81ed4

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81ed4

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Prime factorization	34
Step size	29.6296¢
Octave	41\81ed4 (1214.81¢)
Twelfth	64\81ed4 (1896.3¢)
Consistency limit	1
Distinct consistency limit	1

81 equal divisions of the 4th harmonic (abbreviated 81ed4) is a nonoctave tuning system that divides the interval of 4/1 into 81 equal parts of about 29.6 ¢ each. Each step represents a frequency ratio of 4^1/81, or the 81st root of 4.

Intervals

Steps	Cents	Approximate Ratios
0	0	1/1
1	29.63
2	59.259	30/29, 31/30
3	88.889	39/37
4	118.519	31/29
5	148.148
6	177.778
7	207.407	35/31
8	237.037
9	266.667	7/6
10	296.296
11	325.926	35/29
12	355.556	38/31
13	385.185
14	414.815
15	444.444	22/17
16	474.074	25/19
17	503.704
18	533.333
19	562.963	18/13
20	592.593	31/22
21	622.222	33/23
22	651.852
23	681.481	37/25
24	711.111
25	740.741	23/15
26	770.37	39/25
27	800
28	829.63	21/13
29	859.259
30	888.889
31	918.519	17/10
32	948.148	19/11
33	977.778	37/21
34	1007.407
35	1037.037	31/17
36	1066.667
37	1096.296
38	1125.926
39	1155.556	37/19
40	1185.185
41	1214.815
42	1244.444	37/18, 39/19
43	1274.074	23/11
44	1303.704
45	1333.333
46	1362.963	11/5
47	1392.593	38/17
48	1422.222	25/11
49	1451.852
50	1481.481
51	1511.111
52	1540.741
53	1570.37
54	1600
55	1629.63
56	1659.259
57	1688.889
58	1718.519
59	1748.148
60	1777.778
61	1807.407
62	1837.037
63	1866.667
64	1896.296
65	1925.926
66	1955.556	31/10
67	1985.185	22/7
68	2014.815
69	2044.444
70	2074.074
71	2103.704
72	2133.333
73	2162.963
74	2192.593	39/11
75	2222.222
76	2251.852	11/3
77	2281.481
78	2311.111	19/5
79	2340.741
80	2370.37
81	2400

Harmonics

Approximation of harmonics in 81ed4

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+14.8	-5.7	+0.0	-1.1	+9.2	+9.0	+14.8	-11.3	+13.7	-3.2	-5.7
	Relative (%)	+50.0	-19.1	+0.0	-3.8	+30.9	+30.2	+50.0	-38.2	+46.2	-10.7	-19.1
Steps (reduced)		41 (41)	64 (64)	81 (0)	94 (13)	105 (24)	114 (33)	122 (41)	128 (47)	135 (54)	140 (59)	145 (64)

Approximation of harmonics in 81ed4

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	+3.9	-5.9	-6.8	+0.0	+13.6	+3.5	-1.2	-1.1	+3.3	+11.6	-6.1
	Relative (%)	+13.2	-19.8	-22.9	+0.0	+45.8	+11.8	-4.1	-3.8	+11.1	+39.3	-20.4
Steps (reduced)		150 (69)	154 (73)	158 (77)	162 (0)	166 (4)	169 (7)	172 (10)	175 (13)	178 (16)	181 (19)	183 (21)