97ed9: Difference between revisions
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97ed9 sets a height record on the [[The Riemann zeta function and tuning|Riemann zeta function]] with [[The Riemann zeta function and tuning#Removing primes|primes 2 and 3 removed]], approximating 30.59745 EDO. This record remains unbeaten until approximately 41.3478 EDO. | 97ed9 sets a height record on the [[The Riemann zeta function and tuning|Riemann zeta function]] with [[The Riemann zeta function and tuning#Removing primes|primes 2 and 3 removed]], approximating 30.59745 EDO. This record remains unbeaten until approximately 41.3478 EDO. | ||
Additionally, 97ed9 is close to [[125zpi]]. | Additionally, 97ed9 is close to [[125zpi]] (see [[Zeta peak index]]). | ||
== Harmonic series == | == Harmonic series == | ||
=== 2 to 15 === | |||
{{Harmonics in equal|97|9| | {{Harmonics in equal|97|9|1|columns=7}} | ||
{{Harmonics in equal|97|9|1| | {{Harmonics in equal|97|9|1|columns=7|title=(contd.)|start=8}} | ||
=== 38 to 52 === | |||
{{Harmonics in equal|97|9|1|columns=7|collapsed=1|start=37}} | |||
{{Harmonics in equal|97|9|1|columns=7|title=(contd.)|collapsed=1|start=45}} | |||
{{todo|expand}} | |||
[[Category:Ed9]][[Category:Equal-step tuning]] | [[Category:Ed9]][[Category:Equal-step tuning]] | ||
Revision as of 01:00, 23 December 2024
Theory
97ed9 is an equal-step tuning system created by dividing the interval of 9/1 into 97 equal parts.
This system can be approximated as 30.6001 EDO, meaning each step of 97ed9 corresponds closely to five steps of 153edo.
This non-octave, non-tritave scale features a well-balanced harmonic series segment from 4 to 9 and another from 39 to 50. It performs well across all prime harmonics from 5 to 19, with the exception of 13, which is slightly flat.
97ed9 sets a height record on the Riemann zeta function with primes 2 and 3 removed, approximating 30.59745 EDO. This record remains unbeaten until approximately 41.3478 EDO.
Additionally, 97ed9 is close to 125zpi (see Zeta peak index).
Harmonic series
2 to 15
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.7 | +19.6 | -7.9 | -2.0 | -3.9 | +3.7 | +7.8 |
| Relative (%) | +40.0 | +50.0 | -20.0 | -5.1 | -10.0 | +9.5 | +20.0 | |
| Steps (reduced) |
31 (31) |
49 (49) |
61 (61) |
71 (71) |
79 (79) |
86 (86) |
92 (92) | |
| Harmonic | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | +13.7 | +5.5 | +11.8 | -9.2 | +19.4 | +17.6 |
| Relative (%) | +0.0 | +34.9 | +14.1 | +30.0 | -23.4 | +49.5 | +44.9 | |
| Steps (reduced) |
97 (0) |
102 (5) |
106 (9) |
110 (13) |
113 (16) |
117 (20) |
120 (23) | |
38 to 52
| Harmonic | 38 | 39 | 40 | 41 | 42 | 43 | 44 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.2 | +10.4 | +5.8 | +2.3 | -0.2 | -1.7 | -2.3 |
| Relative (%) | +41.3 | +26.6 | +14.9 | +5.8 | -0.5 | -4.4 | -5.9 | |
| Steps (reduced) |
161 (64) |
162 (65) |
163 (66) |
164 (67) |
165 (68) |
166 (69) |
167 (70) | |
| Harmonic | 46 | 47 | 48 | 49 | 50 | 51 | 52 | |
|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.8 | +1.1 | +3.9 | +7.4 | +11.7 | +16.6 | -17.0 |
| Relative (%) | -2.2 | +2.9 | +10.0 | +18.9 | +29.7 | +42.3 | -43.4 | |
| Steps (reduced) |
169 (72) |
170 (73) |
171 (74) |
172 (75) |
173 (76) |
174 (77) |
174 (77) | |