69edo: Difference between revisions

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Saving my progress on 69edo's list of approximate intervals. I'm around 1/3-1/2 done. I'm going to save now so I don't lose any progress and hopefully finish this chart sometime today.
I just about finished the 69edo interval approximation chart. There are some missing names I couldn't find, so I left them blank, but I pretty much did my best with this chart.
Line 5: Line 5:
|-
|-
!Degree
!Degree
!Name and Abbreviation
!Name
!Cents
!Cents
!Approximate Ratios*
!Approximate Ratios*
Line 17: Line 17:
|-
|-
|1
|1
|
|Ptolemy's comma
|17.391
|17.391
|
|[[100/99]]
|
| -0.008
|-
|-
|2
|2
|
|Jubilisma, _____
|34.783
|34.783
|
|[[50/49]], [[101/99]]
|
| -0.193, 0.157
|-
|-
|3
|3
|
|_____, _____
|52.174
|52.174
|20/19
|[[34/33]], [[101/98]]
| -1.844
| 0.491, -0.028
|-
|-
|4
|4
|
|_____
|69.565
|69.565
|
|[[76/73]]
|
| -0.158
|-
|-
|5
|5
|
|Small undevicesimal semitone
|86.957
|86.957
|
|[[20/19]]
|
| -1.844
|-
|-
|6
|6
|
|Large septendecimal semitone
|104.348
|104.348
|17/16
|[[17/16]]
| -0.608
| -0.608
|-
|-
|7
|7
|
|Septimal diatonic semitone
|121.739
|121.739
|15/14
|[[15/14]]
|2.296
|2.296
|-
|-
|8
|8
|
|Tridecimal neutral second
|139.130
|139.130
|13/12
|[[13/12]]
|0.558
|0.558
|-
|-
|9
|9
|
|Vicesimotertial neutral second
|156.522
|156.522
|
|[[23/21]]
|
| -0.972
|-
|-
|10
|10
|
|Large neutral undevicesimal second
|173.913
|173.913
|
|[[21/19]]
|
|0.645
|-
|-
|11
|11
|
|Quasi-meantone
|191.304
|191.304
|19/17
|[[19/17]]
| -1.253
| -1.253
|-
|-
|12
|12
|
|Whole tone
|208.696
|208.696
|9/8
|[[9/8]]
|4.786
|4.786
|-
|-
|13
|13
|
|Septimal whole tone
|226.087
|226.087
|8/7
|[[8/7]]
| -5.087
| -5.087
|-
|-
|14
|14
|
|Vicesimotertial subminor third
|243.478
|243.478
|23/20
|[[23/20]]
|1.518
|1.518
|-
|-
|15
|15
|
|Subminor third, _____
|260.870
|260.870
|7/6, 29/25
|[[7/6]], [[29/25]]
| -6.001, 3.920
| -6.001, 3.920
|-
|-
|16
|16
|
|_____
|278.261
|278.261
|27/23
|[[27/23]]
|0.670
|0.670
|-
|-
|17
|17
|
|Pythagorean minor third
|295.652
|295.652
|32/27
|[[32/27]]
|1.517
|1.517
|-
|-
|18
|18
|
|Classic minor third
|313.043
|313.043
|[[6/5]]
|[[6/5]]
Line 125: Line 125:
|-
|-
|19
|19
|
|Vicesimotertial supraminor third
|330.435
|330.435
|23/19
|[[23/19]]
| -0.327
| -0.327
|-
|-
|20
|20
|
|Undecimal neutral third
|347.826
|347.826
|[[11/9]]
|[[11/9]]
Line 137: Line 137:
|-
|-
|21
|21
|
|Septendecimal submajor third
|365.217
|365.217
|21/17
|[[21/17]]
| -0.608
| -0.608
|-
|-
|22
|22
|
|Classic major third
|382.609
|382.609
|[[5/4]]
