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'''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_Syntonic_Comma_Meantone|1/6 comma meantone]] (and is almost exactly 10/57 comma meantone.) [http://en.wikipedia.org/wiki/Georg_Philipp_Telemann Telemann] suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by [http://en.wikipedia.org/wiki/Leopold_Mozart Leopold] and [http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart Wolfgang Mozart]. It can also be used for [[Meantone_family|mohajira and liese]] temperaments.
'''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_Syntonic_Comma_Meantone|1/6 comma meantone]] (and is almost exactly 10/57 comma meantone.) [http://en.wikipedia.org/wiki/Georg_Philipp_Telemann Telemann] suggested it as a theoretical basis for analyzing the intervals of meantone, in which he was followed by [http://en.wikipedia.org/wiki/Leopold_Mozart Leopold] and [http://en.wikipedia.org/wiki/Wolfgang_Amadeus_Mozart Wolfgang Mozart]. It can also be used for [[Meantone_family|mohajira and liese]] temperaments.


5-limit commas: 81/80, <31 1 -14|
5-limit commas: 81/80, &lt;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
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 26411/25920 26411/26244 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
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
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==
==Intervals==
Line 15: Line 15:
| | Degrees of 55-EDO
| | Degrees of 55-EDO
| | Cents value
| | Cents value
|pions
|7mus
| | Ratios it approximates
| | Ratios it approximates
|-
|-
| | 0
| | 0
| | 0
| colspan="3"| 0
| | 1/1
| | 1/1
|-
|-
| | 1
| | 1
| | 21.818
| | 21.818
| | 128/125
|23.127
|27.927 (1B.ED6<sub>16</sub>)
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
|-
|-
| | 2
| | 2
| | 43.636
| | 43.636
| |  
|46.2545
|55.8545 (37.DAC<sub>16</sub>)
| |36/35
|-
|-
| | 3
| | 3
| | 65.455
| | 65.455
| |  
|69.382
|83.782 (53.C82<sub>16</sub>)
| |28/27, ''25/24''
|-
|-
| | 4
| | 4
| | 87.273
| | 87.273
| | 25/24
|92.509
|111.709 (6F.B58<sub>16</sub>)
| | 25/24, 21/20
|-
|-
| | 5
| | 5
| | 109.091
| | 109.091
|115.636
|139.636 (8B.A2E8<sub>16</sub>)
| | 16/15
| | 16/15
|-
|-
| | 6
| | 6
| | 130.909
| | 130.909
| |
|138.764
|167.564 (A7.905<sub>16</sub>)
| |14/13, ''13/12''
|-
|-
| | 7
| | 7
| | 152.727
| | 152.727
| |
|161.891
|195.491 (C3.7DA<sub>16</sub>)
| |13/12, 12/11
|-
|-
| | 8
| | 8
| | 174.545
| | 174.5455
| |
|185.018
|223.418 (DF.6B1<sub>16</sub>)
| |11/10, ''10/9''
|-
|-
| | 9
| | 9
| | 196.364
| | 196.364
|208.1455
|251.3455 (FB.587<sub>16</sub>)
| | 9/8, 10/9
| | 9/8, 10/9
|-
|-
| | 10
| | 10
| | 218.182
| | 218.182
|231.273
|279.273 (117.467<sub>16</sub>)
|17/15
|-
|-
| | 11
| | 11
| | 240.000
| | 240
|254.4
|307.2 (133.333<sub>16</sub>)
|8/7, 15/13
|-
|-
| | 12
| | 12
| | 261.818
| | 261.818
|277.527
|335.127 (14F.209<sub>16</sub>)
|7/6
|-
|-
| | 13
| | 13
| | 283.636
| | 283.636
|300.6545
|363.0545 (16B.0DF<sub>16</sub>)
|13/11
|-
|-
| | 14
| | 14
| | 305.455
| | 305.4545
|323.782
|400.982 (190.FB48<sub>16</sub>)
|6/5-
|-
|-
| | 15
| | 15
| | 327.273
| | 327.273
|346.909
|418.909 (1A2.E8C<sub>16</sub>)
|6/5+
|-
|-
| | 16
| | 16
| | 349.091
| | 349.091
|370.036
|446.836 (1BE.D62<sub>16</sub>)
|11/9, 27/22
|-
|-
| | 17
| | 17
| | 370.909
| | 370.909
|393.164
|474.763 (1DA.C56<sub>16</sub>)
|16/13
|-
|-
| | 18
| | 18
| | 392.727
| | 392.727
|416.291
|502.691 (1F4.A0E<sub>16</sub>)
|5/4
|-
|-
| | 19
| | 19
| | 414.545
| | 414.5455
|439.418
|530.618 (212.9ED<sub>16</sub>)
|14/11
|-
|-
| | 20
| | 20
| | 436.364
| | 436.364
|462.5455
|558.5455 (22E.8BA<sub>16</sub>)
|9/7
|-
|-
| | 21
| | 21
| | 458.182
| | 458.182
|485.673
|586.473 (54A.79<sub>16</sub>)
|13/10
|-
|-
| | 22
| | 22
| | 480.000
| | 480
|508.8
|614.4 (266.666<sub>16</sub>)
|21/16
|-
|-
