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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 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|prec=2|columns=14}}
{{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)}}
{{Harmonics in equal|36|intervals=odd|columns=14|prec=2|start=15|collapsed=true|title=Approximation of odd harmonics in 36edo (continued)}}
Line 1,015: Line 1,015:
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct


== Scales ==
== Octave stretch or compression ==
{{main|List of MOS scales in 36edo}}
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
 
'''Catler'''
* [[Lost spirit]] (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'''
 
[[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'''
; [[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)}}


== Tuning by ear ==
; [[57edt]]
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.
* 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)}}


Tuning file: [[9x4just]]
; 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)}}


== Related equal tunings ==
; [[TE|36et, 13-limit TE tuning]]
What follows is a comparison of stretched- and compressed-octave 36edo tunings.  
* 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)}}


Of the stretched-octave tunings listed, ''36ed513/256'', 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 center-all mw-collapsible mw-collapsed"
 
|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings
Of the compressed-octave tunings listed, ''36ed511/256'', 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).
 
The [[edonoi]] scales of [[57edt]] and [[101ed7]] are almost exactly the same as 36edo. They are 36edo with the octave stretched by less than 1{{c}}. Their 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, 57edt or 101ed7 may be worth considering.
 
{| class="wikitable sortable center-all"
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(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
! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents)
| 33.547 || +23 || +31 || -5 || -41 || +27 || 36 || 57 || 83 || 100 || 124
! rowspan="2" | Mapping of primes 2–13 (steps)
|-
|-
! 36ed257/128
! 2 !! 3 !! 5 !! 7 !! 11 !! 13
| 33.521 || +20 || +26 || -12 || +50 || +15 || 36 || 57 || 83 || 101 || 124
|-
|-
! 36ed513/256
! 21edf
| 33.427 || +10 || +11 || -35 || +23 || -18 || 36 || 57 || 83 || 101 || 124
| 1203.351
| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
| 36, 57, 83, 101, 124, 133
|-
|-
! 57edt
! 57edt
| 33.368 || +4 || 0 || +50 || +5 || -40 || 36 || 57 || 83 || 101 || 124
| 1201.235
|-
| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
! 101ed7
| 36, 57, 84, 101, 124, 133
| 33.355 || +2 || -2 || +46 || 0 || -46 || 36 || 57 || 84 || 101 || 124
|-
|-
! 155zpi
! 155zpi
| 33.350 || +2 || -3 || +45 || -1 || -48 || 36 || 57 || 84 || 101 || 124
| 1200.587
| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0
| 36, 57, 83, 101, 124, 133
|-
|-
! 36edo
! 36edo
| '''33.333''' || '''0'''|| '''-6''' || '''+40''' || '''-6''' || '''+46''' || '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125'''
| '''1200.000'''
|-
| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
! 36ed511/256
| '''36, 57, 84, 101, 125, 133'''
| 33.239 || -10 || -22 || +18 || -35 || +11 || 36 || 57 || 84 || 101 || 125
|-
|-
! 156zpi
! 13-limit TE
| 33.152 || -20 || -37 || -5 || +38 || -23 || 36 || 57 || 84 || 102 || 125
| 1198.929
| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1
| 36, 57, 84, 101, 125, 133
|-
|-
! 36ed255/128
! 11-limit TE
| 33.145 || -21 || -38 || -6 || +36 || -25 || 36 || 57 || 84 || 102 || 125
| 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 ==
== Instruments ==
Retrieved from "https://en.xen.wiki/w/36edo"