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{{ | {{Mathematical interest}} | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | {{ED intro}} | ||
== Theory == | == Theory == | ||
The 96th harmonic is too wide to be a useful 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 stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo. | The 96th harmonic is far too wide to be a useful 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 stretched by 0.837{{c}}), making 204ed96 extremely close to optimal for 31edo. | ||
=== Harmonics === | === Harmonics === | ||
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{{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}} | {{Harmonics in equal|204|96|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 204ed96 (continued)}} | ||
* [[18edf]] – relative edf | * [[18edf]] – relative edf | ||
* [[31edo]] – relative edo | * [[31edo]] – relative edo | ||
| Line 16: | Line 15: | ||
* [[72ed5]] – relative ed5 | * [[72ed5]] – relative ed5 | ||
* [[80ed6]] – relative ed6 | * [[80ed6]] – relative ed6 | ||
* [[87ed7]] – relative ed7 | |||
* [[107ed11]] – relative ed11 | |||
* [[111ed12]] – relative ed12 | * [[111ed12]] – relative ed12 | ||
* [[138ed22]] – relative ed22 | |||
* [[39cET]] | |||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category:Zeta-optimized tunings]] | [[Category:Zeta-optimized tunings]] | ||
Latest revision as of 15:03, 16 July 2025
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
| ← 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 is far too wide to be a useful 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 stretched by 0.837 ¢), 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) | |