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 | 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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