User:BudjarnLambeth/Draft related tunings section

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The guidelines

These are draft guidelines for what a standard "related tunings"-type section should look like on edo pages, using 36edo as an example.


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 subgroup TE- or WE-optimal tunings, based on the best choice(s) of subgroup for the edo
  • 1 other equal tuning of any kind at all (optional)

Additional guidelines for selecting tunings:

  • In total, roughly 6 tunings should be listed, give or take a few.
  • The selection of tunings should cover a range of meaningfully different tunings (eg they cover a range of different mappings and/or they approximate different harmonics well or badly).
Further instructions
  • Adding the comparison table at the end is optional.
  • The number of decimal places to use in the comparison table is up to the user's discretion, as long as it is self-consistent within the table.
Where this section should be placed on an edo page

  • Synopsis & infobox
  • (Any foundational introductory subsections)
  • Theory
    • Harmonics
    • (Any short subsections about theory unique to the edo)
    • Additional properties
    • Subsets and supersets
  • Interval table
  • Notation
  • (Any long subsections about theory unique to the edo)
  • Approximation to JI
  • Regular temperament properties
    • Uniform maps
    • Commas
    • Rank-2 temperaments
  • OCTAVE STRETCH OR COMPRESSION
  • Scales
  • (Any subsections about practice unique to the edo)
  • Instruments
  • Music
  • See also
  • Notes
  • Further reading
  • External links

Note: This particular set of headings in this order is only how most edo pages look at the moment, but it might be replaced with a more intuitive standard in the future. If and when that happens, this guideline should be modified to adopt that new standard.

Useful links for working on this

Example (36edo)

Octave stretch or compression

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

21edf
  • Step size: 33.426 ¢, octave size: 1203.351 ¢

Stretching the octave of 36edo by a little over 3 ¢ results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all harmonics up to 16 within 13.4 ¢. The tuning 21edf does this.

Approximation of harmonics in 21edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 -11.9 +6.7 +7.2 +10.1 +6.7 -8.6 -6.4 +10.1
Relative (%) +10.0 +10.0 +20.1 -35.7 +20.1 +21.7 +30.1 +20.1 -25.6 -19.3 +30.1
Steps
(reduced) 		36
(15) 		57
(15) 		72
(9) 		83
(20) 		93
(9) 		101
(17) 		108
(3) 		114
(9) 		119
(14) 		124
(19) 		129
(3)
Approximation of harmonics in 21edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.2 +10.6 -8.6 +13.4 +8.7 +10.1 -16.7 -5.2 +10.6 -3.1 -13.2 +13.4
Relative (%) +15.5 +31.7 -25.6 +40.1 +26.1 +30.1 -49.9 -15.6 +31.7 -9.2 -39.5 +40.1
Steps
(reduced) 		133
(7) 		137
(11) 		140
(14) 		144
(18) 		147
(0) 		150
(3) 		152
(5) 		155
(8) 		158
(11) 		160
(13) 		162
(15) 		165
(18)
57edt
  • Step size: 33.368 ¢, octave size: 1201.235 ¢

If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 ¢ optimises 36edo for that dual-5 usage, while also making slight improvements to primes 3, 7, 11, and 13. This approximates all harmonics up to 16 within 16.6 ¢. Several almost-identical tunings do this: 57edt, 93ed6, 101ed7, 155zpi, and the 2.3.7.13-subgroup TE and WE tunings of 36et.

Approximation of harmonics in 57edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +1.3 +3.7 +0.0 -15.6 -13.7 +2.5
Relative (%) +3.7 +0.0 +7.4 +49.7 +3.7 +3.9 +11.1 +0.0 -46.6 -41.2 +7.4
Steps
(reduced) 		36
(36) 		57
(0) 		72
(15) 		84
(27) 		93
(36) 		101
(44) 		108
(51) 		114
(0) 		119
(5) 		124
(10) 		129
(15)
Approximation of harmonics in 57edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 +2.5 +16.6 +4.9 +0.1 +1.2 +7.7 -14.3 +1.3 -12.5 +10.6 +3.7
Relative (%) -7.9 +7.6 +49.7 +14.8 +0.3 +3.7 +23.2 -42.9 +3.9 -37.5 +31.9 +11.1
Steps
(reduced) 		133
(19) 		137
(23) 		141
(27) 		144
(30) 		147
(33) 		150
(36) 		153
(39) 		155
(41) 		158
(44) 		160
(46) 		163
(49) 		165
(51)
36edo
  • Step size: 33.333 ¢, octave size: 1200.000 ¢

Pure-octaves 36edo approximates all harmonics up to 16 within 15.3 ¢.

