55edo: Difference between revisions
→Intervals: Removed 7mus column |
|||
Line 15: | Line 15: | ||
| | Degrees of 55-EDO | | | Degrees of 55-EDO | ||
| | Cents value | | | Cents value | ||
| | Ratios it approximates | | | Ratios it approximates | ||
|- | |- | ||
| 0 | | 0 | ||
|0.000 | |0.000 | ||
| | 1/1 | | | 1/1 | ||
|- | |- | ||
| | 1 | | | 1 | ||
| | 21.818 | | | 21.818 | ||
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80'' | | | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80'' | ||
|- | |- | ||
| | 2 | | | 2 | ||
| | 43.636 | | | 43.636 | ||
| |36/35 | | |36/35 | ||
|- | |- | ||
| | 3 | | | 3 | ||
| | 65.4545 | | | 65.4545 | ||
| |28/27, ''25/24'' | | |28/27, ''25/24'' | ||
|- | |- | ||
| | 4 | | | 4 | ||
| | 87.273 | | | 87.273 | ||
| | 25/24, 21/20 | | | 25/24, 21/20 | ||
|- | |- | ||
| | 5 | | | 5 | ||
| | 109.091 | | | 109.091 | ||
| | 16/15 | | | 16/15 | ||
|- | |- | ||
| | 6 | | | 6 | ||
| | 130.909 | | | 130.909 | ||
| |14/13, ''13/12'' | | |14/13, ''13/12'' | ||
|- | |- | ||
| | 7 | | | 7 | ||
| | 152.727 | | | 152.727 | ||
| |13/12, 12/11 | | |13/12, 12/11 | ||
|- | |- | ||
| | 8 | | | 8 | ||
| | 174.5455 | | | 174.5455 | ||
| |11/10, ''10/9'' | | |11/10, ''10/9'' | ||
|- | |- | ||
| | 9 | | | 9 | ||
| | 196.364 | | | 196.364 | ||
| | 9/8, 10/9 | | | 9/8, 10/9 | ||
|- | |- | ||
| | 10 | | | 10 | ||
| | 218.182 | | | 218.182 | ||
|17/15 | |17/15 | ||
|- | |- | ||
| | 11 | | | 11 | ||
| | 240 | | | 240 | ||
|8/7, 15/13 | |8/7, 15/13 | ||
|- | |- | ||
| | 12 | | | 12 | ||
| | 261.818 | | | 261.818 | ||
|7/6 | |7/6 | ||
|- | |- | ||
| | 13 | | | 13 | ||
| | 283.636 | | | 283.636 | ||
|13/11 | |13/11 | ||
|- | |- | ||
| | 14 | | | 14 | ||
| | 305.4545 | | | 305.4545 | ||
|6/5- | |6/5- | ||
|- | |- | ||
| | 15 | | | 15 | ||
| | 327.273 | | | 327.273 | ||
|6/5+ | |6/5+ | ||
|- | |- | ||
| | 16 | | | 16 | ||
| | 349.091 | | | 349.091 | ||
|11/9, 27/22 | |11/9, 27/22 | ||
|- | |- | ||
| | 17 | | | 17 | ||
| | 370.909 | | | 370.909 | ||
|16/13 | |16/13 | ||
|- | |- | ||
| | 18 | | | 18 | ||
| | 392.727 | | | 392.727 | ||
|5/4 | |5/4 | ||
|- | |- | ||
| | 19 | | | 19 | ||
| | 414.5455 | | | 414.5455 | ||
|14/11 | |14/11 | ||
|- | |- | ||
| | 20 | | | 20 | ||
| | 436.364 | | | 436.364 | ||
|9/7 | |9/7 | ||
|- | |- | ||
| | 21 | | | 21 | ||
| | 458.182 | | | 458.182 | ||
|13/10 | |13/10 | ||
|- | |- | ||
| | 22 | | | 22 | ||
