36edo: Difference between revisions

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That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33{{c}}, one can arrive at a 24-tone subset of 36edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.

=== ~~Harmonics~~ ===

=== Odd harmonics ===

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} [[49/48]] × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17 (since the 25th harmonic is more accurate than the 5th harmonic, and the 55th harmonic is more accurate than the 5th and 11th harmonics), and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

36edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).

36edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo #Octave stretch or compression|octave stretch or compression]].

{{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}

=== 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 ~~three~~ temperament [[~~Didymus rank three family|~~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.

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=== Additional properties ===

36edo ~~also~~ offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

36edo offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.

The [[edonoi]] scales of [[57edt]] and [[101ed7]] are almost exactly the same as 36edo. It is 36edo with the [[stretched octave|octave stretched]] by less than 1{{c}}. Its main usage is to optimise 36edo for use as a [[dual-n|dual-5]] tuning, while also making slight improvements to 3/1 and 7/1 as well. So if one intends to use both 36edo’s vals for 5/1 at once, 101ed7 may be worth considering.

Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]].

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! Additional ratios of 2.3.7.13.17.19<ref group="note" name="subg" />

! Additional ratios of 2.3.5.7<ref group="note" name="incons">Inconsistent intervals are in ''italics.''</ref>

! colspan="3" | [[Ups and ~~Downs Notation|Ups~~ and downs ~~notation]]~~

! colspan="3" | [[Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and d2)

|-

| 0

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| 17

| 566.7

|

| [[112/81]]

| [[18/13]]

| [[7/5]]

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| 18

| 600.0

|

| [[729/512]], [[1024/729]]

| [[17/12]], [[24/17]]

| [[45/32]], [[64/45]]

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| 19

| 633.3

|

| 81/56

| [[13/9]]

| [[10/7]]

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| 20

| 666.7

| 72/49

| [[72/49]]

|

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| D

|}

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].

== 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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== Approximation to JI ==

~~=== Interval mappings ===~~

~~{{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}}~~

=== 3-limit (Pythagorean) approximations (same as 12edo): ===

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63/32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.

=== 15-odd-limit approximations ===

{{Q-odd-limit intervals|35.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36ce val mapping}}

~~=== Zeta peak index ===~~

{{~~ZPI~~

~~| zpi = 155~~

~~| steps = 35.9823877000425~~

~~| step size = 33.3496490006021~~

~~| tempered height = 6.027497~~

~~| pure height = 5.885059~~

~~| integral = 1.028887~~

~~| gap = 14.706508~~

~~| octave = 1200.58736402167~~

~~| consistent = 8~~

~~| distinct = 8~~

}}

== Regular temperament properties ==

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=== Uniform maps ===

=== Commas ===

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| 0.42

| Sathurugu

| ~~Schismina~~

| Minisma

|-

| 17

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| Go comma

|}

=== Rank-2 temperaments ===

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| 166.67

| 10/9

| [[Squirrel]] (36), [[coendou]] (36c)

| [[Squirrel]] (36) / [[coendou]] (36c)

| [[1L 6s]], [[7L 1s]], [[7L 8s]], [[7L 15s]], [[7L 22s]]

|-

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| 233.33

| 8/7

| [[Slendric]] / [[~~mothra~~]] / [[~~guiron~~]]

| [[Slendric]] (36) / [[guiron]] (36e) / [[mothra]]

| [[1L 4s]], [[1L 5s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], [[5L 16s]], [[5L 21s]], [[5L 26s]]

|-

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| 566.67

| 7/5

| [[Liese]], [[pycnic]] (36c)

| [[Liese]] (36) / [[pycnic]] (36c)

| [[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], …, [[2L 15s]], [[17L 2s]]

|-

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| 166.67

| 10/9

| [[~~Hedgehog~~]] (~~36ceff~~), [[~~echidna~~]] (36)

| [[Echidna]] (36) / [[hedgehog]] (36ceff)

| [[2L 4s]], [[6L 2s]], [[8L 6s]], [[14L 8s]]

|-

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| 233.33

| 8/7

| [[Baladic]] / [[echidnic]]

| [[Baladic]] (36) / [[echidnic]] (36e)

| [[4L 2s]], [[6L 4s]], [[10L 6s]], [[10L 16s]]

|-

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| 500.00 (33.33)

| 4/3 (36/35)

| [[Niner]]

| [[Niner]] (36)

| [[9L 9s]]

|-

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| 233.33 (33.33)

| 8/7 (64/63)

| [[~~Catler~~]]

| [[Catnip]] (36) / [[catler]] (36e) / [[compton]] (36ce)

| [[12L 12s]]

|-

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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 ==

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.

