View source for 36edo

{{Infobox ET}}
{{Wikipedia|Sixth tone}}
{{ED intro}}

== Theory ==
Since 36 is divisible by 12, it contains the overly-familiar [[12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33{{c}}, which could be called "sixth tones." 36edo also contains [[18edo]] ("third tones") and [[9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo]] whole tone scale, [[4edo]] full-diminished seventh chord, and the [[3edo]] augmented triad, all of which are present in 12edo.

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.

=== 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 (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|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 ===
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.

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.
Another 5-limit alternative val is {{val| 36 57 83 }} (36c-edo), which is similar to the patent val but has 5/4 mapped to the 367{{c}} submajor third rather than the major third. This mapping supports very sharp [[porcupine]] temperament using 5\36 as a generator.

=== Additional properties ===
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.

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

=== Subsets and supersets ===
36edo is the 7th [[highly composite edo]], with subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18 }}. 72edo, which doubles it, provides correction for its approximated harmonics 5 and 11.

== Intervals  ==
{| class="wikitable center-1 right-2 center-6 center-7 center-8"
|-
! #
! [[Cent]]s
! Approximate<br>ratios of 2.3.7<ref group="note" name="subg">{{sg|limit=2.3.7 or 2.3.7.13.17.19 subgroup}}</ref>
! 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]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>3</sup>A1 and d2)
|-
| 0
| 0.0
| 1/1
| 
| 
| P1
| perfect unison
| D
|-
| 1
| 33.3
| [[49/48]], [[64/63]]
| 
| 
| ^1
| up unison
| ^D
|-
| 2
| 66.7
| [[28/27]]
| 
| 
| vm2
| downminor 2nd
| vEb
|-
| 3
| 100.0
| 256/243
| [[17/16]], [[18/17]]
| [[16/15]]
| m2
| minor 2nd
| Eb
|-
| 4
| 133.3
| 243/224
| [[14/13]], [[13/12]]
| 
| ^m2
| upminor 2nd
| ^Eb
|-
| 5
| 166.7
| [[54/49]]
| 
| 
| vM2
| downmajor 2nd
| vE
|-
| 6
| 200.0
| [[9/8]]
| [[19/17]]
| ''[[10/9]]''
| M2
| major 2nd
| E
|-
| 7
| 233.3
| [[8/7]]
| 
| 
| ^M2
| upmajor 2nd
| ^E
|-
| 8
| 266.7
| [[7/6]]
| 
| 
| vm3
| downminor 3rd
| vF
|-
| 9
| 300.0
| [[32/27]]
| [[19/16]]
| [[6/5]]
| m3
| minor 3rd
| F
|-
| 10
| 333.3
| 98/81
| [[17/14]]
| 
| ^m3
| upminor 3rd
| ^F
|-
| 11
| 366.7
| 243/196
| [[16/13]], [[26/21]], [[21/17]]
| 
| vM3
| downmajor 3rd
| vF#
|-
| 12
| 400.0
| [[81/64]]
| [[24/19]]
| [[5/4]], ''[[32/25]]''
| M3
| major 3rd
| F#
|-
| 13
| 433.3
| [[9/7]]
| 
| 
| ^M3
| upmajor 3rd
| ^F#
|-
| 14
| 466.7
| [[64/49]], [[21/16]]
| [[17/13]]
| 
| v4
| down 4th
| vG
|-
| 15
| 500.0
| [[4/3]]
| 
| 
| P4
| 4th
| G
|-
| 16
| 533.3
| [[49/36]]
| 
| 
| ^4
| up 4th
| ^G
|-
| 17
| 566.7
| 
| [[18/13]]
| [[7/5]]
| vA4
| downaug 4th
| vG#
|-
| 18
| 600.0
| 
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
| A4, d5
| aug 4th, dim 5th
| G#, Ab
|-
| 19
| 633.3
| 
| [[13/9]]
| [[10/7]]
| ^d5
| updim 5th
| ^Ab
|-
| 20
| 666.7
| 72/49
| 
| 
| v5
| down 5th
| vA
|-
| 21
| 700.0
| [[3/2]]
| 
| 
| P5
| 5th
| A
|-
| 22
| 733.3
| [[49/32]], [[32/21]]
| [[26/17]]
| 
| ^5
| up fifth
| ^A
|-
| 23
| 766.7
| [[14/9]]
| 
| 
| vm6
| downminor 6th
| vBb
|-
| 24
| 800.0
| [[128/81]]
| [[19/12]]
| [[8/5]], ''[[25/16]]''
| m6
| minor 6th
| Bb
|-
| 25
| 833.3
| 392/243
| [[13/8]], [[21/13]], [[34/21]]
| 
| ^m6
| upminor 6th
| ^Bb
|-
| 26
| 866.7
| 81/49
| [[28/17]]
| 
| vM6
| downmajor 6th
| vB
|-
| 27
| 900.0
| [[27/16]]
| [[32/19]]
| [[5/3]]
| M6
| major 6th
| B
|-
| 28
| 933.3
| [[12/7]]
| 
| 
| ^M6
| upmajor 6th
| ^B
|-
| 29
| 966.7
| [[7/4]]
| 
| 
| vm7
| downminor 7th
| vC
|-
| 30
| 1000.0
| [[16/9]]
| [[34/19]]
| ''[[9/5]]''
| m7
| minor 7th
| C
|-
| 31
| 1033.3
| 49/27
| 
| 
| ^m7
| upminor 7th
| ^C
|-
| 32
| 1066.7
| 448/243
| [[13/7]], [[24/13]]
| 
| vM7
| downmajor 7th
| vC#
|-
| 33
| 1100.0
| [[243/128]]
| [[32/17]], [[17/9]]
| [[15/8]]
| M7
| major 7th
| C#
|-
| 34
| 1133.3
| [[27/14]]
| 
| 
| ^M7
| upmajor 7th
| ^C#
|-
| 35
| 1166.7
| 63/32, 96/49
| 
| 
| v8
| down 8ve
| vD
|-
| 36
| 1200.0
| 2/1
| 
| 
| P8
| 8ve
| 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 ===
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).

