28edt
| ← 27edt | 28edt | 29edt → |
28edt divides the third harmonic into 28 equal parts of 67.9 cents each. It is the simplest edt which intelligibly doubles the Anti-BP harmonic chain[clarification needed].
Theory
From a no-twos perspective, 28edt has a nearly-pure 5th harmonic (which tritave-reduces to 5/3), with 41/28 being a continued fraction convergent to log3(5). It poorly approximates harmonic 7, but has an acceptable approximation of harmonic 11, so it makes sense to consider it as a 3.5.11 subgroup system. In the 3.5.11 subgroup, it supports the deneb temperament, which equates 3 flat 11/9s to 5/3, and the delta centauri temperament, which equates 9 near-just 5/3s tritave-reduced to 11/9 (although delta centauri is a microtemperament and larger EDTs like 71edt offer much better tunings of it).
28edt has a very accurate 8th harmonic, and as such, one step is very close to every third step of 53edo, and the tritave-reduced 8th harmonic 8/3 and tritave-reduced 8th subharmonic 9/8 are equally well-approximated. It tempers out the schisma in the 8.3.5 subgroup, albeit a really strange tuning due to being a multiple of 4edt and being index-2 in BP.
Similar to how 14edt can be seen as 9edo with significantly stretched octaves (~23 ¢), 28edt can be seen as 18edo with the same stretched octave. Note that using the stretched octave will map 8/1 to an inconsistent approximation rather than the aforementioned accurate 8/1. Considering that 29edt is a compressed 18edo, neither 28edt or 29edt are truly equivalent to 18edo (these two basically correspond to 18edo's dual fifths).
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +22.7 | +0.0 | -22.6 | -1.3 | +22.7 | +27.5 | +0.1 | +0.0 | +21.4 | -7.8 | -22.6 |
| Relative (%) | +33.4 | +0.0 | -33.2 | -1.9 | +33.4 | +40.5 | +0.2 | +0.0 | +31.5 | -11.4 | -33.2 | |
| Steps (reduced) |
18 (18) |
28 (0) |
35 (7) |
41 (13) |
46 (18) |
50 (22) |
53 (25) |
56 (0) |
59 (3) |
61 (5) |
63 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -25.3 | -17.7 | -1.3 | +22.8 | -14.2 | +22.7 | -3.0 | -23.9 | +27.5 | +14.9 | +5.9 |
| Relative (%) | -37.2 | -26.1 | -1.9 | +33.6 | -20.9 | +33.4 | -4.4 | -35.1 | +40.5 | +22.0 | +8.7 | |
| Steps (reduced) |
65 (9) |
67 (11) |
69 (13) |
71 (15) |
72 (16) |
74 (18) |
75 (19) |
76 (20) |
78 (22) |
79 (23) |
80 (24) | |
Intervals
| Steps | cents | hekts | notation |
|---|---|---|---|
| 1 | 67.927 | 46.429 | Cq/D\\y |
| 2 | 135.854 | 92.857 | Cp/D\\ |
| 3 | 203.781 | 139.286 | Cpq/Dy |
| 4 | 271.708 | 185.714 | D |
| 5 | 339.635 | 232.143 | Dp/Ey |
| 6 | 407.562 | 278.571 | E |
| 7 | 475.489 | 325.000 | Epq/F\\y |
| 8 | 543.416 | 371.429 | Ep/F\\ |
| 9 | 611.343 | 417.857 | Eq/Fy |
| 10 | 679.270 | 464.286 | F |
| 11 | 747.197 | 510.714 | Fq/Gy |
| 12 | 815.124 | 557.143 | G |
| 13 | 883.051 | 603.571 | Gq/H\\y |
| 14 | 950.978 | 650.000 | Gp/H\\ |
| 15 | 1018.904 | 696.429 | Gpq/Hy |
| 16 | 1086.831 | 742.857 | H |
| 17 | 1154.758 | 789.286 | Hq/Jy |
| 18 | 1222.685 | 835.714 | J |
| 19 | 1290.612 | 882.143 | Jq/A\\y |
| 20 | 1358.539 | 928.571 | Jp/A\\ |
| 21 | 1426.466 | 975.000 | Jpq/Ay |
| 22 | 1494.393 | 1021.429 | A |
| 23 | 1562.320 | 1067.857 | Aq/B\\y |
| 24 | 1630.247 | 1114.286 | Ap/B\\ |
| 25 | 1698.174 | 1160.714 | Apq/By |
| 26 | 1766.101 | 1207.143 | B |
| 27 | 1834.028 | 1253.571 | Bq/Cy |
| 28 | 1901.955 | 1300.000 | C |