View source for User:BudjarnLambeth/Draft related tunings section

{{Editable user page}}

== 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:
* [https://sintel.pythonanywhere.com/ Temperament Calculator] by [[User:Sintel|Sintel]] (calculates WE & TE)
* [http://x31eq.com/temper/ x31eq Temperament Finder] by [[Graham Breed]] (calculates TE)

; 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.''

= Example (36edo) =
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 36edo tunings.

; [[21edf]] 
* Step size: 33.426{{c}}, octave size: 1203.351{{c}}
Stretching the octave of 36edo by a little over 3{{c}} 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{{c}}. The tuning 21edf does this.
{{Harmonics in equal|21|3|2|columns=11|collapsed=true}}
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}

; [[57edt]]
* Step size: 33.368{{c}}, octave size: 1201.235{{c}}
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1{{c}} 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{{c}}. Several almost-identical tunings do this: 57edt, [[93ed6]], [[101ed7]], [[zpi|155zpi]], and the 2.3.7.13-subgroup [[TE]] and [[WE]] tunings of 36et.
{{Harmonics in equal|57|3|1|columns=11|collapsed=true}}
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}

; 36edo
* Step size: 33.333{{c}}, octave size: 1200.000{{c}} 
Pure-octaves 36edo approximates all harmonics up to 16 within 15.3{{c}}.
{{Harmonics in equal|36|2|1|intervals=integer|columns=11|collapsed=true}}
{{Harmonics in equal|36|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36edo (continued)}}

; [[TE|36et, 13-limit TE tuning]]
* Step size: 33.304{{c}}, octave size: 1198.929{{c}}
Compressing the octave of 36edo by about 2{{c}} 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{{c}}. The 11- and 13-limit TE tunings of 36et both do this, as do their respective WE tunings.
{{Harmonics in cet|33.303596|columns=11|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et}}
{{Harmonics in cet|33.303596|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 13-limit TE tuning of 36et (continued)}}

{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings
|-
! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents) 
! rowspan="2" | 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.

; [[zpi|ZPINAME]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}

; [[EDONOI]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}

; [[WE|ETNAME, SUBGROUP WE tuning]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; EDONAME
* Step size: NNN{{c}}, octave size: NNN{{c}} 
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONAME (continued)}}

; [[WE|ETNAME, SUBGROUP WE tuning]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}

; [[EDONOI]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in EDONOI (continued)}}

; [[zpi|ZPINAME]] 
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this.
{{Harmonics in cet|100|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in ZPINAME (continued)}}

= Plan for roll-out =
Edo pages which currently have  an "octave stretch", "related tunings", "zeta properties", etc. section:
* Done (with table): {{EDOs|36 edo}}.
* Done (table not added yet): {{EDOs|7, 12, 17 edos}}.
--
* High priority pages: {{EDOs|19, 22, 27, 31, 41, 58, 72 edos}}.
* Medium-high priority pages: {{EDOs|8, 13, 14, 16, 23, 60, 99 edos}}.
* Low-medium priority pages: {{EDOs|32, 33, 39, 42, 45, 54, 59, 64, 103, 118, 152 edos}}.
* Low priority pages: {{EDOs|111, 125, 145, 159, 166, 182, 198, 212, 243, 247 edos}}.

; 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: <nowiki>"{{See also|36edo and octave stretch}}"</nowiki>.

=== 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

19edo
* 49ed6
* 30ed3
* 2.3.5.11 WE (63.192c)
* 13-limit WE (63.291c)
* 65zpi (63.331c)

22edo
* 1ed54.5c
* 11-limit WE (54.494c)
* 13-limit WE (54.546c)
* 80zpi (54.483c)

27edo
* 43edt
* 70ed6
* 90ed10
* 97ed12
* 7-limit WE (44.306c)
* 13-limit WE (44.375c)
* 105zpi (44.674c)
* 106zpi (44.302c)

31edo
* 80ed6
* 111ed12
* 25ed7/4 (replaces 229ed169)
* 11-limit WE (38.748c)
* 13-limit WE (38.725c)
* 127zpi (38.737c)

41edo
* 65edt
* 106ed6
* 147ed12
* 11-limit WE (29.277c)
* 13-limit WE (29.267c)
* 184zpi (29.277c)

58edo
* 92edt
* 150ed6
* 7-limit WE (20.667c)
* 13-limit WE (20.663c)
* 288zpi (20.736c)
* 289zpi (20.666c)

72edo
* 144edt
* 186ed6
* 11-limit WE (	16.677c)
* 13-limit WE (16.680c)
* 380zpi (16.678c)