=== 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}}

Alternatively, one can use sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp3|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.

=== 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]].

==== Evo flavor ====
<imagemap>
File:36-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
default [[File:36-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:36-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
default [[File:36-EDO_Revo_Sagittal.svg]]
</imagemap>

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

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

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&nbsp;4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

The "red unison" and "blue unison" are in fact the same interval (33.333{{c}}), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50{{c}} step to sound much more discordant when used as a subminor second.

People with perfect (absolute) pitch often have a difficult time listening to xenharmonic and non-12edo scales, since their ability to memorize and become accustomed to the pitches and intervals of 12edo results in other pitches and intervals sounding out of tune. This is not as much of a problem with 36edo, due to its similarity to 12. With practice, it might even be possible to extend one's perfect pitch to be able to recognize blue and red notes.

=== "Quark" ===
In particle physics, [https://en.wikipedia.org/wiki/Baryon baryons], which are the main building blocks of atomic nuclei, are always comprised of three quarks. One could draw an analogy between baryons and semitones (the main building block of 12edo); the baryon is comprised of three quarks and the semitone of three sixth-tones. The number of quarks in a [https://en.wikipedia.org/wiki/Color_charge colorless] particle is always a multiple of three; similarly, the width of "colorless" intervals (the 12-edo intervals, which are neither red nor blue), expressed in terms of sixth-tones, is always a multiple of three. Because of this amusing coincidence, Mason Green proposes referring to the 33.333{{c}} sixth-tone interval as a "quark".

== Approximation to JI ==
[[File:36ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 36edo]]

=== 3-limit (Pythagorean) approximations (same as 12edo): ===
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.

4/3 = 498.045... cents; 15 degrees of 36edo = 500 cents.

9/8 = 203.910... cents; 6 degrees of 36edo = 200 cents.

