55edo: Difference between revisions

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→Selected just intervals by error: 55edo, not 43edo, although both are cute
→Selected just intervals by error: completed with the help of https://scratch.mit.edu/projects/247892491/
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! | Error (abs., in [[cent|cents]])
! | Error (abs., in [[cent|cents]])
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" |  
| style="text-align:center;" | 1.280
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" |  
| style="text-align:center;" | 1.683
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 2.090
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" |  
| style="text-align:center;" | 2.640
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 2.963
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" |  
| style="text-align:center;" | 3.773
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" |  
| style="text-align:center;" | 3.968
|-
|-
| 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;" |  
| style="text-align:center;" | 5.053
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 5.863
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 6.414
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" |  
| style="text-align:center;" | 7.546
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" |  
| style="text-align:center;" | 7.741
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" |  
| style="text-align:center;" | 8.504
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" |  
| style="text-align:center;" | 8.826
|-
|-
| 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;" |  
| style="text-align:center;" | 10.187
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" |  
| style="text-align:center;" | 10.381
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" |
| style="text-align:center;" | 11.466
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" |  
| style="text-align:center;" | 12.277
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" |
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" |
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" |  
| style="text-align:center;" | 13.960
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" |  
| style="text-align:center;" | 14.155
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" |  
| style="text-align:center;" | 15.239
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" |  
| style="text-align:center;" | 16.245
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" |  
| style="text-align:center;" | 17.928
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" |  
| style="text-align:center;" | 19.207
|}
|}


Revision as of 17:07, 22 September 2018

55 tone equal temperament

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: 81/80, 686/675, 6144/6125

11-limit commas: 81/80, 121/120, 176/175, 686/675

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
16/15, 15/8 2.640
14/11, 11/7 2.963
4/3, 3/2 3.773
13/10, 20/13 3.968
7/6, 12/7 5.053
11/8, 16/11 5.863
5/4, 8/5 6.414
9/8, 16/9 7.546
15/13, 26/15 7.741
15/11, 22/15 8.504
8/7, 7/4 8.826
6/5, 5/3 10.187
16/13, 13/8 10.381
15/14, 28/15 11.466
11/10, 20/11 12.277
10/9, 9/5 13.960
13/12, 24/13 14.155
7/5, 10/7 15.239
13/11, 22/13 16.245
18/13, 13/9 17.928
14/13, 13/7 19.207


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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