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__FORCETOC__
__FORCETOC__


= Theory =
== Theory ==
<b>36edo</b>, by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.
'''36edo''', by definition, divides the 2:1 octave into 36 equal steps, each of which is exactly 33⅓ cents.


36 is a highly composite number, factoring into 2×2×3×3. Since 36 is divisible by 12, it contains the overly-familiar [[12edo|12edo]] as a subset. It divides 12edo's 100-cent half step into three microtonal steps of approximately 33 cents, which could be called "sixth tones." 36edo also contains [[18edo|18edo]] ("third tones") and [[9edo|9edo]] ("two-thirds tones") as subsets, not to mention the [[6edo|6edo]] whole tone scale, [[4edo|4edo]] full-diminished seventh chord, and the [[3edo|3edo]] augmented triad, all of which are present in 12edo.
36 is a highly composite number, factoring into 2×2×3×3. 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 cents, 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 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']). Three 12edo instruments could play the entire gamut.
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 cents, one can arrive at a 24-tone subset of 36 edo (see, for instance, Jacob Barton's piece for two clarinets, [http://www.jacobbarton.net/2010/02/de-quinin-for-two-clarinets/ De-quinin']). Three 12edo instruments could play the entire gamut.


For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [http://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [http://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [http://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 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|72edo]] does in the full [[17-limit|17-limit]].
For those interested in approximations to just intonation, 36edo offers no improvement over 12edo in the 5-limit, since its nearest approximation to 5:4 is the overly-familiar 400-cent sharp third. However, it excels at approximations involving 3 and 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[just intonation subgroup]], 36edo's single degree of around 33 cents serves a double function as 49:48, the so-called [http://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36 cents, and as 64:63, the so-called [http://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27 cents. Meanwhile, its second degree functions as 28:27, the so-called [http://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (which = 49:48 x 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]].


The 36edo 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 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|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, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.
The 36edo 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 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, in the 17-limit 51/50, and in the 19-limit 76/75, 91/90 and 96/95.


As a 5-limit temperament, the patent val for 36edo is [[Wedgies_and_Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9&gt; is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.
As a 5-limit temperament, the patent val for 36edo is [[Wedgies and Multivals|contorted]], meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is &lt;36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9&gt; is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a [[transversal]] for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val.


Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.
Heinz Bohlen proposed it as a suitable temperament for approximating his 833-cents scale.


= Intervals =
== Intervals ==
{| class="wikitable"
{| class="wikitable center-all right-2"
|-
|-
! Degree
! Degree
![[cent|cents]]
! [[cent|cents]]
! Approximate
! Approximate <br> ratios of 2.3.7
 
! Additional ratios <br> of 2.3.7.13.17
ratios of 2.3.7
! colspan="3" | [[Ups and Downs Notation|ups and downs <br> notation]]
! Additional ratios
 
