25edt
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| ← 24edt | 25edt | 26edt → |
25 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 25edt or 25ed3), is a nonoctave tuning system that divides the interval of 3/1 into 25 equal parts of about 76.1 ¢ each. Each step represents a frequency ratio of 31/25, or the 25th root of 3.
Theory
25edt corresponds to 15.7732…edo, or 16 equal divisions of a stretched octave (1217.25 ¢) and a tritave twin of the Armodue/Hornbostel flat third-tone system:
- 6th = 1065.095 ¢
- squared = 2130.19 ¢ → 228.235 ¢
- cubed = 1293.33 ¢
- fourth power = 2358.425 ¢ → 456.47 ¢
It can be used as a tuning for mavila and has an antidiatonic (2L 5s) scale which approximates Pelog tunings in Indonesian gamelan music.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.3 | +0.0 | +34.5 | +28.6 | +17.3 | -21.4 | -24.3 | +0.0 | -30.2 | +33.0 | +34.5 |
| Relative (%) | +22.7 | +0.0 | +45.4 | +37.6 | +22.7 | -28.1 | -32.0 | +0.0 | -39.8 | +43.4 | +45.4 | |
| Steps (reduced) |
16 (16) |
25 (0) |
32 (7) |
37 (12) |
41 (16) |
44 (19) |
47 (22) |
50 (0) |
52 (2) |
55 (5) |
57 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -28.0 | -4.1 | +28.6 | -7.1 | -36.0 | +17.3 | -0.3 | -13.0 | -21.4 | -25.8 | -26.7 |
| Relative (%) | -36.8 | -5.4 | +37.6 | -9.3 | -47.3 | +22.7 | -0.4 | -17.1 | -28.1 | -34.0 | -35.1 | |
| Steps (reduced) |
58 (8) |
60 (10) |
62 (12) |
63 (13) |
64 (14) |
66 (16) |
67 (17) |
68 (18) |
69 (19) |
70 (20) |
71 (21) | |
Subsets and supersets
Since 25 factors into primes as 52, 25edt contains 5edt as its only nontrivial subset edt.
Intervals
| # | Cents | Hekts | Armodue name |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 1 |
| 1 | 76.1 | 52.0 | 1#/2bb |
| 2 | 152.2 | 104.0 | 1x/2b |
| 3 | 228.2 | 156.0 | 2 |
| 4 | 304.3 | 208.0 | 2#/3bb |
| 5 | 380.4 | 260.0 | 2x/3b |
| 6 | 456.5 | 312.0 | 3 |
| 7 | 532.5 | 364.0 | 3#/4b |
| 8 | 608.6 | 416.0 | 4 |
| 9 | 684.7 | 468.0 | 4#/5bb |
| 10 | 760.8 | 520.0 | 4x/5b |
| 11 | 836.9 | 572.0 | 5 |
| 12 | 912.9 | 624.0 | 5#/6bb |
| 13 | 989.0 | 676.0 | 5x/6b |
| 14 | 1065.1 | 728.0 | 6 |
| 15 | 1141.2 | 780.0 | 6#/7bb |
| 16 | 1217.3 | 832.0 | 6x/7b |
| 17 | 1293.3 | 884.0 | 7 |
| 18 | 1369.4 | 936.0 | 7#/8b |
| 19 | 1445.5 | 988.0 | 8 |
| 20 | 1521.6 | 1040.0 | 8#/9bb |
| 21 | 1597.6 | 1092.0 | 8x/9b |
| 22 | 1673.7 | 1144.0 | 9 |
| 23 | 1749.8 | 1196.0 | 9#/1bb |
| 24 | 1825.9 | 1248.0 | 9x/1b |
| 25 | 1902.0 | 1300.0 | 1 |