5ed5/4
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 |
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.
| generator | cents value a (octave-reduced) |
17-limit ratio (octave-reduced) |
|---|---|---|
| 1 | 77.2 | 117/112, 256/245, 68/65 |
| 2 | 154.4 | 130/119, 35/32 |
| 3 | 231.6 | 8/7 |
| 4 | 308.8 | 117/98, 140/117 |
| 5 | 386.0 | 5/4 |
| 6 | 463.2 | 17/13 |
| 7 | 540.4 | 175/128 |
| 8 | 617.6 | 10/7 |
| 9 | 694.8 | 112/75 |
| 10 | 772.0 | 25/16 |
| 11 | 849.2 | 44/27, 18/11 |
| 12 | 926.4 | 128/75 |
| 13 | 1003.6 | 25/14 |
| 14 | 1080.8 | 28/15 |
| 15 | 1158.0 | 39/20 |
| 16 | 35.2 | 55/54, 52/51, 51/50, 50/49, 49/48, 45/44 |
| 17 | 112.4 | 16/15 |
| 18 | 189.6 | 39/35 |
| 19 | 266.8 | 7/6 |
| 20 | 344.0 | 39/32 |
| 21 | 421.2 | 51/40 |
| 22 | 498.4 | 4/3 |
| 23 | 575.6 | 39/28 |
| 24 | 652.8 | 35/24 |
| 25 | 730.0 | 32/21 |
| 26 | 807.2 | 51/32 |
| 27 | 884.4 | 5/3 |
| 28 | 961.6 | 68/39 |
| 29 | 1038.8 | 51/28 |
| 30 | 1116.0 | 40/21, 21/11 |
| 31 | 1193.2 | |
| 32 | 70.4 | 26/25, 25/24 |
| 33 | 147.6 | 12/11 |
| 34 | 224.8 | 91/80 |
| 35 | 302.0 | 25/21 |
| 36 | 379.3 | 96/77 |
| 37 | 456.5 | 13/10 |
| 38 | 533.7 | 34/25, 15/11 |
| 39 | 610.9 | 64/45 |
| 40 | 688.1 | 52/35 |
| 41 | 765.3 | 14/9 |
| 42 | 842.5 | 13/8 |
| 43 | 919.7 | 17/10 |
| 44 | 996.9 | 16/9 |
| 45 | 1074.1 | 13/7 |
| 46 | 1151.3 | 68/35, 35/18 |
| 47 | 28.5 | 65/64, 64/63, 56/55 |
| 48 | 105.7 | 17/16 |
| 49 | 182.9 | 10/9 |
| 50 | 260.1 | 64/55 |
| 51 | 337.3 | 17/14 |
| 52 | 414.5 | 14/11 |
| 53 | 491.7 | 65/49 |
| 54 | 568.9 | 25/18 |
| 55 | 646.1 | 16/11 |
| 56 | 723.3 | 85/56 |
| 57 | 800.5 | 35/22 |
| 58 | 877.7 | 128/77 |
| 59 | 954.9 | 26/15 |
| 60 | 1032.1 | 20/11 |
| 61 | 1109.3 | 91/48, 256/135 |
| 62 | 1186.5 | 208/105 |
| 63 | 63.7 | 28/27 |
| 64 | 140.9 | 13/12 |
| 65 | 218.1 | 17/15, 25/22 |
| 66 | 295.3 | 32/27 |
| 67 | 372.5 | 26/21 |
| 68 | 449.7 | 35/27 |
| 69 | 526.9 | 65/48 |
| 70 | 604.1 | 17/12 |
| 71 | 681.3 | 40/27 |
| 72 | 758.5 | 65/42 |
| 73 | 835.7 | 34/21 |
| 74 | 912.9 | 56/33 |
| 75 | 990.1 | 39/22 |
| 76 | 1067.3 | 50/27 |
| 77 | 1144.5 | 64/33 |
| 78 | 21.7 | 91/90, 85/84, 78/77 |
| 79 | 98.9 | 35/33 |
| 80 | 176.1 | 195/176 |
| 81 | 253.3 | 52/45 |
| 82 | 330.5 | 40/33 |
| 83 | 407.7 | 91/72 |
| 84 | 484.9 | 119/90 |
| 85 | 562.1 | 112/81 |
| 86 | 639.3 | 13/9 |
| 87 | 716.5 | 68/45, 50/33 |
| 88 | 793.7 | 128/81 |
| 89 | 870.9 | 119/72 |
| 90 | 948.1 | 140/81 |
| 91 | 1025.3 | 65/36 |
| 92 | 1102.5 | 17/9 |
| 93 | 1179.7 | 160/81, 196/99, 240/121 |
| 94 | 56.9 | 91/88 |
| 95 | 134.1 | 68/63 |
| 96 | 211.3 | 112/99 |
| 97 | 288.5 | 13/11 |
| 98 | 365.7 | 68/55 |
| 99 | 442.9 | 128/99 |
| 100 | 520.1 | 104/77 |
| 101 | 597.3 | |
| 102 | 674.5 | |
| 103 | 751.7 | 17/11 |
| 104 | 828.9 | 160/99 |
| 105 | 906.1 | |
| 106 | 983.3 | 136/77 |
| 107 | 1060.6 | |
| 108 | 1137.8 | 52/27, 85/44 |
| 109 | 15.0 | 100/99 |
| 110 | 92.2 | 128/121, 256/243 |
| 111 | 169.4 | |
| 112 | 246.6 | |
| 113 | 323.8 | |
| 114 | 401.0 | 34/27 |
| 115 | 478.2 | |
| 116 | 555.4 | |
| 117 | 632.6 | |
| 118 | 709.8 | |
| 119 | 787.0 | 52/33 |
| 120 | 864.2 | |
| 121 | 941.4 | |
| 122 | 1018.6 | |
| 123 | 1095.8 | |
| 124 | 1173.0 | 65/33 |
| 125 | 50.2 | 34/33 |
a in 17-limit POTE tuning
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.
