55edo

From Xenharmonic Wiki
Jump to: navigation, search

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 pions 7mus Ratios it approximates
0 0 1/1
1 21.818 23.127 27.927 (1B.ED616) 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, 81/80
2 43.636 46.2545 55.8545 (37.DAC16) 36/35
3 65.455 69.382 83.782 (53.C8216) 28/27, 25/24
4 87.273 92.509 111.709 (6F.B5816) 25/24, 21/20
5 109.091 115.636 139.636 (8B.A2E816) 16/15
6 130.909 138.764 167.564 (A7.90516) 14/13, 13/12
7 152.727 161.891 195.491 (C3.7DA16) 13/12, 12/11
8 174.5455 185.018 223.418 (DF.6B116) 11/10, 10/9
9 196.364 208.1455 251.3455 (FB.58716) 9/8, 10/9
10 218.182 231.273 279.273 (117.46716) 17/15
11 240 254.4 307.2 (133.33316) 8/7, 15/13
12 261.818 277.527 335.127 (14F.20916) 7/6
13 283.636 300.6545 363.0545 (16B.0DF16) 13/11
14 305.4545 323.782 390.982 (186.FB4816) 6/5-
15 327.273 346.909 418.909 (1A2.E8C16) 6/5+
16 349.091 370.036 446.836 (1BE.D6216) 11/9, 27/22
17 370.909 393.164 474.763 (1DA.C5616) 16/13
18 392.727 416.291 502.691 (1F6.A0E16) 5/4
19 414.5455 439.418 530.618 (212.9ED16) 14/11
20 436.364 462.5455 558.5455 (22E.8BA16) 9/7
21 458.182 485.673 586.473 (24A.7916) 13/10
22 480 508.8 614.4 (266.66616) 21/16
23 501.818 531.927 642.327 (282.550816) 4/3, 27/20
24 523.636 555.0545 670.2545 (29E.41316) 27/20
25 545.4545 578.182 698.182 (2BA.2E916) 11/8
26 567.273 601.309 726.109 (2D6.1BF16) 18/13, 25/18
27 589.091 624.436 754.036 (2F2.09516) 7/5
28 610.909 647.564 781.944 (30D.F7B16) 10/7
29 632.727 670.691 809.891 (329.E4116) 13/9, 36/25
30 654.5455 693.818 837.818 (345.D1716) 16/11
31 676.364 716.9455 865.7455 (361.BEC16) 40/27
32 698.182 740.073 893.673 (37D.AAF816) 3/2, 40/27
33 720 763.2 921.6 (399.99A16) 32/21
34 741.818 786.327 949.527 (3B5.8716) 20/13
35 763.636 809.4545 977.4545 (3D1.74616) 14/9
36 785.4545 832.582 1005.382 (FED.61216) 11/7
37 807.273 855.709 1033.309 (409.5F216) 8/5
38 829.091 878.836 1061.327 (425.3A916) 13/8
39 850.909 901.964 1089.164 (441.19E16) 18/11, 44/27
40 872.727 925.091 1117.091 (45D.17416) 5/3-
41 894.5455 948.218 1145.018 (479.04A816) 5/3+
42 916.364 971.3455 1172.9455 (494.F2116) 22/13
43 938.182 994.473 1200.873 (4B0.DF716) 12/7
44 960 1017.6 1228.8 (4CC.CCD16) 7/4, 26/15
45 981.818 1040.727 1256.727 (4E8.B9916) 30/17
46 1003.636 1063.8545 1284.6545 (504.A7916) 16/9, 9/5
47 1025.4545 1058.982 1312.582 (520.94F16) 9/5, 20/11
48 1047.273 1110.109 1340.509 (53C.81616) 11/6, 24/13
49 1069.091 1133.236 1368.436 (558.6FB16) 24/13, 13/7
50 1090.909 1156.364 1396.364 (574.5D1816) 15/8
51 1112.727 1179.491 1424.291 (590.4A716) 40/21, 48/25
52 1134.5455 1202.618 1452.218 (FAC.37D16) 56/27, 48/25
53 1156.364 1225.7455 1480.1455 (FC8.25316) 35/18
54 1178.182 1248.873 1508.073 (FE4.12A16) 125/64, 63/32, 128/65, 77/39, 180/91, 196/99, 160/81
55 1200 1272 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