Xenharmonic Wiki

User:BudjarnLambeth/Draft related tunings section

< User:BudjarnLambeth
Revision as of 08:48, 21 August 2025 by BudjarnLambeth (talk | contribs) (Add a blank version for ease of use)
This user page is editable by any wiki editor.

As a general rule, most users expect their user space to be edited only by themselves, except for minor edits (e.g. maintenance), undoing obviously harmful edits such as vandalism or disruptive editing, and user talk pages.

However, by including this message box, the author of this user page has indicated that this page is open to contributions from other users (e.g. content-related edits).

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.


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 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, 3 to 8 tunings should be listed.
  • The selection of tunings should cover a range of meaningfully different tunings (eg with a range of different mappings).
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.

Plan for roll-out

Edo pages which currently have an "octave stretch", "related tunings", "zeta properties", etc. section:

This standard will need to be rolled out to those above pages.

It can optionally be rolled out to other edo pages later.

Things to note
  • When rolling it 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, eg: "{{See also|36edo and octave stretch}}".

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.

High-priority

7edo

  • 11edt
  • 18ed6
  • 2.3.5.11.13 TE
  • 2.3.11.13 TE
  • Best nearby ZPI(s)

12edo (too many edonoi)

  • 40ed10
  • 7edf
  • 19edt
  • 31ed6
  • 5-limit TE
  • 7-limit TE
  • 2.3.5.17.19 TE
  • Best nearby ZPI(s)

17edo

  • 27edt
  • 44ed6
  • 2.3.7.11 TE
  • 2.3.7.11.13 TE
  • Best nearby ZPI(s)

19edo

  • 49ed6
  • 30ed3
  • 11edf
  • 2.3.5.11 TE
  • 13-limit TE
  • Best nearby ZPI(s)

22edo

  • 123ed48 (try to find a simpler but similar tuning)
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

27edo

  • 43edt
  • 70ed6
  • 90ed10
  • 97ed12
  • 7-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

31edo

  • 80ed6
  • 111ed12
  • 229ed169 (try to find a simpler but similar tuning)
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

41edo

  • 65edt
  • 106ed6
  • 147ed12
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

58edo

  • 92edt
  • 150ed6
  • 7-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

72edo

  • 144edt
  • 186ed6
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)
Medium-high priority

8edo

  • 1ed148.5c
  • Best nearby ZPI(s)

13edo

  • 2.5.11.13 TE
  • 2.3.5.11.13 TE
  • Best nearby ZPI(s)

14edo

  • 22edt
  • 36ed6
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

16edo

  • 25edt
  • 41ed6
  • 57ed12
  • 2.5.7.13 TE
  • 13-limit TE
  • Best nearby ZPI(s)

23edo (too many edonoi)

  • Main: "23edo and octave stretching"
  • 36edt
  • 59ed6
  • 60ed6
  • 68ed8
  • 11ed7/5
  • 1ed33/32
  • 2.3.5.13 TE
  • 2.7.11 TE
  • 13-limit TE
  • Best nearby ZPI(s)

60edo (too many edonoi)

  • 95edt
  • 139ed5
  • 155ed6
  • 208ed11
  • 255ed19
  • 27ed23 (best for catnip temperament)
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

99edo

  • 157edt
  • 256ed6
  • 7-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

103edo (too many edonoi)

  • 163edt
  • 239ed5
  • 289ed7
  • 356ed11
  • 381ed13
  • 421ed17
  • 466ed23
  • 7-limit TE
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

118edo

  • 187edt
  • 69edf
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

152edo

  • 241edt
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)
Low-medium priority

32edo

  • 90ed7
  • 51edt
  • 75ed5
  • 1ed46/45
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

33edo (too many edonoi)

  • 76ed5
  • 92ed7
  • 52edt
  • 1ed47/46
  • 114ed11
  • 122ed13
  • 93ed7
  • 23edPhi
  • 77ed5
  • 123ed13
  • 115ed11
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

39edo

  • 62edt
  • 101ed6
  • 39ed255/128 (replace with something similar but simpler)
  • 2.3.5.11 TE
  • 2.3.7.11.13 TE
  • 13-limit TE
  • Best nearby ZPI(s)

42edo (^replace w something similar but simpler)

  • 42ed257/128^
  • 42ed255/128^
  • APS720jot^
  • APS715jot^
  • 7-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

45edo

  • 126ed7
  • APS3.21farab (replace with something similar but simpler)
  • 7-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

54edo

  • 86edt
  • 126ed5
  • 152ed7
  • APS4/5méride (replace with something similar but simpler)
  • 2.3.7.11.13 TE
  • 13-limit TE
  • Best nearby ZPI(s)

59edo

  • 93edt
  • 166ed7
  • 203ed11
  • 7-limit TE
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)

64edo

  • 149ed5
  • 180ed7
  • 222ed11
  • 64ed257/128 (replace with something similar but simpler)
  • 11-limit TE
  • 13-limit TE
  • Best nearby ZPI(s)
Low priority

(add brainstorm list here)

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.

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

_ing the octave of EDONAME by a little over 3 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning TUNING does this.

Approximation of prime harmonics in 12edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Relative (%) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Steps
(reduced) 		12
(0) 		19
(7) 		28
(4) 		34
(10) 		42
(6) 		44
(8) 		49
(1) 		51
(3) 		54
(6) 		58
(10) 		59
(11)
Approximation of harmonics in TUNING (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79 83
Error Absolute (¢) +48.7 -29.1 -11.5 +34.5 +26.5 +40.8 -16.9 +20.7 +20.3 -27.8 +35.5 +50.0
Relative (%) +48.7 -29.1 -11.5 +34.5 +26.5 +40.8 -16.9 +20.7 +20.3 -27.8 +35.5 +50.0
Steps
(reduced) 		63
(3) 		64
(4) 		65
(5) 		67
(7) 		69
(9) 		71
(11) 		71
(11) 		73
(1) 		74
(2) 		74
(2) 		76
(4) 		77
(5)
EDONAME
  • Step size: NNN ¢, octave size: NNN ¢

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

Approximation of harmonics in 12edo
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)
TUNING
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by a little over 3 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning TUNING does this.

Approximation of prime harmonics in 12edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Relative (%) +0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5 -28.3 -29.6 -45.0
Steps
(reduced) 		12
(0) 		19
(7) 		28
(4) 		34
(10) 		42
(6) 		44
(8) 		49
(1) 		51
(3) 		54
(6) 		58
(10) 		59
(11)
Approximation of harmonics in TUNING (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79 83
Error Absolute (¢) +48.7 -29.1 -11.5 +34.5 +26.5 +40.8 -16.9 +20.7 +20.3 -27.8 +35.5 +50.0
Relative (%) +48.7 -29.1 -11.5 +34.5 +26.5 +40.8 -16.9 +20.7 +20.3 -27.8 +35.5 +50.0
Steps
(reduced) 		63
(3) 		64
(4) 		65
(5) 		67
(7) 		69
(9) 		71
(11) 		71
(11) 		73
(1) 		74
(2) 		74
(2) 		76
(4) 		77
(5)
Retrieved from "https://en.xen.wiki/index.php?title=User:BudjarnLambeth/Draft_related_tunings_section&oldid=207848"