4ed5/4
| ← 3ed5/4 | 4ed5/4 | 5ed5/4 → |
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 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -41.1 | +29.6 | +14.5 | +14.5 | -11.4 | +11.4 | -26.6 | -37.4 | -26.6 | +1.6 | +44.1 |
| Relative (%) | -42.5 | +30.7 | +15.0 | +15.0 | -11.9 | +11.8 | -27.5 | -38.7 | -27.5 | +1.6 | +45.6 | |
| Steps (reduced) |
12 (0) |
20 (0) |
25 (1) |
29 (1) |
32 (0) |
35 (3) |
37 (1) |
39 (3) |
41 (1) |
43 (3) |
45 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.1 | -29.6 | +44.1 | +28.9 | +20.5 | +18.2 | +21.1 | +28.9 | +41.0 | -39.5 | -19.9 |
| Relative (%) | +2.2 | -30.7 | +45.6 | +29.9 | +21.3 | +18.8 | +21.9 | +29.9 | +42.5 | -40.9 | -20.6 | |
| Steps (reduced) |
46 (2) |
47 (3) |
49 (1) |
50 (2) |
51 (3) |
52 (0) |
53 (1) |
54 (2) |
55 (3) |
55 (3) |
56 (0) | |
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 sequence: 87, 236, 323, 410, 733
Badness: 0.179304
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 sequence: 87, 236, 323, 410
Badness: 0.100077
13-limit
Comma list: 676/675, 1001/1000, 4096/4095, 14641/14580
Mapping: [⟨1 4 2 -5 7 7], ⟨0 -30 4 97 -44 -41]]
POTE generator: ~143/135 = 96.586
Optimal ET sequence: 87, 236, 323, 410
Badness: 0.046151