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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|69}} Nice.
{{ED intro}}
 
== Theory ==
== 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.
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.


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.
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.


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 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[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. 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 22 & 69, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit.
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 ===
=== Odd harmonics ===
Line 26: Line 25:
! Error (abs, [[cent|¢]])
! Error (abs, [[cent|¢]])
|-
|-
|0
| 0
|Natural Unison, 1
| Natural Unison, 1
|0.000
| 0.000
|[[1/1]]
| [[1/1]]
|0.000
| 0.000
|-
|-
|1
| 1
|Ptolemy's comma
| Ptolemy's comma
|17.391
| 17.391
|[[100/99]]
| [[100/99]]
| -0.008
| −0.008
|-
|-
|2
| 2
|Jubilisma, lesser septimal sixth tone
| Jubilisma, lesser septimal sixth tone
|34.783
| 34.783
|[[50/49]], [[101/99]]
| [[50/49]], [[101/99]]
| -0.193, 0.157
| −0.193, 0.157
|-
|-
|3
| 3
|lesser septendecimal quartertone, _____
| lesser septendecimal quartertone, _____
|52.174
| 52.174
|[[34/33]], [[101/98]]
| [[34/33]], [[101/98]]
| 0.491, -0.028
| 0.491, −0.028
|-
|-
|4
| 4
|_____
| _____
|69.565
| 69.565
|[[76/73]]
| [[76/73]]
| -0.158
| −0.158
|-
|-
|5
| 5
|Small undevicesimal semitone
| Small undevicesimal semitone
|86.957
| 86.957
|[[20/19]]
| [[20/19]]
| -1.844
| −1.844
|-
|-
|6
| 6
|Large septendecimal semitone
| Large septendecimal semitone
|104.348
| 104.348
|[[17/16]]
| [[17/16]]
| -0.608
| −0.608
|-
|-
|7
| 7
|Septimal diatonic semitone
| Septimal diatonic semitone
|121.739
| 121.739
|[[15/14]]
| [[15/14]]
|2.296
| 2.296
|-
|-
|8
| 8
|Tridecimal neutral second
| Tridecimal neutral second
|139.130
| 139.130
|[[13/12]]
| [[13/12]]
|0.558
| 0.558
|-
|-
|9
| 9
|Vicesimotertial neutral second
| Vicesimotertial neutral second
|156.522
| 156.522
|[[23/21]]
| [[23/21]]
| -0.972
| −0.972
|-
|-
|10
| 10
| Undevicesimal large neutral second, undevicesimal whole tone
| Undevicesimal large neutral second, undevicesimal whole tone
|173.913
| 173.913
|[[21/19]]
| [[21/19]]
|0.645
| 0.645
|-
|-
|11
| 11
|Quasi-meantone
| Quasi-meantone
|191.304
| 191.304
|[[19/17]]
| [[19/17]]
| -1.253
| −1.253
|-
|-
|12
| 12
|Whole tone
| Whole tone
|208.696
| 208.696
|[[9/8]]
| [[9/8]]
|4.786
| 4.786
|-
|-
|13
| 13
|Septimal whole tone
| Septimal whole tone
|226.087
| 226.087
|[[8/7]]
| [[8/7]]
| -5.087
| −5.087
|-
|-
|14
| 14
|Vicesimotertial semifourth
| Vicesimotertial semifourth
|243.478
| 243.478
|[[23/20]]
| [[23/20]]
|1.518
| 1.518
|-
|-
|15
| 15
|Subminor third, undetricesimal subminor third
| Subminor third, undetricesimal subminor third
|260.870
| 260.870
|[[7/6]], [[29/25]]
| [[7/6]], [[29/25]]
| -6.001, 3.920
| −6.001, 3.920
|-
|-
|16
| 16
| Vicesimotertial subminor third
| Vicesimotertial subminor third
|278.261
| 278.261
|[[27/23]]
| [[27/23]]
|0.670
| 0.670
|-
|-
|17
| 17
|Pythagorean minor third
| Pythagorean minor third
|295.652
| 295.652
|[[32/27]]
| [[32/27]]
|1.517
| 1.517
|-
|-
|18
| 18
|Classic minor third
| Classic minor third
|313.043
| 313.043
|[[6/5]]
| [[6/5]]
| -2.598
| −2.598
|-
|-
|19
| 19
|Vicesimotertial supraminor third
| Vicesimotertial supraminor third
|330.435
| 330.435
|[[23/19]]
| [[23/19]]
| -0.327
| −0.327
|-
|-
|20
| 20
|Undecimal neutral third
