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.

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)}}

Line 1,018:

What follows is a comparison of stretched- and compressed-octave 36edo tunings.

; [[21edf]]

* Step size: 33.426{{c}}, octave size: 1203.351{{c}}

* Step size: 33.426{{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.

* Octave size: 1203.3{{c}}

{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}

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 primes up to 11 within ''12.0{{c}}''. The tuning 21edf does this.

; [[57edt]]

* Step size: 33.368{{c}}

* Step size: 33.368{{c}}, octave size: 1201.235{{c}}

* Octave size: 1201.2{{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)}}

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 primes up to 11 within ''16.6{{c}}''. Five almost-identical tunings do this: 57edt, [[101ed7]], [[zpi|155zpi]], and the [[TE]] and [[WE]] 2.3.7.13 subgroup WE tunings of 36edo.

; 36edo

* Step size: 33.333{{c}}, octave size: 1200.000{{c}}

Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.

; ~~Pure-octaves~~ 36edo

{{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}

* Step size: 33.333{{c}}

{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}

* Octave size: 1200.0{{c}}

~~{{Harmonics in equal|36|2|1|columns=12|collapsed=true}}~~

~~{{Harmonics in equal|36|2|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}~~

Pure-octaves 36edo approximates all ~~primes~~ up to 11 within ''15.3{{c}}''.

~~; [[TE|11-limit TE 36edo]]~~

* Step size: 33.287{{c}}

* Octave size: 1198.3{{c}}

{{Harmonics in ~~cet~~|~~33.287~~|columns=12|collapsed=true~~|title=Approximation of harmonics in 11lim WE-tuned 36edo~~}}

{{Harmonics in ~~cet~~|~~33.287~~|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ~~11lim WE-tuned~~ 36edo (continued)}}

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 primes up to 11 within ''9.7{{c}}''. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings.

; [[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"

|+ ~~Stretched/~~compressed tunings ~~comparison table~~

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

|-

! rowspan="2" | Tuning !! rowspan="2" | ~~Step~~ size<br>(cents) !! colspan="6" | Prime error (cents)

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

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

~~! rowspan="2" | Octave stretch~~

|-

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

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

! 13

|-

! 21edf

| 33.~~426~~

| 1203.351

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

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

~~| +0.275%~~

|-

! 57edt

| 33.~~368 |~~| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6

| 1201.235

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

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

~~| +0.001%~~

|-

! 155zpi

| 33.~~346~~

| 1200.587

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

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

~~| +0.0005%~~

|-

! 36edo

| '''33.~~333~~'''

| '''1200.000'''

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

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

~~| '''0%'''~~

|-

! 13-limit WE

! 13-limit TE

| 33.~~302~~

| 1198.929

| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4

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

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

~~| -0.0009%~~

|-

! 11-limit WE

! 11-limit TE

| 33.~~286~~

| 1198.330

| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5

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

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

~~| -0.00142%~~

|}

@@ Line 13: / Line 13: @@
 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.
+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)}}
@@ Line 1,018: / Line 1,018: @@
 What follows is a comparison of stretched- and compressed-octave 36edo tunings.
+; [[21edf]]
-; [[21edf]]
+* Step size: 33.426{{c}}, octave size: 1203.351{{c}}
-* Step size: 33.426{{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.
-* Octave size: 1203.3{{c}}
+{{Harmonics in equal|21|3|2|columns=11|collapsed=true}}
-{{Harmonics in equal|21|3|2|columns=12|collapsed=true}}
 {{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
-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 primes up to 11 within ''12.0{{c}}''. The tuning 21edf does this.
 ; [[57edt]]
-* Step size: 33.368{{c}}
+* Step size: 33.368{{c}}, octave size: 1201.235{{c}}
-* Octave size: 1201.2{{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|collapsed=true}}
+{{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)}}
-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 primes up to 11 within ''16.6{{c}}''. Five almost-identical tunings do this: 57edt, [[101ed7]], [[zpi|155zpi]], and the [[TE]] and [[WE]] 2.3.7.13 subgroup WE tunings of 36edo.
+; 36edo
+* Step size: 33.333{{c}}, octave size: 1200.000{{c}}
+Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.
-; Pure-octaves 36edo
+{{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}
-* Step size: 33.333{{c}}
+{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}
-* Octave size: 1200.0{{c}}
-{{Harmonics in equal|36|2|1|columns=12|collapsed=true}}
-{{Harmonics in equal|36|2|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
-Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''.
-; [[TE|11-limit TE 36edo]]
-* Step size: 33.287{{c}}
-* Octave size: 1198.3{{c}}
-{{Harmonics in cet|33.287|columns=12|collapsed=true|title=Approximation of harmonics in 11lim WE-tuned 36edo}}
-{{Harmonics in cet|33.287|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11lim WE-tuned 36edo (continued)}}
-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 primes up to 11 within ''9.7{{c}}''. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings.
+; [[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"
-|+ Stretched/compressed tunings comparison table
+|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings
 |-
-! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)
+! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents)
 ! rowspan="2" | Mapping of primes 2–13 (steps)
-! rowspan="2" | Octave stretch
 |-
-! 2 !! 3 !! 5 !! 7 !! 11
+! 2 !! 3 !! 5 !! 7 !! 11 !! 13
-! 13
 |-
 ! 21edf
-| 33.426
+| 1203.351
 | +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
 | 36, 57, 83, 101, 124, 133
-| +0.275%
 |-
 ! 57edt
-| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
+| 1201.235
+| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
 | 36, 57, 84, 101, 124, 133
-| +0.001%
 |-
 ! 155zpi
-| 33.346
+| 1200.587
 | +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0
 | 36, 57, 83, 101, 124, 133
-| +0.0005%
 |-
 ! 36edo
-| '''33.333'''
+| '''1200.000'''
 | '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
 | '''36, 57, 84, 101, 125, 133'''
-| '''0%'''
 |-
-! 13-limit WE
+! 13-limit TE
-| 33.302
+| 1198.929
-| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4
+| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1
 | 36, 57, 84, 101, 125, 133
-| -0.0009%
 |-
-! 11-limit WE
+! 11-limit TE
-| 33.286
+| 1198.330
-| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5
+| −1.7 || −4.6 || +9.8 || −6.8 || +9.5 || −13.4
 | 36, 57, 84, 101, 125, 133
-| -0.00142%
 |}