55edo

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55edo divides the octave into 55 parts of 21.818 ¢. 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.

Theory

Approximation of odd harmonics in 55edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -3.77 +6.41 -8.83 -7.55 -5.86 +10.38 +2.64 +4.14 +7.94 +9.22 +4.45
Relative (%) -17.3 +29.4 -40.5 -34.6 -26.9 +47.6 +12.1 +19.0 +36.4 +42.3 +20.4
Steps
(reduced) 		87
(32) 		128
(18) 		154
(44) 		174
(9) 		190
(25) 		204
(39) 		215
(50) 		225
(5) 		234
(14) 		242
(22) 		249
(29)

5-limit commas: 81/80, [31 1 -14⟩, [27 5 -15⟩

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

# Cents Approximate ratios ups and downs notation
0 0.000 1/1 P1 perfect 1sn D
1 21.818 128/125, 64/63, 65/64, 78/77,

91/90, 99/98, 81/80

^1 up 1sn ^D
2 43.636 36/35 ^^1 dup 1sn ^^D
3 65.4545 28/27, 25/24 vvm2 dudminor 2nd vvEb
4 87.273 25/24, 21/20 vm2 downminor 2nd vEb
5 109.091 16/15 m2 minor 2nd Eb
6 130.909 14/13, 13/12 ^m2 upminor 2nd ^Eb
7 152.727 13/12, 12/11 ~2 mid 2nd vvE
8 174.5455 11/10, 10/9 vM2 downmajor 2nd vE
9 196.364 9/8, 10/9 M2 major 2nd E
10 218.182 17/15 ^M2 upmajor 2nd ^E
11 240 8/7, 15/13 ^^M2 dupmajor 2nd ^^E
12 261.818 7/6 vvm3 dudminor 3rd vvF
13 283.636 13/11 vm3 downminor 3rd vF
14 305.4545 6/5- m3 minor 3rd F
15 327.273 6/5+ ^m3 upminor 3rd ^F
16 349.091 11/9, 27/22 ~3 mid 3rd ^^F
17 370.909 16/13 vM3 downmajor 3rd vF#
18 392.727 5/4 M3 major 3rd F#
19 414.5455 14/11 ^M3 upmajor 3rd ^F#
20 436.364 9/7 ^^M3 dupmajor 3rd ^^F#
21 458.182 13/10 vv4 dud 4th vvG
22 480 21/16 v4 down 4th vG
23 501.818 4/3, 27/20 P4 perfect 4th G
24 523.636 27/20 ^4 up 4th ^G
25 545.4545 11/8 ~4 mid 4th ^^G
26 567.273 18/13, 25/18 vA4 downaug 4th vG#
27 589.091 7/5 A4, vd5 aug 4th, downdim 5th G#, vAb
28 610.909 10/7 ^A4, d5 upaug 4th, dim 5th ^G#, Ab
29 632.727 13/9, 36/25 ^d5 updim 5th ^Ab
30 654.5455 16/11 ~5 mid 5th vvA
31 676.364 40/27 v5 down 5th vA
32 698.182 3/2, 40/27 P5 perfect 5th A
33 720 32/21 ^5 up 5th ^A
34 741.818 20/13 ^^5 dup 5th ^^A
35 763.636 14/9 vvm6 dudminor 6th vvBb
36 785.4545 11/7 vm6 downminor 6th vBb
37 807.273 8/5 m6 minor 6th Bb
38 829.091 13/8 ^m6 upminor 6th ^Bb
39 850.909 18/11, 44/27 ~6 mid 6th vvB
40 872.727 5/3- vM6 downmajor 6th vB
41 894.5455 5/3+ M6 major 6th B
42 916.364 22/13 ^M6 upmajor 6th ^B
43 938.182 12/7 ^^M6 dupmajor 6th ^^B
44 960 7/4, 26/15 vvm7 dudminor 7th vvC
45 981.818 30/17 vm7 downminor 7th vC
46 1003.636 16/9, 9/5 m7 minor 7th C
47 1025.4545 9/5, 20/11 ^m7 upminor 7th ^C
48 1047.273 11/6, 24/13 ~7 mid 7th ^^C
49 1069.091 24/13, 13/7 vM7 downmajor 7th vC#
50 1090.909 15/8 M7 major 7th C#
51 1112.727 40/21, 48/25 ^M7 upmajor 7th ^C#
52 1134.5455 56/27, 48/25 ^^M7 dupmajor 7th ^^C#
53 1156.364 35/18 vv8 dud 8ve vvD
54 1178.182 125/64, 63/32, 128/65, 77/39,

180/91, 196/99, 160/81

v8 down 8ve vD
55 1200 2/1 P8 perfect 8ve D

Selected just intervals by error

The following table shows how 15-odd-limit just intervals are represented in 55edo (ordered by absolute error). The following tables show how 15-odd-limit intervals are represented in 55edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 55edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
13/7, 14/13 2.611 12.0
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/9, 18/13 3.890 17.8
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
13/11, 22/13 5.573 25.5
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
7/5, 10/7 6.579 30.2
9/8, 16/9 7.546 34.6
13/12, 24/13 7.664 35.1
15/13, 26/15 7.741 35.5
9/5, 10/9 7.858 36.0
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
11/10, 20/11 9.541 43.7
5/3, 6/5 10.187 46.7
15/14, 28/15 10.352 47.4
13/8, 16/13 10.381 47.6
15-odd-limit intervals in 55edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
9/7, 14/9 1.280 5.9
11/9, 18/11 1.683 7.7
11/6, 12/11 2.090 9.6
15/8, 16/15 2.640 12.1
11/7, 14/11 2.963 13.6
3/2, 4/3 3.773 17.3
13/10, 20/13 3.968 18.2
7/6, 12/7 5.053 23.2
11/8, 16/11 5.863 26.9
5/4, 8/5 6.414 29.4
9/8, 16/9 7.546 34.6
15/13, 26/15 7.741 35.5
15/11, 22/15 8.504 39.0
7/4, 8/7 8.826 40.5
5/3, 6/5 10.187 46.7
13/8, 16/13 10.381 47.6
15/14, 28/15 11.466 52.6
11/10, 20/11 12.277 56.3
9/5, 10/9 13.960 64.0
13/12, 24/13 14.155 64.9
7/5, 10/7 15.239 69.8
13/11, 22/13 16.245 74.5
13/9, 18/13 17.928 82.2
13/7, 14/13 19.207 88.0

Music

External links

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