55edo
55edo divides the octave into 55 parts of 21.818 cents. It can be used for a meantone tuning, and is close to 1/6 comma meantone (and is almost exactly 10/57 comma meantone.) Telemann suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by Leopold and Wolfgang Mozart. It can also be used for mohajira and liese temperaments.
5-limit commas: 81/80, <31 1 -14|, <-165 220 55|
7-limit commas: 31104/30625, 6144/6125, 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, 686/675, 10976/10935, 16807/16384, 84035/82944
11-limit commas: 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, 243/242, 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, 176/175, 2079/2048, 385/384, 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, 121/120, 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, 14641/14580
13-limit commas: 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, 512/507, 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, 1575/1573, 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, 144/143, 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, 4096/4095, 1701/1664, 105/104, 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, 66/65, 352/351, 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, 351/350, 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, 31213/31104, 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125
Intervals
| Degrees of 55-EDO | Cents value | 7mus | Ratios it approximates |
| 0 | 1/1 | ||
| 1 | 21.818 | 27.927 (1B.ED616) | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, 81/80 |
| 2 | 43.636 | 55.8545 (37.DAC16) | 36/35 |
| 3 | 65.4545 | 83.782 (53.C8216) | 28/27, 25/24 |
| 4 | 87.273 | 111.709 (6F.B5816) | 25/24, 21/20 |
| 5 | 109.091 | 139.636 (8B.A2E816) | 16/15 |
| 6 | 130.909 | 167.564 (A7.90516) | 14/13, 13/12 |
| 7 | 152.727 | 195.491 (C3.7DA16) | 13/12, 12/11 |
| 8 | 174.5455 | 223.418 (DF.6B116) | 11/10, 10/9 |
| 9 | 196.364 | 251.3455 (FB.58716) | 9/8, 10/9 |
| 10 | 218.182 | 279.273 (117.46716) | 17/15 |
| 11 | 240 | 307.2 (133.33316) | 8/7, 15/13 |
| 12 | 261.818 | 335.127 (14F.20916) | 7/6 |
| 13 | 283.636 | 363.0545 (16B.0DF16) | 13/11 |
| 14 | 305.4545 | 390.982 (186.FB4816) | 6/5- |
| 15 | 327.273 | 418.909 (1A2.E8C16) | 6/5+ |
| 16 | 349.091 | 446.836 (1BE.D6216) | 11/9, 27/22 |
| 17 | 370.909 | 474.763 (1DA.C5616) | 16/13 |
| 18 | 392.727 | 502.691 (1F6.A0E16) | 5/4 |
| 19 | 414.5455 | 530.618 (212.9ED16) | 14/11 |
| 20 | 436.364 | 558.5455 (22E.8BA16) | 9/7 |
| 21 | 458.182 | 586.473 (24A.7916) | 13/10 |
| 22 | 480 | 614.4 (266.66616) | 21/16 |
| 23 | 501.818 | 642.327 (282.550816) | 4/3, 27/20 |
| 24 | 523.636 | 670.2545 (29E.41316) | 27/20 |
| 25 | 545.4545 | 698.182 (2BA.2E916) | 11/8 |
| 26 | 567.273 | 726.109 (2D6.1BF16) | 18/13, 25/18 |
| 27 | 589.091 | 754.036 (2F2.09516) | 7/5 |
| 28 | 610.909 | 781.944 (30D.F7B16) | 10/7 |
| 29 | 632.727 | 809.891 (329.E4116) | 13/9, 36/25 |
| 30 | 654.5455 | 837.818 (345.D1716) | 16/11 |
| 31 | 676.364 | 865.7455 (361.BEC16) | 40/27 |
| 32 | 698.182 | 893.673 (37D.AAF816) | 3/2, 40/27 |
| 33 | 720 | 921.6 (399.99A16) | 32/21 |
| 34 | 741.818 | 949.527 (3B5.8716) | 20/13 |
| 35 | 763.636 | 977.4545 (3D1.74616) | 14/9 |
| 36 | 785.4545 | 1005.382 (FED.61216) | 11/7 |
| 37 | 807.273 | 1033.309 (409.5F216) | 8/5 |
| 38 | 829.091 | 1061.327 (425.3A916) | 13/8 |
| 39 | 850.909 | 1089.164 (441.19E16) | 18/11, 44/27 |
| 40 | 872.727 | 1117.091 (45D.17416) | 5/3- |
| 41 | 894.5455 | 1145.018 (479.04A816) | 5/3+ |
| 42 | 916.364 | 1172.9455 (494.F2116) | 22/13 |
| 43 | 938.182 | 1200.873 (4B0.DF716) | 12/7 |
| 44 | 960 | 1228.8 (4CC.CCD16) | 7/4, 26/15 |
| 45 | 981.818 | 1256.727 (4E8.B9916) | 30/17 |
| 46 | 1003.636 | 1284.6545 (504.A7916) | 16/9, 9/5 |
| 47 | 1025.4545 | 1312.582 (520.94F16) | 9/5, 20/11 |
| 48 | 1047.273 | 1340.509 (53C.81616) | 11/6, 24/13 |
| 49 | 1069.091 | 1368.436 (558.6FB16) | 24/13, 13/7 |
| 50 | 1090.909 | 1396.364 (574.5D1816) | 15/8 |
| 51 | 1112.727 | 1424.291 (590.4A716) | 40/21, 48/25 |
| 52 | 1134.5455 | 1452.218 (FAC.37D16) | 56/27, 48/25 |
| 53 | 1156.364 | 1480.1455 (FC8.25316) | 35/18 |
| 54 | 1178.182 | 1508.073 (FE4.12A16) | 125/64, 63/32, 128/65, 77/39, 180/91, 196/99, 160/81 |
| 55 | 1200 | 1536 (60016) | 2/1 |
Selected just intervals by error
The following table shows how some prominent just intervals are represented in 55edo (ordered by absolute error).
| Interval, complement | Error (abs., in cents) |
|---|---|
| 9/7, 14/9 | 1.280 |
| 11/9, 18/11 | 1.683 |
| 12/11, 11/6 | 2.090 |
| 14/13, 13/7 | 2.611 |
| 16/15, 15/8 | 2.640 |
| 14/11, 11/7 | 2.963 |
| 4/3, 3/2 | 3.773 |
| 18/13, 13/9 | 3.890 |
| 13/10, 20/13 | 3.968 |
| 7/6, 12/7 | 5.053 |
| 13/11, 22/13 | 5.573 |
| 11/8, 16/11 | 5.863 |
| 5/4, 8/5 | 6.414 |
| 7/5, 10/7 | 6.579 |
| 9/8, 16/9 | 7.546 |
| 13/12, 24/13 | 7.664 |
| 15/13, 26/15 | 7.741 |
| 10/9, 9/5 | 7.858 |
| 15/11, 22/15 | 8.504 |
| 8/7, 7/4 | 8.826 |
| 11/10, 20/11 | 9.541 |
| 6/5, 5/3 | 10.187 |
| 15/14, 28/15 | 10.352 |
| 16/13, 13/8 | 10.381 |
Mozart - Adagio in B minor KV 540 by Carlo Serafini (blog entry)
"Mozart's tuning: 55edo" (containing another listening example) in the tonalsoft encyclopedia