| | 23
| | 23
| | 501.818
| | 501.818
|531.927
|642.327 (282.5508<sub>16</sub>)
|4/3, 27/20
|-
|-
| | 24
| | 24
| | 523.636
| | 523.636
|555.0545
|670.2545 (29E.413<sub>16</sub>)
|''27/20''
|-
|-
| | 25
| | 25
| | 545.455
| | 545.4545
|578.182
|698.182 (2BA.2E9<sub>16</sub>)
|11/8
|-
|-
| | 26
| | 26
| | 567.273
| | 567.273
|601.309
|726.109 (2D6.1BF<sub>16</sub>)
|18/13, 25/18
|-
|-
| | 27
| | 27
| | 589.091
| | 589.091
|624.436
|754.036 (2F2.095<sub>16</sub>)
|7/5
|-
|-
| | 28
| | 28
| | 610.909
| | 610.909
|647.564
|
|10/7
|-
|-
| | 29
| | 29
| | 632.727
| | 632.727
|670.691
|
|13/9, 36/25
|-
|-
| | 30
| | 30
| | 654.545
| | 654.5455
|693.818
|837.818 (345.D17<sub>16</sub>)
|16/11
|-
|-
| | 31
| | 31
| | 676.364
| | 676.364
|716.9455
|
|''40/27''
|-
|-
| | 32
| | 32
| | 698.182
| | 698.182
|740.073
|
|3/2, 40/27
|-
|-
| | 33
| | 33
| | 720.000
| | 720
|763.2
|921.6 (399.99A<sub>16</sub>)
|32/21
|-
|-
| | 34
| | 34
| | 741.818
| | 741.818
|786.327
|
|20/13
|-
|-
| | 35
| | 35
| | 763.636
| | 763.636
|809.4545
|977.4545 (3D1.746<sub>16</sub>)
|14/9
|-
|-
| | 36
| | 36
| | 785.455
| | 785.4545
|832.582
|
|11/7
|-
|-
| | 37
| | 37
| | 807.273
| | 807.273
|855.709
|
|8/5
|-
|-
| | 38
| | 38
| | 829.091
| | 829.091
|878.836
|
|13/8
|-
|-
| | 39
| | 39
| | 850.909
| | 850.909
|901.964
|
|18/11, 44/27
|-
|-
| | 40
| | 40
| | 872.727
| | 872.727
|925.091
|1117.091 (45D.174<sub>16</sub>)
|5/3-
|-
|-
| | 41
| | 41
| | 894.545
| | 894.5455
|948.218
|
|5/3+
|-
|-
| | 42
| | 42
| | 916.364
| | 916.364
|971.3455
|
|22/13
|-
|-
| | 43
| | 43
| | 938.182
| | 938.182
|994.473
|
|12/7
|-
|-
| | 44
| | 44
| | 960.000
| | 960
|1017.6
|1228.8 (4CC.CCD<sub>16</sub>)
|7/4, 26/15
|-
|-
| | 45
| | 45
| | 981.818
| | 981.818
|1040.727
|1256.727 (4E8.B99<sub>16</sub>)
|30/17
|-
|-
| | 46
| | 46
| | 1003.636
| | 1003.636
|1063.8545
|
|16/9, 9/5
|-
|-
| | 47
| | 47
| | 1025.455
| | 1025.4545
|1058.982
|
|''9/5'', 20/11
|-
|-
| | 48
| | 48
| | 1047.273
| | 1047.273
|1110.109
|
|11/6, 24/13
|-
|-
| | 49
| | 49
| | 1069.091
| | 1069.091
|1133.236
|
|''24/13'', 13/7
|-
|-
| | 50
| | 50
| | 1090.909
| | 1090.909
|1156.364
|1396.364 (574.5D18<sub>16</sub>)
|15/8
|-
|-
| | 51
| | 51
| | 1112.727
| | 1112.727
|1179.491
|
|40/21, 48/25
|-
|-
| | 52
| | 52
| | 1134.545
| | 1134.5455
|1202.618
|
|56/27, ''48/25''
|-
|-
| | 53
| | 53
| | 1156.364
| | 1156.364
|1225.7455
|
|35/18
|-
|-
| | 54
| | 54
| | 1178.182
| | 1178.182
|1248.873
|
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
|-
|-
| | 55
| | 55
| | 1200.000
| | 1200
|1272
|1536 (600<sub>16</sub>)
|2/1
|}
|}


Revision as of 02:12, 4 April 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 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 323.782 400.982 (190.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 (1F4.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 (54A.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 10/7
29 632.727 670.691 13/9, 36/25
30 654.5455 693.818 837.818 (345.D1716) 16/11
31 676.364 716.9455 40/27
32 698.182 740.073 3/2, 40/27
33 720 763.2 921.6 (399.99A16) 32/21
34 741.818 786.327 20/13
35 763.636 809.4545 977.4545 (3D1.74616) 14/9
36 785.4545 832.582 11/7
37 807.273 855.709 8/5
38 829.091 878.836 13/8
39 850.909 901.964 18/11, 44/27
40 872.727 925.091 1117.091 (45D.17416) 5/3-
41 894.5455 948.218 5/3+
42 916.364 971.3455 22/13
43 938.182 994.473 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 16/9, 9/5
47 1025.4545 1058.982 9/5, 20/11
48 1047.273 1110.109 11/6, 24/13
49 1069.091 1133.236 24/13, 13/7
50 1090.909 1156.364 1396.364 (574.5D1816) 15/8
51 1112.727 1179.491 40/21, 48/25
52 1134.5455 1202.618 56/27, 48/25
53 1156.364 1225.7455 35/18
54 1178.182 1248.873 125/64, 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

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