55edo: Difference between revisions

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→Intervals: Removed pions column
Line 15: Line 15:
| | Degrees of 55-EDO
| | Degrees of 55-EDO
| | Cents value
| | Cents value
|pions
|7mus
|7mus
| | Ratios it approximates
| | Ratios it approximates
|-
|-
| colspan="4" | 0
| colspan="3" | 0
| | 1/1
| | 1/1
|-
|-
| | 1
| | 1
| | 21.818
| | 21.818
|23.127
|27.927 (1B.ED6<sub>16</sub>)
|27.927 (1B.ED6<sub>16</sub>)
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
| | 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, ''81/80''
Line 30: Line 28:
| | 2
| | 2
| | 43.636
| | 43.636
|46.2545
|55.8545 (37.DAC<sub>16</sub>)
|55.8545 (37.DAC<sub>16</sub>)
| |36/35
| |36/35
Line 36: Line 33:
| | 3
| | 3
| | 65.4545
| | 65.4545
|69.382
|83.782 (53.C82<sub>16</sub>)
|83.782 (53.C82<sub>16</sub>)
| |28/27, ''25/24''
| |28/27, ''25/24''
Line 42: Line 38:
| | 4
| | 4
| | 87.273
| | 87.273
|92.509
|111.709 (6F.B58<sub>16</sub>)
|111.709 (6F.B58<sub>16</sub>)
| | 25/24, 21/20
| | 25/24, 21/20
Line 48: Line 43:
| | 5
| | 5
| | 109.091
| | 109.091
|115.636
|139.636 (8B.A2E8<sub>16</sub>)
|139.636 (8B.A2E8<sub>16</sub>)
| | 16/15
| | 16/15
Line 54: Line 48:
| | 6
| | 6
| | 130.909
| | 130.909
|138.764
|167.564 (A7.905<sub>16</sub>)
|167.564 (A7.905<sub>16</sub>)
| |14/13, ''13/12''
| |14/13, ''13/12''
Line 60: Line 53:
| | 7
| | 7
| | 152.727
| | 152.727
|161.891
|195.491 (C3.7DA<sub>16</sub>)
|195.491 (C3.7DA<sub>16</sub>)
| |13/12, 12/11
| |13/12, 12/11
Line 66: Line 58:
| | 8
| | 8
| | 174.5455
| | 174.5455
|185.018
|223.418 (DF.6B1<sub>16</sub>)
|223.418 (DF.6B1<sub>16</sub>)
| |11/10, ''10/9''
| |11/10, ''10/9''
Line 72: Line 63:
| | 9
| | 9
| | 196.364
| | 196.364
|208.1455
|251.3455 (FB.587<sub>16</sub>)
|251.3455 (FB.587<sub>16</sub>)
| | 9/8, 10/9
| | 9/8, 10/9
Line 78: Line 68:
| | 10
| | 10
| | 218.182
| | 218.182
|231.273
|279.273 (117.467<sub>16</sub>)
|279.273 (117.467<sub>16</sub>)
|17/15
|17/15
Line 84: Line 73:
| | 11
| | 11
| | 240
| | 240
|254.4
|307.2 (133.333<sub>16</sub>)
|307.2 (133.333<sub>16</sub>)
|8/7, 15/13
|8/7, 15/13
Line 90: Line 78:
| | 12
| | 12
| | 261.818
| | 261.818
|277.527
|335.127 (14F.209<sub>16</sub>)
|335.127 (14F.209<sub>16</sub>)
|7/6
|7/6
Line 96: Line 83:
| | 13
| | 13
| | 283.636
| | 283.636
|300.6545
|363.0545 (16B.0DF<sub>16</sub>)
|363.0545 (16B.0DF<sub>16</sub>)
|13/11
|13/11
Line 102: Line 88:
| | 14
| | 14
| | 305.4545
| | 305.4545
|323.782
|390.982 (186.FB48<sub>16</sub>)
|390.982 (186.FB48<sub>16</sub>)
|6/5-
|6/5-
Line 108: Line 93:
| | 15
| | 15
| | 327.273
| | 327.273
|346.909
|418.909 (1A2.E8C<sub>16</sub>)
|418.909 (1A2.E8C<sub>16</sub>)
|6/5+
|6/5+
Line 114: Line 98:
| | 16
| | 16
| | 349.091
| | 349.091
|370.036
|446.836 (1BE.D62<sub>16</sub>)
|446.836 (1BE.D62<sub>16</sub>)
