5ed5/4
| ← 4ed5/4 | 5ed5/4 | 6ed5/4 → |
5ED5/4 is the equal division of the just major third into five parts of 77.2627 cents each, corresponding to every second step of 31edo. It is related to Carlos Alpha and the 7-limit temperaments which temper out 2100875/2097152 (including the tertiaseptal temperament and the valentine temperament).
Intervals
| degree | cents value | ratio |
|---|---|---|
| 0 | 0.0000 | 1/1 |
| 1 | 77.2627 | (5/4)1/5 |
| 2 | 154.5255 | (5/4)2/5 |
| 3 | 231.7882 | (5/4)3/5 |
| 4 | 309.0510 | (5/4)4/5 |
| 5 | 386.3137 | 5/4 |
| 6 | 463.5765 | (5/4)6/5 |
| 7 | 540.8392 | (5/4)7/5 |
| 8 | 618.1019 | (5/4)8/5 |
| 9 | 695.3647 | (5/4)9/5 |
| 10 | 772.6274 | (5/4)2 = 25/16 |
| 11 | 849.8902 | (5/4)11/5 |
| 12 | 927.1529 | (5/4)12/5 |
| 13 | 1004.4157 | (5/4)13/5 |
| 14 | 1081.6784 | (5/4)14/5 |
| 15 | 1158.9411 | (5/4)3 = 125/64 |
| 16 | 1236.2039 | (5/4)16/5 |
| 17 | 1313.4666 | (5/4)17/5 |
| 18 | 1390.7294 | (5/4)18/5 |
| 19 | 1467.9921 | (5/4)19/5 |
| 20 | 1545.2549 | (5/4)4 = 625/256 |
| 21 | 1622.5176 | (5/4)21/5 |
| 22 | 1699.7803 | (5/4)22/5 |
| 23 | 1777.0431 | (5/4)23/5 |
| 24 | 1854.3058 | (5/4)24/5 |
| 25 | 1931.5686 | (5/4)5 = 3125/1024 |
| 26 | 2008.8313 | (5/4)26/5 |
| 27 | 2086.0941 | (5/4)27/5 |
| 28 | 2163.3568 | (5/4)28/5 |
| 29 | 2240.6195 | (5/4)29/5 |
| 30 | 2317.8823 | (5/4)6 = 15625/4096 |
| 31 | 2395.1450 | (5/4)31/5 |
| 32 | 2472.4078 | (5/4)32/5 |
| 33 | 2549.6705 | (5/4)33/5 |
| 34 | 2626.9333 | (5/4)34/5 |
| 35 | 2704.1960 | (5/4)7 = 78125/16384 |
| 36 | 2781.4587 | (5/4)36/5 |
| 37 | 2858.7215 | (5/4)37/5 |
| 38 | 2935.9842 | (5/4)38/5 |
| 39 | 3013.2470 | (5/4)39/5 |
| 40 | 3090.5097 | (5/4)8 = 390625/65536 |
| 41 | 3167.7725 | (5/4)41/5 |
| 42 | 3245.0352 | (5/4)42/5 |
| 43 | 3322.2979 | (5/4)43/5 |
| 44 | 3399.5607 | (5/4)44/5 |
| 45 | 3476.8234 | (5/4)9 = 1953125/262144 |
| 46 | 3554.0862 | (5/4)46/5 |
| 47 | 3631.3489 | (5/4)47/5 |
| 48 | 3708.6117 | (5/4)48/5 |
| 49 | 3785.8744 | (5/4)49/5 |
| 50 | 3863.1371 | (5/4)10 = 9765625/1048576 |
| 51 | 3940.3999 | (5/4)51/5 |
| 52 | 4017.6626 | (5/4)52/5 |
| 53 | 4094.9254 | (5/4)53/5 |
| 54 | 4172.1881 | (5/4)54/5 |
| 55 | 4249.4509 | (5/4)11 = 48828125/4194304 |
| 56 | 4326.7136 | (5/4)56/5 |
| 57 | 4403.9763 | (5/4)57/5 |
| 58 | 4481.2391 | (5/4)58/5 |
| 59 | 4558.5018 | (5/4)59/5 |
| 60 | 4635.7646 | (5/4)12 = 244140625/16777216 |
| 61 | 4713.0273 | (5/4)61/5 |
| 62 | 4790.2901 | (5/4)62/5 |
| 63 | 4867.5528 | (5/4)63/5 |
| 64 | 4944.8155 | (5/4)64/5 |
| 65 | 5022.0783 | (5/4)13 = 1220703125/67108864 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +36.2 | +29.6 | -4.9 | -4.9 | -11.4 | +30.7 | +31.3 | -18.0 | +31.3 | +20.9 | +24.8 |
| Relative (%) | +46.9 | +38.3 | -6.3 | -6.3 | -14.8 | +39.8 | +40.6 | -23.3 | +40.6 | +27.0 | +32.0 | |
| Steps (reduced) |
16 (1) |
25 (0) |
31 (1) |
36 (1) |
40 (0) |
44 (4) |
47 (2) |
49 (4) |
52 (2) |
54 (4) |
56 (1) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -36.6 | -10.3 | +24.8 | -9.7 | -37.4 | +18.2 | +1.8 | -9.7 | -16.9 | -20.2 | -19.9 |
| Relative (%) | -47.3 | -13.4 | +32.0 | -12.6 | -48.4 | +23.5 | +2.4 | -12.6 | -21.9 | -26.1 | -25.7 | |
| Steps (reduced) |
57 (2) |
59 (4) |
61 (1) |
62 (2) |
63 (3) |
65 (0) |
66 (1) |
67 (2) |
68 (3) |
69 (4) |
70 (0) | |
5ED5/4 as a generator
Valentine
Aside from 2100875/2097152, valentine temperament tempers out 126/125, 1029/1024, 6144/6125, and 64827/64000 in the 7-limit. It can be described as the 31&46 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 22/21) can serve as its generator. In the 11-limit, it tempers out 121/120, 176/175, and 441/440.
Tertiaseptal
Aside from 2100875/2097152, tertiaseptal temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.
Tertia
Aside from 2100875/2097152, tertia temperament tempers out 385/384, 1331/1323, and 1375/1372 in the 11-limit. It can be described as the 31&140 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 352/351, 385/384, 561/560, 625/624, and 715/714.