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36edo: Difference between revisions

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Relegate much of the octave stretch section to a related tunings section since some of these aren't even suitable for a stretched or compressed tuning. Unify precision. Style
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=== Octave stretch ===
=== 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).
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 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).
 
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"
! 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 & 101ed7 ^
| 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
|}
 
^ Identical within 0.1 cents.


=== Additional properties ===
=== Additional properties ===
Line 1,077: Line 1,034:


Tuning file: [[9x4just]]
Tuning file: [[9x4just]]
== Related equal tunings ==
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
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).
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
| 33.547 || 23 || 31 || 5 || 41 || 27 || 36 || 57 || 83 || 100 || 124
|-
! 36ed257/128
| 33.521 || 20 || 26 || 12 || 50 || 15 || 36 || 57 || 83 || 101 || 124
|-
! 36ed513/256
| 33.427 || 10 || 11 || 35 || 23 || 18 || 36 || 57 || 83 || 101 || 124
|-
! 57edt
| 33.368 || 4 || 0 || 50 || 5 || 40 || 36 || 57 || 83 || 101 || 124
|-
! 155zpi
| 33.350 || 2 || 3 || 45 || 1 || 48 || 36 || 57 || 84 || 101 || 124
|-
! 101ed7
| 33.355 || 2 || 2 || 46 || 0 || 46 || 36 || 57 || 84 || 101 || 124
|-
! 36edo
| '''33.333''' || '''0'''|| '''6'''|| '''40'''|| '''8'''|| '''45'''|| '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125'''
|-
! 36ed511/256
| 33.239 || 10 || 22 || 18 || 35 || 11 || 36 || 57 || 84 || 101 || 125
|-
! 156zpi
| 33.152 || 20 || 37 || 5 || 38 || 23 || 36 || 57 || 84 || 102 || 125
|-
! 36ed255/128
| 33.145 || 21 || 38 || 6 || 36 || 25 || 36 || 57 || 84 || 102 || 125
|}


== Instruments ==
== Instruments ==
Retrieved from "https://en.xen.wiki/w/36edo"