78ed7/3

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78ed7/3

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Prime factorization	2 × 3 × 13
Step size	18.806¢
Octave	64\78ed7/3 (1203.59¢) (→32\39ed7/3)
Twelfth	101\78ed7/3 (1899.41¢)
Consistency limit	3
Distinct consistency limit	3

78 equal divisions of 7/3 (abbreviated 78ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 78 equal parts of about 18.8 ¢ each. Each step represents a frequency ratio of (7/3)^1/78, or the 78th root of 7/3.

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	18.806
2	37.612
3	56.418	30/29, 31/30
4	75.224	23/22, 24/23
5	94.03	19/18, 37/35, 39/37
6	112.836	31/29
7	131.642	27/25
8	150.448	12/11
9	169.254
10	188.06	29/26, 39/35
11	206.866	35/31
12	225.672	33/29
13	244.478	15/13, 23/20, 38/33
14	263.285
15	282.091	20/17, 33/28
16	300.897	25/21
17	319.703
18	338.509	17/14, 28/23
19	357.315
20	376.121	36/29
21	394.927	39/31
22	413.733	33/26
23	432.539	9/7
24	451.345	13/10, 35/27
25	470.151	38/29
26	488.957
27	507.763
28	526.569	19/14, 23/17
29	545.375	26/19, 37/27
30	564.181	18/13
31	582.987	7/5
32	601.793	17/12
33	620.599	10/7
34	639.405	13/9
35	658.211	19/13
36	677.017	31/21, 34/23, 37/25
37	695.823
38	714.629
39	733.435	26/17, 29/19
40	752.241	17/11
41	771.048	39/25
42	789.854	30/19
43	808.66
44	827.466	21/13, 29/18
45	846.272	31/19
46	865.078	28/17, 33/20
47	883.884	5/3
48	902.69
49	921.496	17/10, 29/17
50	940.302	31/18
51	959.108
52	977.914	37/21
53	996.72
54	1015.526	9/5
55	1034.332	20/11
56	1053.138
57	1071.944	13/7
58	1090.75
59	1109.556	19/10, 36/19
60	1128.362	23/12
61	1147.168	33/17
62	1165.974
63	1184.78
64	1203.586
65	1222.392
66	1241.198
67	1260.004	29/14, 31/15
68	1278.811	23/11
69	1297.617	36/17
70	1316.423	15/7
71	1335.229
72	1354.035
73	1372.841
74	1391.647	29/13, 38/17
75	1410.453
76	1429.259
77	1448.065	30/13
78	1466.871	7/3

Harmonics

Approximation of harmonics in 78ed7/3

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	+3.59	-2.55	+7.17	-3.02	+1.04	-2.55	-8.05	-5.09	+0.57	+4.82	+4.63
	Relative (%)	+19.1	-13.5	+38.1	-16.1	+5.5	-13.5	-42.8	-27.1	+3.0	+25.6	+24.6
Steps (reduced)		64 (64)	101 (23)	128 (50)	148 (70)	165 (9)	179 (23)	191 (35)	202 (46)	212 (56)	221 (65)	229 (73)

Approximation of harmonics in 78ed7/3

Harmonic		13	14	15	16	17	18	19	20	21	22	23
Error	Absolute (¢)	-2.30	+1.04	-5.57	-4.46	+3.42	-1.50	-1.08	+4.15	-5.09	+8.40	+6.67
	Relative (%)	-12.2	+5.5	-29.6	-23.7	+18.2	-8.0	-5.7	+22.1	-27.1	+44.7	+35.5
Steps (reduced)		236 (2)	243 (9)	249 (15)	255 (21)	261 (27)	266 (32)	271 (37)	276 (42)	280 (46)	285 (51)	289 (55)