76ed80: Difference between revisions
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{{Mathematical interest}} | |||
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== Theory == | |||
The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}, making 76ed80 extremely close to optimal for 12edo. | |||
=== Harmonics === | |||
{{Harmonics in equal|76|80|1|intervals=integer|columns=11}} | |||
{{Harmonics in equal|76|80|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 76ed80 (continued)}} | |||
== See also == | |||
* [[12edo]] – relative edo | |||
* [[19edt]] – relative edt | |||
* [[28ed5]] – relative ed5 | |||
* [[31ed6]] – relative ed6 | |||
* [[40ed10]] – relative ed10 | |||
* [[43ed12]] – relative ed12 | |||
[[Category:12edo]] | |||
[[Category:Zeta-optimized tunings]] | |||
Latest revision as of 11:57, 26 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. |
| ← 75ed80 | 76ed80 | 77ed80 → |
76 equal divisions of the 80th harmonic (abbreviated 76ed80) is a nonoctave tuning system that divides the interval of 80/1 into 76 equal parts of about 99.8 ¢ each. Each step represents a frequency ratio of 801/76, or the 76th root of 80.
Theory
The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous 12edo. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being compressed by about 2.16 ¢. The local zeta peak around 12 is located at 12.023183, which has a step size of 99.807 ¢ and an octave of 1197.686 ¢, making 76ed80 extremely close to optimal for 12edo.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.2 | -5.4 | -4.3 | +8.6 | -7.5 | +25.1 | -6.5 | -10.8 | +6.5 | +41.1 | -9.7 |
| Relative (%) | -2.2 | -5.4 | -4.3 | +8.7 | -7.6 | +25.1 | -6.5 | -10.8 | +6.5 | +41.2 | -9.7 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (28) |
31 (31) |
34 (34) |
36 (36) |
38 (38) |
40 (40) |
42 (42) |
43 (43) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -48.5 | +22.9 | +3.3 | -8.6 | -13.8 | -12.9 | -6.7 | +4.3 | +19.7 | +39.0 | -38.0 | -11.9 |
| Relative (%) | -48.5 | +22.9 | +3.3 | -8.7 | -13.8 | -12.9 | -6.7 | +4.3 | +19.7 | +39.0 | -38.1 | -11.9 | |
| Steps (reduced) |
44 (44) |
46 (46) |
47 (47) |
48 (48) |
49 (49) |
50 (50) |
51 (51) |
52 (52) |
53 (53) |
54 (54) |
54 (54) |
55 (55) | |