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

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:

• 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)	Prime error (% step size)						Prime mapping (steps)
		2	3	5	7	11	13	2	3	5	7	11	13
154zpi	33.547	+23	+31	-5	-41	+27		36	57	83	100	124
57edt	33.368	+4	0	+50	+5	-40		36	57	83	101	124
2.3.7.13 WE	33.361
101ed7	33.355	+2	-2	+46	0	-46		36	57	84	101	124
36edo	33.333	0	-6	+40	-6	+46		36	57	84	101	125
13lim WE	33.302
11lim WE	33.286
156zpi	33.152	-20	-37	-5	+38	-23		36	57	84	102	125