36edo: Difference between revisions

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=== Mappings ===
=== Mappings ===
36edo's patent val, like 12, tempers out 81/80, 128/125, and 648/625 in the 5-limit. It departs from 12 in the 7-limit, tempering out 686/675 and 1029/1000, and as a no-fives temperament, 1029/1024 and 118098/117649. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out 56/55, 245/242, and 540/539, and is the [[optimal patent val]] for the rank four temperament tempering out 56/55, as well as the rank-3 temperament [[melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out 78/77 and 91/90, in the 17-limit 51/50, and in the 19-limit 76/75 and 96/95.
36edo's patent val, like 12, tempers out [[81/80]], [[128/125]], and [[648/625]] in the 5-limit. It departs from 12 in the 7-limit, tempering out [[686/675]] and [[1029/1000]], and as a no-fives temperament, [[1029/1024]] and [[118098/117649]]. The no-fives temperament tempering out 1029/1024, [[slendric]], is well supported by 36edo, its generator of ~8/7 represented by 7 steps of 36edo. In the 11-limit, the patent val tempers out [[56/55]], [[245/242]], and [[540/539]], and is the [[optimal patent val]] for the rank four temperament tempering out [[56/55]], as well as the rank-3 temperament [[melpomene]] tempering out 81/80 and 56/55. In the 13-limit, it tempers out [[78/77]] and [[91/90]], in the 17-limit [[51/50]], and in the 19-limit [[76/75]] and [[96/95]].


As a 5-limit temperament, the patent val for 36edo is [[contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals.
As a 5-limit temperament, the patent val for 36edo is [[contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals.
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| 17
| 17
| 566.7
| 566.7
|  
| [[112/81]]
| [[18/13]]
| [[18/13]]
| [[7/5]]
| [[7/5]]
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| 18
| 18
| 600.0
| 600.0
|  
| [[729/512]], [[1024/729]]
| [[17/12]], [[24/17]]
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
| [[45/32]], [[64/45]]
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| 19
| 19
| 633.3
| 633.3
|  
| 81/56
| [[13/9]]
| [[13/9]]
| [[10/7]]
| [[10/7]]
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| 20
| 20
| 666.7
| 666.7
| 72/49
| [[72/49]]
|  
|  
|  
|  
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== Notation ==
== 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.
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{sharpness-sharp3a|36}}
{{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 ===
=== 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 ====
==== Evo flavor ====
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=== 15-odd-limit approximations ===
=== 15-odd-limit approximations ===
{{Q-odd-limit intervals|36}}
{{Q-odd-limit intervals|36}}
{{Q-odd-limit intervals|35.98|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36e val mapping}}
{{Q-odd-limit intervals|35.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36ce val mapping}}


{{clear}}
{{Clear}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 0.42
| 0.42
| Sathurugu
| Sathurugu
| Schismina
| Minisma
|-
|-
| 17
| 17
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|  
|  
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: [[57edt]], [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.
 
; [[21edf]]
* Step size: 33.426{{c}}, octave size: 1203.351{{c}}
Stretching the octave of 36edo by a little over 3{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4{{c}}. The tuning 21edf does this.
{{Harmonics in equal|21|3|2|columns=11|collapsed=true}}
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
 
; [[57edt]]
* Step size: 33.368{{c}}, octave size: 1201.235{{c}}
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.
{{Harmonics in equal|57|3|1|columns=11|collapsed=true}}
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
 
; 36edo
* Step size: 33.333{{c}}, octave size: 1200.000{{c}}
Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.
{{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}
{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}
 
; [[TE|36et, 13-limit TE tuning]]
* Step size: 33.304{{c}}, octave size: 1198.929{{c}}
Compressing the octave of 36edo by about 1{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.
{{Harmonics in cet|33.303596|columns=11|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et}}
{{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et (continued)}}
 
; [[TE|36et, 11-limit TE tuning]]
* Step size: 33.287{{c}}, octave size: 1198.330{{c}}
{{Harmonics in cet|33.287|columns=11|collapsed=true|title=Approximation of harmonics in 11lim TE-tuned 36edo}}
{{Harmonics in cet|33.287|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11lim TE-tuned 36edo (continued)}}


Compressing the octave of 36edo by about 2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within 9.7{{c}}. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings.
Compressing the octave of 36edo by 1–2{{c}} results in much improved primes 5 and 11, but much worse primes 7 and 13. The 11- and 13-limit [[TE]] tunings of 36et both do this, as do their respective [[WE]] tunings.


