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{{mathematical interest}} | |||
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== Theory == | |||
The 304th harmonic is far too wide to be a useful equivalence, so 198ed304 is better thought of as a compressed version of [[24edo]]. Indeed, tuning the 304/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.301{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 24 is located at 24.005742, which has a step size of 49.98804{{c}} and an octave of 1199.713{{c}} (which is compressed by 0.287{{c}}), making 198ed304 extremely close to optimal for 24edo. | |||
=== Harmonics === | |||
{{Harmonics in equal|298|304|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|198|304|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 198ed304 (continued)}} | |||
== See also == | |||
* [[14edf]] – relative edf | |||
* [[24edo]] – relative edo | |||
* [[38edt]] – relative edt | |||
* [[56ed5]] – relative ed5 | |||
* [[62ed6]] – relative ed6 | |||
* [[86ed12]] – relative ed12 | |||
[[Category:24edo]] | |||
[[Category:Zeta-optimized tunings]] | |||
Revision as of 18:05, 18 June 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. |
| ← 197ed304 | 198ed304 | 199ed304 → |
198 equal divisions of the 304th harmonic (abbreviated 198ed304) is a nonoctave tuning system that divides the interval of 304/1 into 198 equal parts of about 50 ¢ each. Each step represents a frequency ratio of 3041/198, or the 198th root of 304.
Theory
The 304th harmonic is far too wide to be a useful equivalence, so 198ed304 is better thought of as a compressed version of 24edo. Indeed, tuning the 304/1 ratio just instead of 2/1 results in octaves being stretched by about 0.301 ¢. The local zeta peak around 24 is located at 24.005742, which has a step size of 49.98804 ¢ and an octave of 1199.713 ¢ (which is compressed by 0.287 ¢), making 198ed304 extremely close to optimal for 24edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.3 | -8.8 | -8.7 | +3.6 | -13.1 | -14.3 | -13.0 | +15.6 | -0.7 | +0.3 | +15.8 |
| Relative (%) | -13.0 | -26.5 | -26.1 | +10.8 | -39.5 | -43.1 | -39.1 | +47.0 | -2.2 | +1.0 | +47.4 | |
| Steps (reduced) |
36 (36) |
57 (57) |
72 (72) |
84 (84) |
93 (93) |
101 (101) |
108 (108) |
115 (115) |
120 (120) |
125 (125) |
130 (130) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.4 | -20.0 | +10.6 | -1.2 | -6.2 | -5.2 | +1.2 | +12.4 | -22.1 | -2.7 | +20.4 | -3.3 |
| Relative (%) | +16.7 | -39.9 | +21.1 | -2.4 | -12.4 | -10.3 | +2.4 | +24.8 | -44.2 | -5.3 | +40.7 | -6.7 | |
| Steps (reduced) |
89 (89) |
91 (91) |
94 (94) |
96 (96) |
98 (98) |
100 (100) |
102 (102) |
104 (104) |
105 (105) |
107 (107) |
109 (109) |
110 (110) | |