| Undecimal neutral third
|347.826
| 347.826
|[[11/9]]
| [[11/9]]
|0.418
| 0.418
|-
|-
|21
| 21
|Septendecimal submajor third
| Septendecimal submajor third
|365.217
| 365.217
|[[21/17]]
| [[21/17]]
| -0.608
| −0.608
|-
|-
|22
| 22
|Classic major third
| Classic major third
|382.609
| 382.609
|[[5/4]]
| [[5/4]]
| -3.705
| −3.705
|-
|-
|23
| 23
| Undetricesimal major third, Septendecimal major third
| Undetricesimal major third, Septendecimal major third
|400.000
| 400.000
|[[29/23]], [[34/27]]
| [[29/23]], [[34/27]]
| -1.303, 0.910
| −1.303, 0.910
|-
|-
|24
| 24
|Undecimal major third
| Undecimal major third
|417.391
| 417.391
|[[14/11]]
| [[14/11]]
| -0.117
| −0.117
|-
|-
|25
| 25
|Supermajor third
| Supermajor third
|434.783
| 434.783
|[[9/7]]
| [[9/7]]
| -0.301
| −0.301
|-
|-
|26
| 26
|Barbados third
| Barbados third
|452.174
| 452.174
|[[13/10]]
| [[13/10]]
| -2.040
| −2.040
|-
|-
|27
| 27
|Septimal sub-fourth
| Septimal sub-fourth
|469.565
| 469.565
|[[21/16]]
| [[21/16]]
| -1.216
| −1.216
|-
|-
|28
| 28
|_____
| _____
|486.957
| 486.957
|[[53/40]]
| [[53/40]]
| -0.234
| −0.234
|-
|-
|29
| 29
|Just perfect fourth
| Just perfect fourth
|504.348
| 504.348
|[[4/3]]
| [[4/3]]
|6.303
| 6.303
|-
|-
|30
| 30
|Vicesimotertial acute fourth
| Vicesimotertial acute fourth
|521.739
| 521.739
|[[23/17]]
| [[23/17]]
| -1.580
| −1.580
|-
|-
|31
| 31
|Undecimal augmented fourth
| Undecimal augmented fourth
|539.130
| 539.130
|[[15/11]]
| [[15/11]]
|2.180
| 2.180
|-
|-
|32
| 32
|Undecimal superfourth, undetricesimal superfourth
| Undecimal superfourth, undetricesimal 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
| 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, undevicesimal tritone
| High tritone, undevicesimal 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
| Undetricesimal tritone
| Undetricesimal tritone
|643.478
| 643.478
|[[29/20]]
| [[29/20]]
|0.215
| 0.215
|-
|-
|38
| 38
| Undevicesimal diminished fifth, undecimal diminished fifth
| Undevicesimal diminished fifth, 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, _____
| 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
| Just perfect fifth
|695.652
| 695.652
|[[3/2]]
| [[3/2]]
| -6.303
| −6.303
|-
|-
|41
| 41
|_____
| _____
|713.043
| 713.043
|[[80/53]]
| [[80/53]]
|0.234
| 0.234
|-
|-
|42
| 42
|Super-fifth, undetricesimal super-fifth
| Super-fifth, undetricesimal 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
| Septendecimal subminor sixth
|747.826
| 747.826
|[[17/11]]
| [[17/11]]
| -5.811
| −5.811
|-
|-
|44
| 44
|Subminor sixth
| Subminor sixth
|765.217
| 765.217
|[[14/9]]
| [[14/9]]
|0.301
| 0.301
|-
|-
|45
| 45
|Undecimal minor sixth
| Undecimal minor sixth
|782.609
| 782.609
|[[11/7]]
| [[11/7]]
|0.117
| 0.117
|-
|-
|46
| 46
| Septendecimal subminor sixth
| Septendecimal subminor sixth
|800.000
| 800.000
|[[27/17]]
| [[27/17]]
| -0.910
| −0.910
|-
|-
|47
| 47
|Classic minor sixth
| Classic minor sixth
|817.391
| 817.391
|[[8/5]]
| [[8/5]]
|3.705
| 3.705
|-
|-
|48
| 48
|Septendecimal supraminor sixth
| Septendecimal supraminor sixth
|834.783
| 834.783
|[[34/21]]
| [[34/21]]
|0.608
| 0.608
|-
|-
|49
| 49
|Undecimal neutral sixth
| Undecimal neutral sixth
|852.174
| 852.174
|[[18/11]]
| [[18/11]]
| -0.418
| −0.418
|-
|-
|50
| 50
|Vicesimotertial submajor sixth
| Vicesimotertial submajor sixth
|869.565
| 869.565
|[[38/23]]
| [[38/23]]
|0.327
| 0.327
|-
|-
|51
| 51
|Classic major sixth
| Classic major sixth
|886.957
| 886.957
|[[5/3]]
| [[5/3]]
|2.598
| 2.598
|-
|-
|52
| 52
|Pythagorean major sixth