{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings
|+ style="font-size: 105%; white-space: nowrap;" | Comparison of stretched and compressed tunings
|-
|-
! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents)  
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; [[Polymicrotonal]] scales
; [[Polymicrotonal]] scales
* 12-tone 4&9edo polymicrotonal scale: 4 4 1 3 4 2 2 4 3 1 4 4
* [[Werntz Nocturne scale]]: 4 2 2 4 4 2 2 4 4 2 2 4
* 12-tone 6&9edo polymicrotonal scale: 4 2 2 4 4 2 2 4 4 2 2 4
* 12-tone 4&9edo scale: 4 4 1 3 4 2 2 4 3 1 4 4
* 12-tone 9&12edo polymicrotonal scale: 4 2 2 4 3 3 3 3 3 3 2 4
* 12-tone 9&12edo scale: 4 2 2 4 3 3 3 3 3 3 2 4
* 12-tone 12&18edo polymicrotonal scale: 4 4 2 2 3 3 3 3 2 4 2 4
* 12-tone 12&18edo scale: 4 4 2 2 3 3 3 3 2 4 2 4
* 18-tone 9&12edo polymicrotonal scale: 3 1 2 2 1 3 3 1 2 2 1 3 3 1 2 2 1 3
* 18-tone 9&12edo scale: 3 1 2 2 1 3 3 1 2 2 1 3 3 1 2 2 1 3
* 24-tone 12&18edo polymicrotonal scale: 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
* 24-tone 12&18edo scale: 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2


; [[Baladic]][16] subsets
; [[Baladic]][16] subsets
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* 12-tone subset: 4 3 1 3 4 3 3 4 1 3 3 4
* 12-tone subset: 4 3 1 3 4 3 3 4 1 3 3 4


{| class="wikitable mw-collapsible mw-collapsed"
; [[Catnip]][24] subsets
|+[[Catnip]][24] subsets
* Bright catnip[24] MOS: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1  
|Bright catnip[24] MOS: 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1  
** 12edo plus 1 extra min7 note: 3 3 3 3 3 3 3 3 3 2 1 3 3
12edo plus 1 extra min7 note: 3 3 3 3 3 3 3 3 3 2 1 3 3
** 12edo with 7/4 replacing 9/5: 3 3 3 3 3 3 3 3 3 2 4 3
 
** 12edo with 7/4 replacing 9/5 & 7/6 replacing 6/5 3 3 2 4 3 3 3 3 3 2 4 3
12edo with 7/4 replacing 9/5: 3 3 3 3 3 3 3 3 3 2 4 3
** 12-tone chord 30:34:35:36:37:38:40:35:47:52:53:56 approximated from [[30afdo]]^: 6 2 1 2 1 3 6 2 6 1 2 4  
 
*** Rotated [[5afdo]]: 6 6 9 8 7
12edo with 7/4 replacing 9/5 & 7/6 replacing 6/5 3 3 2 4 3 3 3 3 3 2 4 3
*** Flattened Ionian pentatonic: 11 4 6 11 4
 
*** Flattened blues Aeolian pentatonic I: 8 7 6 2 13
12-tone chord 30:34:35:36:37:38:40:35:47:52:53:56 approximated from [[30afdo]]^: 6 2 1 2 1 3 6 2 6 1 2 4  
*** Flattened cosmic: 15 6 2 7 6
 
