28ed5

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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.

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

Regular temperaments

Main article: Quindromeda family

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 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