139ed5: Difference between revisions

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It is similar to [[60edo]], but with the octave (2/1) being [[octave stretch|stretched]] by about 2.7 cents, and with the interval [[5/1]] being [[just]], instead of 2/1 being just.
== Theory ==
139ed5 is similar to [[60edo]], but with the 5th harmonic being [[just]], instead of the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|stretched]] by about 2.73 cents.  


== Harmonics ==
On the harmonics 2, [[3/1|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.
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.
=== Harmonics ===
 
{{Harmonics in equal|139|5|1|intervals=integer|columns=11}}
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.
{{Harmonics in equal|139|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 139ed5 (continued)}}
{{Harmonics in equal|139|5|1|intervals=prime}}
 
 
60edo for comparison:
{{Harmonics in equal|60|2|1|intervals=prime}}


== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}


{{todo|expand}}
 
[[Category:60edo]]
[[Category:60edo]]

Revision as of 12:41, 28 May 2025

← 138ed5 139ed5 140ed5 →
Prime factorization 139 (prime)
Step size 20.0454 ¢ 
Octave 60\139ed5 (1202.73 ¢)
Twelfth 95\139ed5 (1904.32 ¢)
Consistency limit 10
Distinct consistency limit 10

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 51/139, or the 139th root of 5.

Theory

139ed5 is similar to 60edo, but with the 5th harmonic being just, instead of the octave being just. The octave is stretched by about 2.73 cents.

On the 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.

Harmonics

Approximation of harmonics in 139ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.73 +2.36 +5.45 +0.00 +5.09 -1.19 +8.18 +4.72 +2.73 -1.92 +7.81
Relative (%) +13.6 +11.8 +27.2 +0.0 +25.4 -6.0 +40.8 +23.5 +13.6 -9.6 +39.0
Steps
(reduced) 		60
(60) 		95
(95) 		120
(120) 		139
(0) 		155
(16) 		168
(29) 		180
(41) 		190
(51) 		199
(60) 		207
(68) 		215
(76)
Approximation of harmonics in 139ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.56 +1.53 +2.36 -9.14 +6.17 +7.45 -5.98 +5.45 +1.17 +0.81 +4.04 -9.51
Relative (%) +47.7 +7.6 +11.8 -45.6 +30.8 +37.1 -29.8 +27.2 +5.8 +4.0 +20.1 -47.4
Steps
(reduced) 		222
(83) 		228
(89) 		234
(95) 		239
(100) 		245
(106) 		250
(111) 		254
(115) 		259
(120) 		263
(124) 		267
(128) 		271
(132) 		274
(135)

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
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