# 4ed5/4

 ← 3ed5/4 4ed5/4 5ed5/4 →
Prime factorization 22
Step size 96.5784¢
Octave 12\4ed5/4 (1158.94¢) (→3\1ed5/4)
Twelfth 20\4ed5/4 (1931.57¢) (→5\1ed5/4)
Consistency limit 2
Distinct consistency limit 1
Special properties

4ED5/4 is the equal division of the just major third into four parts of 96.5784 cents each, corresponding to 12.4251 EDO. It is related to the temperament which tempers out 48828125/48771072 and 67108864/66976875 in the 7-limit (hemiluna temperament), which is supported by 87EDO, 323EDO, 410EDO, and 733EDO among others.

## Intervals

degree cents value ratio
0 0.0000 1/1
1 96.5784 (5/4)1/4
2 193.1569 (5/4)1/2
3 289.7353 (5/4)3/4
4 386.3137 5/4
5 482.8921 (5/4)5/4
6 579.4706 (5/4)3/2
7 676.0490 (5/4)7/4
8 772.6274 (5/4)2 = 25/16
9 869.2059 (5/4)9/4
10 965.7843 (5/4)5/2
11 1062.3627 (5/4)11/4
12 1158.9411 (5/4)3 = 125/64
13 1255.5196 (5/4)13/4
14 1352.0980 (5/4)7/2
15 1448.6764 (5/4)15/4
16 1545.2549 (5/4)4 = 625/256
17 1641.8333 (5/4)17/4
18 1738.4117 (5/4)9/2
19 1834.9901 (5/4)19/4
20 1931.5686 (5/4)5 = 3125/1024
21 2028.1470 (5/4)21/4
22 2124.7254 (5/4)11/2
23 2221.3039 (5/4)23/4
24 2317.8823 (5/4)6 = 15625/4096
25 2414.4607 (5/4)25/4
26 2511.0391 (5/4)13/2
27 2607.6176 (5/4)27/4
28 2704.1960 (5/4)7 = 78125/16384
29 2800.7744 (5/4)29/4
30 2897.3529 (5/4)15/2
31 2993.9313 (5/4)31/4
32 3090.5097 (5/4)8 = 390625/65536
33 3187.0881 (5/4)33/4
34 3283.6666 (5/4)17/2
35 3380.2450 (5/4)35/4
36 3476.8234 (5/4)9 = 1953125/262144
37 3573.4019 (5/4)37/4
38 3669.9803 (5/4)19/2
39 3766.5587 (5/4)39/4
40 3863.1371 (5/4)10 = 9765625/1048576
41 3959.7156 (5/4)41/4
42 4056.2940 (5/4)21/2
43 4152.8724 (5/4)43/4
44 4249.4509 (5/4)11 = 48828125/4194304
45 4346.0293 (5/4)45/4
46 4442.6077 (5/4)23/2
47 4539.1861 (5/4)47/4
48 4635.7646 (5/4)12 = 244140625/16777216
49 4732.3430 (5/4)49/4
50 4828.9214 (5/4)25/2
51 4925.4999 (5/4)51/4
52 5022.0783 (5/4)13 = 1220703125/67108864

## Related regular temperament

4ED5/4 tuning is closely related to the Hemiluna temperament (87&323), which tempers out 48828125/48771072 (neptunisma, laruruleyo) and 67108864/66976875 (decovulture comma, sasabirugugu) in the 7-limit. It splits a hemithird interval into two equal parts.

Hemiluna (87 & 323)

7-limit
Comma list: 48828125/48771072, 67108864/66976875
Mapping: [1 4 2 -5], 0 -30 4 97]]
POTE generator: ~200/189 = 96.591
Optimal ET sequence87, 236, 323, 410, 733

11-limit
Comma list: 5632/5625, 14641/14580, 131072/130977
Mapping: [1 4 2 -5 7], 0 -30 4 97 -44]]
POTE generator: ~200/189 = 96.587
Optimal ET sequence87, 236, 323, 410