55edo: Difference between revisions

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Line 212: Line 212:
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 2.090
| style="text-align:center;" | 2.090
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" |2.611
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
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| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 3.773
| style="text-align:center;" | 3.773
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" |3.890
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
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| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 5.053
| style="text-align:center;" | 5.053
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" |5.573
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
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| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 6.414
| style="text-align:center;" | 6.414
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" |6.579
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 7.546
| style="text-align:center;" | 7.546
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" |7.664
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 7.741
| style="text-align:center;" | 7.741
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" |7.858
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
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| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" |9.541
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 10.187
| style="text-align:center;" | 10.187
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 10.352
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 10.381
| style="text-align:center;" | 10.381
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 11.466
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 12.277
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 13.960
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 14.155
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 15.239
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 16.245
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 17.928
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 19.207
|}
|}


Revision as of 22:56, 29 January 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|

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

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 Ratios it approximates
0 0 1/1
1 21.818 128/125
2 43.636
3 65.455
4 87.273 25/24
5 109.091 16/15
6 130.909
7 152.727
8 174.545
9 196.364 9/8, 10/9
10 218.182
11 240.000
12 261.818
13 283.636
14 305.455
15 327.273
16 349.091
17 370.909
18 392.727
19 414.545
20 436.364
21 458.182
22 480.000
23 501.818
24 523.636
25 545.455
26 567.273
27 589.091
28 610.909
29 632.727
30 654.545
31 676.364
32 698.182
33 720.000
34 741.818
35 763.636
36 785.455
37 807.273
38 829.091
39 850.909
40 872.727
41 894.545
42 916.364
43 938.182
44 960.000
45 981.818
46 1003.636
47 1025.455
48 1047.273
49 1069.091
50 1090.909
51 1112.727
52 1134.545
53 1156.364
54 1178.182
55 1200.000

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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