{| class="wikitable sortable center-all mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | Comparison of stretched and compressed tunings

|-

! rowspan="2" | Tuning !! rowspan="2" | Octave size (cents) !! colspan="6" | Prime error (cents)

! rowspan="2" | Mapping of primes 2–13 (steps)

|-

! 2 !! 3 !! 5 !! 7 !! 11 !! 13

|-

! 21edf

| 1203.351

| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1

| 36, 57, 83, 101, 124, 133

|-

! 57edt

| 1201.235

| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6

| 36, 57, 84, 101, 124, 133

|-

! 155zpi

| 1200.587

| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0

| 36, 57, 84, 101, 124, 133

|-

! 36edo

| '''1200.000'''

| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''

| '''36, 57, 84, 101, 125, 133'''

|-

! 13-limit TE

| 1198.929

| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1

| 36, 57, 84, 101, 125, 133

|-

! 11-limit TE

| 1198.330

| −1.7 || −4.6 || +9.8 || −6.8 || +9.5 || −13.4

| 36, 57, 84, 101, 125, 133

|}

== Scales ==

~~{{main|~~List of MOS scales in 36edo}}

''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

* [[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

~~'''Catler'''~~

; [[Catnip]][24] subsets

* [[~~Lost spirit~~]] (approximated from [[~~Meantone~~]] in [[~~31edo~~]]): ~~'''~~9 6 2 4 7 2 ~~6'''~~

* 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

~~'''Hedgehog'''~~

; [[Liese]][19] subsets

* [[~~Hedgehog~~]][14] MOS: ~~'''~~3 2 3 2 3 2 ~~3 3~~ 2 3 2 3 2 ~~3'''~~

Liese[19] MOS: 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2

* ~~Palace (subset of Hedgehog~~[14]): ~~'''~~5 5 ~~5 6 5 5 5'''~~

* [[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

[[~~833 Cent Golden Scale (Bohlen)~~]]: ~~'''3~~ 4 4 3 4 4 ~~3'''~~

; [[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

833 Cent Golden Scale MOS [11]: '''3 1 3 3 1 3 1 3 3 1 3'''

; [[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

; 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'')

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

== Tuning by ear ==

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== 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).

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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; [[Ivan Bratt]]

* [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]]

* [https://www.youtube.com/shorts/SXspsdNtxQg ''36edo''] (2023)

* [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]]

* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 ~~"Elements -~~ Lightning"] from ''Elements'' (2020)

* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 ''Lightning''] from ''Elements'' (2020)

; [[User:Francium|Francium]]

Line 1,069:

Line 1,203:

; [[Herman Klein]]

* [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]]

Line 1,075:

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)

; [[NullPointerException Music]]

* [https://www.youtube.com/watch?v=i5Pa5R7PKu4 ''Jungle Hillocks''] (2020)

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

; [[Juhani Nuorvala]]

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

* [https://m.youtube.com/watch?v=wy3qlby0Yiw ''Prelude from 'Suite 36' for Lumatone and piano''] (2025)

; [[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)

; [[Chris Vaisvil]]

Line 1,090:

Line 1,235:

; [[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

~~== Notes ==~~

~~<references group="note" />~~

[[Category:Listen]]

