36edo: Difference between revisions

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== Notation ==

=== ~~Colored notes~~ ===

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

~~One way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, {{colored note|blue|A}} is 33{{frac|3}}{{c}} below {{colored note|A}}~~ and {{~~colored note|red~~|A}} ~~is 33{{frac|3}} cents above {{colored note|A}}. Or~~, ~~the colors could~~ be ~~written out (red A, blue C♯, etc.) or abbreviated as rA, bC♯, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo)~~.

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

If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.

=== ~~Ups~~ and downs notation ===

=== Kite's ups and downs notation ===

Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.

~~Alternatively~~, ~~one can~~ use ~~sharps~~ and ~~flats~~ with ~~arrows, borrowed from extended~~ [[~~Helmholtz–Ellis~~ notation]]:

=== Colored notes ===

~~{{Sharpness-sharp3|36}}~~

One way of notating 36edo (at least for people who are not colorblind) is to use colors. For example, {{colored note|blue|A}} is 33{{frac|3}}{{c}} below {{colored note|A}} and {{colored note|red|A}} is 33{{frac|3}} cents above {{colored note|A}}. Or, the colors could be written out (red A, blue C♯, etc.) or abbreviated as rA, bC♯, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo).

~~If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows~~.

=== Sagittal notation ===

This notation uses the same sagittal sequence as [[43edo#Sagittal notation|~~43-EDO~~]], is a subset of the notation for [[72edo#Sagittal notation|~~72-EDO~~]], and is a superset of the notations for ~~EDOs~~ [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], and [[6edo#Sagittal notation|6]].

This notation uses the same sagittal sequence as [[43edo #Sagittal notation|43edo]], is a subset of the notation for [[72edo #Sagittal notation|72edo]], and is a superset of the notations for edos [[18edo #Sagittal notation|18]], [[12edo #Sagittal notation|12]], and [[6edo #Sagittal notation|6]].

==== Evo flavor ====

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For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250{{c}} (halfway between a tone and a third) and 450{{c}} (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 33{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250{{c}} (halfway between a tone and a third) and 450{{c}} (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

36edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a very soft [[1L 4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

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== Scales ==

''See also: [[List of MOS scales in 36edo]]''

{{Idiosyncratic terms|[[Budjarn Lambeth]] coined and is so far the only known user of the "N&M-tone scale"-type names, the names of the catnip subsets, and the name "baobab".}}

; [[Polymicrotonal]] scales

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* Niner[18] MOS: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 (''9/18 keys have a 3/2, 0/18 keys have both a 4/3 and a 3/2'')

* Niner[18] [[modmos]]: 1 1 3 1 3 5 1 1 1 3 1 3 1 1 3 1 3 3 (''11/18 keys have a 3/2, 6/18 keys have both a 4/3 and a 3/2'')

* Baobab: 9 6 6 6 4 5 (''approximated from [[30afdo]]'')

== Tuning by ear ==

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36edo can also be played using three instruments tuned to 12edo with different root notes (that is, a sixth-tone apart).

At least one 36edo harmonium has been built (the "sixth-tone harmonium" used by 20th century Czech microtonal composer {{W|Alois Habá}}), having effectively two quasi-Halberstadt keyboards (but with the bottom one duplicated at the top), each having 18 notes per octave organized in 6 groups of 2 natural-position notes and 1 flat/sharp-position note — see the Music section below for a video demonstration.

== Music ==

=== 20th century ===

; [[Alois Hába]]

* ''Six compositions for sixth-tone harmonium, op. 37'' (1928)

** [https://www.youtube.com/watch?v=UJwLdLMvdu8 String quartet version] — performed by Stamitz Quartet (unknown recording date) (currently the video title erroneously says this is for sixteenth-tone harmonium)

** [https://www.youtube.com/watch?v=-Ktbo7EfhEA Sixth-tone harmonium version] — performed by Miroslav Beinhauer ([https://miroslavbeinhauer.com/?page_id=139&lang=en site]) (2019, with much of the video showing keyboard views of the sixth-tone harmonium being played)

; {{W|Henri Pousseur}}

* [https://www.youtube.com/watch?v=tiWbbbU6ZPw ''Prospection''] (1952-1953) – for three pianos detuned by a sixth-tone from each other

=== 21st century ===

; [[Ambient Esoterica]]

* [https://www.youtube.com/watch?v=nadDPD1m-1Y ''A Trip to PRS with a Circle of Augmented 5ths''] (2024)

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* [https://www.youtube.com/shorts/3la1W-_-ceA ''36edo cowbell''] (2025)

* [https://www.youtube.com/shorts/MjUdMnUA-2k ''18 tone unequal improv''] (2026) (this is tuned as two rings of [[9edo]] offset by 35{{c}}, to make a good approximation of [[3/2]] available, for a tuning that is an 18 note subset of a well-tempered derivative of 36edo)

* [https://www.youtube.com/shorts/KCtEYSkEK8U ''36edo improv''] (2026)

; [[E8 Heterotic]]

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; [[Joseph Monzo]]

* [https://m.youtube.com/watch?v=7t79lBI_4_s ''African Rhythms in 36-ET''] (2005)

