204ed96: Difference between revisions
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== Theory == | == Theory == | ||
The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around | The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 31 is located at 30.978382, which has a step size of 38.737{{c}} and an octave of 1200.837{{c}} (which is compressed by 2.31{{c}}), making 204ed96 extremely close to optimal for 31edo. | ||
=== Harmonics === | === Harmonics === | ||
{{Harmonics in equal| | {{Harmonics in equal|204|96|1|intervals=integer|columns=11}} | ||
{{Harmonics in equal| | {{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}} | ||
== See also == | == See also == | ||
Revision as of 03:49, 9 June 2025
| ← 203ed96 | 204ed96 | 205ed96 → |
204 equal divisions of the 96th harmonic (abbreviated 204ed96) is a nonoctave tuning system that divides the interval of 96/1 into 204 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 961/204, or the 204th root of 96.
Theory
The 96th harmonic would be extremely wide for an equivalence, so 204ed96 is better thought of as a stretched version of 31edo. Indeed, tuning the 96/1 ratio just instead of 2/1 results in octaves being stretched by about 0.79 ¢. The local zeta peak around 31 is located at 30.978382, which has a step size of 38.737 ¢ and an octave of 1200.837 ¢ (which is compressed by 2.31 ¢), making 204ed96 extremely close to optimal for 31edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.8 | -3.9 | +1.6 | +2.6 | -3.1 | +1.1 | +2.4 | -7.9 | +3.4 | -6.7 | -2.4 |
| Relative (%) | +2.0 | -10.2 | +4.1 | +6.7 | -8.1 | +2.9 | +6.1 | -20.3 | +8.8 | -17.2 | -6.1 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (72) |
80 (80) |
87 (87) |
93 (93) |
98 (98) |
103 (103) |
107 (107) |
111 (111) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +14.0 | +1.9 | -1.3 | +3.1 | +14.4 | -7.1 | +15.5 | +4.2 | -2.8 | -5.9 | -5.4 | -1.6 |
| Relative (%) | +36.2 | +4.9 | -3.4 | +8.1 | +37.2 | -18.3 | +40.1 | +10.8 | -7.3 | -15.2 | -13.8 | -4.1 | |
| Steps (reduced) |
115 (115) |
118 (118) |
121 (121) |
124 (124) |
127 (127) |
129 (129) |
132 (132) |
134 (134) |
136 (136) |
138 (138) |
140 (140) |
142 (142) | |