of 2.3.7.13.17
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:right;" | 0.00
| 0.00
| style="text-align:center;" | 1/1
| 1/1
| style="text-align:center;" |  
|  
| style="text-align:center;" | P1
| P1
| style="text-align:center;" | perfect unison
| perfect unison
| style="text-align:center;" | D
| D
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:right;" | 33.33
| 33.33
| style="text-align:center;" | 64/63, [[49/48|49/48]]
| 64/63, [[49/48]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^1
| ^1
| style="text-align:center;" | up unison
| up unison
| style="text-align:center;" | ^D
| ^D
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:right;" |66.67
| style="text-align:right;" |66.67
| style="text-align:center;" | [[28/27|28/27]]
| [[28/27]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | vm2
| vm2
| style="text-align:center;" | downminor 2nd
| downminor 2nd
| style="text-align:center;" | vEb
| vEb
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:right;" | 100
| 100
| style="text-align:center;" | 256/243
| 256/243
| style="text-align:center;" | [[17/16|17/16]], [[18/17|18/17]]
| [[17/16]], [[18/17]]
| style="text-align:center;" | m2
| m2
| style="text-align:center;" | minor 2nd
| minor 2nd
| style="text-align:center;" | Eb
| Eb
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:right;" | 133.33
| 133.33
| style="text-align:center;" | 243/224
| 243/224
| style="text-align:center;" | [[14/13|14/13]], [[13/12|13/12]]
| [[14/13]], [[13/12]]
| style="text-align:center;" | ^m2
| ^m2
| style="text-align:center;" | upminor 2nd
| upminor 2nd
| style="text-align:center;" | ^Eb
| ^Eb
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:right;" | 166.67
| 166.67
| style="text-align:center;" | [[54/49|54/49]]
| [[54/49]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vE
| vE
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:right;" | 200
| 200
| style="text-align:center;" | [[9/8|9/8]]
| [[9/8]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | M2
| M2
| style="text-align:center;" | major 2nd
| major 2nd
| style="text-align:center;" | E
| E
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:right;" | 233.33
| 233.33
| style="text-align:center;" | [[8/7|8/7]]
| [[8/7]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^M2
| ^M2
| style="text-align:center;" | upmajor 2nd
| upmajor 2nd
| style="text-align:center;" | ^E
| ^E
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:right;" | 266.67
| 266.67
| style="text-align:center;" | [[7/6|7/6]]
| [[7/6]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | vm3
| vm3
| style="text-align:center;" | downminor 3rd
| downminor 3rd
| style="text-align:center;" | vF
| vF
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:right;" | 300
| 300
| style="text-align:center;" | [[32/27|32/27]]
| [[32/27]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | m3
| m3
| style="text-align:center;" | minor 3rd
| minor 3rd
| style="text-align:center;" | F
| F
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:right;" | 333.33
| 333.33
| style="text-align:center;" | 98/81
| 98/81
| style="text-align:center;" | [[17/14|17/14]]
| [[17/14]]
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^F
| ^F
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:right;" | 366.67
| 366.67
| style="text-align:center;" | 243/196
| 243/196
| style="text-align:center;" | [[16/13|16/13]], [[26/21|26/21]], [[21/17|21/17]]
| [[16/13]], [[26/21]], [[21/17]]
| style="text-align:center;" | vM3
| vM3
| style="text-align:center;" | downmajor 3rd
| downmajor 3rd
| style="text-align:center;" | vF#
| vF#
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:right;" | 400
| 400
| style="text-align:center;" | [[81/64|81/64]]
| [[81/64]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | M3
| M3
| style="text-align:center;" | major 3rd
| major 3rd
| style="text-align:center;" | F#
| F#
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:right;" | 433.33
| 433.33
| style="text-align:center;" | [[9/7|9/7]]
| [[9/7]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^M3
| ^M3
| style="text-align:center;" | upmajor 3rd
| upmajor 3rd
| style="text-align:center;" | ^F#
| ^F#
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:right;" | 466.67
| 466.67
| style="text-align:center;" | [[64/49|64/49]], [[21/16|21/16]]
| [[64/49]], [[21/16]]
| style="text-align:center;" | [[17/13|17/13]]
| [[17/13]]
| style="text-align:center;" | v4
| v4
| style="text-align:center;" | down 4th
| down 4th
| style="text-align:center;" | vG
| vG
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:right;" | 500.00
| 500.00
| style="text-align:center;" | [[4/3|4/3]]
| [[4/3]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | P4
| P4
| style="text-align:center;" | 4th
| 4th
| style="text-align:center;" | G
| G
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:right;" | 533.33
| 533.33
| style="text-align:center;" | [[49/36|49/36]]
| [[49/36]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^4
| ^4
| style="text-align:center;" | up 4th
| up 4th
| style="text-align:center;" | ^G
| ^G
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:right;" | 566.67
| 566.67
| style="text-align:center;" |  
|  
| style="text-align:center;" | [[18/13|18/13]]
| [[18/13]]
| style="text-align:center;" | vA4
| vA4
| style="text-align:center;" | downaug 4th
| downaug 4th
| style="text-align:center;" | vG#
| vG#
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:right;" | 600
| 600
| style="text-align:center;" |  
|  
| style="text-align:center;" |  
|  
| style="text-align:center;" | A4, d5
| A4, d5
| style="text-align:center;" | aug 4th, dim 5th
| aug 4th, dim 5th
| style="text-align:center;" | G#, Ab
| G#, Ab
