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 for any given edo:

  • The edo's pure-octaves tuning
  • 1 to 3 nearby edonoi (eg an edt, an edf, an ed5, an ed7, an ed4/3, anything like that)
  • 1 to 2 nearby ZPIs (or any other "infinite harmonics" optimised tuning other than ZPI)
  • 1 to 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
  • 0 to 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
  • 1 other equal tuning of any kind at all (optional)


Additional guidelines for selecting tunings:

  • In total, 5 to 8 tunings should be listed.
  • The selection of tunings should include at least one stretched tuning and at least one compressed tuning.
  • The most stretched tuning should have +15% to +25% relative error on prime 2, the most compressed should have -15 to -25%, and all the other tunings should cover a wide range of possible tunings in between.



Octave stretch or compression

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

Tuning Step size
(cents)		 Prime error (cents) Mapping of primes 2-13 (steps) Stretch
2 3 5 7 11 13
154zpi 33.547 +7.7 +10.2 -1.9 -14.1 +8.5 -12.3 36, 57, 83, 100, 124, 132 +23.1%
36ed513/256 33.427 +3.4 +3.4 -11.9 +7.3 -6.4 +5.3 36, 57, 83, 101, 124, 133 +10.2%
57edt 33.368 +1.2 0 +16.6 +1.3 -13.7 -2.6 36, 57, 84, 101, 124, 133 +3.6%
36edo 33.333 0 -2.0 +13.7 -2.2 +15.3 -7.2 36, 57, 84, 101, 125, 133 0%
13lim WE 33.302 -1.1 -3.7 +11.1 -5.3 +11.4 -11.4 36, 57, 84, 101, 125, 133 -3.3%
11lim WE 33.286 -1.7 -4.7 +9.7 -6.9 +9.4 -13.5 36, 57, 84, 101, 125, 133 -5.1%
156zpi 33.152 -6.5 -12.3 -1.5 +12.7 -7.3 +1.8 36, 57, 84, 102, 125, 134 -19.5%
36ed513/256

Stretching the octave of 36edo by about 3.5 cents results in much improved primes 5 and 11, but a much worse prime 7. This approximates all primes up to 11 within 12 cents. The tuning 36ed513/256 does this.

57edt / 101ed7 / 155zpi / 2.3.7.13 WE-tuned 36edo

If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 cent 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 17 cents. Four almost identical tunings do this: 57edt, 101ed7, 155zpi, and the 2.3.7.13 subgroup WE tuning of 36edo.

Pure-octaves 36edo

Pure-octaves 36edo approximates all primes up to 11 within 16 cents.

11-limit WE 36edo / 13-limit WE 36edo

Compressing the octave of 36edo by about 2 cents results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all primes up to 11 within 10 cents. The 11- and 13-limit WE tunings of 36edo both do this, as do their respective TE tunings.

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