16/9 = 996.090... cents; 30 degrees of 36edo = 1000 cents.

27/16 = 905.865... cents; 27 degrees of 36edo = 900 cents.

32/27 = 294.135... cents; 9 degrees of 36edo = 300 cents.

81/64 = 407.820... cents; 12 degrees of 36edo = 400 cents.

128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.

=== 7-limit approximations: ===
==== 7 only: ====
7/4 = 968.826... cents; 29 degrees of 36edo = 966.666... cents.

8/7 = 231.174... cents; 7 degrees of 36edo = 233.333... cents.

49/32 = 737.652... cents; 22 degrees of 36edo = 733.333... cents.

64/49 = 462.348... cents; 14 degrees of 36edo = 466.666... cents.

==== 3 and 7: ====
7/6 = 266.871... cents; 8 degrees of 36edo = 266.666... cents.

12/7 = 933.129... cents; 28 degrees of 36 = 933.333... cents.

9/7 = 435.084... cents; 13 degrees of 36edo = 433.333... cents.

14/9 = 764.916... cents; 23 degrees of 36edo = 766.666... cents.

28/27 = 62.961... cents; 2 degrees of 36edo = 66.666... cents.

27/14 = 1137.039... cents; 34 degrees of 36edo = 1133.333... cents.

21/16 = 470.781... cents; 14 degrees of 36edo = 466.666... cents.

32/21 = 729.219... cents; 22 degrees of 36edo = 733.333... cents.

49/48 = 35.697... cents; 1 degree of 36edo = 33.333... cents.

96/49 = 1164.303... cents; 35 degrees of 36edo = 1166.666... cents.

49/36 = 533.742... cents; 16 degrees of 36edo = 533.333... cents.

72/49 = 666.258... cents; 20 degrees of 36edo = 666.666... cents.

64/63 = 27.264... cents; 1 degree of 36edo = 33.333... cents.

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

=== 15-odd-limit approximations ===
{{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}}

{{clear}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.7
| 1029/1024, 118098/117649
| {{Mapping| 36 57 101 }}
| +0.67
| 0.51
| 1.53
|-
| 2.3.7.13
| 169/168, 729/728, 1029/1024
| {{Mapping| 36 57 101 133 }}
| +0.99
| 0.71
| 2.12
|-
| 2.3.7.13.17
| 169/168, 273/272, 289/288, 729/728
| {{Mapping| 36 57 101 133 147 }}
| +1.03
| 0.64
| 1.92
|-
| 2.3.7.13.17.19
| 153/152, 169/168, 273/272, 289/288, 343/342
| {{Mapping| 36 57 101 133 147 153 }}
| +0.76
| 0.84
| 2.52
|- style="border-top: double;"
| 2.3.5.7
| 81/80, 128/125, 686/675
| {{Mapping| 36 57 84 101 }}
| −0.98
| 2.87
| 8.63
|-
| 2.3.5.7.11
| 56/55, 81/80, 128/125, 540/539
| {{Mapping| 36	57 84 101 125 }}
| −1.67
| 2.92
| 8.76
|-
| 2.3.5.7.11.13
| 56/55, 78/77, 81/80, 91/90, 128/125
| {{Mapping| 36	57 84 101 125 133 }}
| −1.07
| 2.98
| 8.96
|-
| 2.3.5.7.11.13.17
| 51/50, 56/55, 78/77, 81/80, 91/90, 128/125
| {{Mapping| 36 57 84 101 125 133 147 }}
| −0.75
| 2.88
| 8.63
|-
| 2.3.5.7.11.13.17.19
| 51/50, 56/55, 76/75, 78/77, 81/80, 91/90, 96/95
| {{Mapping| 36 57 84 101 125 133 147 153 }}
| −0.73
| 2.69
| 8.08
|}

