69edo: Difference between revisions

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**Imported revision 629810669 - Original comment: **
Music: Add Bryan Deister's ''Compass - Mili (microtonal cover in 69edo)'' (2025)
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:kai.lugheidh|kai.lugheidh]] and made on <tt>2018-05-15 17:21:24 UTC</tt>.<br>
== Theory ==
: The original revision id was <tt>629810669</tt>.<br>
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.
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 69 equal division, 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]]". As a meantone system, it is on the flat side, with a fifth of 695.652 cents. It is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to "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&amp;69 temperament tempering out 3125/3087 along with 81/80. In the 11-limit it tempers out 99/98, and supports the 31&amp;69 variant of mohajira, identical to the standard 11-limit mohajira in 31 but not in 69.</pre></div>
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.
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;69edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 69 equal division, which divides the octave into 69 equal parts of 17.391 cents each, has been called  &amp;quot;the love-child of &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt; and &lt;a class="wiki_link" href="/quarter-comma%20meantone"&gt;quarter-comma meantone&lt;/a&gt;&amp;quot;. As a meantone system, it is on the flat side, with a fifth of 695.652 cents. It is closer to 2/7-comma meantone than 1/4-comma, and is nearly identical to &amp;quot;Synch-Meantone&amp;quot;, 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.&lt;br /&gt;
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 &amp; 69}} variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69.
&lt;br /&gt;
 
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&amp;amp;69 temperament tempering out 3125/3087 along with 81/80. In the 11-limit it tempers out 99/98, and supports the 31&amp;amp;69 variant of mohajira, identical to the standard 11-limit mohajira in 31 but not in 69.&lt;/body&gt;&lt;/html&gt;</pre></div>
The [[concoctic scale]] for 69edo is 22\69, and the corresponding rank two temperament is {{nowrap|22 &amp; 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
|-
| 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
|}
<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"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -109 69 }}
| {{mapping| 69 109 }}
| +1.99
| 1.99
| 11.43
|-
| 2.3.5
| 81/80, {{monzo| -41 1 17 }}
| {{mapping| 69 109 160 }}
| +1.86
| 1.64
| 9.40
|-
| 2.3.5.7
| 81/80, 126/125, 4117715/3981312
| {{mapping| 69 109 160 193 }} (69d)
| +2.49
| 1.79
| 10.28
|-
| 2.3.5.7
| 81/80, 3125/3087, 6144/6125
| {{mapping| 69 109 160 194 }} (69)
| +0.94
| 2.13
| 12.23
|}
 
=== Rank 2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator
! Temperaments
|-
| 1
| 2\69
| [[Gammy]] (69de)
|-
|1
|5\69
|[[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>
|-
| 1
| 19\69
| [[Rarity]]
|-
| 1
| 20\69
| [[Mohaha]] (69e)
|-
| 1
| 22\69
| [[Caleb]] (69)<br>[[marveltri]] (69)
|-
| 1
| 29\69
| [[Meantone]] (69d)
|-
| 3
| 5\69
| [[Augmented family #Ogene|Ogene]] (69bceef)
|-
| 3
| 6\69
| [[August]] (7-limit, 69cdd)<br>[[Lithium]] (69)
|-
| 3
| 9\69
| [[Nessafof]] (69e)
|}
<references group="note" />
 
== 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 }}