138edt

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← 137edt

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.

Theory

138edt is related to 87edo, but with the perfect twelfth instead of the octave tuned just. Like 87edo, it is consistent to the 16-integer-limit.

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 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	-2.63	-6.89	-2.29	-3.77	+1.54	-0.94	+1.93	-4.18	-5.95	-3.79	+1.95	-2.82
	Relative (%)	-19.1	-50.0	-16.6	-27.3	+11.2	-6.8	+14.0	-30.3	-43.2	-27.5	+14.1	-20.5
Steps (reduced)		322 (46)	331 (55)	340 (64)	348 (72)	356 (80)	363 (87)	370 (94)	376 (100)	382 (106)	388 (112)	394 (118)	399 (123)

Subsets and supersets

Since 138 factors into primes as 2 × 69, 138edt contains 2edt and 69edt as subset edts.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	13.8	9.4
2	27.6	18.8
3	41.3	28.3	41/40, 42/41, 43/42, 44/43
4	55.1	37.7	31/30, 32/31
5	68.9	47.1	26/25
6	82.7	56.5	43/41
7	96.5	65.9
8	110.3	75.4	16/15
9	124	84.8	29/27, 43/40
10	137.8	94.2	13/12
11	151.6	103.6	12/11
12	165.4	113	11/10
13	179.2	122.5	51/46
14	193	131.9	19/17
15	206.7	141.3
16	220.5	150.7	25/22
17	234.3	160.1
18	248.1	169.6	15/13
19	261.9	179	50/43
20	275.6	188.4	34/29
21	289.4	197.8	13/11
22	303.2	207.2	25/21, 31/26
23	317	216.7	6/5
24	330.8	226.1	23/19
25	344.6	235.5	50/41
26	358.3	244.9	16/13
27	372.1	254.3	31/25
28	385.9	263.8	5/4
29	399.7	273.2	29/23, 34/27
30	413.5	282.6	33/26, 47/37
31	427.3	292	32/25
32	441	301.4	40/31
33	454.8	310.9	13/10
34	468.6	320.3	38/29
35	482.4	329.7	41/31
36	496.2	339.1
37	509.9	348.6	43/32, 51/38
38	523.7	358	23/17
39	537.5	367.4	15/11
40	551.3	376.8	11/8
41	565.1	386.2	18/13, 43/31
42	578.9	395.7
43	592.6	405.1	31/22, 38/27
44	606.4	414.5	44/31
45	620.2	423.9
46	634	433.3
47	647.8	442.8	16/11
48	661.5	452.2	22/15, 41/28
49	675.3	461.6	31/21, 34/23
50	689.1	471
51	702.9	480.4	3/2
52	716.7	489.9
53	730.5	499.3	29/19, 32/21
54	744.2	508.7	20/13, 43/28
55	758	518.1	31/20, 48/31
56	771.8	527.5	25/16
57	785.6	537	52/33
58	799.4	546.4	27/17, 46/29
59	813.2	555.8	8/5
60	826.9	565.2	29/18, 50/31
61	840.7	574.6	13/8
62	854.5	584.1
63	868.3	593.5	33/20, 38/23
64	882.1	602.9
65	895.8	612.3	52/31
66	909.6	621.7	22/13
67	923.4	631.2	29/17, 46/27
68	937.2	640.6	43/25
69	951	650	26/15, 45/26
70	964.8	659.4
71	978.5	668.8	44/25, 51/29
72	992.3	678.3	39/22
73	1006.1	687.7	34/19
74	1019.9	697.1
75	1033.7	706.5	20/11
76	1047.5	715.9
77	1061.2	725.4	24/13
78	1075	734.8	54/29
79	1088.8	744.2	15/8
80	1102.6	753.6	17/9
81	1116.4	763	40/21
82	1130.1	772.5	48/25
83	1143.9	781.9	31/16
84	1157.7	791.3	39/20, 41/21
85	1171.5	800.7
86	1185.3	810.1
87	1199.1	819.6	2/1
88	1212.8	829
89	1226.6	838.4
90	1240.4	847.8	43/21, 45/22
91	1254.2	857.2	33/16
92	1268	866.7	52/25
93	1281.8	876.1	44/21
94	1295.5	885.5
95	1309.3	894.9
96	1323.1	904.3
97	1336.9	913.8	13/6
98	1350.7	923.2	24/11
99	1364.4	932.6	11/5
100	1378.2	942	51/23
101	1392	951.4	38/17
102	1405.8	960.9
103	1419.6	970.3
104	1433.4	979.7
105	1447.1	989.1	30/13
106	1460.9	998.6
107	1474.7	1008
108	1488.5	1017.4	26/11
109	1502.3	1026.8	50/21
110	1516.1	1036.2	12/5
111	1529.8	1045.7	46/19
112	1543.6	1055.1	39/16
113	1557.4	1064.5
114	1571.2	1073.9	52/21
115	1585	1083.3	5/2
116	1598.7	1092.8
117	1612.5	1102.2	33/13
118	1626.3	1111.6
119	1640.1	1121
120	1653.9	1130.4	13/5
121	1667.7	1139.9
122	1681.4	1149.3
123	1695.2	1158.7
124	1709	1168.1	51/19
125	1722.8	1177.5	46/17
126	1736.6	1187	30/11
127	1750.3	1196.4	11/4
128	1764.1	1205.8	36/13
129	1777.9	1215.2
130	1791.7	1224.6	45/16
131	1805.5	1234.1
132	1819.3	1243.5
133	1833	1252.9
134	1846.8	1262.3
135	1860.6	1271.7	41/14
136	1874.4	1281.2
137	1888.2	1290.6
138	1902	1300	3/1