=== Uniform maps ===
{{Uniform map|min=35.8|max=36.2}}

=== Commas ===
This is a partial list of the [[comma]]s that 36et [[tempering out|tempers out]] with its patent [[val]], {{val| 36 57 84 101 125 133 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| <abbr title="531441/524288">(12 digits)</abbr>
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| <abbr title="262144/253125">(12 digits)</abbr>
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| <abbr title="67108864/66430125">(16 digits)</abbr>
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadbigu
| [[Kirnberger's atom]]
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[1728/1715]]
| {{monzo| 6 3 -1 -3 }}
| 13.07
| Triru-agu
| Orwellisma
|-
| 7
| <abbr title="2097152/2083725">(14 digits)</abbr>
| {{Monzo| 21 -5 -2 -3}}
| 11.12
| Satriru-agugu
| [[Bronzisma]]
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| <abbr title="2147483648/2144153025">(20 digits)</abbr>
| {{monzo| 31 -6 -2 -6 }}
| 2.69
| Sasa-tribiru-agugu
| [[Pessoalisma]]
|-
| 7
| <abbr title="40353607/40310784">(16 digits)</abbr>
| {{monzo| -11 -9 0 9 }}
| 1.84
| Tritrizo
| [[Septimal ennealimma]]
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[1350/1331]]
| {{monzo| 1 3 2 0 -3}}
| 24.54
| Trilu-ayoyo
| Large tetracot diesis
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3}}
| 3.03
| Triluyo
| Wizardharry comma, pine comma
|-
| 11
| [[9801/9800]]
| {{monzo|-3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[78/77]]
| {{monzo| 1 1 0 -1 -1 1}}
| 22.34
| Tholuru
| Negustma
|-
| 13
| [[91/90]]
| {{monzo|-1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[169/168]]
| {{monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Thothoru
| Buzurgisma, dhanvantarisma
|-
| 13
| [[196/195]]
| {{monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[2197/2187]]
| {{monzo| 0 -7 0 0 0 3 }}
| 7.90
| Satritho
| Threedie comma
|-
| 13
| [[2080/2079]]
| {{monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Tholuruyo
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[273/272]]
| {{monzo| -4 1 0 1 0 1 -1 }}
| 6.35
| Suthozo
| Tannisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[76/75]]
| {{monzo| 2 -1 -2 0 0 0 0 1 }}
| 22.93
| Nogugu
| Large undevicesimal ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[133/132]]
| {{monzo| -2 -1 0 1 -1 0 0 1 }}
| 13.07
| Noluzo
| Minithirdma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
<references group="note" />

=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 12et are not included. 

{| class="wikitable center-all left-5 left-6"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
! Mos scales
|-
| 1
| 5\36
| 166.67
| 10/9
| [[Squirrel]] (36), [[coendou]] (36c) 
| [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]], [[7L&nbsp;15s]], [[7L&nbsp;22s]]
|-
| 1
| 7\36
| 233.33
| 8/7
| [[Slendric]] / [[mothra]] / [[guiron]]
| [[1L&nbsp;4s]], [[1L&nbsp;5s]], [[5L&nbsp;1s]], [[5L&nbsp;6s]], [[5L&nbsp;11s]], [[5L&nbsp;16s]], [[5L&nbsp;21s]], [[5L&nbsp;26s]]
|-
| 1
| 11\36
| 366.67
| 
| 
| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[3L&nbsp;7s]], [[10L&nbsp;3s]], [[13L&nbsp;10s]]
|-
| 1
| 13\36
| 433.33
| 
| 
| [[2L&nbsp;1s]], [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[3L&nbsp;8s]], [[11L&nbsp;3s]], [[11L&nbsp;14s]]
|-
| 1
| 17\36
| 566.67
| 7/5
| [[Liese]], [[pycnic]] (36c)
| [[2L&nbsp;1s]], [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], [[2L&nbsp;9s]], …, [[2L&nbsp;15s]], [[17L&nbsp;2s]]
|-
| 2
| 5\36
| 166.67
| 10/9
| [[Hedgehog]] (36ceff), [[echidna]] (36)
| [[2L&nbsp;4s]], [[6L&nbsp;2s]], [[8L&nbsp;6s]], [[14L&nbsp;8s]]
|-
| 2
| 7\36
| 233.33
| 8/7
| [[Baladic]] / [[echidnic]]
| [[4L&nbsp;2s]], [[6L&nbsp;4s]], [[10L&nbsp;6s]], [[10L&nbsp;16s]]
|-
| 3
| 5\36
| 166.67
| 
| 
| [[6L&nbsp;3s]], [[6L&nbsp;9s]], [[15L&nbsp;6s]]
|-
| 4
| 2\36
| 66.67
| 
| 
| [[4L&nbsp;4s]], [[4L&nbsp;8s]], [[4L&nbsp;12s]], [[16L&nbsp;4s]]
|-
| 4
| 4\36
| 133.33
| 
| 
| [[4L&nbsp;4s]], [[8L&nbsp;4s]], [[8L&nbsp;12s]], [[8L&nbsp;20s]] 
|-
| 6
| 1\36
| 33.33
| 
| 
| 
|-
| 9
| 15\36<br>(1\36)
| 500.00<br>(33.33)
| 4/3<br>(36/35)
| [[Niner]]
| [[9L&nbsp;9s]]
|-
| 12
| 7\36<br>(1\36)
| 233.33<br>(33.33)
| 8/7<br>(64/63)
| [[Catler]]
| [[12L&nbsp;12s]]
|-
| 18
| 1\36
| 33.33
| 
| 
| 
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 36edo tunings.