; [[Norokusi]]

* [https://www.youtube.com/watch?v=JPpjYzddGSg&t=495s ''Symphony for String Orchestra''] (2024)

** [https://www.youtube.com/watch?v=l5SQOI1kTHc ''Arranged for orchestra''] (2026)

* [https://www.youtube.com/watch?v=_DQKj0afWwE ''Hiroki Naito - Symphony: Moratorium for Large Orchestra in a 36EDO Subset''] (2025 - 2026) (also has a [[24edo]] vibraphone part)

* [https://www.youtube.com/watch?v=Sno4HVCegF8 ''Septet in a 36-EDO Subset''] (2026)

** [https://www.youtube.com/watch?v=dZgwqWYzGNc ''Pieces for Orchestra nr.103b in a 36-EDO Subset''] (this is the orchestral arrangement of the Septet)

; [[NullPointerException Music]]

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; [[Chris Orphal]]

* [https://www.youtube.com/watch?v=qbOOzC4360M ''Vademecum - Vadetecum (36-EDO) - Perf. New Music New Mexico''] (2023) (Saxophone: Edwin Anthony; Horn: Samuel Lutz; Trumpet: Doug Falk; Guitar: Carlos Arellano; Guitar tuned -31¢: Chris Orphal; Piano: Axel Retif)

~~; {{W|Henri Pousseur}}~~

* [https://www.youtube.com/watch?v=tiWbbbU6ZPw ''Prospection''] (1952-1953) – for three pianos detuned by a sixth-tone from each other

; [[Chris Vaisvil]]

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; [[Stephen Weigel]]

* [https://soundcloud.com/overtoneshock/exponentially-more-lost-and-forgetful-36-edo ''Exponentially More Lost and Forgetful''] (2017) played by flautists Orlando Cela and Wei Zhao

; [[Wusuinianmai Wanjunqushou]] (吾虽年迈万军取shou)

* ''[https://www.bilibili.com/video/BV1874P6hEmn G is the seventh letter]'' (2026)

[[Category:Listen]]

