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:

• Temperament Calculator by Sintel
• x31eq Temperament Finder by Graham Breed

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
• 1 other equal tuning of any kind at all (optional, can be left out)

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.

Todo:

• Make the mapping info take up less horizontal space
• Add a column for amount of stretch (maybe as a %?)

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)
		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
36ed513/256	33.427	+3.4	+3.4	-11.9	+7.3	-6.4	+5.3	36, 57, 83, 101, 124, 133
57edt	33.368	+1.2	0	+16.6	+1.3	-13.7	-2.6	36, 57, 84, 101, 124, 133
2.3.7.13 WE	33.361	+1.0	-0.4	+16.0	+0.6	-14.6	-3.5	36, 57, 84, 101, 124, 133
36edo	33.333	0	-2.0	+13.7	-2.2	+15.3	-7.2	36, 57, 84, 101, 125, 133
13lim WE	33.302	-1.1	-3.7	+11.1	-5.3	+11.4	-11.4	36, 57, 84, 101, 125, 133
11lim WE	33.286	-1.7	-4.7	+9.7	-6.9	+9.4	-13.5	36, 57, 84, 101, 125, 133
156zpi	33.152	-6.5	-12.3	-1.5	+12.7	-7.3	+1.8	36, 57, 84, 102, 125, 134

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.

57edt or 2.3.7.13 WE 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.

Pure-octaves 36edo

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

11-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.