; [[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 2{{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)}}

{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
|+ style="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, 83, 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}}

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

'''Hedgehog'''
* [[Hedgehog]][14] MOS: '''3 2 3 2 3 2 3 3 2 3 2 3 2 3'''
* Palace{{idio}} (subset of Hedgehog[14]): '''5 5 5 6 5 5 5'''

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

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

== Tuning by ear ==
After [[9edo]] and [[19edo]], 36edo is one of the easiest tuning systems to recreate tuning solely by ear, due to the way its purest intervals are arranged. If you view it as a lattice of 4 9edo's (which can be formed by stacking 9 [[7/6]]'s with a comma of less than 2 cents) connected by [[3/2]]'s, you get a tonal diamond with a wolf note between the extreme corners of approximately 7.7 cents, about the same relative error as stacking 12 perfect 3/2's compared to [[12edo]] and obviously a much smaller absolute error. It's a good system to use on large organs that have three manuals.

Tuning file: [[9x4just]]

== 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).

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

; [[Miroslav Beinhauer]]
* [https://subrosalabel.bandcamp.com/album/pieces-for-sixth-tone-harmonium ''Pieces For Sixth-Tone Harmonium''] (2024) Full album in 36edo

; [[Ivan Bratt]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/boomers.mp3 ''Boomers'']

; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=psvrsa10-Wo ''36edo jam''] (2025)

; [[E8 Heterotic]]
* [https://youtu.be/VRbXae4L00A?si=YLuyLAP6mxzbBC45 "Elements - Lightning"] from ''Elements'' (2020)

; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=cnmbAz-BF8A ''Playing Children''] (2022) – for three pianos detuned by a sixth-tone from each other

; [[Joe Hayseed]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/hay.mp3 ''Hay'']

; [[Herman Klein]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3 ''Something''] (2022)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=hTolMEjqRq0 ''Arietta with 5 Variations'', for Organ] (2024)

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

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

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

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

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/36edo/20120418-36edo.mp3 ''Thoughts in Legolas Tuning'']

; [[Nick Vuci]]
* [https://en.xen.wiki/w/File:NickVuci-20220903-36edo-Fantasy_for_Sixth_Tone_Harmonium.mp3 ''Fantasy for Sixth-Tone Harmonium''] (2022) – [https://en.xen.wiki/images/9/91/NickVuci-20220903-36edo-Fantasy_for_Sixth_Tone_Harmonium.mp3 play]

; [[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

[[Category:Listen]]
[[Category:Meantone]]
[[Category:Melpomene]]
[[Category:Subgroup temperaments]]