96ed5
| ← 95ed5 | 96ed5 | 97ed5 → |
96 equal divisions of the 5th harmonic (abbreviated 96ed5) is a nonoctave tuning system that divides the interval of 5/1 into 96 equal parts of about 29 ¢ each. Each step represents a frequency ratio of 51/96, or the 96th root of 5.
Theory
This non-octave, non-tritave scale features a well-balanced harmonic series segment from 5 to 9, and performs exceptionally well across all prime harmonics from 5 to 23, with the exception of 19.
This system can be approximated as 41.34495 EDO, meaning each step of 96ed5 corresponds roughly to three steps of 124edo.
96ed5 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 41.3478 EDO. This record remains unbeaten until approximately 98.62575 EDO.
Additionally, 96ed5 is related to 186zpi.
Harmonic series
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.0 | +13.6 | +9.0 | +0.0 | +3.6 | -2.0 | -1.0 | -1.8 | -10.0 | -0.9 | -6.4 | +0.2 | -12.0 | +13.6 | -11.0 |
| Relative (%) | -34.5 | +47.0 | +31.0 | +0.0 | +12.5 | -7.0 | -3.5 | -6.0 | -34.5 | -3.0 | -22.0 | +0.6 | -41.5 | +47.0 | -38.0 | |
| Steps (reduced) |
41 (41) |
66 (66) |
83 (83) |
96 (0) |
107 (11) |
116 (20) |
124 (28) |
131 (35) |
137 (41) |
143 (47) |
148 (52) |
153 (57) |
157 (61) |
162 (66) |
165 (69) | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.1 | -11.8 | +10.7 | +9.0 | +11.6 | -10.9 | -0.8 | +12.6 | +0.0 | -9.9 | +11.9 | +7.0 | +4.3 | +3.6 | +4.9 | +8.0 |
| Relative (%) | +0.4 | -40.5 | +37.0 | +31.0 | +40.0 | -37.5 | -2.6 | +43.5 | +0.0 | -33.9 | +40.9 | +24.0 | +14.7 | +12.5 | +16.9 | +27.5 | |
| Steps (reduced) |
169 (73) |
172 (76) |
176 (80) |
179 (83) |
182 (86) |
184 (88) |
187 (91) |
190 (94) |
192 (0) |
194 (2) |
197 (5) |
199 (7) |
201 (9) |
203 (11) |
205 (13) |
207 (15) | |
Optimization
In the 32-integer-limit and 5.7.11.13.17.23 subgroup, the lowest relative error is 41.346437627379-edo, or 41<1189.94532112775>, or 29.023056612872 cents.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -10.1 | +13.6 | +8.9 | -0.1 | +3.5 | -2.2 | -1.1 | -1.9 | -10.2 | -1.0 | -6.5 | +0.0 | -12.2 | +13.5 | -11.2 |
| Relative (%) | -34.6 | +46.7 | +30.7 | -0.3 | +12.1 | -7.4 | -3.9 | -6.5 | -35.0 | -3.5 | -22.5 | +0.0 | -42.1 | +46.4 | -38.6 | |
| Steps | 41 | 66 | 83 | 96 | 107 | 116 | 124 | 131 | 137 | 143 | 148 | 153 | 157 | 162 | 165 | |
| Harmonic | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.1 | -11.9 | +10.5 | +8.8 | +11.4 | -11.1 | -1.0 | +12.4 | -0.2 | -10.1 | +11.7 | +6.8 | +4.1 | +3.4 | +4.7 | +7.8 |