; Medium-high priority

8edo
* 29ed12
* No-7s 17-limit WE (147.895c)
* No-7s 19-limit WE (148.148c)
* 18zpi (153.463c)
* 19zpi (147.467c)

13edo
* 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)

14edo
* 22edt
* 36ed6
* 11-limit WE (85.842c)
* 13-limit WE (85.759c)
* 42zpi (86.329c)

16edo
* 25edt
* 41ed6
* 57ed12
* 2.5.7.13 WE (75.105c)
* 13-limit WE (75.315c)
* 15zpi (75.262c)

23edo (too many edonoi, too many ZPIs)
* Main: "23edo and octave stretching"
* 36edt
* 59ed6
* 60ed6
* 68ed8
* 11ed7/5
* 1ed33/32
* 2.3.5.13 WE (52.447c)
* 2.7.11 WE (51.962c)
* 13-limit WE (52.237c)
* 83zpi (53.105c)
* 84zpi (52.615c)
* 85zpi (52.114c)
* 86zpi (	51.653c)
* 87zpi (51.201c)

60edo (too many edonoi, too many zpis)
* 95edt
* 139ed5
* 155ed6
* 208ed11
* 255ed19
* 272ed23 (great for catnip temperament)
* 13-limit WE (20.013c)
* 299zpi (20.128c)
* 300zpi (20.093c)
* 301zpi (20.027c)
* 302zpi (19.962c)
* 303zpi (19.913c)
* 304zpi (19.869c)

99edo
* 157edt
* 256ed6
* 7-limit WE (12.117c)
* 13-limit WE (12.123c)
* 567zpi (12.138c)
* 568zpi (12.115c)

; Low-medium priority

32edo (too many edonoi, too many zpis)
* 90ed7
* 51edt
* 75ed5
* 1ed46/45
* 11-limit WE (37.453c)
* 13-limit WE (37.481c)
* 131zpi (37.862c)
* 132zpi (37.662c)
* 133zpi (37.418c)
* 134zpi (37.176c)

33edo (too many edonoi)
* 76ed5
* 92ed7
* 52edt
* 1ed47/46
* 114ed11
* 122ed13
* 93ed7
* 23edPhi
* 77ed5
* 123ed13
* 115ed11
* 11-limit WE (36.349c)
* 13-limit WE (36.357c)
* 137zpi (36.628c)
* 138zpi (36.394c)
* 139zpi (36.179c)

39edo
* 62edt
* 101ed6
* 18ed11/8
* 2.3.5.11 WE (30.703c)
* 2.3.7.11.13 WE (30.787c)
* 13-limit WE (30.757c)
* 171zpi (30.973c)
* 172zpi (30.836c)
* 173zpi (30.672c)

42edo
* 42ed257/128 (replace w something similar but simpler)
* AS123/121 (1ed123/121)
* 11ed6/5
* 34ed7/4
* 7-limit WE (28.484c)
* 13-limit WE (28.534c)
* 189zpi (28.689c)
* 190zpi (28.572c)
* 191zpi (28.444c)

45edo
* 126ed7
* 13ed11/9
* 7-limit WE (26.745c)
* 13-limit WE (26.695c)
* 207zpi (26.762)
* 208zpi (26.646)
* 209zpi (26.550)

54edo
* 86edt
* 126ed5
* 152ed7
* 38ed5/3
* 40ed5/3
* 2.3.7.11.13 WE (22.180c)
* 13-limit WE (22.198c)
* 262zpi (22.313c)
* 263zpi (22.243c)
* 264zpi (22.175c)

59edo (too many ZPIs)
* 93edt
* 166ed7
* 203ed11
* 7-limit WE (20.301c)
* 11-limit WE (20.310c)
* 13-limit WE (20.320c)
* 293zpi (20.454c)
* 294zpi (20.399c)
* 295zpi (20.342c)
* 296zpi (20.282c)
* 297zpi (20.229c)

64edo (too many ZPIs, too many edonoi)
* 149ed5
* 180ed7
* 222ed11
* 47ed5/3
* 11-limit WE (18.755c)
* 13-limit WE (18.752c)
* 325zpi (18.868c)
* 326zpi (18.816c)
* 327zpi (18.767c)
* 328zpi (18.721c)
* 329zpi (18.672c)
* 330zpi (18.630c)

103edo (too many edonoi)
* 163edt
* 239ed5
* 289ed7
* 356ed11
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)

118edo
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)

152edo
* 241edt
* 13-limit WE (	7.894c)
* Best nearby ZPI(s)

; Low priority

(add brainstorm list here)