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	Approximate ratios
0	0	1/1
1	27.565
2	55.129	31/30, 34/33
3	82.694
4	110.258	33/31
5	137.823
6	165.387	11/10
7	192.952	19/17
8	220.517	17/15
9	248.081	15/13
10	275.646	27/23
11	303.21	25/21, 31/26, 37/31
12	330.775	23/19
13	358.339
14	385.904
15	413.468	33/26
16	441.033
17	468.598
18	496.162
19	523.727	23/17
20	551.291
21	578.856	7/5
22	606.42	27/19
23	633.985	13/9
24	661.55
25	689.114
26	716.679
27	744.243
28	771.808
29	799.372	27/17
30	826.937	21/13
31	854.502	18/11
32	882.066	5/3
33	909.631
34	937.195
35	964.76
36	992.324
37	1019.889	9/5
38	1047.453	11/6
39	1075.018	13/7
40	1102.583	17/9
41	1130.147	25/13
42	1157.712
43	1185.276
44	1212.841
45	1240.405
46	1267.97	27/13
47	1295.535	19/9
48	1323.099	15/7
49	1350.664
50	1378.228	31/14
51	1405.793
52	1433.357
53	1460.922
54	1488.487	26/11
55	1516.051
56	1543.616
57	1571.18
58	1598.745
59	1626.309	23/9
60	1653.874	13/5
61	1681.438	37/14
62	1709.003
63	1736.568	30/11
64	1764.132
65	1791.697	31/11
66	1819.261
67	1846.826
68	1874.39
69	1901.955	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)