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User:BudjarnLambeth/Draft related tunings section: Difference between revisions

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== Introduction ==
== The guidelines ==
'''This is a draft of what a standard "related tunings" section might look like on edo pages, using [[36edo]] as an example.'''
'''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]]
* [http://x31eq.com/temper/ x31eq Temperament Finder] by [[Graham Breed]]


Which tunings should be listed for any given edo:
; 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
* 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)
Line 16: Line 17:


Additional guidelines for selecting tunings:
Additional guidelines for selecting tunings:
* In total, 5 to 8 tunings should be listed.
* In total, 3 to 8 tunings should be listed.
* The selection of tunings should include at least one stretched tuning and at least one compressed tuning.
* The selection of tunings should cover a range of meaningfully different tunings (eg with a range of different mappings).
* 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.


=== Plan for roll-out ===
; Further instructions
Edo pages which currently have  an "octave stretch", "related tunings", "zeta properties", etc. section:
* Adding the comparison table at the end is optional.
* EDOS: {{EDOs|7, 8, 12, 13, 14, 16, 17, 19, 22, 23, 31, 32, 33, 36, 39, 41, 42, 45, 54, 58, 59, 60, 64, 72, 99, 103, 111, 118, 125, 145, 152, 159, 166, 182, 198, 212, 243, 247}}.
* 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.


High-priority pages:
; Where this section should be placed on an edo page:
* EDOS: {{EDOs|7, 12, 17, 19, 22, 27, 31, 36, 41, 58, 72, 99, 103, 118, 152}}.
* 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.''


This standard will need to be rolled out to those above pages ''once this standard is ready''. (Not yet!!)
= Example (36edo) =
== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 36edo tunings.


It can optionally be rolled out to more edo pages later.
; [[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)}}


When rolling it out try not to delete existing body text but instead rework it where possible.
; [[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)}}


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.
; 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)}}


== Octave stretch or compression ==
; [[TE|36et, 13-limit TE tuning]]
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
* 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"
|+ style="white-space: nowrap;" | Comparison of stretched and compressed tunings
|-
|-
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" | Tuning !! rowspan="2" | Octave size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" | Mapping of primes 2–13 (steps)
! rowspan="2" | Mapping of primes 2–13 (steps)
! rowspan="2" | Stretch
|-
|-
! 2 !! 3 !! 5 !! 7 !! 11  
! 2 !! 3 !! 5 !! 7 !! 11 !! 13
! 13
|-
! 154zpi
| 33.547
| +7.7 || +10.2 || −1.9 || −14.1 || +8.5 || −12.3
| 36, 57, 83, 100, 124, 132
| +23.1%
|-
|-
! 21edf
! 21edf
| 33.426
| 1203.351
| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
| +3.3 || +3.3 || −12.0 || +7.2 || −6.5 || +5.1
| 36, 57, 83, 101, 124, 133
| 36, 57, 83, 101, 124, 133
| +10.2%
|-
|-
! 57edt
! 57edt
| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
| 1201.235
| +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
| 36, 57, 84, 101, 124, 133
| 36, 57, 84, 101, 124, 133
| +3.6%
|-
|-
! 155zpi
! 155zpi
| 33.346
| 1200.587
| +0.587 || −1.025 || +15.057 || −0.511 || −15.961 || −5.024
| +0.6 || −1.0 || +15.1 || −0.5 || −16.0|| −5.0
| 36, 57, 83, 101, 124, 133
| 36, 57, 83, 101, 124, 133
| +1.761%
|-
|-
! 36edo
! 36edo
| '''33.333'''
| '''1200.000'''
| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
| '''0.0''' || '''−2.0''' || '''+13.7''' || '''−2.2''' || '''+15.3''' || '''−7.2'''
| '''36, 57, 84, 101, 125, 133'''
| '''36, 57, 84, 101, 125, 133'''
| '''0%'''
|-
|-
! 13-limit WE
! 13-limit TE
| 33.302
| 1198.929
| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4
| −1.1 || −3.7 || +11.2 || −5.2 || +11.6 || −11.1
| 36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| −3.3%
|-
|-
! 11-limit WE
! 11-limit TE
| 33.286
| 1198.330
| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5
| −1.7 || −4.6 || +9.8 || −6.8 || +9.5 || −13.4
| 36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| −5.1%
|-
! 156zpi
| 33.152
| −6.5 || −12.3 || −1.5 || +12.7 || −7.3 || +1.8
| 36, 57, 84, 102, 125, 134
| −19.5%
|}
|}


; [[21edf]]
= Blank template =
{{Harmonics in equal|21|3|2|columns=12|collapsed=true}}
== Octave stretch or compression ==
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
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, 19 edos}}.
--
* High priority pages: {{EDOs|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>.
 
=== 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)
 
; High-priority
 
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)


Stretching the octave of 36edo by about 3.5{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all primes up to 11 within ''12 cents''. The tuning 21edf does this.
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)


; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]
103edo (too many edonoi)
{{Harmonics in equal|57|3|1|columns=12|collapsed=true}}
* 163edt
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
* 239ed5
* 289ed7
* 356ed11
* 381ed13
* 421ed17
* 466ed23
* 13-limit WE (11.658c)
* Best nearby ZPI(s)


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 primes up to 11 within ''16.6{{c}}''. Four almost identical tunings do this: 57edt, 101ed7, 155zpi, and the 2.3.7.13 subgroup WE tuning of 36edo.
118edo
* 187edt
* 69edf
* 13-limit WE (10.171c)
* Best nearby ZPI(s)


; Pure-octaves 36edo
152edo
Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''.
* 241edt
* 13-limit WE ( 7.894c)
* Best nearby ZPI(s)


; [[WE|11-limit WE 36edo / 13-limit WE 36edo]]
; Low priority
{{Harmonics in cet|33.302|columns=12|collapsed=true}}
{{Harmonics in cet|33.302|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed33.302c (continued)}}


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 primes up to 11 within ''9.7{{c}}''. The 11- and 13-limit WE tunings of 36edo both do this, as do their respective [[TE]] tunings.
(add brainstorm list here)
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Draft_related_tunings_section"