10edt
| ← 9edt | 10edt | 11edt → |
10 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 10edt or 10ed3), is a nonoctave tuning system that divides the interval of 3/1 into 10 equal parts of about 190 ¢ each. Each step represents a frequency ratio of 31/10, or the 10th root of 3.
Theory
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the 5/4 inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as 7/5. We can use this to readily construct chords such as 4:5:7:12, although the 7/4, being 18 cents flat, does considerable damage to the consonance of this chord.
10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
One step of 10edt can serve as the generator for pocus temperament, a merge of sensamagic and 2.3.5.7.13 catakleismic, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -58.8 | +0.0 | +72.5 | +66.6 | -58.8 | +54.7 | +13.7 | +0.0 | +7.8 | +33.0 | +72.5 |
| relative (%) | -31 | +0 | +38 | +35 | -31 | +29 | +7 | +0 | +4 | +17 | +38 | |
| Steps (reduced) |
6 (6) |
10 (0) |
13 (3) |
15 (5) |
16 (6) |
18 (8) |
19 (9) |
20 (0) |
21 (1) |
22 (2) |
23 (3) | |
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -58.8 | +0.0 | +66.6 | +54.7 | +33.0 | -66.0 | +40.1 | +37.8 | +87.4 | +66.5 | -49.0 |
| relative (%) | -31 | +0 | +35 | +29 | +17 | -35 | +21 | +20 | +46 | +35 | -26 | |
| Steps (reduced) |
6 (6) |
10 (0) |
15 (5) |
18 (8) |
22 (2) |
23 (3) |
26 (6) |
27 (7) |
29 (9) |
31 (1) |
31 (1) | |
Interval table
| Degrees | Cents | Hekts | Approximate Ratios |
| 0 | 1/1 | ||
|---|---|---|---|
| 1 | 190.196 | 130 | 10/9, 28/25 |
| 2 | 380.391 | 260 | 5/4 |
| 3 | 570.587 | 390 | 7/5 |
| 4 | 760.782 | 520 | 14/9 |
| 5 | 950.978 | 650 | 45/26, 26/15 |
| 6 | 1141.173 | 780 | 27/14 |
| 7 | 1331.369 | 910 | 15/7 (15/14 plus an octave) |
| 8 | 1521.564 | 1040 | 12/5 (6/5 plus an octave) |
| 9 | 1711.760 | 1170 | 27/10 |
| 10 | 1901.955 | 1300 | 3/1 |