44ed5: Difference between revisions
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== Theory == | == Theory == | ||
44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the | 44ed5 is very similar to [[19edo]], but with the [[5/1]] rather than the 2/1 being just. It is extremely close to the [[The Riemann zeta function and tuning|zeta peak]] near 19; the local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877{{c}}, and the octave of 44ed5 differs by only {{sfrac|1|9}}{{c}}, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy. | ||
Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are [[30edt]], [[49ed6]], and [[93ed30]]. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp. | |||
Tunings in this range are a promising option for pianos and harpsichords since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has. | |||
=== Harmonics === | |||
{{Harmonics in equal|44|5|1}} | |||
{{Harmonics in equal|44|5|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in {{susbt:PAGENAME}} (continued)}} | |||
[[Category:19edo]] | |||
[[Category:Zeta-optimized tunings]] | [[Category:Zeta-optimized tunings]] | ||
Revision as of 13:36, 18 June 2025
| ← 43ed5 | 44ed5 | 45ed5 → |
(semiconvergent)
44 equal divisions of the 5th harmonic (abbreviated 44ed5) is a nonoctave tuning system that divides the interval of 5/1 into 44 equal parts of about 63.3 ¢ each. Each step represents a frequency ratio of 51/44, or the 44th root of 5.
Theory
44ed5 is very similar to 19edo, but with the 5/1 rather than the 2/1 being just. It is extremely close to the zeta peak near 19; the local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877 ¢, and the octave of 44ed5 differs by only 1/9 ¢, thus minimizing relative error as much as possible. Because 19edo itself is a flat-tending system, stretching the octave improves the overall tuning accuracy.
Other variants (which stretch the octave slightly more, but the differences are probably imperceptible) are 30edt, 49ed6, and 93ed30. The latter of the two optimizes the accuracy of the 1:5:6 triad, since the 5 is as flat as the 6 is sharp.
Tunings in this range are a promising option for pianos and harpsichords since they have stretched partials, and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 cents or so (depending on string length), which coincidentally is very close to what the zeta-optimal stretched version of 19edo has.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.2 | -2.2 | +6.4 | +0.0 | +1.0 | -12.6 | +9.5 | -4.4 | +3.2 | +28.2 | +4.2 |
| Relative (%) | +5.0 | -3.5 | +10.0 | +0.0 | +1.6 | -19.9 | +15.1 | -6.9 | +5.0 | +44.5 | +6.6 | |
| Steps (reduced) |
19 (19) |
30 (30) |
38 (38) |
44 (0) |
49 (5) |
53 (9) |
57 (13) |
60 (16) |
63 (19) |
66 (22) |
68 (24) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -7.8 | -9.4 | -2.2 | +12.7 | -28.9 | -1.2 | -31.5 | +6.4 | -14.8 | +31.3 | +17.7 | +7.3 |
| Relative (%) | -12.2 | -14.8 | -3.5 | +20.1 | -45.6 | -1.9 | -49.7 | +10.0 | -23.3 | +49.5 | +28.0 | +11.6 | |
| Steps (reduced) |
70 (26) |
72 (28) |
74 (30) |
76 (32) |
77 (33) |
79 (35) |
80 (36) |
82 (38) |
83 (39) |
85 (41) |
86 (42) |
87 (43) | |