67edo: Difference between revisions

(30 intermediate revisions by 13 users not shown)

Line 1:

== Theory ==

67edo tempers out 81/80, [[support~~|supporting~~]] meantone ~~temperament~~, with a tuning which is ~~approximately~~ [[1/~~6_syntonic_comma_meantone~~|1/6 comma]] (the tuning favored by Mozart and contemporaries)~~, or~~ 0.16 comma, meantone. In the 7-limit the patent val tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[~~Meantone_family|~~mothra ~~temperament~~]]. 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 [[~~Orgonia|~~orgone ~~temperament~~]].

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.

A promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the ~~first~~ edo to have both meantone ~~(26 could be called meantone, but it's more of a flattone)~~ and an orgone temperament. It has relatively good approximations of the 3rd, 7th, 11th, 13th, 15th, 17th ~~harmonics~~, although the 5th, 9th, and 19th as well as certain higher ones are workable as well. 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.~~

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]].

~~67edo is the 19th~~ [[~~prime EDO~~]].

=== Prime harmonics ===

== Intervals==

=== Subsets and supersets ===

67edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]].

== Intervals ==

==Scales==

== Notation ==

* [[5- to 10~~-tone scales~~ in 67edo]]

=== Stein–Zimmermann–Gould notation ===

[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:

=== 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).

=== Sagittal notation ===

==== Evo flavor ====

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 ====

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 ==

=== Mos scales ===

* 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

* Formicarium (original/default tuning): 14 7 18 14 14

* Negri Blues (original/default tuning): 14 14 3 8 14 14

* Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14

* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7

* 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

* Blues Aeolian Hexatonic: 17 11 6 5 6 22

* Blues Aeolian Pentatonic I: 17 11 11 6 22

* Blues Aeolian Pentatonic II: 17 22 6 11 11

* Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5

* Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6

* Blues Dorian Hexatonic: 17 11 11 11 6 11

* Blues Dorian Pentatonic: 17 22 11 6 11

* Blues Dorian Septatonic: 17 11 6 5 11 6 11

* Blues Harmonic Hexatonic: 11 6 11 11 22 6

* Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6

* Blues Leading: 17 11 6 5 17 6 5

* Blues Minor: 17 11 6 5 17 11

* Blues Minor Maj7: 17 11 6 5 22 6

* Blues Pentachordal: 11 6 11 5 6 28

* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5

* Akebono I: 11 6 11 11 17

* Augmented: 17 6 16 6 16 6

* Dominant Pentatonic: 11 11 17 17 11

* Hirajoshi: 11 6 12 6 22

* Javanese Pentachordal: 6 11 17 4 29

=== Others ===

* 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

* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11

* Mechanical (approximated from [[16afdo]]): 17 5 17 15 13

* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6

* Springwater (approximated from [[8afdo]]): 11 11 17 15 13

* Volcanic (approximated from [[16afdo]]): 6 16 17 15 13

* Deja Vu (approximated from [[101afdo]]): 18 21 6 12 10

* Freeway (approximated from [[6afdo]]): 15 12 11 11 9 8

* Mushroom (approximated from [[30afdo]]): 15 12 11 4 24

* Underpass (approximated from [[10afdo]]): 18 21 12 6 10

* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13

* 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

* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11

* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 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}}

~~==Music==~~

; [[Budjarn Lambeth]]

* [~~http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone],~~ [~~[Peter Kosmorsky~~]] ~~{{dead link}}~~

* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] (2024)

* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation]~~, [[Budjarn Lambeth]]~~

[[Category:Equal divisions of the octave|##]]

~~[[Category:Prime EDO]]~~

[[Category:Meantone]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{Infobox ET}}{{EDO intro|67}}
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-edo tempers out 81/80, [[support|supporting]] meantone temperament, with a tuning which is approximately [[1/6_syntonic_comma_meantone|1/6 comma]] (the tuning favored by Mozart and contemporaries), or 0.16 comma, meantone. In the 7-limit the patent val tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[Meantone_family|mothra temperament]]. 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 [[Orgonia|orgone temperament]].
+edo [[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.
-A promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the first edo to have both meantone (26 could be called meantone, but it's more of a flattone) and an orgone temperament. It has relatively good approximations of the 3rd, 7th, 11th, 13th, 15th, 17th harmonics, although the 5th, 9th, and 19th as well as certain higher ones are workable as well. 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.
+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]].
-edo is the 19th [[prime EDO]].
 === Prime harmonics ===
 {{Harmonics in equal|67|columns=13}}
-== Intervals==
+=== Subsets and supersets ===
+edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]].
+== Intervals ==
 {{Interval table}}
-==Scales==
+== Notation ==
-* [[5- to 10-tone scales in 67edo]]
+=== Stein–Zimmermann–Gould notation ===
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
+{{Sharpness-sharp5-szg}}
+=== Kite's ups and downs notation ===
+edo 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 ==
+{{Idiosyncratic terms}}
+=== Mos scales ===
+* 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
+* Formicarium (original/default tuning): 14 7 18 14 14
+* Negri Blues (original/default tuning): 14 14 3 8 14 14
+* Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
+* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
+* 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
+* Blues Aeolian Hexatonic: 17 11 6 5 6 22
+* Blues Aeolian Pentatonic I: 17 11 11 6 22
+* Blues Aeolian Pentatonic II: 17 22 6 11 11
+* Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
+* Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
+* Blues Dorian Hexatonic: 17 11 11 11 6 11
+* Blues Dorian Pentatonic: 17 22 11 6 11
+* Blues Dorian Septatonic: 17 11 6 5 11 6 11
+* Blues Harmonic Hexatonic: 11 6 11 11 22 6
+* Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
+* Blues Leading: 17 11 6 5 17 6 5
+* Blues Minor: 17 11 6 5 17 11
+* Blues Minor Maj7: 17 11 6 5 22 6
+* Blues Pentachordal: 11 6 11 5 6 28
+* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5
+* Akebono I: 11 6 11 11 17
+* Augmented: 17 6 16 6 16 6
+* Dominant Pentatonic: 11 11 17 17 11
+* Hirajoshi: 11 6 12 6 22
+* Javanese Pentachordal: 6 11 17 4 29
+=== Others ===
+* 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
+* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11
+* Mechanical (approximated from [[16afdo]]): 17 5 17 15 13
+* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6
+* Springwater (approximated from [[8afdo]]): 11 11 17 15 13
+* Volcanic (approximated from [[16afdo]]): 6 16 17 15 13
+* Deja Vu (approximated from [[101afdo]]): 18 21 6 12 10
+* Freeway (approximated from [[6afdo]]): 15 12 11 11 9 8
+* Mushroom (approximated from [[30afdo]]): 15 12 11 4 24
+* Underpass (approximated from [[10afdo]]): 18 21 12 6 10
+* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13
+* 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
+* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11
+* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 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}}
-==Music==
+; [[Budjarn Lambeth]]
-* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone], [[Peter Kosmorsky]] {{dead link}}
+* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation] (2024)
-* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation], [[Budjarn Lambeth]]
 [[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
-[[Category:Prime EDO]]
 [[Category:Meantone]]
+[[Category:Listen]]