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	13.8
2	27.6
3	41.3	41/40, 42/41, 43/42, 44/43
4	55.1	31/30, 32/31
5	68.9	26/25
6	82.7	43/41
7	96.5
8	110.3	16/15
9	124	29/27, 43/40
10	137.8	13/12
11	151.6	12/11
12	165.4	11/10
13	179.2	51/46
14	193	19/17
15	206.7
16	220.5	25/22
17	234.3
18	248.1	15/13
19	261.9	50/43
20	275.6	34/29
21	289.4	13/11
22	303.2	25/21, 31/26
23	317	6/5
24	330.8	23/19
25	344.6	50/41
26	358.3	16/13
27	372.1	31/25
28	385.9	5/4
29	399.7	29/23, 34/27
30	413.5	33/26, 47/37
31	427.3	32/25
32	441	40/31
33	454.8	13/10
34	468.6	38/29
35	482.4	41/31
36	496.2
37	509.9	43/32, 51/38
38	523.7	23/17
39	537.5	15/11
40	551.3	11/8
41	565.1	18/13, 43/31
42	578.9
43	592.6	31/22, 38/27
44	606.4	44/31
45	620.2
46	634
47	647.8	16/11
48	661.5	22/15, 41/28
49	675.3	31/21, 34/23
50	689.1
51	702.9	3/2
52	716.7
53	730.5	29/19, 32/21
54	744.2	20/13, 43/28
55	758	31/20, 48/31
56	771.8	25/16
57	785.6	52/33
58	799.4	27/17, 46/29
59	813.2	8/5
60	826.9	29/18, 50/31
61	840.7	13/8
62	854.5
63	868.3	33/20, 38/23
64	882.1
65	895.8	52/31
66	909.6	22/13
67	923.4	29/17, 46/27
68	937.2	43/25
69	951	26/15, 45/26
70	964.8
71	978.5	44/25, 51/29
72	992.3	39/22
73	1006.1	34/19
74	1019.9
75	1033.7	20/11
76	1047.5
77	1061.2	24/13
78	1075	54/29
79	1088.8	15/8
80	1102.6	17/9
81	1116.4	40/21
82	1130.1	48/25
83	1143.9	31/16
84	1157.7	39/20, 41/21
85	1171.5
86	1185.3
87	1199.1	2/1
88	1212.8
89	1226.6
90	1240.4	43/21, 45/22
91	1254.2	33/16
92	1268	52/25
93	1281.8	44/21
94	1295.5
95	1309.3
96	1323.1
97	1336.9	13/6
98	1350.7	24/11
99	1364.4	11/5
100	1378.2	51/23
101	1392	38/17
102	1405.8
103	1419.6
104	1433.4
105	1447.1	30/13
106	1460.9
107	1474.7
108	1488.5	26/11
109	1502.3	50/21
110	1516.1	12/5
111	1529.8	46/19
112	1543.6	39/16
113	1557.4
114	1571.2	52/21
115	1585	5/2
116	1598.7
117	1612.5	33/13
118	1626.3
119	1640.1
120	1653.9	13/5
121	1667.7
122	1681.4
123	1695.2
124	1709	51/19
125	1722.8	46/17
126	1736.6	30/11
127	1750.3	11/4
128	1764.1	36/13
129	1777.9
130	1791.7	45/16
131	1805.5
132	1819.3
133	1833
134	1846.8
135	1860.6	41/14
136	1874.4
137	1888.2
138	1902	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)