28ed5

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← 27ed5 28ed5 29ed5 →
Prime factorization 22 × 7
Step size 99.5112¢ 
Octave 12\28ed5 (1194.13¢) (→3\7ed5)
Twelfth 19\28ed5 (1890.71¢)
Consistency limit 9
Distinct consistency limit 6

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

Approximation of harmonics in 28ed5
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)
Approximation of harmonics in 28ed5
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

External links