*** Catnip moonbeam: 6 3 12 11 4  
    •  Rotated [[6afdo]]: 6 6 9 8 7
** 12-tone chord 24:25:27:28:30:32:33:36:38:39:42:45 approximated from [[24afdo]]: 2 4 2 4 3 2 4 3 2 3 4 3
 
** 12-tone chord 18:19:20:22:23:24:26:27:29:31:32:35 approximated from [[18afdo]]: 3 3 4 3 2 4 2 4 3 2 4 2
    • Flattened Ionian pentatonic: 11 4 6 11 4
* Dark catnip[24] MOS: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2  
 
** 12edo plus 1 extra maj7 note: 3 3 3 3 3 3 3 3 3 3 1 2 3
    • Flattened blues Aeolian pentatonic I: 8 7 6 2 13
** 12edo plus 1 extra maj2 note: 3 3 1 2 3 3 3 3 3 3 3 3 3
 
** 12edo but 6/5, 8/5 & 9/5 are sharp not flat: 3 3 4 2 3 3 3 4 2 4 2 3
    • Flattened cosmic: 15 6 2 7 6
** 12edo but 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 3 3 4 2 3 4 2 4 2 4 2 3
 
** 12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat: 4 2 4 2 3 3 3 4 2 4 2 3
    • Catnip moonbeam: 6 3 12 11 4  
** 12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 4 2 4 2 3 4 2 4 2 4 2 3
 
** 12-tone, approximates the chord 42:45:47:48:51:56:59:63:68:71:76:78 from [[42afdo]]: 4 2 1 3 5 3 3 4 2 4 2 3 (from catnip[24] in [[60edo]])
12-tone chord 24:25:27:28:30:32:33:36:38:39:42:45 approximated from [[24afdo]]: 2 4 2 4 3 2 4 3 2 3 4 3
*** Sharpened minor: 7 3 5 6 4 6 5
 
*** Sharpened minor pentatonic: 10 5 6 10 5
12-tone chord 18:19:20:22:23:24:26:27:29:31:32:35 approximated from [[18afdo]]: 3 3 4 3 2 4 2 4 3 2 4 2
*** Sharpened minor harmonic pentatonic I: 7 3 11 12 3
 
*** Sharpened Phyrgian pentatonic: 4 6 11 4 11
 
*** Sharpened blues Aeolian pentatonic I: 10 5 6 4 11
Dark catnip[24] MOS: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2  
*** Sharpened blues Aeolian hexatonic: 10 5 3 3 4 11
 
*** Sharpened blues Dorian hexatonic: 10 5 6 6 4 5
12edo plus 1 extra maj7 note: 3 3 3 3 3 3 3 3 3 3 1 2 3
*** Sharpened blues pentachordal I: 6 4 5 3 3 15
 
*** Sharpened akebono I: 6 4 11 6 9
12edo plus 1 extra maj2 note: 3 3 1 2 3 3 3 3 3 3 3 3 3
*** Sharpened hirajoshi: 6 4 11 4 11
 
*** Extra sharpened hirajoshi: 7 3 11 4 11
12edo but 6/5, 8/5 & 9/5 are sharp not flat: 3 3 4 2 3 3 3 4 2 4 2 3
*** Catnip Deja Vu: 10 11 4 6 5
 
*** Catnip underpass: 10 11 6 4 5  
12edo but 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 3 3 4 2 3 4 2 4 2 4 2 3
** 12-tone chord 18:19:20:21:22:24:25:27:28:30:32:34 approximated from [[18afdo]]: 3 3 2 3 4 2 4 2 3 4 3 3
 
12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat: 4 2 4 2 3 3 3 4 2 4 2 3
 
12edo but 16/15, 6/5, 8/5 & 9/5 are sharp not flat, with 10/7 replacing 7/5: 4 2 4 2 3 4 2 4 2 4 2 3
 
12-tone, approximates the chord 42:45:47:48:51:56:59:63:68:71:76:78 from [[42afdo]]^: 4 2 1 3 5 3 3 4 2 4 2 3
 