|11/9, 27/22
|11/9, 27/22
Line 120: Line 103:
| | 17
| | 17
| | 370.909
| | 370.909
|393.164
|474.763 (1DA.C56<sub>16</sub>)
|474.763 (1DA.C56<sub>16</sub>)
|16/13
|16/13
Line 126: Line 108:
| | 18
| | 18
| | 392.727
| | 392.727
|416.291
|502.691 (1F6.A0E<sub>16</sub>)
|502.691 (1F6.A0E<sub>16</sub>)
|5/4
|5/4
Line 132: Line 113:
| | 19
| | 19
| | 414.5455
| | 414.5455
|439.418
|530.618 (212.9ED<sub>16</sub>)
|530.618 (212.9ED<sub>16</sub>)
|14/11
|14/11
Line 138: Line 118:
| | 20
| | 20
| | 436.364
| | 436.364
|462.5455
|558.5455 (22E.8BA<sub>16</sub>)
|558.5455 (22E.8BA<sub>16</sub>)
|9/7
|9/7
Line 144: Line 123:
| | 21
| | 21
| | 458.182
| | 458.182
|485.673
|586.473 (24A.79<sub>16</sub>)
|586.473 (24A.79<sub>16</sub>)
|13/10
|13/10
Line 150: Line 128:
| | 22
| | 22
| | 480
| | 480
|508.8
|614.4 (266.666<sub>16</sub>)
|614.4 (266.666<sub>16</sub>)
|21/16
|21/16
Line 156: Line 133:
| | 23
| | 23
| | 501.818
| | 501.818
|531.927
|642.327 (282.5508<sub>16</sub>)
|642.327 (282.5508<sub>16</sub>)
|4/3, 27/20
|4/3, 27/20
Line 162: Line 138:
| | 24
| | 24
| | 523.636
| | 523.636
|555.0545
|670.2545 (29E.413<sub>16</sub>)
|670.2545 (29E.413<sub>16</sub>)
|''27/20''
|''27/20''
Line 168: Line 143:
| | 25
| | 25
| | 545.4545
| | 545.4545
|578.182
|698.182 (2BA.2E9<sub>16</sub>)
|698.182 (2BA.2E9<sub>16</sub>)
|11/8
|11/8
Line 174: Line 148:
| | 26
| | 26
| | 567.273
| | 567.273
|601.309
|726.109 (2D6.1BF<sub>16</sub>)
|726.109 (2D6.1BF<sub>16</sub>)
|18/13, 25/18
|18/13, 25/18
Line 180: Line 153:
| | 27
| | 27
| | 589.091
| | 589.091
|624.436
|754.036 (2F2.095<sub>16</sub>)
|754.036 (2F2.095<sub>16</sub>)
|7/5
|7/5
Line 186: Line 158:
| | 28
| | 28
| | 610.909
| | 610.909
|647.564
|781.944 (30D.F7B<sub>16</sub>)
|781.944 (30D.F7B<sub>16</sub>)
|10/7
|10/7
Line 192: Line 163:
| | 29
| | 29
| | 632.727
| | 632.727
|670.691
|809.891 (329.E41<sub>16</sub>)
|809.891 (329.E41<sub>16</sub>)
|13/9, 36/25
|13/9, 36/25
Line 198: Line 168:
| | 30
| | 30
| | 654.5455
| | 654.5455
|693.818
|837.818 (345.D17<sub>16</sub>)
|837.818 (345.D17<sub>16</sub>)
|16/11
|16/11
Line 204: Line 173:
| | 31
| | 31
| | 676.364
| | 676.364
|716.9455
|865.7455 (361.BEC<sub>16</sub>)
|865.7455 (361.BEC<sub>16</sub>)
|''40/27''
|''40/27''
Line 210: Line 178:
| | 32
| | 32
| | 698.182
| | 698.182
|740.073
|893.673 (37D.AAF8<sub>16</sub>)
|893.673 (37D.AAF8<sub>16</sub>)
|3/2, 40/27
|3/2, 40/27
Line 216: Line 183:
| | 33
| | 33
| | 720
| | 720
|763.2
|921.6 (399.99A<sub>16</sub>)
|921.6 (399.99A<sub>16</sub>)
|32/21
|32/21
Line 222: Line 188:
| | 34
| | 34
| | 741.818
| | 741.818
|786.327
|949.527 (3B5.87<sub>16</sub>)
|949.527 (3B5.87<sub>16</sub>)
|20/13
|20/13
Line 228: Line 193:
| | 35
| | 35
| | 763.636
| | 763.636
|809.4545
|977.4545 (3D1.746<sub>16</sub>)
|977.4545 (3D1.746<sub>16</sub>)