@@ Line 380: / Line 380: @@
 == Notation ==
-=== Colored notes ===
+=== Stein–Zimmermann–Gould notation ===
-One way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, {{colored note|blue|A}} is 33{{frac|3}}{{c}} below {{colored note|A}} and {{colored note|red|A}} is 33{{frac|3}} cents above {{colored note|A}}. Or, the colors could be written out (red A, blue C♯, etc.) or abbreviated as rA, bC♯, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo).
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
+{{Sharpness-sharp3-szg|36}}
+If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.
-=== Ups and downs notation ===
+=== Kite's ups and downs notation ===
 Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
 {{Ups and downs sharpness}}
-Alternatively, one can use sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+=== Colored notes ===
-{{Sharpness-sharp3|36}}
+One way of notating 36edo (at least for people who are not colorblind) is to use colors. For example, {{colored note|blue|A}} is 33{{frac|3}}{{c}} below {{colored note|A}} and {{colored note|red|A}} is 33{{frac|3}} cents above {{colored note|A}}. Or, the colors could be written out (red A, blue C♯, etc.) or abbreviated as rA, bC♯, etc. This use of red and blue is consistent with [[Kite's_color_notation|color notation]] (ru and zo).
-If the arrows are taken to have their own layer of enharmonic spellings, then in some cases notes may be best denoted using double arrows.
 === Sagittal notation ===
-This notation uses the same sagittal sequence as [[43edo#Sagittal notation|43-EDO]], is a subset of the notation for [[72edo#Sagittal notation|72-EDO]], and is a superset of the notations for EDOs [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], and [[6edo#Sagittal notation|6]].
+This notation uses the same sagittal sequence as [[43edo #Sagittal notation|43edo]], is a subset of the notation for [[72edo #Sagittal notation|72edo]], and is a superset of the notations for edos [[18edo #Sagittal notation|18]], [[12edo #Sagittal notation|12]], and [[6edo #Sagittal notation|6]].
 ==== Evo flavor ====
@@ Line 418: / Line 419: @@
 For people accustomed to 12edo, 36edo is one of the easiest (if not ''the'' easiest) higher edo to become accustomed to. This is because one way to envision it is as an extended 12edo to which [https://en.wikipedia.org/wiki/Blue_note blue notes] (which are a sixth-tone lower than normal) and "red notes" (a sixth-tone higher) have been added.
-The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250{{c}} (halfway between a tone and a third) and 450{{c}} (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 33{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.
+The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250{{c}} (halfway between a tone and a third) and 450{{c}} (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 333{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.
 edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a very soft [[1L&nbsp;4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.
@@ Line 1,061: / Line 1,062: @@
 == Scales ==
 ''See also: [[List of MOS scales in 36edo]]''
-{{Idiosyncratic terms}}
+{{Idiosyncratic terms|[[Budjarn Lambeth]] coined and is so far the only known user of the "N&M-tone scale"-type names, the names of the catnip subsets, and the name "baobab".}}
 ; [[Polymicrotonal]] scales
@@ Line 1,147: / Line 1,148: @@
 * Niner[18] MOS: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 (''9/18 keys have a 3/2, 0/18 keys have both a 4/3 and a 3/2'')
 * Niner[18] [[modmos]]: 1 1 3 1 3 5 1 1 1 3 1 3 1 1 3 1 3 3  (''11/18 keys have a 3/2, 6/18 keys have both a 4/3 and a 3/2'')
+* Baobab: 9 6 6 6 4 5 (''approximated from [[30afdo]]'')
 == Tuning by ear ==
@@ Line 1,157: / Line 1,159: @@
 edo can also be played using three instruments tuned to 12edo with different root notes (that is, a sixth-tone apart).
+At least one 36edo harmonium has been built (the "sixth-tone harmonium" used by 20th century Czech microtonal composer {{W|Alois Habá}}), having effectively two quasi-Halberstadt keyboards (but with the bottom one duplicated at the top), each having 18 notes per octave organized in 6 groups of 2 natural-position notes and 1 flat/sharp-position note — see the Music section below for a video demonstration.
 == Music ==
+=== 20th century ===
+; [[Alois Hába]]
+* ''Six compositions for sixth-tone harmonium, op. 37'' (1928)
+** [https://www.youtube.com/watch?v=UJwLdLMvdu8 String quartet version] — performed by Stamitz Quartet (unknown recording date) (currently the video title erroneously says this is for sixteenth-tone harmonium)
+** [https://www.youtube.com/watch?v=-Ktbo7EfhEA Sixth-tone harmonium version] — performed by Miroslav Beinhauer ([https://miroslavbeinhauer.com/?page_id=139&lang=en site]) (2019, with much of the video showing keyboard views of the sixth-tone harmonium being played)
+; {{W|Henri Pousseur}}
+* [https://www.youtube.com/watch?v=tiWbbbU6ZPw ''Prospection''] (1952-1953) – for three pianos detuned by a sixth-tone from each other
+=== 21st century ===
 ; [[Ambient Esoterica]]
 * [https://www.youtube.com/watch?v=nadDPD1m-1Y ''A Trip to PRS with a Circle of Augmented 5ths''] (2024)
@@ Line 1,176: / Line 1,190: @@
 * [https://www.youtube.com/shorts/3la1W-_-ceA ''36edo cowbell''] (2025)
 * [https://www.youtube.com/shorts/MjUdMnUA-2k ''18 tone unequal improv''] (2026) (this is tuned as two rings of [[9edo]] offset by 35{{c}}, to make a good approximation of [[3/2]] available, for a tuning that is an 18 note subset of a well-tempered derivative of 36edo)
+* [https://www.youtube.com/shorts/KCtEYSkEK8U ''36edo improv''] (2026)
 ; [[E8 Heterotic]]
@@ Line 1,197: / Line 1,212: @@
 ; [[Joseph Monzo]]
 * [https://m.youtube.com/watch?v=7t79lBI_4_s ''African Rhythms in 36-ET''] (2005)
+; [[Norokusi]]
+* [https://www.youtube.com/watch?v=JPpjYzddGSg&t=495s ''Symphony for String Orchestra''] (2024)
+** [https://www.youtube.com/watch?v=l5SQOI1kTHc ''Arranged for orchestra''] (2026)
+* [https://www.youtube.com/watch?v=_DQKj0afWwE ''Hiroki Naito - Symphony: Moratorium for Large Orchestra in a 36EDO Subset''] (2025 - 2026) (also has a [[24edo]] vibraphone part)
+* [https://www.youtube.com/watch?v=Sno4HVCegF8 ''Septet in a 36-EDO Subset''] (2026)
+** [https://www.youtube.com/watch?v=dZgwqWYzGNc ''Pieces for Orchestra nr.103b in a 36-EDO Subset''] (this is the orchestral arrangement of the Septet)
 ; [[NullPointerException Music]]
@@ Line 1,206: / Line 1,228: @@
 ; [[Chris Orphal]]
 * [https://www.youtube.com/watch?v=qbOOzC4360M ''Vademecum - Vadetecum (36-EDO) - Perf. New Music New Mexico''] (2023) (Saxophone: Edwin Anthony; Horn: Samuel Lutz; Trumpet: Doug Falk; Guitar: Carlos Arellano; Guitar tuned -31¢: Chris Orphal; Piano: Axel Retif)
-; {{W|Henri Pousseur}}
-* [https://www.youtube.com/watch?v=tiWbbbU6ZPw ''Prospection''] (1952-1953) – for three pianos detuned by a sixth-tone from each other
 ; [[Chris Vaisvil]]
@@ Line 1,218: / Line 1,237: @@
 ; [[Stephen Weigel]]
 * [https://soundcloud.com/overtoneshock/exponentially-more-lost-and-forgetful-36-edo ''Exponentially More Lost and Forgetful''] (2017) played by flautists Orlando Cela and Wei Zhao
+; [[Wusuinianmai Wanjunqushou]] (吾虽年迈万军取shou)
+* ''[https://www.bilibili.com/video/BV1874P6hEmn G is the seventh letter]'' (2026)
 [[Category:Listen]]

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