55edo
55 tone equal temperament[edit]
Welcome to 55edo. It is a relatively generic meantone system, utilizing a different 7 than in septimal meantone. It also doesn't temper the 3 and 9 as much as some other meantones. Therefore it may be considered one of the purest, most generic forms of tempered meantone harmony possible. This makes it a wonderful meantone system.
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|
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 26411/25920 26411/26244 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[edit]
| Degrees of 55-EDO | Cents value | Ratios it approximates |
| 0 | 0 | 1/1 |
| 1 | 21.818 | |
| 2 | 43.636 | |
| 3 | 65.455 | |
| 4 | 87.273 | |
| 5 | 109.091 | 14/13, 16/15 |
| 6 | 130.909 | 15/14 |
| 7 | 152.727 | 12/11, 11/10, 13/12 |
| 8 | 174.545 | |
| 9 | 196.364 | 9/8, 10/9 |
| 10 | 218.182 | |
| 11 | 240.000 | 8/7, 15/13 |
| 12 | 261.818 | 7/6 |
| 13 | 283.636 | |
| 14 | 305.455 | 6/5, 13/11 |
| 15 | 327.273 | |
| 16 | 349.091 | 11/9, 16/13 |
| 17 | 370.909 | |
| 18 | 392.727 | 5/4 |
| 19 | 414.545 | 14/11 |
| 20 | 436.364 | 9/7 |
| 21 | 458.182 | 13/10 |
| 22 | 480.000 | |
| 23 | 501.818 | 4/3 |
| 24 | 523.636 | |
| 25 | 545.455 | 11/8, 18/13, 15/11 |
| 26 | 567.273 | 7/5 |
| 27 | 589.091 | |
| 28 | 610.909 | |
| 29 | 632.727 | 10/7 |
| 30 | 654.545 | 16/11, 13/9, 22/15 |
| 31 | 676.364 | |
| 32 | 698.182 | 3/2 |
| 33 | 720.000 | |
| 34 | 741.818 | 20/13 |
| 35 | 763.636 | 14/9 |
| 36 | 785.455 | 11/7 |
| 37 | 807.273 | 8/5 |
| 38 | 829.091 | |
| 39 | 850.909 | 18/11, 13/8 |
| 40 | 872.727 | |
| 41 | 894.545 | 5/3, 22/13 |
| 42 | 916.364 | |
| 43 | 938.182 | 12/7 |
| 44 | 960.000 | 7/4, 26/15 |
| 45 | 981.818 | |
| 46 | 1003.636 | 16/9, 9/5 |
| 47 | 1025.455 | |
| 48 | 1047.273 | 11/6, 20/11, 24/13 |
| 49 | 1069.091 | 28/15 |
| 50 | 1090.909 | 13/7, 15/8 |
| 51 | 1112.727 | |
| 52 | 1134.545 | |
| 53 | 1156.364 | |
| 54 | 1178.182 | |
| 55 | 1200.000 | 2/1 |
Selected just intervals by error[edit]
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 |
| 16/15, 15/8 | 2.640 |
| 14/11, 11/7 | 2.963 |
| 4/3, 3/2 | 3.773 |
| 13/10, 20/13 | 3.968 |
| 7/6, 12/7 | 5.053 |
| 11/8, 16/11 | 5.863 |
| 5/4, 8/5 | 6.414 |
| 9/8, 16/9 | 7.546 |
| 15/13, 26/15 | 7.741 |
| 15/11, 22/15 | 8.504 |
| 8/7, 7/4 | 8.826 |
| 6/5, 5/3 | 10.187 |
| 16/13, 13/8 | 10.381 |
| 15/14, 28/15 | 11.466 |
| 11/10, 20/11 | 12.277 |
| 10/9, 9/5 | 13.960 |
| 13/12, 24/13 | 14.155 |
| 7/5, 10/7 | 15.239 |
| 13/11, 22/13 | 16.245 |
| 18/13, 13/9 | 17.928 |
| 14/13, 13/7 | 19.207 |
Mozart - Adagio in B minor KV 540 by Carlo Serafini (blog entry)
"Mozart's tuning: 55edo" (containing another listening example) in the tonalsoft encyclopedia