5ed7: Difference between revisions
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== Theory == | == Theory == | ||
5ed7 can be treated as an [[11-limit]] [[exotemperament]]. It performs very well at approximating the 11-limit for its size, having under 25 [[relative cent]]s of relative error on harmonics 2, 3, 5, 7 and 11. Of course, that's only treating things as relative. It still has roughly 100 cents of absolute error on all those harmonics except 7, and worst of all is the octave having 148 cents of absolute error. But this could still be theoretically interesting since it does a lot better than other sub-[[5edo]] equal temperaments. | 5ed7 can be treated as an [[11-limit]] [[exotemperament]]. It performs very well at approximating the 11-limit for its size, having under 25 [[relative cent]]s of relative error on harmonics 2, 3, 5, 7 and 11. Of course, that's only treating things as relative. It still has roughly 100 cents of absolute error on all those harmonics (except 7 obviously), and worst of all is the octave having 148 cents of absolute error. But this could still be theoretically interesting since it does a lot better than other sub-[[5edo]] equal temperaments. | ||
The [[armodue]] temperament can be seen as 5ed7 with an independent octave generator and with the single step equated to [[3/2]] (also mapping the 5th harmonic as 7 octaves - 3 generators). | |||
== Intervals == | == Intervals == | ||
Latest revision as of 22:43, 18 October 2026
| ← 4ed7 | 5ed7 | 6ed7 → |
(semiconvergent)
(semiconvergent)
5 equal divisions of the 7th harmonic (abbreviated 5ed7) is a nonoctave tuning system that divides the interval of 7/1 into 5 equal parts of about 674 ¢ each. Each step represents a frequency ratio of 71/5, or the 5th root of 7.
Theory
5ed7 can be treated as an 11-limit exotemperament. It performs very well at approximating the 11-limit for its size, having under 25 relative cents of relative error on harmonics 2, 3, 5, 7 and 11. Of course, that's only treating things as relative. It still has roughly 100 cents of absolute error on all those harmonics (except 7 obviously), and worst of all is the octave having 148 cents of absolute error. But this could still be theoretically interesting since it does a lot better than other sub-5edo equal temperaments.
The armodue temperament can be seen as 5ed7 with an independent octave generator and with the single step equated to 3/2 (also mapping the 5th harmonic as 7 octaves - 3 generators).
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 673.8 | 3/2, 10/7, 13/9, 19/13, 20/13 |
| 2 | 1347.5 | 9/4, 11/5, 13/6, 15/7, 19/9, 20/9, 21/10 |
| 3 | 2021.3 | 10/3, 13/4, 19/6 |
| 4 | 2695.1 | 14/3, 19/4 |
| 5 | 3368.8 | 7/1 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +148 | +119 | -91 | +0 | -109 | +276 | -189 | +293 | -38 | +234 | +119 |
| Relative (%) | +21.9 | +17.7 | -13.5 | +0.0 | -16.1 | +40.9 | -28.0 | +43.4 | -5.7 | +34.8 | +17.6 | |
| Steps (reduced) |
2 (2) |
3 (3) |
4 (4) |
5 (0) |
6 (1) |
7 (2) |
7 (2) |
8 (3) |
8 (3) |
9 (4) |
9 (4) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -187 | +309 | +226 | +72 | -136 | -322 | +295 | +132 | +32 | -16 | -153 |
| Relative (%) | -27.8 | +45.8 | +33.6 | +10.7 | -20.2 | -47.7 | +43.7 | +19.6 | +4.7 | -2.4 | -22.7 | |
| Steps (reduced) |
9 (4) |
10 (0) |
10 (0) |
10 (0) |
10 (0) |
10 (0) |
11 (1) |
11 (1) |
11 (1) |
11 (1) |
11 (1) | |
|
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