| | 480 | | | 480 | ||
|21/16 | |21/16 | ||
|- | |- | ||
| | 23 | | | 23 | ||
| | 501.818 | | | 501.818 | ||
|4/3, 27/20 | |4/3, 27/20 | ||
|- | |- | ||
| | 24 | | | 24 | ||
| | 523.636 | | | 523.636 | ||
|''27/20'' | |''27/20'' | ||
|- | |- | ||
| | 25 | | | 25 | ||
| | 545.4545 | | | 545.4545 | ||
|11/8 | |11/8 | ||
|- | |- | ||
| | 26 | | | 26 | ||
| | 567.273 | | | 567.273 | ||
|18/13, 25/18 | |18/13, 25/18 | ||
|- | |- | ||
| | 27 | | | 27 | ||
| | 589.091 | | | 589.091 | ||
|7/5 | |7/5 | ||
|- | |- | ||
| | 28 | | | 28 | ||
| | 610.909 | | | 610.909 | ||
|10/7 | |10/7 | ||
|- | |- | ||
| | 29 | | | 29 | ||
| | 632.727 | | | 632.727 | ||
|13/9, 36/25 | |13/9, 36/25 | ||
|- | |- | ||
| | 30 | | | 30 | ||
| | 654.5455 | | | 654.5455 | ||
|16/11 | |16/11 | ||
|- | |- | ||
| | 31 | | | 31 | ||
| | 676.364 | | | 676.364 | ||
|''40/27'' | |''40/27'' | ||
|- | |- | ||
| | 32 | | | 32 | ||
| | 698.182 | | | 698.182 | ||
|3/2, 40/27 | |3/2, 40/27 | ||
|- | |- | ||
| | 33 | | | 33 | ||
| | 720 | | | 720 | ||
|32/21 | |32/21 | ||
|- | |- | ||
| | 34 | | | 34 | ||
| | 741.818 | | | 741.818 | ||
|20/13 | |20/13 | ||
|- | |- | ||
| | 35 | | | 35 | ||
| | 763.636 | | | 763.636 | ||
|14/9 | |14/9 | ||
|- | |- | ||
| | 36 | | | 36 | ||
| | 785.4545 | | | 785.4545 | ||
|11/7 | |11/7 | ||
|- | |- | ||
| | 37 | | | 37 | ||
| | 807.273 | | | 807.273 | ||
|8/5 | |8/5 | ||
|- | |- | ||
| | 38 | | | 38 | ||
| | 829.091 | | | 829.091 | ||
|13/8 | |13/8 | ||
|- | |- | ||
| | 39 | | | 39 | ||
| | 850.909 | | | 850.909 | ||
|18/11, 44/27 | |18/11, 44/27 | ||
|- | |- | ||
| | 40 | | | 40 | ||
| | 872.727 | | | 872.727 | ||
|5/3- | |5/3- | ||
|- | |- | ||
| | 41 | | | 41 | ||
| | 894.5455 | | | 894.5455 | ||
|5/3+ | |5/3+ | ||
|- | |- | ||
| | 42 | | | 42 | ||
| | 916.364 | | | 916.364 | ||
|22/13 | |22/13 | ||
|- | |- | ||
| | 43 | | | 43 | ||
| | 938.182 | | | 938.182 | ||
|12/7 | |12/7 | ||
|- | |- | ||
| | 44 | | | 44 | ||
| | 960 | | | 960 | ||
|7/4, 26/15 | |7/4, 26/15 | ||
|- | |- | ||
| | 45 | | | 45 | ||
| | 981.818 | | | 981.818 | ||
|30/17 | |30/17 | ||
|- | |- | ||
| | 46 | | | 46 | ||
| | 1003.636 | | | 1003.636 | ||
|16/9, 9/5 | |16/9, 9/5 | ||
|- | |- | ||
| | 47 | | | 47 | ||
| | 1025.4545 | | | 1025.4545 | ||
|''9/5'', 20/11 | |''9/5'', 20/11 | ||
|- | |- | ||
| | 48 | | | 48 | ||
| | 1047.273 | | | 1047.273 | ||
|11/6, 24/13 | |11/6, 24/13 | ||
|- | |- | ||