@@ Line 8: / Line 8: @@
 That 36edo contains 12edo as a subset makes it compatible with traditional instruments tuned to 12edo. By tuning one 12-edo instrument up or down about 33{{c}}, one can arrive at a 24-tone subset of 36edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']{{Dead link}}). Three 12edo instruments could play the entire gamut.
-=== Harmonics ===
+=== Odd harmonics ===
-In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} [[49/48]] × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17 (since the 25th harmonic is more accurate than the 5th harmonic, and the 55th harmonic is more accurate than the 5th and 11th harmonics), and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).
-{{harmonics in equal|36|prec=2}}
+edo has almost 50% relative error on harmonics 5/1 and 11/1. This means that whether one [[octave stretch|stretches]] or [[octave shrinking|compresses]] the octave, either way it will improve 36edo's approximations of [[JI]], but in opposite directions, as long as it is done by the right amount, as discussed in more detail in [[36edo #Octave stretch or compression|octave stretch or compression]].
+{{Harmonics in equal|36|intervals=odd|prec=2|columns=14}}
+{{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}
 === Mappings ===
-edo'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 three temperament [[Didymus rank three family|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.
+edo'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.
@@ Line 21: / Line 24: @@
 === Additional properties ===
-edo also offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.
+edo offers a good approximation to the [[acoustic phi|acoustic golden ratio]], as 25\36. [[Heinz Bohlen]] proposed 36edo as a suitable temperament for approximating his 833-cents scale. The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 36edo could be treated as a 2.3.7.ϕ.17 temperament.
-The [[edonoi]] scales of [[57edt]] and [[101ed7]] are almost exactly the same as 36edo. It is 36edo with the [[stretched octave|octave stretched]] by less than 1{{c}}. Its main usage is to optimise 36edo for use as a [[dual-n|dual-5]] tuning, while also making slight improvements to 3/1 and 7/1 as well. So if one intends to use both 36edo’s vals for 5/1 at once, 101ed7 may be worth considering.
 Thanks to its sevenths, 36edo is an ideal tuning for its size for [[metallic harmony]].
@@ Line 38: / Line 39: @@
 ! Additional ratios<br>of 2.3.7.13.17.19<ref group="note" name="subg" />
 ! Additional ratios<br>of 2.3.5.7<ref group="note" name="incons">Inconsistent intervals are in ''italics.''</ref>
-! colspan="3" | [[Ups and Downs Notation|Ups and downs<br />notation]]
+! colspan="3" | [[Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and d2)
 |-
 | 0
@@ Line 195: / Line 197: @@
 | 17
 | 566.7
-|
+| [[112/81]]
 | [[18/13]]
 | [[7/5]]
@@ Line 204: / Line 206: @@
 | 18
 | 600.0
-|
+| [[729/512]], [[1024/729]]
 | [[17/12]], [[24/17]]
 | [[45/32]], [[64/45]]
@@ Line 213: / Line 215: @@
 | 19
 | 633.3
-|
+| 81/56
 | [[13/9]]
 | [[10/7]]
@@ Line 222: / Line 224: @@
 | 20
 | 666.7
-| 72/49
+| [[72/49]]
 |
 |
@@ Line 373: / Line 375: @@
 | D
 |}
+<references group="note" />
 Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].
 == 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.
-{{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 ===
-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 428: / Line 432: @@
 == Approximation to JI ==
 [[File:36ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 36edo]]
-=== Interval mappings ===
-{{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}}
 === 3-limit (Pythagorean) approximations (same as 12edo): ===
@@ Line 489: / Line 489: @@
 /32 = 1172.736... cents; 35 degrees of 36edo = 1166.666... cents.
-{{clear}}
+=== 15-odd-limit approximations ===
+{{Q-odd-limit intervals|36}}
+{{Q-odd-limit intervals|35.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36ce val mapping}}
-=== Zeta peak index ===
+{{Clear}}
-{{ZPI
-| zpi = 155
-| steps = 35.9823877000425
-| step size = 33.3496490006021
-| tempered height = 6.027497
-| pure height = 5.885059
-| integral = 1.028887
-| gap = 14.706508
-| octave = 1200.58736402167
-| consistent = 8
-| distinct = 8
-}}
 == Regular temperament properties ==
@@ Line 582: / Line 572: @@
 === Uniform maps ===
-{{Uniform map|edo=36}}
+{{Uniform map|min=35.8|max=36.2}}
 === Commas ===
@@ Line 825: / Line 815: @@
 | 0.42
 | Sathurugu
-| Schismina
+| Minisma
 |-
 | 17
@@ Line 911: / Line 901: @@
 | Go comma
 |}
+<references group="note" />
 === Rank-2 temperaments ===
@@ Line 929: / Line 920: @@
 | 166.67
 | 10/9
-| [[Squirrel]] (36), [[coendou]] (36c)
+| [[Squirrel]] (36) / [[coendou]] (36c)
 | [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]], [[7L&nbsp;15s]], [[7L&nbsp;22s]]
