36edo: Difference between revisions
m →Octave stretch or compression: Some primes are noticeably better or worse in the 11lim tuning vs the 13lim so I think it's worth including both |
→Music: Add Bryan Deister's ''36edo improv'' (2026) |
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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. | ||
| Line 197: | Line 197: | ||
| 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]] | ||
| Line 224: | Line 224: | ||
| 20 | | 20 | ||
| 666.7 | | 666.7 | ||
| 72/49 | | [[72/49]] | ||
| | | | ||
| | | | ||
| Line 380: | Line 380: | ||
== 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. | ||
{{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 ==== | ||
| Line 490: | Line 491: | ||
=== 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 919: | Line 920: | ||
| 166.67 | | 166.67 | ||
| 10/9 | | 10/9 | ||
| [[Squirrel]] (36) | | [[Squirrel]] (36) / [[coendou]] (36c) | ||
| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]] | | [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]] | ||
|- | |- | ||
| Line 926: | Line 927: | ||
| 233.33 | | 233.33 | ||
| 8/7 | | 8/7 | ||
| [[Slendric]] / [[ | | [[Slendric]] (36) / [[guiron]] (36e) / [[mothra]] | ||
| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]] | | [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]] | ||
|- | |- | ||
| Line 947: | Line 948: | ||
| 566.67 | | 566.67 | ||
| 7/5 | | 7/5 | ||
| [[Liese]] | | [[Liese]] (36) / [[pycnic]] (36c) | ||
| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]] | | [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]] | ||
|- | |- | ||
| Line 954: | Line 955: | ||
| 166.67 | | 166.67 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Echidna]] (36) / [[hedgehog]] (36ceff) | ||
| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]] | | [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]] | ||
|- | |- | ||
| Line 961: | Line 962: | ||
| 233.33 | | 233.33 | ||
| 8/7 | | 8/7 | ||
| [[Baladic]] / [[echidnic]] | | [[Baladic]] (36) / [[echidnic]] (36e) | ||
| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]] | | [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]] | ||
|- | |- | ||
| Line 996: | Line 997: | ||
| 500.00<br>(33.33) | | 500.00<br>(33.33) | ||
| 4/3<br>(36/35) | | 4/3<br>(36/35) | ||
| [[Niner]] | | [[Niner]] (36) | ||
| [[9L 9s]] | | [[9L 9s]] | ||
|- | |- | ||
| Line 1,003: | Line 1,004: | ||
| 233.33<br>(33.33) | | 233.33<br>(33.33) | ||
| 8/7<br>(64/63) | | 8/7<br>(64/63) | ||
| [[ | | [[Catnip]] (36) / [[catler]] (36e) / [[compton]] (36ce) | ||
| [[12L 12s]] | | [[12L 12s]] | ||
|- | |- | ||
| 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 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. | |||
Compressing the octave of 36edo by | |||
{| 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 1,071: | Line 1,042: | ||
| 1200.587 | | 1200.587 | ||
| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0 | | +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0 | ||
| 36, 57, | | 36, 57, 84, 101, 124, 133 | ||
|- | |- | ||
! 36edo | ! 36edo | ||
| Line 1,090: | Line 1,061: | ||
== Scales == | == Scales == | ||
''See also: [[List of MOS scales in 36edo]]'' | |||
{{Idiosyncratic terms}} | |||
; [[Polymicrotonal]] scales | |||
* [[Werntz Nocturne 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 scale: 4 2 2 4 3 3 3 3 3 3 2 4 | |||
* 12-tone 12&18edo scale: 4 4 2 2 3 3 3 3 2 4 2 4 | |||
* 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 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] MOS: 3 1 3 1 3 3 1 3 3 1 3 1 3 3 1 3 | |||
* 12-tone subset: 3 4 1 3 4 3 3 4 1 3 4 3 | |||
* 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 | |||
*** 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] MOS: 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 1 2 1 2 2 1 2 | |||
* Fennec ''(approx. from [[14edo]])'': 5 5 5 6 2 11 2 | |||
* Echidna[14] MOS: 3 2 3 2 3 2 3 3 2 3 2 3 2 3 | |||
** ''(the squirrel[6] & [7] MOSes occur as subsets of Echidna[14])'' | |||
** 12-tone subset: 3 2 3 2 5 3 3 5 2 3 2 3 | |||
; [[Liese]][19] subsets | |||
Liese[19] MOS: 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2 | |||
* [[Lost spirit]]: 9 6 2 4 7 2 6 | |||
* Liese[17] MOS: 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 | |||
** Liese[15] MOS: 2 2 2 5 2 2 2 2 2 2 2 5 2 2 2 | |||
*** Liese[13] MOS: 2 2 2 7 2 2 2 2 2 7 2 2 2 | |||
**** Liese[11] MOS: 2 2 9 2 2 2 2 2 9 2 2 | |||
***** Liese[9] MOS: 2 2 11 2 2 2 11 2 2 | |||
; [[Slendric]][21] subsets | |||
Slendric[21] MOS: 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 1 4 1 1 | |||
* | * Slendric[16] MOS: 1 5 1 1 5 1 1 1 5 1 1 5 1 1 5 1 | ||
** 12-tone subset: 6 1 1 5 2 6 2 5 1 1 5 1 | |||
** Slendric[11] MOS: 1 6 1 6 1 6 1 6 1 6 1 | |||
[[ | ; [[Squirrel]][22] subsets | ||
Squirrel[22] MOS: 1 3 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 3 1 | |||
* Squirrel[15] MOS: 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 | |||
** 12-tone subset: 5 1 4 1 4 6 4 1 4 1 4 1 | |||
** Squirrel[8] MOS: 5 5 5 1 5 5 5 5 | |||
*** Squirrel[7] MOS: 5 5 5 6 5 5 5 | |||
**** Squirrel[6] MOS: 5 5 11 5 5 5 | |||
833 Cent Golden Scale MOS [11]: '''3 1 3 3 1 3 1 3 3 | ; Other scales | ||
* 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 | |||
* 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'') | |||
== Tuning by ear == | == Tuning by ear == | ||
| Line 1,109: | Line 1,155: | ||
== Instruments == | == Instruments == | ||
36edo can be played on the [[Lumatone]] | 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 1,120: | 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]] | ||
* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 | * [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 ''Lightning''] from ''Elements'' (2020) | ||
; [[User:Francium|Francium]] | ; [[User:Francium|Francium]] | ||
| Line 1,135: | 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]] | ||
| Line 1,141: | Line 1,199: | ||
; [[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]] | ||