21ed5/2

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← 20ed5/221ed5/222ed5/2 →
Prime factorization 3 × 7
Step size 75.5387¢ 
Octave 16\21ed5/2 (1208.62¢)
(semiconvergent)
Twelfth 25\21ed5/2 (1888.47¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 6

21ed5/2 is the equal division of the 5/2 interval into 21 parts of approximately 75.539 cents each. It roughly corresponds to 16edo.

Theory

From a no-threes point of view, 21ed5/2 tempers out 50/49 in the 7-limit (being a jubilic system similar to 13ed5/2), 625/616 and 176/175 in the 11-limit, and 143/140, 715/686 and 847/845 in the 13-limit. It is not particularly excellent as a no-threes system with the 5/4 and 7/4 being noticeably off, but can work for 5/2-equivalent jubilic.

Harmonics

Approximation of harmonics in 21ed5/2
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.6 -13.5 +17.2 +8.6 -4.9 +30.4 +25.9 -27.0 +17.2 +3.3 +3.8
Relative (%) +11.4 -17.9 +22.8 +11.4 -6.4 +40.3 +34.2 -35.7 +22.8 +4.4 +5.0
Steps
(reduced)
16
(16)
25
(4)
32
(11)
37
(16)
41
(20)
45
(3)
48
(6)
50
(8)
53
(11)
55
(13)
57
(15)
Approximation of harmonics in 21ed5/2
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +16.3 -36.5 -4.9 +34.5 +5.1 -18.4 -36.4 +25.9 +16.9 +11.9 +10.5
Relative (%) +21.5 -48.3 -6.4 +45.6 +6.7 -24.3 -48.2 +34.2 +22.4 +15.8 +13.9
Steps
(reduced)
59
(17)
60
(18)
62
(20)
64
(1)
65
(2)
66
(3)
67
(4)
69
(6)
70
(7)
71
(8)
72
(9)

Interval table

Steps Cents Jubilic[8] notation Approximate ratios*
0 0.000 J 1/1
1 75.539 J& 26/25
2 151.078 K@ 35/32
3 226.616 K 8/7, 28/25
4 302.155 K&, L@ 13/11, 77/64
5 377.694 L 5/4, 11/9, 16/13, 49/40
6 453.233 L& 13/10, 32/25
7 528.771 M@ 11/8, 35/26
8 604.310 M 7/5, 10/7
9 679.849 M& 16/11, 52/35
10 755.388 N@ 11/7, 20/13, 25/16, 49/32
11 830.926 N 8/5, 13/8
12 906.465 N&, O@ 22/13, 55/32
13 982.004 O 7/4, 25/14
14 1057.543 O& 13/7, 20/11
15 1133.081 P@ 25/13
16 1208.620 P 2/1
17 1284.159 P&, Q@ 52/25
18 1359.698 Q 11/5
19 1435.236 Q& 16/7
20 1510.775 J@ 26/11
21 1586.314 J 5/2

* Based on treating 21ed5/2 as a no-threes 13-limit temperament