67edo: Difference between revisions
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{{Infobox ET}}{{ | {{Infobox ET}} | ||
{{ED intro}} | |||
== Theory == | == Theory == | ||
67edo tempers out 81/80, [[support | 67edo [[tempering out|tempers out]] [[81/80]], [[support]]ing [[meantone]], with a tuning which is slightly sharp of [[1/6-comma meantone|1/6-comma]] (the tuning favored by {{w|Wolfgang Amadeus Mozart|Mozart}} and contemporaries, though they suggested the flatter and composite [[55edo]] as an approximation). It is indistinguishable from {{nowrap|{{frac|4|25}} {{=}} 0.16-comma}} meantone. In the 7-limit the [[patent val]] tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[mothra]]. In the 11-limit it tempers out [[176/175]] and [[540/539]], supporting [[mosura]], an alternative 11-limit mothra. In the 13-limit it tempers out [[144/143]] and [[196/195]], supporting 13-limit mosura. It tempers out the [[orgonisma]], and on the 2.7.11 subgroup it supports the [[orgone]] temperament. | ||
It is a promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the second edo after [[26edo]] to have both meantone and an orgone temperament. It has relatively good approximations of the [[3/1|3rd]], [[7/1|7th]], [[11/1|11th]], [[13/1|13th]], [[15/1|15th]], [[17/1|17th]] [[harmonic]]s, although the [[5/1|5th]], [[9/1|9th]], and [[19/1|19th]] as well as certain higher ones are workable as well. {{nowrap|33 + 34}} can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 [[subgroup]]. | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|67|columns=13}} | {{Harmonics in equal|67|columns=13}} | ||
== Intervals== | === Subsets and supersets === | ||
67edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]]. | |||
== Intervals == | |||
{{Interval table}} | {{Interval table}} | ||
== Notation == | |||
=== Stein–Zimmermann–Gould notation === | |||
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows: | |||
{{Sharpness-sharp5-szg}} | |||
=== Kite's ups and downs notation === | |||
67edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down). | |||
{{Sharpness-sharp5a}} | |||
=== Sagittal notation === | |||
==== Evo flavor ==== | |||
<imagemap> | |||
File:67-EDO_Evo_Sagittal.svg | |||
desc none | |||
rect 80 0 300 50 [[Sagittal_notation]] | |||
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation] | |||
rect 20 80 160 106 [[896/891]] | |||
rect 160 80 280 106 [[36/35]] | |||
rect 280 80 440 106 [[1053/1024]] | |||
default [[File:67-EDO_Evo_Sagittal.svg]] | |||
</imagemap> | |||
==== Revo flavor ==== | |||
<imagemap> | |||
File:67-EDO_Revo_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 80 160 106 [[896/891]] | |||
rect 160 80 280 106 [[36/35]] | |||
rect 280 80 440 106 [[1053/1024]] | |||
default [[File:67-EDO_Revo_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. | |||
== Scales == | == Scales == | ||
{{Idiosyncratic terms}} | |||
=== | === Mos scales === | ||
* Bustling Docks (original/default tuning): 14 14 11 14 14 | * Meantone[5]: 11 11 17 11 17 | ||
* Meantone[7]: 11 11 6 11 11 11 6 | |||
* Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14 | |||
* Barbados[9]: 11 3 11 3 11 3 11 3 11 | |||
=== Modmos scales === | |||
* Cavernous (original/default tuning): 14 14 11 21 7 | * Cavernous (original/default tuning): 14 14 11 21 7 | ||
* Formicarium (original/default tuning): 14 7 18 14 14 | * Formicarium (original/default tuning): 14 7 18 14 14 | ||
* Negri Blues (original/default tuning): 14 14 3 8 14 14 | * Negri Blues (original/default tuning): 14 14 3 8 14 14 | ||
| Line 25: | Line 68: | ||
* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7 | * Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7 | ||
* Understory (original/default tuning): 14 7 18 7 21 | * Understory (original/default tuning): 14 7 18 7 21 | ||
* Meantone Ionian Pentatonic: 22 6 11 22 6 | |||
* Meantone Minor Melodic: 11 6 11 11 11 11 6 | |||
* Meantone Minor Harmonic: 11 6 11 11 6 16 6 | |||
* Meantone Minor Hexatonic: 11 6 11 11 17 11 | |||
* Meantone Dorian Harmonic: 11 6 16 6 11 6 11 | |||
* Meantone Mixolydian Pentatonic: 22 6 11 17 11 | |||
* Meantone Phrygian Dominant: 6 16 6 11 6 11 11 | |||
* Meantone Phrygian Dominant Hexatonic: 6 16 6 11 6 22 | |||
* Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22 | |||
* Meantone Phrygian Pentatonic: 6 11 22 6 22 | |||
* Meantone Double Harmonic: 6 16 6 11 6 16 6 | |||
=== Blues scales === | |||
=== | * [[Lost spirit]] (approximated from [[31edo]]): 17 11 6 5 13 4 11 | ||
* [[Blackened skies]] (approximated from [[72edo]]): 18 10 5 6 5 18 5 | |||
*[[Lost spirit]] (approximated from [[31edo]]): 17 11 6 5 13 4 11 | |||
*[[Blackened skies]] (approximated from [[72edo]]): 18 10 5 6 5 18 5 | |||
* Blues Aeolian Hexatonic: 17 11 6 5 6 22 | * Blues Aeolian Hexatonic: 17 11 6 5 6 22 | ||
* Blues Aeolian Pentatonic I: 17 11 11 6 22 | * Blues Aeolian Pentatonic I: 17 11 11 6 22 | ||
| Line 76: | Line 98: | ||
* Blues Pentachordal: 11 6 11 5 6 28 | * Blues Pentachordal: 11 6 11 5 6 28 | ||
* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5 | * Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5 | ||
* Akebono I: 11 6 11 11 17 | * Akebono I: 11 6 11 11 17 | ||
* Augmented: 17 6 16 6 16 6 | * Augmented: 17 6 16 6 16 6 | ||
* Dominant Pentatonic: 11 11 17 17 11 | * Dominant Pentatonic: 11 11 17 17 11 | ||
* Hirajoshi: 11 6 12 6 22 | * Hirajoshi: 11 6 12 6 22 | ||
* Javanese Pentachordal: 6 11 17 4 29 | * Javanese Pentachordal: 6 11 17 4 29 | ||
=== Others === | |||
=== | * Approximation of ''[[Pelog]] lima'': 6 10 22 7 22 | ||
*Approximation of [[Pelog]] lima: 6 10 22 7 22 | * [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] ''(octave-reduced: 9 6 23 16 13)'' | ||
* Arcade (approximated from [[32afdo]]): 22 4 13 15 13 | * Arcade (approximated from [[32afdo]]): 22 4 13 15 13 | ||
* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11 | * Cosmic (approximated from [[32afdo]]): 29 10 6 11 11 | ||
* Mechanical (approximated from [[16afdo]] | * Mechanical (approximated from [[16afdo]]): 17 5 17 15 13 | ||
* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6 | * Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6 | ||
* Springwater (approximated from [[8afdo]]): 11 11 17 15 13 | * Springwater (approximated from [[8afdo]]): 11 11 17 15 13 | ||
| Line 102: | Line 119: | ||
* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13 | * Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13 | ||
* Bubblegum/Cola (approximated from [[60afdo]]/[[99afdo]]): 14 13 13 13 14 | * Bubblegum/Cola (approximated from [[60afdo]]/[[99afdo]]): 14 13 13 13 14 | ||
* | * Tropicalpunch/Whitechocolate (approximated from [[62afdo]]/[[90afdo]]): 13 14 13 14 13 | ||
* Lemonade (approximated from [[79afdo]]): 14 13 13 14 13 | * Lemonade (approximated from [[79afdo]]): 14 13 13 14 13 | ||
* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11 | * Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11 | ||
* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 11 | * Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 11 | ||
* Liquorice (approximated from [[101afdo]]): 11 11 12 10 12 11 | * Liquorice (approximated from [[101afdo]]): 11 11 12 10 12 11 | ||
* | * Fishcracker (approximated from [[80afdo]]): 9 11 9 9 10 9 10 | ||
== Instruments == | |||
* [[Lumatone mapping for 67edo]] | |||
== Music == | |||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/uwxey9_jINA ''microtonal improvisation in 67edo''] (2025) | |||
* [https://www.youtube.com/shorts/L6BXGZyvK8Y ''67edo prelude''] (2025) | |||
* [https://www.youtube.com/shorts/za_Ov95HbjQ ''improv in 67edo''] (2025) | |||
; [[Delta Quartz]] | |||
* [https://youtu.be/WOguarC1lEI ''Making microtonality accessible - "Keep It Tight"''] (2026) (also has a small amount of 24edo) | |||
; [[Dolores Catherino]] | |||
* [https://youtu.be/AYHpxeM6o_g ''Moments of Unexpected Beauty''] (2026) | |||
; [[Peter Kosmorsky]] | |||
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone] (2011) {{dead link}} | |||
; [[Budjarn Lambeth]] | |||
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] (2024) | |||
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] | |||
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | [[Category:Equal divisions of the octave|##]] <!-- 2-digit number --> | ||
[[Category:Meantone]] | [[Category:Meantone]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||