28ed5
| ← 27ed5 | 28ed5 | 29ed5 → |
Division of the 5th harmonic into 28 equal parts (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.
Intervals
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 99.5112 | 18/17 | |
| 2 | 199.0224 | 55/49 | |
| 3 | 298.5336 | 19/16 | |
| 4 | 398.0448 | 34/27 | pseudo-5/4 |
| 5 | 497.5560 | 4/3 | |
| 6 | 597.0672 | 24/17 | |
| 7 | 696.5784 | 175/117, 323/216 | meantone fifth (pseudo-3/2) |
| 8 | 796.0896 | 19/12 | |
| 9 | 895.6008 | 57/34 | pseudo-5/3 |
| 10 | 995.1120 | 16/9 | |
| 11 | 1094.6232 | 32/17 | |
| 12 | 1194.1344 | 255/128 | pseudo-octave |
| 13 | 1293.6457 | 19/9 | |
| 14 | 1393.1569 | 38/17, 85/38 | meantone major second plus an octave |
| 15 | 1492.6681 | 45/19 | |
| 16 | 1592.1793 | 128/51 | pseudo-5/2 |
| 17 | 1691.6905 | 85/32 | |
| 18 | 1791.2017 | 45/16 | |
| 19 | 1890.7129 | 170/57 | pseudo-3/1 |
| 20 | 1990.2241 | 60/19 | |
| 21 | 2089.7353 | 117/35 | meantone major sixth plus an octave (pseudo-10/3) |
| 22 | 2189.2465 | 85/24 | |
| 23 | 2288.7577 | 15/4 | |
| 24 | 2388.2689 | 135/34 | pseudo-4/1 |
| 25 | 2487.7801 | 80/19 | |
| 26 | 2587.2913 | 49/11 | |
| 27 | 2686.8025 | 85/18 | |
| 28 | 2786.3137 | exact 5/1 | just major third plus two octaves |
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 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +37.5 | +8.7 | -11.2 | -23.5 | -28.9 | -28.3 | -22.4 | -11.7 | +3.3 | +22.3 | +44.8 |
| Relative (%) | +37.7 | +8.7 | -11.3 | -23.6 | -29.0 | -28.5 | -22.6 | -11.8 | +3.3 | +22.4 | +45.1 | |
| Steps (reduced) |
45 (17) |
46 (18) |
47 (19) |
48 (20) |
49 (21) |
50 (22) |
51 (23) |
52 (24) |
53 (25) |
54 (26) |
55 (27) | |
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 241 EDOs.
Equating 225/224 with 256/255 leads 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 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
- 12EDO - relative EDO
- 19ED3 - relative ED3
- 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