69edo: Difference between revisions

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The '''69 equal division''' or '''69-EDO''', which divides the octave into 69 equal parts of 17.391 [[cent]]s 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.
The '''69 equal division''' or '''69-EDO''', which divides the octave into 69 equal parts of 17.391 [[cent]]s each. Nice.
 
== Theory ==
69edo has been called "the love-child of [[23edo]] and [[quarter-comma meantone]]". 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.
{{primes in edo|69}}
{{primes in edo|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 [[support]]s 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 [[support]]s 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 collapsible mw-collapsed center-1 right-3"
 
== Regular temperament properties ==
{| class="wikitable"
|+
!Subgroup
!Comma list
!Mapping
!Optimal 8ve stretch
(¢)
|-
|2.3
|[-109, 69⟩
|[⟨69 109]]
|1.99
|-
|2.3.5
|81/80, [-41, 1, 17⟩, [-37, -3, 18⟩, [-33, -7, 19⟩
|[⟨69 109 160]]
|1.86
|-
|2.3.5.7
|81/80, 3125/3087, 5625/5488, 6144/6125, 17280/16807, 17496/16807
|[⟨69 109 160 194]]
|0.94
|-
|2.3.5.7.11
|81/80, 99/98, 441/440, 625/616, 864/847, 1344/1331, 2420/2401, 2560/2541
|[⟨69 109 160 194 239]]
|0.44
|}
 
== Table of intervals ==
{| class="wikitable mw-collapsible mw-collapsed collapsible center-1 right-3"
|-
|-
!Degree
!Degree
Line 21: Line 56:
|17.391
|17.391
|[[100/99]]
|[[100/99]]
| -0.008
| -0.008
|-
|-
|2
|2
Line 27: Line 62:
|34.783
|34.783
|[[50/49]], [[101/99]]
|[[50/49]], [[101/99]]
| -0.193, 0.157
| -0.193, 0.157
|-
|-
|3
|3
Line 39: Line 74:
|69.565
|69.565
|[[76/73]]
|[[76/73]]
| -0.158
| -0.158
|-
|-
|5
|5
Line 45: Line 80:
|86.957
|86.957
|[[20/19]]
|[[20/19]]
| -1.844
| -1.844
|-
|-
|6
|6
Line 51: Line 86:
|104.348
|104.348
|[[17/16]]
|[[17/16]]
| -0.608
| -0.608
|-
|-
|7
|7
Line 69: Line 104:
|156.522
|156.522
|[[23/21]]
|[[23/21]]
| -0.972
| -0.972
|-
|-
|10
|10
Line 81: Line 116:
|191.304
|191.304
|[[19/17]]
|[[19/17]]
| -1.253
| -1.253
|-
|-
|12
|12
Line 93: Line 128:
|226.087
|226.087
|[[8/7]]
|[[8/7]]
| -5.087
| -5.087
|-
|-
|14
|14
Line 105: Line 140:
|260.870
|260.870
|[[7/6]], [[29/25]]
|[[7/6]], [[29/25]]
| -6.001, 3.920
| -6.001, 3.920
|-
|-
|16
|16
Line 123: Line 158:
|313.043
|313.043
|[[6/5]]
|[[6/5]]
| -2.598
| -2.598
|-
|-
|19
|19
Line 129: Line 164:
|330.435
|330.435
|[[23/19]]
|[[23/19]]
| -0.327
| -0.327
|-
|-
|20
|20
Line 141: Line 176:
|365.217
|365.217
|[[21/17]]
|[[21/17]]
| -0.608
| -0.608
|-
|-
|22
|22
Line 147: Line 182:
|382.609
|382.609
|[[5/4]]
|[[5/4]]
| -3.705
| -3.705
|-
|-
|23
|23
Line 153: Line 188:
|400.000
|400.000
|[[29/23]], [[34/27]]
|[[29/23]], [[34/27]]
| -1.303, 0.910
| -1.303, 0.910
|-
|-
|24
|24
Line 159: Line 194:
|417.391
|417.391
|[[14/11]]
|[[14/11]]
| -0.117
| -0.117
|-
|-
|25
|25
Line 165: Line 200:
|434.783
|434.783
|[[9/7]]
|[[9/7]]
| -0.301
| -0.301
|-
|-
|26
|26
Line 171: Line 206:
|452.174
|452.174
|[[13/10]]
|[[13/10]]
| -2.040
| -2.040
|-
|-
|27
|27
Line 177: Line 212:
|469.565
|469.565
|[[21/16]]
|[[21/16]]
| -1.216
| -1.216
|-
|-
|28
|28
Line 183: Line 218:
|486.957
|486.957
|[[53/40]]
|[[53/40]]
| -0.234
| -0.234
|-
|-
|29
|29
Line 195: Line 230:
|521.739
|521.739
|[[23/17]]
|[[23/17]]
| -1.580
| -1.580
|-
|-
|31
|31
Line 213: Line 248:
|573.913
|573.913
|[[7/5]], [[25/18]]
|[[7/5]], [[25/18]]
| -8.600, 5.196
| -8.600, 5.196
|-
|-
|34
|34
Line 219: Line 254:
|591.304
|591.304
|[[31/22]]
|[[31/22]]
| -2.413
| -2.413
|-
|-
|35
|35
Line 225: Line 260:
|608.696
|608.696
|[[10/7]], [[27/19]]
|[[10/7]], [[27/19]]
| -8.792, 0.344
| -8.792, 0.344
|-
|-
|36
|36
Line 255: Line 290:
|695.652
|695.652
|[[3/2]]
|[[3/2]]
| -6.303
| -6.303
|-
|-
|41
|41
Line 273: Line 308:
|747.826
|747.826
|[[17/11]]
|[[17/11]]
| -5.811
| -5.811
|-
|-
|44
|44
Line 291: Line 326:
|800.000
|800.000
|[[27/17]]
|[[27/17]]
| -0.910
| -0.910
|-
|-
|47
|47
Line 309: Line 344:
|852.174
|852.174
|[[18/11]]
|[[18/11]]
| -0.418
| -0.418
|-
|-
|50
|50
Line 327: Line 362:
|904.348
|904.348
|[[27/16]]
|[[27/16]]
| -1.517
| -1.517
|-
|-
|53
|53
Line 345: Line 380:
|956.522
|956.522
|[[40/23]]
|[[40/23]]
| -1.518
| -1.518
|-
|-
|56
|56
Line 357: Line 392:
|991.304
|991.304
|[[16/9]]
|[[16/9]]
| -4.786
| -4.786
|-
|-
|58
|58
Line 369: Line 404:
|1026.087
|1026.087
|[[38/21]]
|[[38/21]]
| -0.645
| -0.645
|-
|-
|60
|60
Line 381: Line 416:
|1060.870
|1060.870
|[[24/13]]
|[[24/13]]
| -0.558
| -0.558
|-
|-
|62
|62
Line 387: Line 422:
|1078.261
|1078.261
|[[28/15]]
|[[28/15]]
| -2.296
| -2.296
|-
|-
|63
|63
Line 411: Line 446:
|1147.826
|1147.826
|[[33/17]]
|[[33/17]]
| -0.491
| -0.491
|-
|-
|67
|67
Line 417: Line 452:
|1165.217
|1165.217
|[[49/25]]
|[[49/25]]
| -0.193
| -0.193
|-
|-
|68
|68