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:right;" | 633.33
| 633.33
| style="text-align:center;" |  
|  
| style="text-align:center;" | [[13/9|13/9]]
| [[13/9]]
| style="text-align:center;" | ^d5
| ^d5
| style="text-align:center;" | updim 5th
| updim 5th
| style="text-align:center;" | ^Ab
| ^Ab
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:right;" | 666.67
| 666.67
| style="text-align:center;" | 72/49
| 72/49
| style="text-align:center;" |  
|  
| style="text-align:center;" | v5
| v5
| style="text-align:center;" | down 5th
| down 5th
| style="text-align:center;" | vA
| vA
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:right;" | 700
| 700
| style="text-align:center;" | [[3/2|3/2]]
| [[3/2]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | P5
| P5
| style="text-align:center;" | 5th
| 5th
| style="text-align:center;" | A
| A
|-
|-
| style="text-align:center;" | 22
| 22
| style="text-align:right;" | 733.33
| 733.33
| style="text-align:center;" | [[49/32|49/32]], [[32/21|32/21]]
| [[49/32]], [[32/21]]
| style="text-align:center;" | [[26/17|26/17]]
| [[26/17]]
| style="text-align:center;" | ^5
| ^5
| style="text-align:center;" | up fifth
| up fifth
| style="text-align:center;" | ^A
| ^A
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:right;" | 766.67
| 766.67
| style="text-align:center;" | [[14/9|14/9]]
| [[14/9]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | vm6
| vm6
| style="text-align:center;" | downminor 6th
| downminor 6th
| style="text-align:center;" | vBb
| vBb
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:right;" | 800
| 800
| style="text-align:center;" | [[128/81|128/81]]
| [[128/81]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | m6
| m6
| style="text-align:center;" | minor 6th
| minor 6th
| style="text-align:center;" | Bb
| Bb
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:right;" | 833.33
| 833.33
| style="text-align:center;" | 392/243
| 392/243
| style="text-align:center;" | [[13/8|13/8]], [[21/13|21/13]], [[34/21|34/21]]
| [[13/8]], [[21/13]], [[34/21]]
| style="text-align:center;" | ^m6
| ^m6
| style="text-align:center;" | upminor 6th
| upminor 6th
| style="text-align:center;" | ^Bb
| ^Bb
|-
|-
| style="text-align:center;" | 26
| 26
| style="text-align:right;" | 866.67
| 866.67
| style="text-align:center;" | 81/49
| 81/49
| style="text-align:center;" | [[28/17|28/17]]
| [[28/17]]
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | downmajor 6th
| downmajor 6th
| style="text-align:center;" | vB
| vB
|-
|-
| style="text-align:center;" | 27
| 27
| style="text-align:right;" | 900
| 900
| style="text-align:center;" | [[27/16|27/16]]
| [[27/16]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | M6
| M6
| style="text-align:center;" | major 6th
| major 6th
| style="text-align:center;" | B
| B
|-
|-
| style="text-align:center;" | 28
| 28
| style="text-align:right;" | 933.33
| 933.33
| style="text-align:center;" | [[12/7|12/7]]
| [[12/7]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^M6
| ^M6
| style="text-align:center;" | upmajor 6th
| upmajor 6th
| style="text-align:center;" | ^B
| ^B
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:right;" | 966.67
| 966.67
| style="text-align:center;" | [[7/4|7/4]]
| [[7/4]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | vm7
| vm7
| style="text-align:center;" | downminor 7th
| downminor 7th
| style="text-align:center;" | vC
| vC
|-
|-
| style="text-align:center;" | 30
| 30
| style="text-align:right;" | 1000
| 1000
| style="text-align:center;" | [[16/9|16/9]]
| [[16/9]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | m7
| m7
| style="text-align:center;" | minor 7th
| minor 7th
| style="text-align:center;" | C
| C
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:right;" | 1033.33
| 1033.33
| style="text-align:center;" | 49/27
| 49/27
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | upminor 7th
| upminor 7th
| style="text-align:center;" | ^C
| ^C
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:right;" | 1066.67
| 1066.67
| style="text-align:center;" | 448/243
| 448/243
| style="text-align:center;" | [[13/7|13/7]], [[24/13|24/13]]
| [[13/7]], [[24/13]]
| style="text-align:center;" | vM7
| vM7
| style="text-align:center;" | downmajor 7th
| downmajor 7th
| style="text-align:center;" | vC#
| vC#
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:right;" | 1100
| 1100
| style="text-align:center;" | [[243/128|243/128]]
| [[243/128]]
| style="text-align:center;" | [[32/17|32/17]], [[17/9|17/9]]
| [[32/17]], [[17/9]]
| style="text-align:center;" | M7
| M7
| style="text-align:center;" | major 7th
| major 7th
| style="text-align:center;" | C#
| C#
|-
|-
| style="text-align:center;" | 34
| 34
| style="text-align:right;" | 1133.33
| 1133.33
| style="text-align:center;" | [[27/14|27/14]]
| [[27/14]]
| style="text-align:center;" |  
|  
| style="text-align:center;" | ^M7
| ^M7
| style="text-align:center;" | upmajor 7th
| upmajor 7th
| style="text-align:center;" | ^C#
| ^C#
|-
|-
| style="text-align:center;" | 35
| 35
| style="text-align:right;" | 1166.67
| 1166.67
| style="text-align:center;" | 63/32, 96/49
| 63/32, 96/49
| style="text-align:center;" |  
|  
| style="text-align:center;" | v8
| v8
| style="text-align:center;" | down 8ve
| down 8ve
| style="text-align:center;" | vD
| vD
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:right;" | 1200.00
| 1200.00
| style="text-align:center;" | 2/1
| 2/1
| style="text-align:center;" |  
|  
| style="text-align:center;" | P8
| P8
| style="text-align:center;" | 8ve
| 8ve
| style="text-align:center;" | D
| 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|Ups and Downs Notation - Chords and Chord Progressions]].
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


