96ed5
Theory
96ed5 is an equal-step tuning system created by dividing the interval of 5/1 into 96 equal parts.
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 | |