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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.
{{Infobox ET}}
{{primes in edo|69}}
{{ED intro}}
== 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{{c}}. 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|31EDO]] but not in 69.
69edo offers two kinds of meantone 12-tone scales. One is the raw meantone scale, which has a 7:4 step ratio, and other is period-3 [[Meantone family#Lithium|lithium]] scale, which has a 6:5 step ratio and stems from a temperament tempering out [[3125/3087]] along with [[81/80]]. It should be noted that while the lithium scale has a meantone fifth, it produces a [[3L 6s|tcherepnin]] scale instead of traditional diatonic.
{| class="wikitable center-1 right-3"
 
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. In the 11-limit it tempers out [[99/98]], and supports the {{nowrap|31 & 69}} variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69.
 
The [[concoctic scale]] for 69edo is 22\69, and the corresponding rank two temperament is {{nowrap|22 & 69}}, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit.
 
=== Odd harmonics ===
{{Harmonics in equal|69}}
 
== Intervals ==
{{Interval table}}
 
=== Proposed names ===
{| class="wikitable mw-collapsible mw-collapsed collapsible center-1 right-3"
|-
! Degree
! Carmen's naming system
! Cents
! Approximate Ratios*
! Error (abs, [[cent|¢]])
|-
| 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
|-
|-
!Degree
| 15
!Name
| Subminor third, undetricesimal subminor third
!Cents
| 260.870
!Approximate Ratios*
| [[7/6]], [[29/25]]
!Error (abs, [[cent|¢]])
| −6.001, 3.920
|-
|-
|0
| 16
|Natural Unison, 1
| Vicesimotertial subminor third
|0.000
| 278.261
|[[1/1]]
| [[27/23]]
|0.000
| 0.670
|-
|-
|1
| 17
|Ptolemy's comma
| Pythagorean minor third
|17.391
| 295.652
|[[100/99]]
| [[32/27]]
| -0.008
| 1.517
|-
|-
|2
| 18
|Jubilisma, _____
| Classic minor third
|34.783
| 313.043
|[[50/49]], [[101/99]]
| [[6/5]]
| -0.193, 0.157
| −2.598
|-
|-
|3
| 19
|_____, _____
| Vicesimotertial supraminor third
|52.174
| 330.435
|[[34/33]], [[101/98]]
| [[23/19]]
| 0.491, -0.028
| −0.327
|-
|-
|4
| 20
|_____
| Undecimal neutral third
|69.565
| 347.826
|[[76/73]]
| [[11/9]]
| -0.158
| 0.418
|-
|-
|5
| 21
|Small undevicesimal semitone
| Septendecimal submajor third
|86.957
| 365.217
|[[20/19]]
| [[21/17]]
| -1.844
| −0.608
|-
|-
|6
| 22
|Large septendecimal semitone
| Classic major third
|104.348
| 382.609
|[[17/16]]
| [[5/4]]
| -0.608
| −3.705
|-
|-
|7
| 23
|Septimal diatonic semitone
| Undetricesimal major third, Septendecimal major third
|121.739
| 400.000
|[[15/14]]
| [[29/23]], [[34/27]]
|2.296
| −1.303, 0.910
|-
|-
|8
| 24
|Tridecimal neutral second
| Undecimal major third
|139.130
| 417.391
|[[13/12]]
| [[14/11]]
|0.558
| −0.117
|-
|-
|9
| 25
|Vicesimotertial neutral second
| Supermajor third
|156.522
| 434.783
|[[23/21]]
