139ed5

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139ed5

140ed5 →

Prime factorization	139 (prime)
Step size	20.0454¢
Octave	60\139ed5 (1202.73¢)
Twelfth	95\139ed5 (1904.32¢)
Consistency limit	9
Distinct consistency limit	9

139 equal divisions of the 5th harmonic (abbreviated 139ed5) is a nonoctave tuning system that divides the interval of 5/1 into 139 equal parts of about 20⁠ ⁠¢ each. Each step represents a frequency ratio of 5^1/139, or the 139th root of 5.

It is similar to 60edo, but with the octave (2/1) being stretched by about 2.7 cents, and with the interval 5/1 being just, instead of 2/1 being just.

Harmonics

On harmonics 2.3.5.7.11, 60edo has 0%, 10%, 32%, 44% and 43% relative error.

On those same harmonics, 139ed5 has 14%, 12%, 0%, 6% and 10% relative error.

This is a large improvement relative to the step size of the tuning, and is the main reason why a composer might choose to use 139ed5.

Approximation of prime harmonics in 139ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+2.73	+2.36	+0.00	-1.19	-1.92	+9.56	+6.17	-5.98	+4.04	+3.64	+8.45
	Relative (%)	+13.6	+11.8	+0.0	-6.0	-9.6	+47.7	+30.8	-29.8	+20.1	+18.2	+42.2
Steps (reduced)		60 (60)	95 (95)	139 (0)	168 (29)	207 (68)	222 (83)	245 (106)	254 (115)	271 (132)	291 (13)	297 (19)

60edo for comparison:

Approximation of prime harmonics in 60edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.96	-6.31	-8.83	+8.68	-0.53	-4.96	+2.49	-8.27	-9.58	-5.04
	Relative (%)	+0.0	-9.8	-31.6	-44.1	+43.4	-2.6	-24.8	+12.4	-41.4	-47.9	-25.2
Steps (reduced)		60 (0)	95 (35)	139 (19)	168 (48)	208 (28)	222 (42)	245 (5)	255 (15)	271 (31)	291 (51)	297 (57)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	20
2	40.1	42/41, 43/42, 44/43, 45/44
3	60.1	29/28, 30/29
4	80.2	22/21, 45/43
5	100.2	18/17, 35/33
6	120.3	15/14
7	140.3	51/47
8	160.4	34/31, 45/41
9	180.4
10	200.5	46/41
11	220.5	25/22, 42/37
12	240.5	23/20, 31/27, 54/47
13	260.6	43/37, 50/43
14	280.6	20/17, 47/40
15	300.7	25/21, 44/37
16	320.7
17	340.8	28/23
18	360.8	16/13
19	380.9
20	400.9	29/23
21	421	37/29, 51/40
22	441	40/31, 49/38
23	461	30/23, 47/36
24	481.1	33/25, 37/28
25	501.1
26	521.2	27/20, 50/37
27	541.2	41/30
28	561.3	47/34
29	581.3	7/5
30	601.4	17/12
31	621.4
32	641.5	42/29
33	661.5	22/15, 41/28
34	681.5	40/27, 43/29
35	701.6	3/2
36	721.6	41/27, 44/29, 47/31
37	741.7	23/15, 43/28
38	761.7	45/29
39	781.8	11/7
40	801.8	27/17
41	821.9	37/23, 45/28
42	841.9	13/8
43	862	51/31
44	882
45	902
46	922.1	46/27
47	942.1	31/18, 50/29
48	962.2	54/31
49	982.2	30/17, 37/21
50	1002.3	25/14, 41/23
51	1022.3
52	1042.4	42/23
53	1062.4	24/13
54	1082.5	43/23
55	1102.5	17/9
56	1122.5	44/23
57	1142.6	29/15
58	1162.6	45/23, 47/24
59	1182.7
60	1202.7
61	1222.8
62	1242.8	41/20
63	1262.9
64	1282.9	21/10
65	1303
66	1323
67	1343	50/23
68	1363.1
69	1383.1	20/9
70	1403.2	9/4
71	1423.2
72	1443.3	23/10
73	1463.3
74	1483.4	33/14
75	1503.4	31/13, 50/21
76	1523.5	41/17
77	1543.5	39/16
78	1563.5	37/15
79	1583.6
80	1603.6
81	1623.7	23/9
82	1643.7	31/12
83	1663.8	34/13
84	1683.8	37/14, 45/17
85	1703.9
86	1723.9	46/17
87	1744
88	1764	36/13
89	1784	14/5
90	1804.1	17/6
91	1824.1	43/15
92	1844.2	29/10
93	1864.2	44/15, 47/16
94	1884.3
95	1904.3
96	1924.4
97	1944.4	40/13
98	1964.5	28/9
99	1984.5
100	2004.5	35/11
101	2024.6	29/9
102	2044.6
103	2064.7
104	2084.7	10/3
105	2104.8	27/8
106	2124.8
107	2144.9	38/11
108	2164.9
109	2185
110	2205	25/7
111	2225	47/13
112	2245.1
113	2265.1	37/10
114	2285.2
115	2305.2
116	2325.3	23/6
117	2345.3	31/8
118	2365.4	51/13
119	2385.4
120	2405.5
121	2425.5
122	2445.5
123	2465.6	54/13
124	2485.6	21/5
125	2505.7	17/4
126	2525.7	43/10
127	2545.8
128	2565.8	22/5
129	2585.9	49/11
130	2605.9
131	2626	41/9
132	2646
133	2666	14/3
134	2686.1	33/7
135	2706.1	43/9
136	2726.2	29/6
137	2746.2	44/9
138	2766.3
139	2786.3	5/1