44ed5
| ← 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. The local zeta peak around 19 is located at 18.948087, which has the octave stretched by 3.2877 ¢, and the octave of 44ed5 comes extremely close (differing 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 by different amounts, 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 allow 19edo to be a promising option for pianos and harpsichords with split sharps since they have stretched partials (especially on cheaper spinet pianos, since their short string lengths result in more inharmonicity compared to upright or grand pianos), and the most noticeable partial is the 2nd; thus, a piano tuned to have beatless octaves will actually have them around 1203 ¢ 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) | |