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)}}

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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]].

~~=== Octave stretch ===~~

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).

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

Of the compressed-octave tunings listed, ''36ed513/256'', with a step size about 33.25 cents, performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error).

Of the stretched-octave tunings listed, ''36ed511/256'', with a step size about 33.4 cents, performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error).

~~{| class="wikitable sortable"~~

~~! rowspan="2" | Name of tuning !! rowspan="2" | Step size (cents) !! colspan="5" | Error (% step size) !! colspan="5" | Mapping (# steps)~~

|-

~~! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11 !! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11~~

|-

~~! 154zpi~~

~~| 33.55 || 23 || 31 || 5 || 41 || 27 || 36 || 57 || 83 || 100 || 124~~

|-

~~! 36ed257/128~~

~~| 33.52 || 20 || 26 || 12 || 50 || 15 || 36 || 57 || 83 || 101 || 124~~

|-

~~! 36ed513/256~~

~~| 33.43 || 10 || 11 || 35 || 23 || 18 || 36 || 57 || 83 || 101 || 124~~

|-

~~! 57edt~~

~~| 33.37 || 4 || 0 || 50 || 5 || 40 || 36 || 57 || 83 || 101 || 124~~

|-

~~! 155zpi~~

~~| 33.35 || 2 || 3 || 45 || 1 || 48 || 36 || 57 || 84 || 101 || 124~~

|-

~~! 36EDO~~

~~| '''33.33'''|| '''0'''|| '''6'''|| '''40'''|| '''8'''|| '''45'''|| '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125'''~~

|-

~~! 36ed511/256~~

~~| 33.24 || 10 || 22 || 18 || 35 || 11 || 36 || 57 || 84 || 101 || 125~~

|-

~~! 156zpi~~

~~| 33.15 || 20 || 37 || 5 || 38 || 23 || 36 || 57 || 84 || 102 || 125~~

|-

~~! 36ed255/128~~

~~| 33.145 || 21 || 38 || 6 || 36 || 25 || 36 || 57 || 84 || 102 || 125~~

|}

=== Subsets and supersets ===

Line 1,056:

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=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<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 ==

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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'''

@@ 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)}}
@@ Line 22: / 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]].
-=== Octave stretch ===
-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).
-What follows is a comparison of stretched- and compressed-octave 36edo tunings.
-Of the compressed-octave tunings listed, ''36ed513/256'', with a step size about 33.25 cents, performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error).
-Of the stretched-octave tunings listed, ''36ed511/256'', with a step size about 33.4 cents, performs best in this comparison, approximating all 11-limit primes with less than 36% relative error (<12 cents error).
-{| class="wikitable sortable"
-! rowspan="2" | Name of tuning !! rowspan="2" | Step size (cents) !! colspan="5" | Error (% step size) !! colspan="5" | Mapping (# steps)
-|-
-! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11 !! Prime 2 !! Prime 3 !! Prime 5 !! Prime 7 !! Prime 11
-|-
-! 154zpi
-| 33.55 || 23 || 31 || 5 || 41 || 27 || 36 || 57 || 83 || 100 || 124
-|-
-! 36ed257/128
-| 33.52 || 20 || 26 || 12 || 50 || 15 || 36 || 57 || 83 || 101 || 124
-|-
-! 36ed513/256
-| 33.43 || 10 || 11 || 35 || 23 || 18 || 36 || 57 || 83 || 101 || 124
-|-
-! 57edt
-| 33.37 || 4 || 0 || 50 || 5 || 40 || 36 || 57 || 83 || 101 || 124
-|-
-! 155zpi
-| 33.35 || 2 || 3 || 45 || 1 || 48 || 36 || 57 || 84 || 101 || 124
-|-
-! 36EDO
-| '''33.33'''|| '''0'''|| '''6'''|| '''40'''|| '''8'''|| '''45'''|| '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125'''
-|-
-! 36ed511/256
-| 33.24 || 10 || 22 || 18 || 35 || 11 || 36 || 57 || 84 || 101 || 125
-|-
-! 156zpi
-| 33.15 || 20 || 37 || 5 || 38 || 23 || 36 || 57 || 84 || 102 || 125
-|-
-! 36ed255/128
-| 33.145 || 21 || 38 || 6 || 36 || 25 || 36 || 57 || 84 || 102 || 125
-|}
 === Subsets and supersets ===
@@ Line 1,056: / 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 1,061: / 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'''