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]] | ||
| Line 206: | Line 206: | ||
| 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]] | ||
| Line 215: | Line 215: | ||
| 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 == | ||
=== | === Stein–Zimmermann–Gould notation === | ||
[[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. | |||
=== | === 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. | ||
{{Ups and downs sharpness | {{Ups and downs sharpness}} | ||
=== Colored notes === | |||
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). | |||
=== Sagittal notation === | === Sagittal notation === | ||
This notation uses the same sagittal sequence as [[43edo#Sagittal notation| | 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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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. | 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 | 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. | 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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=== 15-odd-limit approximations === | === 15-odd-limit approximations === | ||
{{Q-odd-limit intervals|36}} | {{Q-odd-limit intervals|36}} | ||
{{Q-odd-limit intervals|35. | {{Q-odd-limit intervals|35.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36ce val mapping}} | ||
{{ | {{Clear}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 814: | Line 815: | ||
| 0.42 | | 0.42 | ||
| Sathurugu | | Sathurugu | ||
| | | Minisma | ||
|- | |- | ||
| 17 | | 17 | ||
| Line 1,013: | Line 1,014: | ||
| | | | ||
|} | |} | ||
<nowiki/>* [[Normal | <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 == | ||
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. | 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. | ||
Compressing the octave of 36edo by | 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) | ||
| Line 1,061: | Line 1,062: | ||
== Scales == | == Scales == | ||
''See also: [[List of MOS scales in 36edo]]'' | ''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 | ; [[Polymicrotonal]] scales | ||
| Line 1,076: | Line 1,077: | ||
* 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 | ||
; [[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 | |||
** 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 | |||
** 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 | |||
*** Flattened Ionian pentatonic: 11 4 6 11 4 | |||
*** Flattened blues Aeolian pentatonic I: 8 7 6 2 13 | |||
*** Flattened cosmic: 15 6 2 7 6 | |||
*** Catnip moonbeam: 6 3 12 11 4 | |||
** 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 | |||
* 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 | |||
** 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 | |||
** 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 | |||
** 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]]) | |||
*** 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 | |||
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 | |||
*** 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 | |||
; [[Echidna]][22] subsets | ; [[Echidna]][22] subsets | ||
| Line 1,185: | 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] 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'') | * 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 == | == 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). | 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 == | == 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]] | ; [[Ambient Esoterica]] | ||
* [https://www.youtube.com/watch?v=nadDPD1m-1Y ''A Trip to PRS with a Circle of Augmented 5ths''] (2024) | * [https://www.youtube.com/watch?v=nadDPD1m-1Y ''A Trip to PRS with a Circle of Augmented 5ths''] (2024) | ||
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; [[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]] | ||
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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) | |||
* [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]] | ; [[NullPointerException Music]] | ||
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; [[Chris Orphal]] | ; [[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) | * [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) | ||
; [[Chris Vaisvil]] | ; [[Chris Vaisvil]] | ||
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; [[Stephen Weigel]] | ; [[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 | * [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]] | [[Category:Listen]] | ||