| [[9/7]]
| -0.972
| −0.301
|-
|-
|10
| 26
|Large neutral undevicesimal second
| Barbados third
|173.913
| 452.174
|[[21/19]]
| [[13/10]]
|0.645
| −2.040
|-
|-
|11
| 27
|Quasi-meantone
| Septimal sub-fourth
|191.304
| 469.565
|[[19/17]]
| [[21/16]]
| -1.253
| −1.216
|-
|-
|12
| 28
|Whole tone
| _____
|208.696
| 486.957
|[[9/8]]
| [[53/40]]
|4.786
| −0.234
|-
|-
|13
| 29
|Septimal whole tone
| Just perfect fourth
|226.087
| 504.348
|[[8/7]]
| [[4/3]]
| -5.087
| 6.303
|-
|-
|14
| 30
|Vicesimotertial subminor third
| Vicesimotertial acute fourth
|243.478
| 521.739
|[[23/20]]
| [[23/17]]
|1.518
| −1.580
|-
|-
|15
| 31
|Subminor third, _____
| Undecimal augmented fourth
|260.870
| 539.130
|[[7/6]], [[29/25]]
| [[15/11]]
| -6.001, 3.920
| 2.180
|-
|-
|16
| 32
|_____
| Undecimal superfourth, undetricesimal superfourth
|278.261
| 556.522
|[[27/23]]
| [[11/8]], [[29/21]]
|0.670
| 5.204, −2.275
|-
|-
|17
| 33
|Pythagorean minor third
| Narrow tritone, classic augmented fourth
|295.652
| 573.913
|[[32/27]]
| [[7/5]], [[25/18]]
|1.517
| −8.600, 5.196
|-
|-
|18
| 34
|Classic minor third
| _____
|313.043
| 591.304
|[[6/5]]
| [[31/22]]
| -2.598
| −2.413
|-
|-
|19
| 35
|Vicesimotertial supraminor third
| High tritone, undevicesimal tritone
|330.435
| 608.696
|[[23/19]]
| [[10/7]], [[27/19]]
| -0.327
| −8.792, 0.344
|-
|-
|20
| 36
|Undecimal neutral third
| _____
|347.826
| 626.087
|[[11/9]]
| [[33/23]]
|0.418
| 1.088
|-
|-
|21
| 37
|Septendecimal submajor third
| Undetricesimal tritone
|365.217
| 643.478
|[[21/17]]
| [[29/20]]
| -0.608
| 0.215
|-
|-
|22
| 38
|Classic major third
| Undevicesimal diminished fifth, undecimal diminished fifth
|382.609
| 660.870
|[[5/4]]
| [[19/13]], [[22/15]]
| -3.705
| 3.884, −2.180
|-
|-
|23
| 39
|_____, _____
| Vicesimotertial grave fifth, _____
|400.000
| 678.261
|[[29/23]], [[34/27]]
| [[34/23]], [[37/25]]
| -1.303, 0.910
| 1.580, −0.456
|-
|-
|24
| 40
|Undecimal major third
| Just perfect fifth
|417.391
| 695.652
|[[14/11]]
| [[3/2]]
| -0.117
| −6.303
|-
|-
|25
| 41
|Supermajor third
| _____
|434.783
| 713.043
|[[9/7]]
| [[80/53]]
| -0.301
| 0.234
|-
|-
|26
| 42
|Barbados third
| Super-fifth, undetricesimal super-fifth
|452.174
| 730.435
|[[13/10]]
| [[32/21]], [[29/19]]
| -2.040
| 1.216, −1.630
|-
|-
|27
| 43
|Septimal sub-fourth
| Septendecimal subminor sixth
|469.565
| 747.826
|[[21/16]]
| [[17/11]]
| -1.216
| −5.811
|-
|-
|28
| 44
|_____
| Subminor sixth
|486.957
| 765.217
|[[53/40]]
| [[14/9]]
| -0.234
| 0.301
|-
|-
|29
| 45
|Just perfect fourth
| Undecimal minor sixth
|504.348
| 782.609
|[[4/3]]
| [[11/7]]
|6.303
| 0.117
|-
|-
|30
| 46
|Vicesimotertial acute fourth
| Septendecimal subminor sixth
|521.739
| 800.000
|[[23/17]]
| [[27/17]]
| -1.580
| −0.910
|-
|-
|31
| 47
|Undecimal augmented fourth
| Classic minor sixth
|539.130
| 817.391
|[[15/11]]
| [[8/5]]
|2.180
| 3.705
|-
|-
|32
| 48
|Undecimal superfourth, _____
