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{{Editable user page}}
'''ABANDONED.'''


== Introduction ==
'''These guidelines have been retracted due to them being largely not liked by most commenters on the XA Discord.'''
 
'''This is a draft of what a standard "related tunings" section might look like on edo pages, using 36edo as an example.'''


{{Editable user page}}


Useful links for working on this:
<pre>
* [https://sintel.pythonanywhere.com/ Temperament Calculator] by [[User:Sintel|Sintel]]
== The guidelines ==
* [http://x31eq.com/temper/ x31eq Temperament Finder] by [[Graham Breed]]
'''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:
; Which tunings should be listed for any given edo:
* The pure-octaves tuning
* 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 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 to 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:
* 1 to 2 subgroup TE- or WE-optimal tunings, based on the best choice(s) of subgroup for the edo
** 13-limit WE and 13-limit TE
* ''1 other equal tuning of any kind at all (optional)''
** 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, 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).


Additional guidelines for selecting tunings:
; Further instructions
* In total, 5 to 7 tunings should be listed.  
* Adding the comparison table at the end is optional.
* The selection of tunings should include at least one stretched tuning and at least one compressed tuning.
* 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.
* The most stretched tuning should have +15% to +25% relative error in 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.


; Where this section should be placed on an edo page:
<small><small>
* 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.''
</small></small>


Todo:
; Useful links for working on this:
* Update the set of tunings to follow the tuning selection rules
* [https://sintel.pythonanywhere.com/ Temperament Calculator] by [[User:Sintel|Sintel]] (calculates WE & TE)
* Make the mapping info take up less horizontal space
* [http://x31eq.com/temper/ x31eq Temperament Finder] by [[Graham Breed]] (calculates TE)
* 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
<br><br>


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


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).
; [[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)}}


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).
; 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)}}