    • Sharpened minor: 7 3 5 6 4 6 5
 
    • Sharpened minor pentatonic: 10 5 6 10 5
 
    • Sharpened minor harmonic pentatonic I: 7 3 11 12 3
 
    • Sharpened Phyrgian pentatonic: 4 6 11 4 11
 
    • Sharpened blues Aeolian pentatonic I: 10 5 6 4 11
 
    • Sharpened blues Aeolian hexatonic: 10 5 3 3 4 11
 
    • Sharpened blues Dorian hexatonic: 10 5 6 6 4 5
 
    • Sharpened blues pentachordal I: 6 4 5 3 3 15
 
    • Sharpened akebono I: 6 4 11 6 9
 
    • Sharpened hirajoshi: 6 4 11 4 11
 
    • Extra sharpened hirajoshi: 7 3 11 4 11
 
    • Catnip Deja Vu: 10 11 4 6 5
 
    • Catnip underpass: 10 11 6 4 5  
 
12-tone chord 18:19:20:21:22:24:25:27:28:30:32:34 approximated from [[18afdo]]: 3 3 2 3 4 2 4 2 3 4 3 3
 
 
^ ''its subsets listed here come from catnip[24] in [[60edo]]''
|}


; [[Echidna]][22] subsets
; [[Echidna]][22] subsets
Line 1,212: Line 1,146:
* 833 Cent Golden Scale MOS[11]: 3 1 3 3 1 3 1 3 3 1 3
* 833 Cent Golden Scale MOS[11]: 3 1 3 3 1 3 1 3 3 1 3
** [[833 Cent Golden Scale (Bohlen)]]: 3 4 4 3 4 4 3
** [[833 Cent Golden Scale (Bohlen)]]: 3 4 4 3 4 4 3
* Niner[18] MOS: 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3
* 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 3 1 3 1 3 3 1 1 3 1 3 1 3 1 3 1 3
* 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'')


== Tuning by ear ==
== Tuning by ear ==
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== Instruments ==
== Instruments ==
36edo can be played on the [[Lumatone]] (see [[Lumatone mapping for 36edo]]) and using three instruments tuned to 12edo with different root notes (that is, a sixth-tone apart).
36edo can be played on the [[Lumatone]]: see [[Lumatone mapping for 36edo]].
 
36edo can also be played using three instruments tuned to 12edo with different root notes (that is, a sixth-tone apart).


== Music ==
== Music ==
Line 1,232: Line 1,168:
; [[Ivan Bratt]]
; [[Ivan Bratt]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3 ''Boomers'']
* [http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3 ''Boomers'']
; [[Stevie Boyes]]
* [https://youtu.be/CUWZfomL-DQ ''Getting in the rhythm]'' (2016)


; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/SXspsdNtxQg ''36edo''] (2023)
* [https://www.youtube.com/watch?v=psvrsa10-Wo ''36edo jam''] (2025)
* [https://www.youtube.com/watch?v=psvrsa10-Wo ''36edo jam''] (2025)
* [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]]
; [[E8 Heterotic]]
Line 1,247: Line 1,190:
; [[Herman Klein]]
; [[Herman Klein]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3 ''Something''] (2022)
* [http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3 ''Something''] (2022)
; [[Budjarn Lambeth]]
* [https://youtu.be/XZKafk-PkPc ''Improvisation in zeta-stretched 36edo (catnip24 scale)''] (2025)


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


; [[NullPointerException Music]]
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=i5Pa5R7PKu4 ''Jungle Hillocks''] (2020)
* [https://www.youtube.com/watch?v=i5Pa5R7PKu4 ''Jungle Hillocks''] (2020)
; [[Juhani Nuorvala]]
* [https://m.youtube.com/watch?v=wy3qlby0Yiw ''Prelude from 'Suite 36' for Lumatone and piano''] (2025)


; [[Chris Orphal]]
; [[Chris Orphal]]