Approximation of harmonics in 36edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -2.2 +0.0 -3.9 +13.7 +15.3 -2.0
Relative (%) +0.0 -5.9 +0.0 +41.1 -5.9 -6.5 +0.0 -11.7 +41.1 +46.0 -5.9
Steps
(reduced) 		36
(0) 		57
(21) 		72
(0) 		84
(12) 		93
(21) 		101
(29) 		108
(0) 		114
(6) 		120
(12) 		125
(17) 		129
(21)
Approximation of harmonics in 36edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.2 -2.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 -4.1 +15.3 +5.1 -2.0
Relative (%) -21.6 -6.5 +35.2 +0.0 -14.9 -11.7 +7.5 +41.1 -12.3 +46.0 +15.2 -5.9
Steps
(reduced) 		133
(25) 		137
(29) 		141
(33) 		144
(0) 		147
(3) 		150
(6) 		153
(9) 		156
(12) 		158
(14) 		161
(17) 		163
(19) 		165
(21)
36et, 13-limit TE tuning
  • Step size: 33.304 ¢, octave size: 1198.929 ¢

Compressing the octave of 36edo by about 2 ¢ results in much improved primes 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 11.6 ¢. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.

Approximation of harmonics in 13-limit TE tuning of 36et
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.1 -3.7 -2.1 +11.2 -4.7 -5.2 -3.2 -7.3 +10.1 +11.6 -5.8
Relative (%) -3.2 -11.0 -6.4 +33.6 -14.2 -15.5 -9.6 -21.9 +30.4 +34.9 -17.4
Step 36 57 72 84 93 101 108 114 120 125 129
Approximation of harmonics in 13-limit TE tuning of 36et (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -11.1 -6.2 +7.5 -4.3 -9.3 -8.4 -2.1 +9.0 -8.8 +10.6 +0.2 -6.9
Relative (%) -33.5 -18.7 +22.6 -12.9 -28.0 -25.1 -6.2 +27.2 -26.5 +31.7 +0.6 -20.6
Step 133 137 141 144 147 150 153 156 158 161 163 165
Comparison of stretched and compressed tunings
Tuning Octave size
(cents)		 Prime error (cents) Mapping of primes 2–13 (steps)
2 3 5 7 11 13
21edf 1203.351 +3.3 +3.3 −12.0 +7.2 −6.5 +5.1 36, 57, 83, 101, 124, 133
57edt 1201.235 +1.2 0.0 +16.6 +1.3 −13.7 −2.6 36, 57, 84, 101, 124, 133
155zpi 1200.587 +0.6 −1.0 +15.1 −0.5 −16.0 −5.0 36, 57, 83, 101, 124, 133
36edo 1200.000 0.0 −2.0 +13.7 −2.2 +15.3 −7.2 36, 57, 84, 101, 125, 133
13-limit TE 1198.929 −1.1 −3.7 +11.2 −5.2 +11.6 −11.1 36, 57, 84, 101, 125, 133
11-limit TE 1198.330 −1.7 −4.6 +9.8 −6.8 +9.5 −13.4 36, 57, 84, 101, 125, 133

Blank template

Octave stretch or compression

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

ZPINAME
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
EDONOI
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
ETNAME, SUBGROUP WE tuning
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
EDONAME
  • Step size: NNN ¢, octave size: NNN ¢

Pure-octaves EDONAME approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in EDONAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
ETNAME, SUBGROUP WE tuning
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55
EDONOI
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
ZPINAME
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Step 44 46 47 48 49 50 51 52 53 54 54 55

Plan for roll-out

Done

To-do

Things to note
  • When rolling this out try not to delete existing body text but instead rework it where possible.
  • This section will not replace any "n-edo and octave stretch" pages. Still, add this section to the relevant edo page, but also link to the "n-edo and octave stretch" page at the top of this section, using the see also template, e.g.: "{{See also|36edo and octave stretch}}".

What to do with edonoi pages that are very close to these edos

  • Edt and edf pages should be permanently kept
  • Other edonoi pages should be temporarily kept until all notable information from their respective pages has been added to:
    • The "octave stretch and compression" section of the edo page.

AND/OR

    • A new "Nedo and octave stretch" page (create one of these if there is too much information to squeeze into the "octave stretch and compression" section).

Possible tunings to be used on each page

You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.

(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)

High-priority

118edo (choose ZPIS)

  • 187edt
  • 69edf
  • 13-limit WE (10.171c)
  • Best nearby ZPI(s)

103edo (narrow down edonoi, choose ZPIS)

  • 163edt
  • 239ed5
  • 266ed6
  • 289ed7
  • 356ed11
  • 369ed12
  • 381ed13
  • 421ed17
  • 466ed23
  • 13-limit WE (11.658c)
  • Best nearby ZPI(s)

111edo (choose ZPIS)

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

13edo

  • Main: "13edo and optimal octave stretching"
  • 2.5.11.13 WE (92.483c)
  • 2.5.7.13 WE (92.804c)
  • 2.3 WE (91.405c) (good for opposite 7 mapping)
  • 38zpi (92.531c)

104edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
Medium-high priority

15edo

  • 52ed11
  • 11lim WE (79.770)
  • 50ed10
  • 47zpi (79.715)
  • 54ed12

15edo's primes 3, 5, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking.

