12/5
111	1529.833	46/19
112	1543.616	39/16
113	1557.398
114	1571.18	52/21
115	1584.963	5/2
116	1598.745
117	1612.527	33/13
118	1626.309
119	1640.092
120	1653.874	13/5
121	1667.656
122	1681.438
123	1695.221
124	1709.003	51/19
125	1722.785	46/17
126	1736.568	30/11
127	1750.35	11/4
128	1764.132	36/13
129	1777.914
130	1791.697	45/16
131	1805.479
132	1819.261
133	1833.044
134	1846.826
135	1860.608	41/14
136	1874.39
137	1888.173
138	1901.955	3/1

Harmonics

Approximation of harmonics in 138edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-0.94	+0.00	-1.88	-2.29	-0.94	-5.95	-2.82	+0.00	-3.23	-2.85	-1.88
	Relative (%)	-6.8	+0.0	-13.7	-16.6	-6.8	-43.2	-20.5	+0.0	-23.5	-20.7	-13.7
Steps (reduced)		87 (87)	138 (0)	174 (36)	202 (64)	225 (87)	244 (106)	261 (123)	276 (0)	289 (13)	301 (25)	312 (36)

Approximation of harmonics in 138edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.63	-6.89	-2.29	-3.77	+1.54	-0.94	+1.93	-4.18	-5.95	-3.79	+1.95
	Relative (%)	-19.1	-50.0	-16.6	-27.3	+11.2	-6.8	+14.0	-30.3	-43.2	-27.5	+14.1
Steps (reduced)		322 (46)	331 (55)	340 (64)	348 (72)	356 (80)	363 (87)	370 (94)	376 (100)	382 (106)	388 (112)	394 (118)