| generator | cents value a (octave-reduced) |
17-limit ratio (octave-reduced) |
|---|---|---|
| 1 | 77.2 | 117/112, 256/245, 68/65, 22/21 |
| 2 | 154.3 | 12/11, 35/32 |
| 3 | 231.5 | 8/7 |
| 4 | 308.6 | 117/98, 140/117 |
| 5 | 385.8 | 5/4 |
| 6 | 463.0 | 17/13 |
| 7 | 540.1 | 15/11 |
| 8 | 617.3 | 10/7 |
| 9 | 694.5 | 112/75 |
| 10 | 771.6 | 25/16 |
| 11 | 848.8 | 85/52, 80/49, 49/30 |
| 12 | 925.9 | 128/75 |
| 13 | 1003.1 | 25/14 |
| 14 | 1080.3 | 28/15 |
| 15 | 1157.4 | 39/20, 88/45 |
| 16 | 34.6 | 56/55, 52/51, 51/50, 50/49, 49/48 |
| 17 | 111.8 | 16/15 |
| 18 | 188.9 | 39/35 |
| 19 | 266.1 | 7/6 |
| 20 | 343.2 | 11/9 |
| 21 | 420.4 | 14/11 |
| 22 | 497.6 | 4/3 |
| 23 | 574.7 | 39/28 |
| 24 | 651.9 | 16/11, 35/24 |
| 25 | 729.1 | 32/21 |
| 26 | 806.2 | 35/22, 51/32 |
| 27 | 883.4 | 5/3 |
| 28 | 960.5 | 68/39 |
| 29 | 1037.7 | 20/11 |
| 30 | 1114.9 | 40/21 |
| 31 | 1192.0 | |
| 32 | 69.2 | 26/25, 25/24 |
| 33 | 146.4 | 49/45 |
| 34 | 223.5 | 25/22 |
| 35 | 300.7 | 25/21 |
| 36 | 377.8 | 56/45 |
| 37 | 455.0 | 13/10 |
| 38 | 532.2 | 34/25 |
| 39 | 609.3 | 64/45 |
| 40 | 686.5 | 52/35 |
| 41 | 763.7 | 14/9 |
| 42 | 840.8 | 13/8, 44/27 |
| 43 | 918.0 | 17/10 |
| 44 | 995.1 | 16/9 |
| 45 | 1072.3 | 13/7 |
| 46 | 1149.5 | 64/33, 68/35, 35/18 |
| 47 | 26.6 | 78/77, 65/64, 64/63, 55/54 |
| 48 | 103.8 | 17/16 |
| 49 | 180.9 | 10/9 |
| 50 | 258.1 | 65/56 |
| 51 | 335.3 | 17/14 |
| 52 | 412.4 | 80/63 |
| 53 | 489.6 | 65/49 |
| 54 | 566.8 | 25/18 |
| 55 | 643.9 | |
| 56 | 721.1 | 50/33, 85/56 |
| 57 | 798.2 | 100/63 |
| 58 | 875.4 | |
| 59 | 952.6 | 26/15 |
| 60 | 1029.7 | 136/75 |
| 61 | 1106.9 | 91/48, 256/135 |
| 62 | 1184.1 | 196/99, 208/105 |
| 63 | 61.2 | 28/27 |
| 64 | 138.4 | 13/12 |
| 65 | 215.5 | 17/15 |
| 66 | 292.7 | 13/11, 32/27 |
| 67 | 369.9 | 26/21 |
| 68 | 447.0 | 35/27 |
| 69 | 524.2 | 65/48 |
| 70 | 601.4 | 17/12 |
| 71 | 678.5 | 40/27 |
| 72 | 755.7 | 17/11 |
| 73 | 832.8 | 34/21 |
| 74 | 910.0 | |
| 75 | 987.2 | 136/77, 85/48 |
| 76 | 1064.3 | 50/27 |
| 77 | 1141.5 | 85/44 |
| 78 | 18.7 | 100/99, 91/90, 85/84 |
| 79 | 95.8 | |
| 80 | 173.0 | |
| 81 | 250.1 | 52/45 |
| 82 | 327.3 | |
| 83 | 404.5 | 91/72 |
| 84 | 481.6 | 119/90 |
| 85 | 558.8 | 112/81 |
| 86 | 636.0 | 13/9 |
| 87 | 713.1 | 68/45 |
| 88 | 790.3 | 52/33, 128/81 |
| 89 | 867.4 | 119/72 |
| 90 | 944.6 | 140/81 |
| 91 | 1021.8 | 65/36 |
| 92 | 1098.9 | 17/9 |
| 93 | 1176.1 | 65/33, 160/81 |
| 94 | 53.2 | 34/33 |
| 95 | 130.4 | 68/63 |
| 96 | 207.6 | |
| 97 | 284.7 | |
| 98 | 361.9 | |
| 99 | 439.1 | |
| 100 | 516.2 | |
| 101 | 593.4 | |
| 102 | 670.5 | |
| 103 | 747.7 | |
| 104 | 824.9 | |
| 105 | 902.0 | |
| 106 | 979.2 | |
| 107 | 1056.4 | |
| 108 | 1133.5 | 52/27 |
| 109 | 10.7 | |
| 110 | 87.8 | 104/99, 256/243 |
| 111 | 165.0 | |
| 112 | 242.2 | |
| 113 | 319.3 | |
| 114 | 396.5 | 34/27 |
a in 17-limit POTE tuning