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{{c}}. Their main usage is to optimise 36edo for use as a [[dual-n|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.
; [[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"
{| class="wikitable sortable center-all mw-collapsible mw-collapsed"
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (% step size) !! colspan="6" | Prime mapping (steps)  
|+ 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  
! 2 !! 3 !! 5 !! 7 !! 11 !! 13
!13!! 2 !! 3 !! 5 !! 7 !! 11
!13
|-
|-
! 154zpi
! 21edf
| 33.547 || +23 || +31 || -5 || -41 || +27
| 1203.351
| || 36 || 57 || 83 || 100 || 124
| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
|
| 36, 57, 83, 101, 124, 133
|-
|-
! 57edt
! 57edt
| 33.368 || +4 || 0 || +50 || +5 || -40
| 1201.235
| || 36 || 57 || 83 || 101 || 124
| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
|
| 36, 57, 84, 101, 124, 133
|-
!2.3.7.13 WE
|33.361
|
|
|
|
|
|
|
|
|
|
|
|
|-
! 101ed7
| 33.355 || +2 || -2 || +46 || 0 || -46
| || 36 || 57 || 84 || 101 || 124
|
|-
|-
! 2.3.7 WE
! 155zpi
| 33.352 || +2 || -3 || +45 || -1 || -48
| 1200.587
| || 36 || 57 || 84 || 101 || 124
| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0
|
| 36, 57, 83, 101, 124, 133
|-
|-
! 36edo
! 36edo
| '''33.333'''|| '''0'''|| '''-6'''|| '''+40'''|| '''-6'''|| '''+46'''
| '''1200.000'''
| || '''36'''|| '''57'''|| '''84'''|| '''101'''|| '''125'''
| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
|
| '''36, 57, 84, 101, 125, 133'''
|-
!13lim WE
|33.302
|
|
|
|
|
|
|
|
|
|
|
|
|-
|-
!11lim WE
! 13-limit TE
|33.286
| 1198.929
|
| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1
|
| 36, 57, 84, 101, 125, 133
|
|
|
|
|
|
|
|
|
|
|-
|-
! 156zpi
! 11-limit TE
| 33.152 || -20 || -37 || -5 || +38 || -23
| 1198.330
| || 36 || 57 || 84 || 102 || 125
| −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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|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|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in cet|100|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}
= Plan for roll-out =
''Done''
* Done (with table): {{EDOs| 36 }}.
* Done (table not added yet): {{EDOs| 7, 8, 9, 12, 14, 16, 17, 19, 22, 23, 27, 28, 31, 32, 38, 41, 58, 60, 72, 99 }}.
* Done (table & some step sizes not added yet): {{EDOs| 15, 18, 33, 39, 42, 45, 54, 59, 64 }}.
* ''Started (brief summary only): {{EDOs| 5, 6, 10, 11, 24, 25, 26, 29, 30, 34, 35, 37, 47, 48, 49, 50, 62, 65, 66 }}.''
''To-do (optional)''
* Highest priority: {{EDOs| 13, 118, 159, 270, 103, 104, 107, 111, 117, 125 }}.
* High priority: {{EDOs| 140, 145, 152, 166, 171, 182, 183, 198, 212, 224, 243, 247, 342 }}.
* Medium-high priority: {{EDOs| 5, 6, 10, 11, 24, 25, 26, 29, 30, 34, 35, 37, 47, 48, 49, 50, 62, 65, 66}}.
* Medium-low priority: {{EDOs| 73, 76, 80, 81, 83, 84, 88, 89, 91, 95, 96, 101, 106, 110, 112, 113, 121, 122, 123, 124 }}.
* Low priority: {{EDOs| 20, 21, 40, 43, 44, 46, 51, 52, 53, 55, 56, 57, 63, 67, 68, 69, 70, 71, 74, 75, 77, 78, 79, 82, 85, 86, 87, 90, 92, 93, 94, 97, 98, 102, 105, 108, 109, 114, 115, 116, 119, 120 }}.
* Lowest priority: All other EDOs above 125.
''Priority is higher if: there is already a section for stretch, compression, nearby tunings or zeta on the page / the edo lends itself to stretch or compression (e.g. of the primes with 20-40% relative error, most tend in the same direction) / the edo gets a lot of attention (it's small, or especially good by some prominent metric, or the page is especially long, or it gets used a lot by composers).''
; 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.: <nowiki>"{{See also|36edo and octave stretch}}"</nowiki>.
=== 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 [[XW:NG|notable]] information from their respective pages has been added to:
** The "octave stretch and compression" section of the edo page.
AND/OR
** A new "''N''edo 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)
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)
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)
Default
* Nearby edt, ed6, ed12 and/or edf
* Nearby ed5, ed10, ed7 and/or ed11 (optional)
* 1-2 WE tunings
* Best nearby ZPI(s)
=== Reasoning for priority levels ===
^^currently has a zeta peak section with only the graph/table and little/no description, or another existing non-standard similar tunings section
* 49edo mostly sharp
* 50edo mostly flat
* 51edo stretch/compression unhelpful
* 52edo stretch/compression unhelpful
* 53edo stretch/compression unhelpful
* 54edo stretch/compression unhelpful
* 55edo stretch/compression unhelpful
* 56edo stretch/compression unhelpful
* 57edo mostly flat (on closer inspection octave stretch doesn't help all that much)
* 58edo mostly sharp
* 59edo mostly sharp
* ''60edo already done''
* 61edo stretch/compression unhelpful
* 62edo mostly flat
* 63edo stretch/compression unhelpful
* 64edo stretch/compression unhelpful
* 65edo mostly sharp
* 66edo mostly flat
* 67edo stretch/compression unhelpful
* 68edo stretch/compression unhelpful
* 69edo stretch/compression unhelpful
* 70edo complicated
* 71edo stretch/compression unhelpful
* ''72edo already done''
* 73edo mostly sharp
* 74edo stretch/compression unhelpful
* 75edo complicated
* 76edo mostly flat
* 77edo stretch/compression unhelpful
* 78edo stretch/compression unhelpful
* 79edo stretch/compression unhelpful
* 80edo mostly sharp
* 81edo mostly flat
* 82edo stretch/compression unhelpful
* 83edo mostly sharp
* 84edo mostly sharp
* 85edo stretch/compression unhelpful
* 86edo stretch/compression unhelpful
* 87edo stretch/compression unhelpful
* 88edo mostly flat
* ^^89edo mostly sharp
* 90edo stretch/compression unhelpful
* 91edo mostly flat
* 92edo stretch/compression unhelpful
* 93edo stretch/compression unhelpful
* 94edo stretch/compression unhelpful
* 95edo mostly sharp
* 96edo mostly flat
* 97edo stretch/compression unhelpful
* 98edo stretch/compression unhelpful
* 99edo mostly sharp
* 100edo complicated
* 101edo mostly sharp
* 102edo stretch/compression unhelpful
* ^^103edo mostly flat
* ^^104edo mostly sharp
* 105edo stretch/compression unhelpful
* 106edo mostly sharp
* ^^107edo mostly flat
* 108edo stretch/compression unhelpful
* 109edo stretch/compression unhelpful
* 110edo mostly flat
* ^^111edo mostly sharp
* 112edo mostly flat
* 113edo mostly flat
* 114edo stretch/compression unhelpful
* 115edo stretch/compression unhelpful
* 116edo stretch/compression unhelpful
* ^^117edo slightly sharp
* ^^118edo stretch/compression unhelpful
* 119edo stretch/compression unhelpful
* 120edo stretch/compression unhelpful
* 121edo mostly sharp
* 122edo mostly flat
* 123edo mostly flat
* 124edo mostly sharp
* ^^125edo mostly flat
</pre>
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