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User:BudjarnLambeth/Draft related tunings section

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Introduction

This is a draft of what a standard "related tunings" section might look like on edo pages, using 36edo as an example.


Useful links for working on this:


Which tunings should be listed:

  • The pure-octaves tuning
  • 1 to 3 nearby edonoi (eg an edt, an edf, an ed5, an ed7, an ed4/3, anything like that)
  • 1 or 2 nearby ZPIs (or any other "infinite harmonics" optimised tuning other than ZPI)
  • 1 or 2 n-limit WE or n-limit TE tunings, valid examples include:
    • 13-limit WE and 13-limit TE
    • 13-limit WE and 11-limit WE
    • 11-limit TE and 7-limit TE
    • 13-limit TE
    • 11-limit WE
    • Anything else along those kind of lines
  • 1 or 2 subgroup WE or TE tunings, valid examples include:
    • 2.3.7 WE and 2.3.7 TE
    • 2.3.5.11 WE and 2.3.5.11.13 WE
    • 2.3.7 TE and 2.3.7.11 TE
    • 2.5.7.11 TE
    • 2.3.7 WE
    • Anything else along those kind of lines


Additional guidelines for selecting tunings:

  • In total, 5 to 7 tunings should be listed.
  • The selection of tunings should include at least one stretched tuning and at least one compressed tuning.
  • The selection of tunings should be different enough from one another that they cover a range of possible mappings.


Todo:

  • Update the set of tunings to follow the tuning selection rules
  • Make the mapping info take up less horizontal space
  • Change error columns to use absolute error (in cents) rather than relative error (%)
  • Add a column for amount of stretch (maybe as a %?)
  • Add columns for prime 13


Octave stretch or compression

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 ¢. Their main usage is to optimise 36edo for use as a 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.

Tuning Step size
(cents)		 Error (% step size) 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
101ed7 33.355 +2 -2 +46 0 -46 36 57 84 101 124
155zpi 33.350 +2 -3 +45 -1 -48 36 57 84 101 124
36edo 33.333 0 -6 +40 -6 +46 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
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