76ed80: Difference between revisions
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== Theory == | |||
76ed80 is closely related to [[12edo]], but with the 80th harmonic rather than the [[2/1|octave]] being just, resulting 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}} (which is compressed by 2.31{{c}}), making 76ed80 very 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 | |||
Revision as of 03:34, 9 June 2025
| ← 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
76ed80 is closely related to 12edo, but with the 80th harmonic rather than the octave being just, resulting 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 ¢ (which is compressed by 2.31 ¢), making 76ed80 very 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) | |