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	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

Best theoretical mapping, even if inconsistent

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

Selected just intervals by error