| Pythagorean major sixth
|904.348
| 904.348
|[[27/16]]
| [[27/16]]
| -1.517
| −1.517
|-
|-
|53
| 53
|Septendecimal major sixth, undetricesimal major sixth
| Septendecimal major sixth, undetricesimal major sixth
|921.739
| 921.739
|[[17/10]], [[29/17]]
| [[17/10]], [[29/17]]
|3.097, -2.883
| 3.097, −2.883
|-
|-
|54
| 54
|Supermajor sixth, undetricesimal supermajor sixth
| Supermajor sixth, undetricesimal supermajor sixth
|939.130
| 939.130
|[[12/7]], [[50/29]]
| [[12/7]], [[50/29]]
|6.001, -3.920
| 6.001, −3.920
|-
|-
|55
| 55
|Vicesimotertial supermajor sixth
| Vicesimotertial supermajor sixth
|956.522
| 956.522
|[[40/23]]
| [[40/23]]
| -1.518
| −1.518
|-
|-
|56
| 56
|Harmonic seventh
| Harmonic seventh
|973.913
| 973.913
|[[7/4]]
| [[7/4]]
|5.087
| 5.087
|-
|-
|57
| 57
|Pythagorean minor seventh
| Pythagorean minor seventh
|991.304
| 991.304
|[[16/9]]
| [[16/9]]
| -4.786
| −4.786
|-
|-
|58
| 58
|Quasi-meantone minor seventh
| Quasi-meantone minor seventh
|1008.696
| 1008.696
|[[34/19]]
| [[34/19]]
|1.253
| 1.253
|-
|-
|59
| 59
|Minor neutral undevicesimal seventh
| Minor neutral undevicesimal seventh
|1026.087
| 1026.087
|[[38/21]]
| [[38/21]]
| -0.645
| −0.645
|-
|-
|60
| 60
|Vicesimotertial neutral seventh
| Vicesimotertial neutral seventh
|1043.478
| 1043.478
|[[42/23]]
| [[42/23]]
|0.972
| 0.972
|-
|-
|61
| 61
|Tridecimal neutral seventh
| Tridecimal neutral seventh
|1060.870
| 1060.870
|[[24/13]]
| [[24/13]]
| -0.558
| −0.558
|-
|-
|62
| 62
|Septimal diatonic major seventh
| Septimal diatonic major seventh
|1078.261
| 1078.261
|[[28/15]]
| [[28/15]]
| -2.296
| −2.296
|-
|-
|63
| 63
|Small septendecimal major seventh
| Small septendecimal major seventh
|1095.652
| 1095.652
|[[32/17]]
| [[32/17]]
|0.608
| 0.608
|-
|-
|64
| 64
|Small undevicesimal semitone
| Small undevicesimal semitone
|1113.043
| 1113.043
|[[20/19]]
| [[20/19]]
|1.844
| 1.844
|-
|-
|65
| 65
|_____
| _____
|1130.435
| 1130.435
|[[73/38]]
| [[73/38]]
|0.158
| 0.158
|-
|-
|66
| 66
| Septendecimal supermajor seventh
| Septendecimal supermajor seventh
|1147.826
| 1147.826
|[[33/17]]
| [[33/17]]
| -0.491
| −0.491
|-
|-
|67
| 67
|_____
| _____
|1165.217
| 1165.217
|[[49/25]]
| [[49/25]]
| -0.193
| −0.193
|-
|-
|68
| 68
|_____
| _____
|1182.609
| 1182.609
|[[99/50]]
| [[99/50]]
|0.008
| 0.008
|-
|-
|69
| 69
|Octave, 8
| Octave, 8
|1200.000
| 1200.000
|[[2/1]]
| [[2/1]]
|0.000
| 0.000
|}
|}
<nowiki>*</nowiki>some simpler ratios listed
<nowiki />* Some simpler ratios listed


== Notation ==
== Notation ==
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals, 69edo can also be notated using [[ups and downs notation]] along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:
{{Sharpness-sharp4}}
Here, a sharp raises by four steps, and a flat lowers by four steps, so arrows can be used to fill in the gap.


===Sagittal notation===
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].
This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].
====Evo flavor====


==== Evo flavor ====
<imagemap>
<imagemap>
File:69-EDO_Evo_Sagittal.svg
File:69-EDO_Evo_Sagittal.svg
Line 464: Line 467:
</imagemap>
</imagemap>


====Revo flavor====
==== Revo flavor ====
 
<imagemap>
<imagemap>
File:69-EDO_Revo_Sagittal.svg
File:69-EDO_Revo_Sagittal.svg
Line 476: Line 478:
</imagemap>
</imagemap>


====Evo-SZ flavor====
==== Evo-SZ flavor ====
 
<imagemap>
<imagemap>
File:69-EDO_Evo-SZ_Sagittal.svg
File:69-EDO_Evo-SZ_Sagittal.svg
Line 492: Line 493:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
Retrieved from "https://en.xen.wiki/w/69edo"