| Septendecimal supraminor sixth
|556.522
| 834.783
|[[11/8]], [[29/21]]
| [[34/21]]
|5.204, -2.275
| 0.608
|-
|-
|33
| 49
|Narrow tritone, classic augmented fourth
| Undecimal neutral sixth
|573.913
| 852.174
|[[7/5]], [[25/18]]
| [[18/11]]
| -8.600, 5.196
| −0.418
|-
|-
|34
| 50
|_____
| Vicesimotertial submajor sixth
|591.304
| 869.565
|[[31/22]]
| [[38/23]]
| -2.413
| 0.327
|-
|-
|35
| 51
|High tritone, _____
| Classic major sixth
|608.696
| 886.957
|[[10/7]], [[27/19]]
| [[5/3]]
| -8.792, 0.344
| 2.598
|-
|-
|36
| 52
|_____
| Pythagorean major sixth
|626.087
| 904.348
|[[33/23]]
| [[27/16]]
|1.088
| −1.517
|-
|-
|37
| 53
|_____
| Septendecimal major sixth, undetricesimal major sixth
|643.478
| 921.739
|[[29/20]]
| [[17/10]], [[29/17]]
|0.215
| 3.097, −2.883
|-
|-
|38
| 54
|_____, undecimal diminished fifth
| Supermajor sixth, undetricesimal supermajor sixth
|660.870
| 939.130
|[[19/13]], [[22/15]]
| [[12/7]], [[50/29]]
|3.884, -2.180
| 6.001, −3.920
|-
|-
|39
| 55
|Vicesimotertial grave fifth, _____
| Vicesimotertial supermajor sixth
|678.261
| 956.522
|[[34/23]], [[37/25]]
| [[40/23]]
|1.580, -0.456
| −1.518
|-
|-
|40
| 56
|Just perfect fifth
| Harmonic seventh
|695.652
| 973.913
|[[3/2]]
| [[7/4]]
| -6.303
| 5.087
|-
|-
|41
| 57
|_____
| Pythagorean minor seventh
|713.043
| 991.304
|[[80/53]]
| [[16/9]]
|0.234
| −4.786
|-
|-
|42
| 58
|Super-fifth, _____
| Quasi-meantone minor seventh
|730.435
| 1008.696
|[[32/21]], [[29/19]]
| [[34/19]]
|1.216, -1.630
| 1.253
|-
|-
|43
| 59
|Septendecimal subminor sixth
| Minor neutral undevicesimal seventh
|747.826
| 1026.087
|[[17/11]]
| [[38/21]]
| -5.811
| −0.645
|-
|-
|44
| 60
|Subminor sixth
| Vicesimotertial neutral seventh
|765.217
| 1043.478
|[[14/9]]
| [[42/23]]
|0.301
| 0.972
|-
|-
|45
| 61
|Undecimal minor sixth
| Tridecimal neutral seventh
|782.609
| 1060.870
|[[11/7]]
| [[24/13]]
|0.117
| −0.558
|-
|-
|46
| 62
|_____
| Septimal diatonic major seventh
|800.000
| 1078.261
|[[27/17]]
| [[28/15]]
| -0.910
| −2.296
|-
|-
|47
| 63
|Classic minor sixth
| Small septendecimal major seventh
|817.391
| 1095.652
|[[8/5]]
| [[32/17]]
|3.705
| 0.608
|-
|-
|48
| 64
|Septendecimal supraminor sixth
| Small undevicesimal semitone
|834.783
| 1113.043
|[[34/21]]
| [[20/19]]
|0.608
| 1.844
|-
|-
|49
| 65
|Undecimal neutral sixth
| _____
|852.174
| 1130.435
|[[18/11]]
| [[73/38]]
| -0.418
| 0.158
|-
|-
|50
| 66
|Vicesimotertial submajor sixth
| Septendecimal supermajor seventh
|869.565
| 1147.826
|[[38/23]]
| [[33/17]]
|0.327
| −0.491
|-
|-
|51
| 67
|Classic major sixth
| _____
|886.957
| 1165.217
|[[5/3]]
| [[49/25]]
|2.598
| −0.193
|-
|-
|52
| 68
|Pythagorean major sixth
| _____
|904.348
| 1182.609
|[[27/16]]
| [[99/50]]
| -1.517
| 0.008
|-
|-
|53
| 69
|Septendecimal major sixth, _____
| Octave, 8
|921.739
| 1200.000
|[[17/10]], [[29/17]]
| [[2/1]]
|3.097, -2.883
| 0.000
|}
<nowiki />* Some simpler ratios listed
 
