97ed9: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | |||
97ed9 corresponds to 30.6001…edo. Each step of 97ed9 corresponds closely to five steps of [[153edo]]. | |||
97ed9 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 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 [[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. | 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]] (see [[Zeta peak index]]). | Additionally, 97ed9 is close to [[125zpi]] (see [[Zeta peak index]]). | ||
== Harmonic series == | === Harmonic series === | ||
=== 2 to 19 === | === 2 to 19 === | ||
| Line 25: | Line 23: | ||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
Revision as of 12:45, 23 May 2025
| ← 95ed9 | 97ed9 | 99ed9 → |
97 equal divisions of the 9th harmonic (abbreviated 97ed9) is a nonoctave tuning system that divides the interval of 9/1 into 97 equal parts of about 39.2 ¢ each. Each step represents a frequency ratio of 91/97, or the 97th root of 9.
Theory
97ed9 corresponds to 30.6001…edo. Each step of 97ed9 corresponds closely to five steps of 153edo.
97ed9 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 19
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +15.7 | +19.6 | -7.9 | -2.0 | -3.9 | +3.7 | +7.8 | +0.0 | +13.7 |
| Relative (%) | +40.0 | +50.0 | -20.0 | -5.1 | -10.0 | +9.5 | +20.0 | +0.0 | +34.9 | |
| Steps (reduced) |
31 (31) |
49 (49) |
61 (61) |
71 (71) |
79 (79) |
86 (86) |
92 (92) |
97 (0) |
102 (5) | |
| Harmonic | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.5 | +11.8 | -9.2 | +19.4 | +17.6 | -15.7 | -3.0 | +15.7 | +0.5 |
| Relative (%) | +14.1 | +30.0 | -23.4 | +49.5 | +44.9 | -40.0 | -7.7 | +40.0 | +1.3 | |
| Steps (reduced) |
106 (9) |
110 (13) |
113 (16) |
117 (20) |
120 (23) |
122 (25) |
125 (28) |
128 (31) |
130 (33) | |
36 to 53
| Harmonic | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.9 | -16.1 | +16.2 | +10.4 | +5.8 | +2.3 | -0.2 | -1.7 | -2.3 |
| Relative (%) | -20.0 | -41.0 | +41.3 | +26.6 | +14.9 | +5.8 | -0.5 | -4.4 | -5.9 | |
| Steps (reduced) |
158 (61) |
159 (62) |
161 (64) |
162 (65) |
163 (66) |
164 (67) |
165 (68) |
166 (69) |
167 (70) | |
| Harmonic | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.0 | -0.8 | +1.1 | +3.9 | +7.4 | +11.7 | +16.6 | -17.0 | -10.8 |
| Relative (%) | -5.1 | -2.2 | +2.9 | +10.0 | +18.9 | +29.7 | +42.3 | -43.4 | -27.5 | |
| Steps (reduced) |
168 (71) |
169 (72) |
170 (73) |
171 (74) |
172 (75) |
173 (76) |
174 (77) |
174 (77) |
175 (78) | |
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 39.2 | |
| 2 | 78.4 | 22/21 |
| 3 | 117.6 | 15/14, 31/29 |
| 4 | 156.9 | |
| 5 | 196.1 | |
| 6 | 235.3 | 39/34 |
| 7 | 274.5 | 34/29, 41/35 |
| 8 | 313.7 | |
| 9 | 352.9 | 38/31 |
| 10 | 392.2 | |
| 11 | 431.4 | |
| 12 | 470.6 | |
| 13 | 509.8 | |
| 14 | 549 | |
| 15 | 588.2 | |
| 16 | 627.4 | |
| 17 | 666.7 | 25/17 |
| 18 | 705.9 | |
| 19 | 745.1 | |
| 20 | 784.3 | 11/7 |
| 21 | 823.5 | 37/23 |
| 22 | 862.7 | |
| 23 | 902 | |
| 24 | 941.2 | 43/25 |
| 25 | 980.4 | |
| 26 | 1019.6 | |
| 27 | 1058.8 | 35/19 |
| 28 | 1098 | |
| 29 | 1137.3 | |
| 30 | 1176.5 | |
| 31 | 1215.7 | |
| 32 | 1254.9 | 31/15 |
| 33 | 1294.1 | |
| 34 | 1333.3 | 41/19 |
| 35 | 1372.5 | |
| 36 | 1411.8 | 43/19 |
| 37 | 1451 | |
| 38 | 1490.2 | 26/11 |
| 39 | 1529.4 | |
| 40 | 1568.6 | |
| 41 | 1607.8 | 38/15, 43/17 |
| 42 | 1647.1 | |
| 43 | 1686.3 | |
| 44 | 1725.5 | |
| 45 | 1764.7 | |
| 46 | 1803.9 | |
| 47 | 1843.1 | 29/10 |
| 48 | 1882.3 | |
| 49 | 1921.6 | |
| 50 | 1960.8 | 31/10 |
| 51 | 2000 | |
| 52 | 2039.2 | |
| 53 | 2078.4 | |
| 54 | 2117.6 | 17/5 |
| 55 | 2156.9 | |
| 56 | 2196.1 | |
| 57 | 2235.3 | |
| 58 | 2274.5 | |
| 59 | 2313.7 | 19/5 |
| 60 | 2352.9 | |
| 61 | 2392.1 | |
| 62 | 2431.4 | |
| 63 | 2470.6 | |
| 64 | 2509.8 | |
| 65 | 2549 | |
| 66 | 2588.2 | |
| 67 | 2627.4 | |
| 68 | 2666.7 | 14/3 |
| 69 | 2705.9 | |
| 70 | 2745.1 | 44/9 |
| 71 | 2784.3 | 5/1 |
| 72 | 2823.5 | |
| 73 | 2862.7 | |
| 74 | 2902 | |
| 75 | 2941.2 | |
| 76 | 2980.4 | |
| 77 | 3019.6 | |
| 78 | 3058.8 | 41/7 |
| 79 | 3098 | |
| 80 | 3137.2 | |
| 81 | 3176.5 | |
| 82 | 3215.7 | |
| 83 | 3254.9 | |
| 84 | 3294.1 | |
| 85 | 3333.3 | |
| 86 | 3372.5 | |
| 87 | 3411.8 | |
| 88 | 3451 | 22/3 |
| 89 | 3490.2 | 15/2 |
| 90 | 3529.4 | |
| 91 | 3568.6 | |
| 92 | 3607.8 | |
| 93 | 3647 | |
| 94 | 3686.3 | |
| 95 | 3725.5 | 43/5 |
| 96 | 3764.7 | |
| 97 | 3803.9 |