Xenharmonic Wiki

55edo

Revision as of 11:54, 17 November 2023 by Francium (talk | contribs) (→Modern renderings: +music)

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.

← 54edo 55edo 56edo →
Prime factorization 5 × 11
Step size 21.8182 ¢ 
Fifth 32\55 (698.182 ¢)
Semitones (A1:m2) 4:5 (87.27 ¢ : 109.1 ¢)
Consistency limit 5
Distinct consistency limit 5

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-87 55⟩ [⟨55 87]] +1.1903 1.1915 5.46
2.3.5 81/80, 6442450944/6103515625 [⟨55 87 128]] -0.1309 2.1012 9.63

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator Temperaments
1 6\55 Twothirdtonic
1 16\55 Vicentino / mohajira
1 23\55 Meantone
1 26\55 Liese
1 27\55 Untriton / aufo
5 6\55 Qintosec
11 3\55 Hendecatonic

Instruments

Lumatone mapping for 55edo

Music

Modern renderings

Johann Sebastian Bach
Wolfgang Amadeus Mozart

21st century

Claudi Meneghin

External links

Retrieved from "https://en.xen.wiki/index.php?title=55edo&oldid=127549"