|14/9
|14/9
Line 234: Line 198:
| | 36
| | 36
| | 785.4545
| | 785.4545
|832.582
|1005.382 (FED.612<sub>16</sub>)
|1005.382 (FED.612<sub>16</sub>)
|11/7
|11/7
Line 240: Line 203:
| | 37
| | 37
| | 807.273
| | 807.273
|855.709
|1033.309 (409.5F2<sub>16</sub>)
|1033.309 (409.5F2<sub>16</sub>)
|8/5
|8/5
Line 246: Line 208:
| | 38
| | 38
| | 829.091
| | 829.091
|878.836
|1061.327 (425.3A9<sub>16</sub>)
|1061.327 (425.3A9<sub>16</sub>)
|13/8
|13/8
Line 252: Line 213:
| | 39
| | 39
| | 850.909
| | 850.909
|901.964
|1089.164 (441.19E<sub>16</sub>)
|1089.164 (441.19E<sub>16</sub>)
|18/11, 44/27
|18/11, 44/27
Line 258: Line 218:
| | 40
| | 40
| | 872.727
| | 872.727
|925.091
|1117.091 (45D.174<sub>16</sub>)
|1117.091 (45D.174<sub>16</sub>)
|5/3-
|5/3-
Line 264: Line 223:
| | 41
| | 41
| | 894.5455
| | 894.5455
|948.218
|1145.018 (479.04A8<sub>16</sub>)
|1145.018 (479.04A8<sub>16</sub>)
|5/3+
|5/3+
Line 270: Line 228:
| | 42
| | 42
| | 916.364
| | 916.364
|971.3455
|1172.9455 (494.F21<sub>16</sub>)
|1172.9455 (494.F21<sub>16</sub>)
|22/13
|22/13
Line 276: Line 233:
| | 43
| | 43
| | 938.182
| | 938.182
|994.473
|1200.873 (4B0.DF7<sub>16</sub>)
|1200.873 (4B0.DF7<sub>16</sub>)
|12/7
|12/7
Line 282: Line 238:
| | 44
| | 44
| | 960
| | 960
|1017.6
|1228.8 (4CC.CCD<sub>16</sub>)
|1228.8 (4CC.CCD<sub>16</sub>)
|7/4, 26/15
|7/4, 26/15
Line 288: Line 243:
| | 45
| | 45
| | 981.818
| | 981.818
|1040.727
|1256.727 (4E8.B99<sub>16</sub>)
|1256.727 (4E8.B99<sub>16</sub>)
|30/17
|30/17
Line 294: Line 248:
| | 46
| | 46
| | 1003.636
| | 1003.636
|1063.8545
|1284.6545 (504.A79<sub>16</sub>)
|1284.6545 (504.A79<sub>16</sub>)
|16/9, 9/5
|16/9, 9/5
Line 300: Line 253:
| | 47
| | 47
| | 1025.4545
| | 1025.4545
|1058.982
|1312.582 (520.94F<sub>16</sub>)
|1312.582 (520.94F<sub>16</sub>)
|''9/5'', 20/11
|''9/5'', 20/11
Line 306: Line 258:
| | 48
| | 48
| | 1047.273
| | 1047.273
|1110.109
|1340.509 (53C.816<sub>16</sub>)
|1340.509 (53C.816<sub>16</sub>)
|11/6, 24/13
|11/6, 24/13
Line 312: Line 263:
| | 49
| | 49
| | 1069.091
| | 1069.091
|1133.236
|1368.436 (558.6FB<sub>16</sub>)
|1368.436 (558.6FB<sub>16</sub>)
|''24/13'', 13/7
|''24/13'', 13/7
Line 318: Line 268:
| | 50
| | 50
| | 1090.909
| | 1090.909
|1156.364
|1396.364 (574.5D18<sub>16</sub>)
|1396.364 (574.5D18<sub>16</sub>)
|15/8
|15/8
Line 324: Line 273:
| | 51
| | 51
| | 1112.727
| | 1112.727
|1179.491
|1424.291 (590.4A7<sub>16</sub>)
|1424.291 (590.4A7<sub>16</sub>)
|40/21, 48/25
|40/21, 48/25
Line 330: Line 278:
| | 52
| | 52
| | 1134.5455
| | 1134.5455
|1202.618
|1452.218 (FAC.37D<sub>16</sub>)
|1452.218 (FAC.37D<sub>16</sub>)
|56/27, ''48/25''
|56/27, ''48/25''
Line 336: Line 283:
| | 53
| | 53
| | 1156.364
| | 1156.364
|1225.7455
|1480.1455 (FC8.253<sub>16</sub>)
|1480.1455 (FC8.253<sub>16</sub>)
|35/18
