28ed5
| ← 27ed5 | 28ed5 | 29ed5 → |
28 equal divisions of the 5th harmonic (abbreviated 28ed5) is a nonoctave tuning system that divides the interval of 5/1 into 28 equal parts of about 99.5 ¢ each. Each step represents a frequency ratio of 51/28, or the 28th root of 5.
Theory
28ed5 is related to 12edo, but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 99.5112 cents. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4. This tuning also has the perfect fourth which is more accurate for 4/3 than that of 12edo, as well as 18/17, 19/16, and 24/17.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.9 | -11.2 | -11.7 | +0.0 | -17.1 | +14.6 | -17.6 | -22.5 | -5.9 | +28.2 | -23.0 |
| Relative (%) | -5.9 | -11.3 | -11.8 | +0.0 | -17.2 | +14.6 | -17.7 | -22.6 | -5.9 | +28.3 | -23.1 | |
| Steps (reduced) |
12 (12) |
19 (19) |
24 (24) |
28 (0) |
31 (3) |
34 (6) |
36 (8) |
38 (10) |
40 (12) |
42 (14) |
43 (15) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +37.5 | +8.7 | -11.2 | -23.5 | -28.9 | -28.3 | -22.4 | -11.7 | +3.3 | +22.3 | +44.8 | -28.8 |
| Relative (%) | +37.7 | +8.7 | -11.3 | -23.6 | -29.0 | -28.5 | -22.6 | -11.8 | +3.3 | +22.4 | +45.1 | -29.0 | |
| Steps (reduced) |
45 (17) |
46 (18) |
47 (19) |
48 (20) |
49 (21) |
50 (22) |
51 (23) |
52 (24) |
53 (25) |
54 (26) |
55 (27) |
55 (27) | |
Subsets and supersets
Since 28 factors into 22 × 7, 28ed5 has subset ed5's 2, 4, 7, and 14.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 99.5 | 18/17 |
| 2 | 199.0 | 9/8 |
| 3 | 298.5 | 6/5 |
| 4 | 398.0 | 5/4 |
| 5 | 497.6 | 4/3 |
| 6 | 597.1 | 7/5 |
| 7 | 696.6 | 3/2 |
| 8 | 796.1 | 8/5 |
| 9 | 895.6 | 5/3 |
| 10 | 995.1 | 7/4 |
| 11 | 1094.6 | 15/8 |
| 12 | 1194.1 | 2/1 |
| 13 | 1293.6 | 17/8 |
| 14 | 1393.2 | 9/4 |
| 15 | 1492.7 | 12/5 |
| 16 | 1592.2 | 5/2 |
| 17 | 1691.7 | 8/3 |
| 18 | 1791.2 | 14/5 |
| 19 | 1890.7 | 3/1 |
| 20 | 1990.2 | 16/5 |
| 21 | 2089.7 | 10/3 |
| 22 | 2189.2 | 7/2 |
| 23 | 2288.8 | 15/4 |
| 24 | 2388.3 | 4/1 |
| 25 | 2487.8 | 17/4 |
| 26 | 2587.3 | 9/2 |
| 27 | 2686.8 | 19/4 |
| 28 | 2786.3 | 5/1 |
Regular temperaments
28ed5 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 1216/1215, 1445/1444, and 6144/6137, which is a cluster temperament with 12 clusters of notes in an octave (quindromeda temperament). This temperament is supported by 12-, 169-, 181-, 193-, 205-, 217-, 229-, and 241edo.
Equating 225/224 with 256/255 leads to quintakwai (12 & 193), which tempers out 400/399 (also equating 20/19 and 21/20) in the 2.3.5.7.17.19 subgroup, and 361/360 with 400/399 leads to quintagar (12 & 217), which tempers out 476/475 (also equating 19/17 with 28/25) in the 2.3.5.7.17.19 subgroup.
See also
- 7edf – relative edf
- 12edo – relative edo
- 19edt – relative edt
- 31ed6 – relative ed6
- 34ed7 – relative ed7
- 40ed10 – relative ed10
- 42ed11 – relative ed11
- AS18/17 – relative ambitonal sequence
External links
- Play 28ed5 – Scale Workshop
- Play 28ed5 – Terpstra Keyboard WebApp