 |-
@@ Line 936: / Line 927: @@
 | 233.33
 | 8/7
-| [[Slendric]] / [[mothra]] / [[guiron]]
+| [[Slendric]] (36) / [[guiron]] (36e) / [[mothra]]
 | [[1L&nbsp;4s]], [[1L&nbsp;5s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], [[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]], [[5L&nbsp;26s]]
 |-
@@ Line 957: / Line 948: @@
 | 566.67
 | 7/5
-| [[Liese]], [[pycnic]] (36c)
+| [[Liese]] (36) / [[pycnic]] (36c)
 | [[2L&nbsp;1s]], [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], …, [[2L&nbsp;15s]], [[17L&nbsp;2s]]
 |-
@@ Line 964: / Line 955: @@
 | 166.67
 | 10/9
-| [[Hedgehog]] (36ceff), [[echidna]] (36)
+| [[Echidna]] (36) / [[hedgehog]] (36ceff)
 | [[2L&nbsp;4s]], [[6L&nbsp;2s]], [[8L&nbsp;6s]], [[14L&nbsp;8s]]
 |-
@@ Line 971: / Line 962: @@
 | 233.33
 | 8/7
-| [[Baladic]] / [[echidnic]]
+| [[Baladic]] (36) / [[echidnic]] (36e)
 | [[4L&nbsp;2s]], [[6L&nbsp;4s]], [[10L&nbsp;6s]], [[10L&nbsp;16s]]
 |-
@@ Line 1,006: / Line 997: @@
 | 500.00<br>(33.33)
 | 4/3<br>(36/35)
-| [[Niner]]
+| [[Niner]] (36)
 | [[9L&nbsp;9s]]
 |-
@@ Line 1,013: / Line 1,004: @@
 | 233.33<br>(33.33)
 | 8/7<br>(64/63)
-| [[Catler]]
+| [[Catnip]] (36) / [[catler]] (36e) / [[compton]] (36ce)
 | [[12L&nbsp;12s]]
 |-
@@ Line 1,023: / Line 1,014: @@
 |
 |}
-<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 ==
+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.
+{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
+|+ 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" | Mapping of primes 2–13 (steps)
+|-
+! 2 !! 3 !! 5 !! 7 !! 11 !! 13
+|-
+! 21edf
+| 1203.351
+| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
+| 36, 57, 83, 101, 124, 133
+|-
+! 57edt
+| 1201.235
+| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
+| 36, 57, 84, 101, 124, 133
+|-
+! 155zpi
+| 1200.587
+| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0
+| 36, 57, 84, 101, 124, 133
+|-
+! 36edo
+| '''1200.000'''
+| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
+| '''36, 57, 84, 101, 125, 133'''
+|-
+! 13-limit TE
+| 1198.929
+| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1
+| 36, 57, 84, 101, 125, 133
+|-
+! 11-limit TE
+| 1198.330
+| −1.7 || −4.6 || +9.8 || −6.8 || +9.5 || −13.4
+| 36, 57, 84, 101, 125, 133
+|}
 == Scales ==
-{{main|List of MOS scales in 36edo}}
+''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
+* [[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
-'''Catler'''
+; [[Catnip]][24] subsets
-* [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]): '''9 6 2 4 7 2 6'''
+* 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
-'''Hedgehog'''
+; [[Liese]][19] subsets
-* [[Hedgehog]][14] MOS: '''3 2 3 2 3 2 3 3 2 3 2 3 2 3'''
+Liese[19] MOS: 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 2 2 2
-* Palace (subset of Hedgehog[14]): '''5 5 5 6 5 5 5'''
+* [[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
-[[833 Cent Golden Scale (Bohlen)]]: '''3 4 4 3 4 4 3'''
+; [[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
-Cent Golden Scale MOS [11]: '''3 1 3 3 1 3 1 3 3 1 3'''
+; [[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
+; 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'')
+* Baobab: 9 6 6 6 4 5 (''approximated from [[30afdo]]'')
 == Tuning by ear ==
@@ Line 1,046: / Line 1,156: @@
 == Instruments ==
-edo 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).
+edo can be played on the [[Lumatone]]: see [[Lumatone mapping for 36edo]].
+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 1,057: / Line 1,181: @@
 ; [[Ivan Bratt]]
 * [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]]
+* [https://www.youtube.com/shorts/SXspsdNtxQg ''36edo''] (2023)
+* [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]]
-* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 "Elements - Lightning"] from ''Elements'' (2020)
+* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 ''Lightning''] from ''Elements'' (2020)
 ; [[User:Francium|Francium]]
@@ Line 1,069: / Line 1,203: @@
 ; [[Herman Klein]]
 * [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]]
@@ Line 1,075: / 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)
 ; [[NullPointerException Music]]
 * [https://www.youtube.com/watch?v=i5Pa5R7PKu4 ''Jungle Hillocks''] (2020)
-; {{W|Henri Pousseur}}
+; [[Juhani Nuorvala]]
-* [https://www.youtube.com/watch?v=tiWbbbU6ZPw ''Prospection''] (1952-1953) – for three pianos detuned by a sixth-tone from each other
+* [https://m.youtube.com/watch?v=wy3qlby0Yiw ''Prelude from 'Suite 36' for Lumatone and piano''] (2025)
+; [[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)
 ; [[Chris Vaisvil]]
@@ Line 1,090: / Line 1,235: @@
 ; [[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
-== Notes ==
-<references group="note" />
 [[Category:Listen]]