139ed5
| ← 138ed5 | 139ed5 | 140ed5 → |
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. Like 60edo, 139ed5 is consistent to the 10-integer-limit.
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 if the focus is on the higher primes, and is the main reason why a composer might choose to use 139ed5.
Harmonics
| 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) | |
| 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) | |
Subsets and supersets
139ed5 is the 34th prime ed5. It does not contain any nontrivial subset ed5's.
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 |