|[[5/4]]
Line 149: Line 149:
|-
|-
|23
|23
|
|_____, _____
|400.000
|400.000
|
|[[29/23]], [[34/27]]
|
| -1.303, 0.910
|-
|-
|24
|24
|
|Undecimal major third
|417.391
|417.391
|14/11
|[[14/11]]
| -0.117
| -0.117
|-
|-
|25
|25
|
|Supermajor third
|434.783
|434.783
|9/7
|[[9/7]]
| -0.301
| -0.301
|-
|-
|26
|26
|
|Barbados third
|452.174
|452.174
|13/10
|[[13/10]]
| -2.040
| -2.040
|-
|-
|27
|27
|
|Septimal sub-fourth
|469.565
|469.565
|21/16
|[[21/16]]
| -1.216
| -1.216
|-
|-
|28
|28
|
|_____
|486.957
|486.957
|
|[[53/40]]
|
| -0.234
|-
|-
|29
|29
|
|Just perfect fourth
|504.348
|504.348
|[[4/3]]
|[[4/3]]
Line 191: Line 191:
|-
|-
|30
|30
|
|Vicesimotertial acute fourth
|521.739
|521.739
|23/17
|[[23/17]]
| -1.580
| -1.580
|-
|-
|31
|31
|
|Undecimal augmented fourth
|539.130
|539.130
|15/11
|[[15/11]]
|2.180
|2.180
|-
|-
|32
|32
|
|Undecimal superfourth, _____
|556.522
|556.522
|11/8, 29/21
|[[11/8]], [[29/21]]
|5.204, -2.275
|5.204, -2.275
|-
|-
|33
|33
|
|Narrow tritone, classic augmented fourth
|573.913
|573.913
|7/5, 25/18
|[[7/5]], [[25/18]]
| -8.600, 5.196
| -8.600, 5.196
|-
|-
|34
|34
|
|_____
|591.304
|591.304
|31/22
|[[31/22]]
| -2.413
| -2.413
|-
|-
|35
|35
|
|High tritone, _____
|608.696
|608.696
|10/7, 27/19
|[[10/7]], [[27/19]]
| -8.792, 0.344
| -8.792, 0.344
|-
|-
|36
|36
|
|_____
|626.087
|626.087
|33/23
|[[33/23]]
|1.088
|1.088
|-
|-
|37
|37
|
|_____
|643.478
|643.478
|29/20
|[[29/20]]
|0.215
|0.215
|-
|-
|38
|38
|
|_____,  undecimal diminished fifth
|660.870
|660.870
|19/13, 22/15
|[[19/13]], [[22/15]]
|3.884, -2.180
|3.884, -2.180
|-
|-
|39
|39
|
|Vicesimotertial grave fifth, _____
|678.261
|678.261
|34/23, 37/25
|[[34/23]], [[37/25]]
|1.580, -0.456
|1.580, -0.456
|-
|-
|40
|40
|
|Just perfect fifth
|695.652
|695.652
|[[3/2]]
|[[3/2]]
Line 257: Line 257:
|-
|-
|41
|41
|
|_____
|713.043
|713.043
|
|[[80/53]]
|
|0.234
|-
|-
|42
|42
|
|Super-fifth, _____
|730.435
|730.435
|32/21, 29/19
|[[32/21]], [[29/19]]
|1.216, -1.630
|1.216, -1.630
|-
|-
|43
|43
|
|Septendecimal subminor sixth
|747.826
|747.826
|17/11
|[[17/11]]
| -5.811
| -5.811
|-
|-
|44
|44
|
|Subminor sixth
|765.217
|765.217
|14/9
|[[14/9]]
|0.301
|0.301
|-
|-
|45
|45
|
|Undecimal minor sixth
|782.609
|782.609
|11/7
|[[11/7]]
|0.117
|0.117
|-
|-
|46
|46
|
|_____
|800.000
|800.000
|27/17
|[[27/17]]
|
| -0.910
|-
|-
|47
|47
|
|Classic minor sixth
|817.391
|817.391
|8/5
|[[8/5]]
|3.705
|3.705
|-
|-
|48
|48
|
|Septendecimal supraminor sixth
|834.783
|834.783
|34/21
|[[34/21]]
|
|
|-
|-
|49
|49
|
|Undecimal neutral sixth
|852.174
|852.174