=Relation to 12edo=
== Relation to 12edo ==


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


The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike [[24edo|24edo]], which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals.
The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. Unlike [[24edo]], which has genuinely foreign intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third), the new intervals in 36edo all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals.


An easy way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, '''A''' is 33.333 cents above '''<span style="background-color: #6ee8e8; color: #071ac7;">A</span>''' and 33.333 cents below '''<span style="background-color: #eda2a2; color: #ff0000;">A</span>'''. 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).
An easy way of notating 36edo (at least for people who aren't colorblind) is to use colors. For example, '''A''' is 33.333 cents above '''<span style="background-color: #6ee8e8; color: #071ac7;">A</span>''' and 33.333 cents below '''<span style="background-color: #eda2a2; color: #ff0000;">A</span>'''. 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).
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People with perfect (absolute) pitch often have a harder time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding bad). 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 too.
People with perfect (absolute) pitch often have a harder time listening to xenharmonic and non-12edo scales, which is due to their ability to memorize and become accustomed to the pitches and intervals of 12edo (which results in other pitches and intervals sounding bad). 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 too.


==="Quark"===
=== "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, I (Mason Green) propose referring to the 33.333-cent sixth-tone interval as a "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, I (Mason Green) propose referring to the 33.333-cent sixth-tone interval as a "quark".


=JI Approximations=
== JI Approximations ==


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


3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.
3/2 = 701.955... cents; 21 degrees of 36edo = 700 cents.
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128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.
128/81 = 792.180... cents; 24 degrees of 36edo = 800 cents.


==7-limit approximations:==
=== 7-limit approximations: ===


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


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


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


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


=Tuning by Ear=
== Tuning by Ear ==
<ul><li>After [[9edo|9edo]] and [[19edo|19edo]], 36edo is one of the easiest tuning systems to recreate tuning solely by ear, due to the way it's purest intervals are arranged. If you view it as a lattice of 4 9edo's (which can be formed by stacking 9 [[7/6|7/6's]] with a comma of less than 2 cents) connected by [[3/2|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|12edo]] and obviously a much smaller absolute error. In addition, this kind of tempering makes major thirds 6.3 cents closer to just in the majority of keys, making them sound sweeter than equal temperament. It's a good system to use on large organs that have three manuals. </li></ul>
After [[9edo]] and [[19edo]], 36edo is one of the easiest tuning systems to recreate tuning solely by ear, due to the way it's 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. In addition, this kind of tempering makes major thirds 6.3 cents closer to just in the majority of keys, making them sound sweeter than equal temperament. It's a good system to use on large organs that have three manuals.


=Music=
== Music ==
* [https://soundcloud.com/overtoneshock/exponentially-more-lost-and-forgetful-36-edo?in=overtoneshock/sets/xenharmonic-microtonal Exponentially More Lost and Forgetful] by [[Stephen Weigel]] played by flautists Orlando Cela and Wei Zhao
* [https://soundcloud.com/overtoneshock/exponentially-more-lost-and-forgetful-36-edo?in=overtoneshock/sets/xenharmonic-microtonal Exponentially More Lost and Forgetful] by [[Stephen Weigel]] played by flautists Orlando Cela and Wei Zhao
* [http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3 Something]</span> by [[Herman Klein]]
* [http://micro.soonlabel.com/gene_ward_smith/36edo/something.mp3 Something]</span> by [[Herman Klein]]
Retrieved from "https://en.xen.wiki/w/36edo"