| | 49 | | | 49 | ||
| | 1069.091 | | | 1069.091 | ||
|''24/13'', 13/7 | |''24/13'', 13/7 | ||
|- | |- | ||
| | 50 | | | 50 | ||
| | 1090.909 | | | 1090.909 | ||
|15/8 | |15/8 | ||
|- | |- | ||
| | 51 | | | 51 | ||
| | 1112.727 | | | 1112.727 | ||
|40/21, 48/25 | |40/21, 48/25 | ||
|- | |- | ||
| | 52 | | | 52 | ||
| | 1134.5455 | | | 1134.5455 | ||
|56/27, ''48/25'' | |56/27, ''48/25'' | ||
|- | |- | ||
| | 53 | | | 53 | ||
| | 1156.364 | | | 1156.364 | ||
|35/18 | |35/18 | ||
|- | |- | ||
| | 54 | | | 54 | ||
| | 1178.182 | | | 1178.182 | ||
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81'' | |125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81'' | ||
|- | |- | ||
| | 55 | | | 55 | ||
| | 1200 | | | 1200 | ||
|2/1 | |2/1 | ||
|} | |} |
Revision as of 15:53, 12 December 2019
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 | Ratios it approximates |
0 | 0.000 | 1/1 |
1 | 21.818 | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, 81/80 |
2 | 43.636 | 36/35 |
3 | 65.4545 | 28/27, 25/24 |
4 | 87.273 | 25/24, 21/20 |
5 | 109.091 | 16/15 |
6 | 130.909 | 14/13, 13/12 |
7 | 152.727 | 13/12, 12/11 |
8 | 174.5455 | 11/10, 10/9 |
9 | 196.364 | 9/8, 10/9 |
10 | 218.182 | 17/15 |
11 | 240 | 8/7, 15/13 |
12 | 261.818 | 7/6 |
13 | 283.636 | 13/11 |
14 | 305.4545 | 6/5- |
15 | 327.273 | 6/5+ |
16 | 349.091 | 11/9, 27/22 |
17 | 370.909 | 16/13 |
18 | 392.727 | 5/4 |
19 | 414.5455 | 14/11 |
20 | 436.364 | 9/7 |
21 | 458.182 | 13/10 |
22 | 480 | 21/16 |
23 | 501.818 | 4/3, 27/20 |
24 | 523.636 | 27/20 |
25 | 545.4545 | 11/8 |
26 | 567.273 | 18/13, 25/18 |
27 | 589.091 | 7/5 |
28 | 610.909 | 10/7 |
29 | 632.727 | 13/9, 36/25 |
30 | 654.5455 | 16/11 |
31 | 676.364 | 40/27 |
32 | 698.182 | 3/2, 40/27 |
33 | 720 | 32/21 |
34 | 741.818 | 20/13 |
35 | 763.636 | 14/9 |
36 | 785.4545 | 11/7 |
37 | 807.273 | 8/5 |
38 | 829.091 | 13/8 |
39 | 850.909 | 18/11, 44/27 |
40 | 872.727 | 5/3- |
41 | 894.5455 | 5/3+ |
42 | 916.364 | 22/13 |
43 | 938.182 | 12/7 |
44 | 960 | 7/4, 26/15 |
45 | 981.818 | 30/17 |
46 | 1003.636 | 16/9, 9/5 |
47 | 1025.4545 | 9/5, 20/11 |
48 | 1047.273 | 11/6, 24/13 |
49 | 1069.091 | 24/13, 13/7 |
50 | 1090.909 | 15/8 |
51 | 1112.727 | 40/21, 48/25 |
52 | 1134.5455 | 56/27, 48/25 |
53 | 1156.364 | 35/18 |
54 | 1178.182 | 125/64, 63/32, 128/65, 77/39, 180/91, 196/99, 160/81 |
55 | 1200 | 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