Revision as of 08:32, 4 March 2022

The 69 equal division or 69-EDO, which divides the octave into 69 equal parts of 17.391 cents each. Nice.

Theory

69edo has been called "the love-child of 23edo and quarter-comma meantone". 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. Script error: No such module "primes_in_edo".

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.

Regular temperament properties

Subgroup Comma list Mapping Optimal 8ve stretch

(¢)

2.3 [-109, 69⟩ [⟨69 109]] 1.99
2.3.5 81/80, [-41, 1, 17⟩, [-37, -3, 18⟩, [-33, -7, 19⟩ [⟨69 109 160]] 1.86
2.3.5.7 81/80, 3125/3087, 5625/5488, 6144/6125, 17280/16807, 17496/16807 [⟨69 109 160 194]] 0.94
2.3.5.7.11 81/80, 99/98, 441/440, 625/616, 864/847, 1344/1331, 2420/2401, 2560/2541 [⟨69 109 160 194 239]] 0.44

Table of intervals

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, lesser septimal sixth tone 34.783 50/49, 101/99 -0.193, 0.157
3 lesser septendecimal quartertone, _____ 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 Undevicesimal large neutral second, undevicesimal whole tone 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 semifourth 243.478 23/20 1.518
15 Subminor third, undetricesimal subminor third 260.870 7/6, 29/25 -6.001, 3.920
16 Vicesimotertial subminor third 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 Undetricesimal major third, Septendecimal major third 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, undetricesimal 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, undevicesimal tritone 608.696 10/7, 27/19 -8.792, 0.344
36 _____ 626.087 33/23 1.088
37 Undetricesimal tritone 643.478 29/20 0.215
38 Undevicesimal diminished fifth, 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, undetricesimal 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 Septendecimal subminor sixth 800.000 27/17 -0.910
47 Classic minor sixth 817.391 8/5 3.705
48 Septendecimal supraminor sixth 834.783 34/21 0.608
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, undetricesimal major sixth 921.739 17/10, 29/17 3.097, -2.883
54 Supermajor sixth, undetricesimal 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 Septendecimal supermajor seventh 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

Scales

  • Mavka[11] - 66676667667
  • MeantoneMajor[7] - 11 11 7 11 11 11 7
  • MeantoneChromatic[12] - 747474774747

Music

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