69edo: Difference between revisions
Wikispaces>kai.lugheidh **Imported revision 629810669 - Original comment: ** |
→Music: Add Bryan Deister's ''Compass - Mili (microtonal cover in 69edo)'' (2025) |
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{{Infobox ET}} | |||
{{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 | 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 {{nowrap|31 & 69}} variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69. | |||
In the | 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 | |||
|- | |||
| 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 — 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 }} |