36edo: Difference between revisions

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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 ===

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

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.

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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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| 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]].

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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|35.98|apx=val|header=none|tag=none|title=15-odd-limit intervals by 36e 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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| Go comma

|}

=== Rank-2 temperaments ===

~~''See [[regular temperament]] for more about what all this means and how to use it.''~~

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

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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

== 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=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=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 (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 ==

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'''Catler'''

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

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

* 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'''

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; [[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]]

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; [[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}}

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; [[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 12: / Line 12: @@
 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).
+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 ===
-''See [[regular temperament]] for more about what all this means and how to use it.''
 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.
@@ Line 24: / 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 41: / 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 376: / 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]].
@@ Line 431: / Line 431: @@
 == 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 491: / Line 487: @@
 /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}}
-=== 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 ==
@@ Line 585: / Line 571: @@
 === Uniform maps ===
-{{Uniform map|edo=36}}
+{{Uniform map|min=35.8|max=36.2}}
 === Commas ===
@@ Line 914: / Line 900: @@
 | Go comma
 |}
+<references group="note" />
 === Rank-2 temperaments ===
-''See [[regular temperament]] for more about what all this means and how to use it.''
 Note: 5-limit temperaments supported by 12et are not included.
@@ Line 1,029: / Line 1,014: @@
 |}
 <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 ==
@@ Line 1,034: / Line 1,085: @@
 '''Catler'''
-* [[Lost spirit]] (approximated from [[Meantone]] in [[31edo]]): '''9 6 2 4 7 2 6'''
+* [[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 (subset of Hedgehog[14]): '''5 5 5 6 5 5 5'''
+* 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'''
@@ Line 1,062: / Line 1,112: @@
 ; [[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]]
@@ Line 1,083: / Line 1,136: @@
 ; [[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}}
@@ Line 1,095: / Line 1,151: @@
 ; [[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]]