| Relative (%) | -0.2 | -41.2 | +36.3 | +30.4 | +39.3 | -38.2 | -3.3 | +42.8 | -0.7 | -34.6 | +40.2 | +23.3 | +14.0 | +11.8 | +16.2 | +26.8 | |
| Steps | 169 | 172 | 176 | 179 | 182 | 184 | 187 | 190 | 192 | 194 | 197 | 199 | 201 | 203 | 205 | 207 | |
Intervals
| Steps | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 29.024 | |
| 2 | 58.048 | 30/29, 31/30 |
| 3 | 87.072 | 41/39 |
| 4 | 116.096 | 31/29 |
| 5 | 145.121 | 25/23, 37/34 |
| 6 | 174.145 | 21/19 |
| 7 | 203.169 | |
| 8 | 232.193 | |
| 9 | 261.217 | 36/31, 43/37 |
| 10 | 290.241 | 13/11 |
| 11 | 319.265 | |
| 12 | 348.289 | |
| 13 | 377.313 | 41/33 |
| 14 | 406.337 | 43/34 |
| 15 | 435.362 | |
| 16 | 464.386 | 17/13 |
| 17 | 493.41 | |
| 18 | 522.434 | 23/17 |
| 19 | 551.458 | |
| 20 | 580.482 | 7/5 |
| 21 | 609.506 | 37/26 |
| 22 | 638.53 | |
| 23 | 667.554 | 25/17 |
| 24 | 696.578 | |
| 25 | 725.603 | 35/23, 38/25 |
| 26 | 754.627 | 17/11 |
| 27 | 783.651 | 11/7 |
| 28 | 812.675 | |
| 29 | 841.699 | |
| 30 | 870.723 | 38/23, 43/26 |
| 31 | 899.747 | 37/22, 42/25 |
| 32 | 928.771 | |
| 33 | 957.795 | 33/19 |
| 34 | 986.819 | 23/13 |
| 35 | 1015.844 | |
| 36 | 1044.868 | 42/23 |
| 37 | 1073.892 | 13/7 |
| 38 | 1102.916 | |
| 39 | 1131.94 | 25/13 |
| 40 | 1160.964 | 43/22 |
| 41 | 1189.988 | |
| 42 | 1219.012 | |
| 43 | 1248.036 | 35/17 |
| 44 | 1277.06 | 23/11 |
| 45 | 1306.085 | |
| 46 | 1335.109 | |
| 47 | 1364.133 | 11/5 |
| 48 | 1393.157 | 38/17 |
| 49 | 1422.181 | 25/11 |
| 50 | 1451.205 | |
| 51 | 1480.229 | |
| 52 | 1509.253 | |
| 53 | 1538.277 | 17/7 |
| 54 | 1567.301 | 42/17 |
| 55 | 1596.326 | |
| 56 | 1625.35 | |
| 57 | 1654.374 | 13/5 |
| 58 | 1683.398 | 37/14 |
| 59 | 1712.422 | 35/13 |
| 60 | 1741.446 | 41/15 |
| 61 | 1770.47 | |
| 62 | 1799.494 | |
| 63 | 1828.518 | |
| 64 | 1857.542 | 38/13 |
| 65 | 1886.567 | |
| 66 | 1915.591 | |
| 67 | 1944.615 | 43/14 |
| 68 | 1973.639 | |
| 69 | 2002.663 | 35/11 |
| 70 | 2031.687 | 42/13 |
| 71 | 2060.711 | 23/7 |
| 72 | 2089.735 | |
| 73 | 2118.759 | 17/5 |
| 74 | 2147.783 | 38/11 |
| 75 | 2176.808 | |
| 76 | 2205.832 | 25/7 |
| 77 | 2234.856 | |
| 78 | 2263.88 | 37/10 |
| 79 | 2292.904 | |
| 80 | 2321.928 | 42/11 |
| 81 | 2350.952 | |
| 82 | 2379.976 | |
| 83 | 2409 | |
| 84 | 2438.024 | |
| 85 | 2467.049 | |
| 86 | 2496.073 | |
| 87 | 2525.097 | 43/10 |
| 88 | 2554.121 | |
| 89 | 2583.145 | |
| 90 | 2612.169 | |
| 91 | 2641.193 | 23/5 |
| 92 | 2670.217 | |
| 93 | 2699.241 | |
| 94 | 2728.266 | 29/6 |
| 95 | 2757.29 | |
| 96 | 2786.314 | 5/1 |