69edo: Difference between revisions

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69edo approximation list is about 1/4-1/3 complete. I'm just saving to make sure I don't lose any progress.
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In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also supports the 12&69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo|31EDO]] but not in 69.
In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. It also supports the 12&69 temperament tempering out 3125/3087 along with [[81/80]]. In the 11-limit it tempers out [[99/98]], and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo|31EDO]] but not in 69.
 
{| class="wikitable center-1 right-3"
|-
!Degree
!Name and Abbreviation
!Cents
!Approximate Ratios*
!Error (abs, [[cent|¢]])
|-
|0
|Natural Unison, 1
|0.000
|[[1/1]]
|0.000
|-
|1
|
|17.391
|
|
|-
|2
|
|34.783
|
|
|-
|3
|
|52.174
|20/19
| -1.844
|-
|4
|
|69.565
|
|
|-
|5
|
|86.957
|
|
|-
|6
|
|104.348
|17/16
| -0.608
|-
|7
|
|121.739
|15/14
|2.296
|-
|8
|
|139.130
|13/12
|0.558
|-
|9
|
|156.522
|
|
|-
|10
|
|173.913
|
|
|-
|11
|
|191.304
|19/17
| -1.253
|-
|12
|
|208.696
|9/8
|4.786
|-
|13
|
|226.087
|8/7
| -5.087
|-
|14
|
|243.478
|23/20
|1.518
|-
|15
|
|260.870
|7/6, 29/25
| -6.001, 3.920
|-
|16
|
|278.261
|27/23
|0.670
|-
|17
|
|295.652
|32/27
|1.517
|-
|18
|
|313.043
|[[6/5]]
| -2.598
|-
|19
|
|330.435
|23/19
| -0.327
|-
|20
|
|347.826
|[[11/9]]
|0.418
|-
|21
|
|365.217
|21/17
| -0.608
|-
|22
|
|382.609
|[[5/4]]
| -3.705
|-
|23
|
|400.000
|
|
|-
|24
|
|417.391
|14/11
| -0.117
|-
|25
|
|434.783
|9/7
| -0.301
|-
|26
|
|452.174
|13/10
| -2.040
|-
|27
|
|469.565
|21/16
| -1.216
|-
|28
|
|486.957
|
|
|-
|29
|
|504.348
|[[4/3]]
|6.303
|-
|30
|
|521.739
|23/17
| -1.580
|-
|31
|
|539.130
|15/11
|2.180
|-
|32
|
|556.522
|11/8, 29/21
|5.204, -2.275
|-
|33
|
|573.913
|7/5, 25/18
| -8.600, 5.196
|-
|34
|
|591.304
|31/22
| -2.413
|-
|35
|
|608.696
|10/7, 27/19
| -8.792, 0.344
|-
|36
|
|626.087
|33/23
|1.088
|-
|37
|
|643.478
|29/20
|0.215
|-
|38
|
|660.870
|19/13, 22/15
|3.884, -2.180
|-
|39
|
|678.261
|34/23, 37/25
|1.580, -0.456
|-
|40
|
|695.652
|[[3/2]]
| -6.303
|-
|41
|
|713.043
|
|
|-
|42
|
|730.435
|
|
|-
|43
|
|747.826
|
|
|-
|44
|
|765.217
|
|
|-
|45
|
|782.609
|
|
|-
|46
|
|800.000
|
|
|-
|47
|
|817.391
|
|
|-
|48
|
|834.783
|
|
|-
|49
|
|852.174
|
|
|-
|50
|
|869.565
|
|
|-
|51
|
|886.957
|
|
|-
|52
|
|904.348
|
|
|-
|53
|
|921.739
|
|
|-
|54
|
|939.130
|
|
|-
|55
|
|956.522
|
|
|-
|56
|
|973.913
|
|
|-
|57
|
|991.304
|
|
|-
|58
|
|1008.696
|
|
|-
|59
|
|1026.087
|
|
|-
|60
|
|1043.478
|
|
|-
|61
|
|1060.870
|
|
|-
|62
|
|1078.261
|
|
|-
|63
|
|1095.652
|
|
|-
|64
|
|1113.043
|
|
|-
|65
|
|1130.435
|
|
|-
|66
|
|1147.826
|
|
|-
|67
|
|1165.217
|
|
|-
|68
|
|1182.609
|
|
|-
|69
|Octave, 8
|1200.000
|[[2/1]]
|0.000
|}
<nowiki>*</nowiki>some simpler ratios listed
[[Category:meantone]]
[[Category:meantone]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 11:06, 21 December 2020

The 69 equal division or 69-EDO, which divides the octave into 69 equal parts of 17.391 cents each, has been called "the love-child of 23edo and quarter-comma meantone". Nice. As a meantone system, it is on the flat side, with a fifth of 695.652 cents. Such a fifth is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to that of "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.

In the 7-limit it is a mohajira system, tempering out 6144/6125, but not a septimal meantone system, as 126/125 maps to one step. It also supports the 12&69 temperament tempering out 3125/3087 along with 81/80. In the 11-limit it tempers out 99/98, and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in 31EDO but not in 69.

Degree Name and Abbreviation Cents Approximate Ratios* Error (abs, ¢)
0 Natural Unison, 1 0.000 1/1 0.000
1 17.391
2 34.783
3 52.174 20/19 -1.844
4 69.565
5 86.957
6 104.348 17/16 -0.608
7 121.739 15/14 2.296
8 139.130 13/12 0.558
9 156.522
10 173.913
11 191.304 19/17 -1.253
12 208.696 9/8 4.786
13 226.087 8/7 -5.087
14 243.478 23/20 1.518
15 260.870 7/6, 29/25 -6.001, 3.920
16 278.261 27/23 0.670
17 295.652 32/27 1.517
18 313.043 6/5 -2.598
19 330.435 23/19 -0.327
20 347.826 11/9 0.418
21 365.217 21/17 -0.608
22 382.609 5/4 -3.705
23 400.000
24 417.391 14/11 -0.117
25 434.783 9/7 -0.301
26 452.174 13/10 -2.040
27 469.565 21/16 -1.216
28 486.957
29 504.348 4/3 6.303
30 521.739 23/17 -1.580
31 539.130 15/11 2.180
32 556.522 11/8, 29/21 5.204, -2.275
33 573.913 7/5, 25/18 -8.600, 5.196
34 591.304 31/22 -2.413
35 608.696 10/7, 27/19 -8.792, 0.344
36 626.087 33/23 1.088
37 643.478 29/20 0.215
38 660.870 19/13, 22/15 3.884, -2.180
39 678.261 34/23, 37/25 1.580, -0.456
40 695.652 3/2 -6.303
41 713.043
42 730.435
43 747.826
44 765.217
45 782.609
46 800.000
47 817.391
48 834.783
49 852.174
50 869.565
51 886.957
52 904.348
53 921.739
54 939.130
55 956.522
56 973.913
57 991.304
58 1008.696
59 1026.087
60 1043.478
61 1060.870
62 1078.261
63 1095.652
64 1113.043
65 1130.435
66 1147.826
67 1165.217
68 1182.609
69 Octave, 8 1200.000 2/1 0.000

*some simpler ratios listed

Retrieved from "https://en.xen.wiki/index.php?title=69edo&oldid=57275"