28ed5: Difference between revisions
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'''Division of the 5th harmonic into 28 equal parts''' (28ed5) is related to [[12edo|12 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size 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. | '''Division of the 5th harmonic into 28 equal parts''' (28ed5) is related to [[12edo|12 edo]], 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. | ||
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Revision as of 01:46, 1 January 2019
Division of the 5th harmonic into 28 equal parts (28ed5) is related to 12 edo, 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.
| 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 | 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 | 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 |
28ed5 as a generator
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. The small chroma interval between adjacent notes in each cluster is very versatile, representing 1088/1083 ~ 256/255 ~ 289/288 ~ 324/323 ~ 361/360 all tempered together. This temperament is supported by 12edo, 205edo, and 217edo among others.