|
|[[18/11]]
|
| -0.418
|-
|-
|50
|50
|
|Vicesimotertial submajor sixth
|869.565
|869.565
|
|[[38/23]]
|
|0.327
|-
|-
|51
|51
|
|Classic major sixth
|886.957
|886.957
|5/3
|[[5/3]]
|2.598
|2.598
|-
|-
|52
|52
|
|Pythagorean major sixth
|904.348
|904.348
|
|[[27/16]]
|
| -1.517
|-
|-
|53
|53
|
|Septendecimal major sixth, _____
|921.739
|921.739
|
|[[17/10]], [[29/17]]
|
|3.097, -2.883
|-
|-
|54
|54
|
|Supermajor sixth, _____
|939.130
|939.130
|
|[[12/7]], [[50/29]]
|
|6.001, -3.920
|-
|-
|55
|55
|
|Vicesimotertial supermajor sixth
|956.522
|956.522
|
|[[40/23]]
|
| -1.518
|-
|-
|56
|56
|
|Harmonic seventh
|973.913
|973.913
|
|[[7/4]]
|
|5.087
|-
|-
|57
|57
|
|Pythagorean minor seventh
|991.304
|991.304
|
|[[16/9]]
|
| -4.786
|-
|-
|58
|58
|
|Quasi-meantone minor seventh
|1008.696
|1008.696
|
|[[34/19]]
|
|1.253
|-
|-
|59
|59
|
|Minor neutral undevicesimal seventh
|1026.087
|1026.087
|
|[[38/21]]
|
| -0.645
|-
|-
|60
|60
|
|Vicesimotertial neutral seventh
|1043.478
|1043.478
|
|[[42/23]]
|
|0.972
|-
|-
|61
|61
|
|Tridecimal neutral seventh
|1060.870
|1060.870
|
|[[24/13]]
|
| -0.558
|-
|-
|62
|62
|
|Septimal diatonic major seventh
|1078.261
|1078.261
|28/15
|[[28/15]]
| -2.296
| -2.296
|-
|-
|63
|63
|
|Small septendecimal major seventh
|1095.652
|1095.652
|
|[[32/17]]
|
|0.608
|-
|-
|64
|64
|
|Small undevicesimal semitone
|1113.043
|1113.043
|
|[[20/19]]
|
|1.844
|-
|-
|65
|65
|
|_____
|1130.435
|1130.435
|
|[[73/38]]
|
|0.158
|-
|-
|66
|66
|
|_____
|1147.826
|1147.826
|
|[[33/17]]
|
| -0.491
|-
|-
|67
|67
|
|_____
|1165.217
|1165.217
|
|[[49/25]]
|
| -0.193
|-
|-
|68
|68
|
|_____
|1182.609
|1182.609
|
|[[99/50]]
|
|0.008
|-
|-
|69
|69

Revision as of 16:01, 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 Cents Approximate Ratios* Error (abs, ¢)
0 Natural Unison, 1 0.000 1/1 0.000
1 Ptolemy's comma 17.391 100/99 -0.008
2 Jubilisma, _____ 34.783 50/49, 101/99 -0.193, 0.157
3 _____, _____ 52.174 34/33, 101/98 0.491, -0.028
4 _____ 69.565 76/73 -0.158
5 Small undevicesimal semitone 86.957 20/19 -1.844
6 Large septendecimal semitone 104.348 17/16 -0.608
7 Septimal diatonic semitone 121.739 15/14 2.296
8 Tridecimal neutral second 139.130 13/12 0.558
9 Vicesimotertial neutral second 156.522 23/21 -0.972
10 Large neutral undevicesimal second 173.913 21/19 0.645
11 Quasi-meantone 191.304 19/17 -1.253
12 Whole tone 208.696 9/8 4.786
13 Septimal whole tone 226.087 8/7 -5.087
14 Vicesimotertial subminor third 243.478 23/20 1.518
15 Subminor third, _____ 260.870 7/6, 29/25 -6.001, 3.920
16 _____ 278.261 27/23 0.670