|35/18
Line 342: Line 288:
| | 54
| | 54
| | 1178.182
| | 1178.182
|1248.873
|1508.073 (FE4.12A<sub>16</sub>)
|1508.073 (FE4.12A<sub>16</sub>)
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
|125/64, 63/32, 128/65, 77/39, 180/91, 196/99, ''160/81''
Line 348: Line 293:
| | 55
| | 55
| | 1200
| | 1200
|1272
|1536 (600<sub>16</sub>)
|1536 (600<sub>16</sub>)
|2/1
|2/1

Revision as of 13:24, 12 December 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 7mus Ratios it approximates
0 1/1
1 21.818 27.927 (1B.ED616) 128/125, 64/63, 65/64, 78/77, 91/90, 99/98, 81/80
2 43.636 55.8545 (37.DAC16) 36/35
3 65.4545 83.782 (53.C8216) 28/27, 25/24
4 87.273 111.709 (6F.B5816) 25/24, 21/20
5 109.091 139.636 (8B.A2E816) 16/15
6 130.909 167.564 (A7.90516) 14/13, 13/12
7 152.727 195.491 (C3.7DA16) 13/12, 12/11
8 174.5455 223.418 (DF.6B116) 11/10, 10/9
9 196.364 251.3455 (FB.58716) 9/8, 10/9
10 218.182 279.273 (117.46716) 17/15
11 240 307.2 (133.33316) 8/7, 15/13
12 261.818 335.127 (14F.20916) 7/6
13 283.636 363.0545 (16B.0DF16) 13/11
14 305.4545 390.982 (186.FB4816) 6/5-
15 327.273 418.909 (1A2.E8C16) 6/5+
16 349.091 446.836 (1BE.D6216) 11/9, 27/22
17 370.909 474.763 (1DA.C5616) 16/13
18 392.727 502.691 (1F6.A0E16) 5/4
19 414.5455 530.618 (212.9ED16) 14/11
20 436.364 558.5455 (22E.8BA16) 9/7
21 458.182 586.473 (24A.7916) 13/10
22 480 614.4 (266.66616) 21/16
23 501.818 642.327 (282.550816) 4/3, 27/20
24 523.636 670.2545 (29E.41316) 27/20
25 545.4545 698.182 (2BA.2E916) 11/8
26 567.273 726.109 (2D6.1BF16) 18/13, 25/18
27 589.091 754.036 (2F2.09516) 7/5
28 610.909 781.944 (30D.F7B16) 10/7
29 632.727 809.891 (329.E4116) 13/9, 36/25
30 654.5455 837.818 (345.D1716) 16/11
31 676.364 865.7455 (361.BEC16) 40/27
32 698.182 893.673 (37D.AAF816) 3/2, 40/27
33 720 921.6 (399.99A16) 32/21
34 741.818 949.527 (3B5.8716) 20/13
35 763.636 977.4545 (3D1.74616) 14/9
36 785.4545 1005.382 (FED.61216) 11/7
37 807.273 1033.309 (409.5F216) 8/5
38 829.091 1061.327 (425.3A916) 13/8
39 850.909 1089.164 (441.19E16) 18/11, 44/27
40 872.727 1117.091 (45D.17416) 5/3-
41 894.5455 1145.018 (479.04A816) 5/3+
42 916.364 1172.9455 (494.F2116) 22/13
43 938.182 1200.873 (4B0.DF716) 12/7
44 960 1228.8 (4CC.CCD16) 7/4, 26/15
45 981.818 1256.727 (4E8.B9916) 30/17
46 1003.636 1284.6545 (504.A7916) 16/9, 9/5
47 1025.4545 1312.582 (520.94F16) 9/5, 20/11
48 1047.273 1340.509 (53C.81616) 11/6, 24/13
49 1069.091 1368.436 (558.6FB16) 24/13, 13/7
50 1090.909 1396.364 (574.5D1816) 15/8
51 1112.727 1424.291 (590.4A716) 40/21, 48/25
52 1134.5455 1452.218 (FAC.37D16) 56/27, 48/25
53 1156.364 1480.1455 (FC8.25316) 35/18
54 1178.182 1508.073 (FE4.12A16) 125/64, 63/32, 128/65, 77/39, 180/91, 196/99, 160/81
55 1200 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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