69edt

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69edt

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Prime factorization	3 × 23
Step size	27.5646¢
Octave	44\69edt (1212.84¢)
Consistency limit	2
Distinct consistency limit	2

69 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 69edt or 69ed3), is a nonoctave tuning system that divides the interval of 3/1 into 69 equal parts of about 27.6 ¢ each. Each step represents a frequency ratio of 3^1/69, or the 69th root of 3.

Intervals

Steps	Cents	Hekts	Approximate ratios
0	0	0	1/1
1	27.6	18.8
2	55.1	37.7	31/30, 34/33
3	82.7	56.5
4	110.3	75.4	33/31
5	137.8	94.2
6	165.4	113	11/10
7	193	131.9	19/17
8	220.5	150.7	17/15
9	248.1	169.6	15/13
10	275.6	188.4	27/23
11	303.2	207.2	25/21, 31/26, 37/31
12	330.8	226.1	23/19
13	358.3	244.9
14	385.9	263.8
15	413.5	282.6	33/26
16	441	301.4
17	468.6	320.3
18	496.2	339.1
19	523.7	358	23/17
20	551.3	376.8
21	578.9	395.7	7/5
22	606.4	414.5	27/19
23	634	433.3	13/9
24	661.5	452.2
25	689.1	471
26	716.7	489.9
27	744.2	508.7
28	771.8	527.5
29	799.4	546.4	27/17
30	826.9	565.2	21/13
31	854.5	584.1	18/11
32	882.1	602.9	5/3
33	909.6	621.7
34	937.2	640.6
35	964.8	659.4
36	992.3	678.3
37	1019.9	697.1	9/5
38	1047.5	715.9	11/6
39	1075	734.8	13/7
40	1102.6	753.6	17/9
41	1130.1	772.5	25/13
42	1157.7	791.3
43	1185.3	810.1
44	1212.8	829
45	1240.4	847.8
46	1268	866.7	27/13
47	1295.5	885.5	19/9
48	1323.1	904.3	15/7
49	1350.7	923.2
50	1378.2	942	31/14
51	1405.8	960.9
52	1433.4	979.7
53	1460.9	998.6
54	1488.5	1017.4	26/11
55	1516.1	1036.2
56	1543.6	1055.1
57	1571.2	1073.9
58	1598.7	1092.8
59	1626.3	1111.6	23/9
60	1653.9	1130.4	13/5
61	1681.4	1149.3	37/14
62	1709	1168.1
63	1736.6	1187	30/11
64	1764.1	1205.8
65	1791.7	1224.6	31/11
66	1819.3	1243.5
67	1846.8	1262.3
68	1874.4	1281.2
69	1902	1300	3/1

Harmonics

Approximation of harmonics in 69edt

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+12.8	+0.0	-1.9	-2.3	+12.8	-5.9	+11.0	+0.0	+10.5	+10.9	-1.9
	Relative (%)	+46.6	+0.0	-6.8	-8.3	+46.6	-21.6	+39.8	+0.0	+38.3	+39.7	-6.8
Steps (reduced)		44 (44)	69 (0)	87 (18)	101 (32)	113 (44)	122 (53)	131 (62)	138 (0)	145 (7)	151 (13)	156 (18)

Approximation of harmonics in 69edt

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.6	+6.9	-2.3	-3.8	+1.5	+12.8	+1.9	-4.2	-5.9	-3.8	+1.9
	Relative (%)	-9.6	+25.0	-8.3	-13.7	+5.6	+46.6	+7.0	-15.1	-21.6	-13.8	+7.1
Steps (reduced)		161 (23)	166 (28)	170 (32)	174 (36)	178 (40)	182 (44)	185 (47)	188 (50)	191 (53)	194 (56)	197 (59)