17 Pythagorean minor third 295.652 32/27 1.517
18 Classic minor third 313.043 6/5 -2.598
19 Vicesimotertial supraminor third 330.435 23/19 -0.327
20 Undecimal neutral third 347.826 11/9 0.418
21 Septendecimal submajor third 365.217 21/17 -0.608
22 Classic major third 382.609 5/4 -3.705
23 _____, _____ 400.000 29/23, 34/27 -1.303, 0.910
24 Undecimal major third 417.391 14/11 -0.117
25 Supermajor third 434.783 9/7 -0.301
26 Barbados third 452.174 13/10 -2.040
27 Septimal sub-fourth 469.565 21/16 -1.216
28 _____ 486.957 53/40 -0.234
29 Just perfect fourth 504.348 4/3 6.303
30 Vicesimotertial acute fourth 521.739 23/17 -1.580
31 Undecimal augmented fourth 539.130 15/11 2.180
32 Undecimal superfourth, _____ 556.522 11/8, 29/21 5.204, -2.275
33 Narrow tritone, classic augmented fourth 573.913 7/5, 25/18 -8.600, 5.196
34 _____ 591.304 31/22 -2.413
35 High tritone, _____ 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 _____, undecimal diminished fifth 660.870 19/13, 22/15 3.884, -2.180
39 Vicesimotertial grave fifth, _____ 678.261 34/23, 37/25 1.580, -0.456
40 Just perfect fifth 695.652 3/2 -6.303
41 _____ 713.043 80/53 0.234
42 Super-fifth, _____ 730.435 32/21, 29/19 1.216, -1.630
43 Septendecimal subminor sixth 747.826 17/11 -5.811
44 Subminor sixth 765.217 14/9 0.301
45 Undecimal minor sixth 782.609 11/7 0.117
46 _____ 800.000 27/17 -0.910
47 Classic minor sixth 817.391 8/5 3.705
48 Septendecimal supraminor sixth 834.783 34/21
49 Undecimal neutral sixth 852.174 18/11 -0.418
50 Vicesimotertial submajor sixth 869.565 38/23 0.327
51 Classic major sixth 886.957 5/3 2.598
52 Pythagorean major sixth 904.348 27/16 -1.517
53 Septendecimal major sixth, _____ 921.739 17/10, 29/17 3.097, -2.883
54 Supermajor sixth, _____ 939.130 12/7, 50/29 6.001, -3.920
55 Vicesimotertial supermajor sixth 956.522 40/23 -1.518
56 Harmonic seventh 973.913 7/4 5.087
57 Pythagorean minor seventh 991.304 16/9 -4.786
58 Quasi-meantone minor seventh 1008.696 34/19 1.253
59 Minor neutral undevicesimal seventh 1026.087 38/21 -0.645
60 Vicesimotertial neutral seventh 1043.478 42/23 0.972
61 Tridecimal neutral seventh 1060.870 24/13 -0.558
62 Septimal diatonic major seventh 1078.261 28/15 -2.296
63 Small septendecimal major seventh 1095.652 32/17 0.608
64 Small undevicesimal semitone 1113.043 20/19 1.844
65 _____ 1130.435 73/38 0.158
66 _____ 1147.826 33/17 -0.491
67 _____ 1165.217 49/25 -0.193
68 _____ 1182.609 99/50 0.008
69 Octave, 8 1200.000 2/1 0.000

*some simpler ratios listed

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