== Notation ==
=== Ups and downs notation ===
69edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals, combined with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].
 
==== Evo flavor ====
<imagemap>
File:69-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 783 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:69-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 751 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:69-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
|54
! rowspan="2" | [[Subgroup]]
|Supermajor sixth, _____
! rowspan="2" | [[Comma list|Comma List]]
|939.130
! rowspan="2" | [[Mapping]]
|[[12/7]], [[50/29]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
|6.001, -3.920
! colspan="2" | Tuning Error
|-
|-
|55
! [[TE error|Absolute]] (¢)
|Vicesimotertial supermajor sixth
! [[TE simple badness|Relative]] (%)
|956.522
|[[40/23]]
| -1.518
|-
|-
|56
| 2.3
|Harmonic seventh
| {{monzo| -109 69 }}
|973.913
| {{mapping| 69 109 }}
|[[7/4]]
| +1.99
|5.087
| 1.99
| 11.43
|-
|-
|57
| 2.3.5
|Pythagorean minor seventh
| 81/80, {{monzo| -41 1 17 }}
|991.304
| {{mapping| 69 109 160 }}
|[[16/9]]
| +1.86
| -4.786
| 1.64
| 9.40
|-
|-
|58
| 2.3.5.7
|Quasi-meantone minor seventh
| 81/80, 126/125, 4117715/3981312
|1008.696
| {{mapping| 69 109 160 193 }} (69d)
|[[34/19]]
| +2.49
|1.253
| 1.79
| 10.28
|-
|-
|59
| 2.3.5.7
|Minor neutral undevicesimal seventh
| 81/80, 3125/3087, 6144/6125
|1026.087
| {{mapping| 69 109 160 194 }} (69)
|[[38/21]]
| +0.94
| -0.645
| 2.13
| 12.23
|}
 
=== Rank 2 temperaments ===
{| class="wikitable center-1 center-2"
|-
|-
|60
! Periods<br>per 8ve
|Vicesimotertial neutral seventh
! Generator
|1043.478
! Temperaments
|[[42/23]]
|0.972
|-
|-
|61
| 1
|Tridecimal neutral seventh
| 2\69
|1060.870
| [[Gammy]] (69de)
|[[24/13]]
| -0.558
|-
|-
|62
|1
|Septimal diatonic major seventh
|5\69
|1078.261
|[[Devichromic chords|Devichromic Octacot]]<ref group="note" name="tempname">Placeholder name, with link to [[Devichromic chords]] article &mdash; no general article currently exists for Devichromic temperament, and this particular incarnation of Devichromic temperament is likely to receive a different permanent name.</ref>
|[[28/15]]
| -2.296
|-
|-
|63
| 1
|Small septendecimal major seventh
| 19\69
|1095.652
| [[Rarity]]
|[[32/17]]
|0.608
|-
|-
|64
| 1
|Small undevicesimal semitone
| 20\69
|1113.043
| [[Mohaha]] (69e)
|[[20/19]]
|1.844
|-
|-
|65
| 1
|_____
| 22\69
|1130.435
| [[Caleb]] (69)<br>[[marveltri]] (69)
|[[73/38]]
|0.158
|-
|-
|66
| 1
|_____
| 29\69
|1147.826
| [[Meantone]] (69d)
|[[33/17]]
| -0.491
|-
|-
|67
| 3
|_____
| 5\69
|1165.217
| [[Augmented family #Ogene|Ogene]] (69bceef)
|[[49/25]]
| -0.193
|-
|-
|68
| 3
|_____
| 6\69
|1182.609
| [[August]] (7-limit, 69cdd)<br>[[Lithium]] (69)
|[[99/50]]
|0.008
|-
|-
|69
| 3
|Octave, 8
| 9\69
|1200.000
| [[Nessafof]] (69e)
|[[2/1]]
|0.000
|}
|}
<nowiki>*</nowiki>some simpler ratios listed
<references group="note" />
[[Category:meantone]]
 
[[Category:Equal divisions of the octave]]
== Scales ==
* Supermajor[11], [[3L 8s]] – 6 6 6 7 6 6 6 7 6 6 7
* Meantone[7], [[5L 2s]] (gen = 40\69) – 11 11 7 11 11 11 7
* Meantone[12], [[7L 5s]] (gen = 40\69) – 7 4 7 4 7 4 7 7 4 7 4 7
* Lithium[9], [[3L 6s]] – 11 6 6 11 6 6 11 6 6
* Lithium[12], [[9L 3s]] – 5 6 6 6 5 6 6 6 5 6 6 6
 
== Instruments ==
 
A [[Lumatone mapping for 69edo]] is available.
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=ZAqPonAHuUM ''microtonal improvisation in 69edo''] (2025)
* [https://www.youtube.com/shorts/4XBELeySMPk ''Compass - Mili (microtonal cover in 69edo)''] (2025)
 
; [[Eliora]]
* [https://www.youtube.com/watch?v=a4vNlDU6Vkw ''Hypergiant Sakura''] (2021)
 
; [[Francium]]
* [https://www.youtube.com/watch?v=Z3m4KqpuKPw ''69 hours before''] (2023)
 
[[Category:Meantone]]
[[Category:Listen]]
 
{{Todo| review }}
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