18edo

  • 42ed5
  • 13lim WE (66.291)
  • 61zpi (66.228)
  • 65ed12
  • 7lim WE (66.148)
  • 47ed6

18edo's primes 3, 5, 7 and 13 are all tuned sharp, so it can benefit from octave shrinking.

25edo

  • 95zpi (48.067)
  • 13lim WE (47.946)
  • 90ed12
  • 65ed6
  • 96zpi (47.642)

25edo's prime 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from octave shrinking.

26edo

  • 13lim WE (46.249)
  • 93ed12
  • 100zpi (46.268)

26edo's simple primes with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from octave stretching.

29edo

  • 46edt
  • 116zpi (41.465)
  • 13lim WE (41.484)
  • 107ed13
  • 100ed11
  • 96ed10

29edo's primes 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from octave stretching.

30edo

  • 39.918zpi (39.918)
  • 13lim WE (39.904)
  • 11lim WE (79.770)
  • 100ed10
  • 108ed12
  • 78ed6

30edo's simple primes with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from octave shrinking.

34edo

  • 11lim WE (35.284)
  • 13lim WE (35.276) (octave identical to 113ed10 within 0.1 ¢)
  • 79ed5
  • 122ed12
  • 88ed6
  • 144zpi (35.248)
  • 126ed13
  • 54edt

34edo's primes 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from octave shrinking.

35edo

  • 11lim WE (35.284)
  • 13lim WE (35.276)
  • 121ed11
  • 149zpi (34.359)
  • 116ed10
  • 98ed7
  • 81ed5
  • 125ed12
  • 90ed6

35edo's primes 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from octave stretching.

37edo

  • 137ed13
  • 161zpi (32.408) (octave identical to 123ed10 within 0.1 ¢)
  • 86ed5
  • 104ed7
  • 13lim WE (32.383)
  • 11lim WE (32.377)
  • 133ed12
  • 96ed6

37edo's primes 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking.

48edo

  • 13lim WE (25.005)
  • 226zpi (25.006)
  • 166ed11
  • 172ed12
  • 124ed6 (octave identical to 11lim WE within 0.1 ¢)
  • 76edt
  • 28edf (octave identical to 159ed10 within 0.1 ¢)

Most of 48edo's simple primes have low error, but its 5 is substantially flat, so 48edo can benefit from slight octave stretching.

Medium-low priority

10edo

  • 2.5.7.13 WE (120.358)
  • 28ed7
  • 37ed13
  • 26zpi (119.899)
  • 2.3.7.13 WE (119.785)
  • 13lim WE (119.776)
  • 36ed12

If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from octave shrinking. If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from octave stretching.

11edo

  • 28ed6
  • 39ed12
  • 2.7.11.13 WE (108.821)
  • 30zpi (108.722)
  • 35ed9
  • 31ed7
  • 41ed13
  • 37ed10

11edo has about equally bad sharp and flat mappings of primes 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight octave shrinking is advisable. To use its primes 3 or 5, extreme octave shrinking or octave stretching can be used, at the cost of making the octaves sound significantly weaker.

24edo ((13lim WE's octave is only 1/10th of a cent different from 24edo))

  • 56ed5
  • 80ed10
  • 89ed13
  • 2.3.5.11.13 WE (49.942)
  • 90zpi (49.988)
  • 11lim WE (50.017)
  • 83ed11
  • 86ed12
  • 62ed6
  • 38edt

If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight octave stretching, mostly to improve its prime 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight octave shrinking, mostly to improve its primes 5 and 13.

5edo

  • 14ed7
  • 2.3.7 WE (239.426)
  • 18ed12

If one wishes to use 5edo as a 2.3.7 subgroup tuning, then it benefits from slight octave shrinking to improve its prime 3.

6edo

  • 19ed9
  • 2.9.5 WE (199.736)
  • 2.9.5.7 WE (199.329)
  • 20ed10
  • 14ed5
  • 12zpi (198.843)
  • 17ed7

If one wishes to use 6edo as a 2.9.5 or 2.9.5.7 subgroup tuning, then it benefits from octave shrinking.

Low-priority

125edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

145edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

152edo

  • 241edt
  • 13-limit WE (7.894c)
  • Best nearby ZPI(s)

159edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

166edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

182edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

198edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

212edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

243edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

247edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
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