# 24ed5

 ← 23ed5 24ed5 25ed5 →
Prime factorization 23 × 3
Step size 116.096¢
Octave 10\24ed5 (1160.96¢) (→5\12ed5)
Twelfth 16\24ed5 (1857.54¢) (→2\3ed5)
Consistency limit 3
Distinct consistency limit 3
Special properties

Division of the 5th harmonic into 24 equal parts (24ed5) is related to the miracle temperament. The step size about 116.0964 cents. It is similar to every third step of 31edo, but with the 5/1 rather than the 2/1 being just. This tuning has a meantone fifth as the number of divisions of the 5th harmonic is multiple of 4.

## Theory

From a no-twos-or-threes point of view, 24ed5 offers a particularly good tuning of the very low-badness 5.7.11 subgroup temperament named as juggernaut, tempering out 125/121. This has a CTE generator of exactly 7/5 (in 24ed5 approximated as 5 steps) and a period of 1\2ed5 or the square root of five (which is equated to 11/5).

## Interval table

degree cents value corresponding
JI intervals
0 0.0000 exact 1/1
1 116.0964 16/15, 15/14
2 232.1928 8/7
3 348.2892 11/9
4 464.3856 17/13
5 580.4820 7/5
6 696.5784 meantone fifth
(pseudo-3/2)
7 812.6748 8/5
8 928.7712 65/38
9 1044.8676 11/6
10 1160.9640 45/23
11 1277.0605 23/11
12 1393.1569 38/17, 85/38 meantone major second plus an octave
13 1509.2533 55/23
14 1625.3497 23/9
15 1741.4461 30/11
16 1857.5425 38/13
17 1973.6389 25/8
18 2089.7353 meantone major sixth plus an octave
(pseudo-10/3)
19 2205.8317 25/7
20 2321.9281 65/17
21 2438.0245 45/11
22 2554.1209 35/8
23 2670.2173 14/3
24 2786.3137 exact 5/1 just major third plus two octaves

## 24ed5 as a generator

24ed5 can also be thought of as a generator of the 2.3.5.7.11.23 subgroup temperament which tempers out 225/224, 243/242, 385/384, and 529/528, which is a cluster temperament with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing 45/44 ~ 46/45 ~ 49/48 ~ 50/49 ~ 55/54 ~ 56/55 ~ 64/63 all tempered together. This temperament is supported by 31edo, 82edo, 113edo, and 144edo.