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== Introduction ==
== Introduction ==
'''This is a draft of what a standard "related tunings" section might look like on edo pages, using [[36edo]] as an example.'''
'''This is a draft of what a standard "related tunings" section might look like on edo pages, using [[36edo]] as an example.'''


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* In total, 5 to 8 tunings should be listed.
* In total, 5 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 include at least one stretched tuning and at least one compressed tuning.
* 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.
* 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 ===
=== Plan for roll-out ===
Edo pages which currently have  an "octave stretch", "related tunings", "zeta properties", etc. section:
Edo pages which currently have  an "octave stretch", "related tunings", "zeta properties", etc. section:
* 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}}.
* 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}}.


Highest priority pages:  
High-priority pages:  
* EDOS: {{EDOs|7, 12, 17, 19, 22, 31, 36, 41, 58, 72, 99, 103, 118, 152}}.
* EDOS: {{EDOs|7, 12, 17, 19, 22, 27, 31, 36, 41, 58, 72, 99, 103, 118, 152}}.


This standard will need to be rolled out to those above pages ''once this standard is ready''. (Not yet!!)
This standard will need to be rolled out to those above pages ''once this standard is ready''. (Not yet!!)


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


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


When a "n-edo and octave stretch" page exists, this section will NOT replace that page. 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.
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.
<br><br>


== Octave stretch or compression ==
== Octave stretch or compression ==
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{| class="wikitable sortable center-all"
{| class="wikitable sortable center-all"
|-
! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)  
! rowspan="2" | Tuning !! rowspan="2" | Step 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
! rowspan="2" | Stretch
|-
|-
Line 50: Line 48:
|-
|-
! 154zpi
! 154zpi
| 33.547 || +7.7 || +10.2 || -1.9 || -14.1 || +8.5
| 33.547
| -12.3
| +7.7 || +10.2 || −1.9 || −14.1 || +8.5 || −12.3
| 36, 57, 83, 100, 124, 132
| 36, 57, 83, 100, 124, 132
| +23.1%
| +23.1%
Line 57: Line 55:
! 21edf
! 21edf
| 33.426
| 33.426
| +3.3
| +3.3 || +3.3 || −12.0 | +7.2 || −6.5 || +5.1
| +3.3
| -12.0
| +7.2
| -6.5
| +5.1
| 36, 57, 83, 101, 124, 133
| 36, 57, 83, 101, 124, 133
| +10.2%
| +10.2%
|-
|-
! 57edt
! 57edt
| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || -13.7
| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6
| -2.6
| 36, 57, 84, 101, 124, 133
| 36, 57, 84, 101, 124, 133
| +3.6%
| +3.6%
|-
! 155zpi
| 33.346
| +0.587 || −1.025 || +15.057 || −0.511 || −15.961 || −5.024
| 36, 57, 83, 101, 124, 133
| +1.761%
|-
|-
! 36edo
! 36edo
| '''33.333''' || '''0.0''' || '''-2.0''' || '''+13.7''' || '''-2.2''' || '''+15.3'''
| '''33.333'''
| '''-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%'''
| '''0%'''
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! 13-limit WE
! 13-limit WE
| 33.302
| 33.302
| -1.1
| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4
| -3.7
| +11.1
| -5.3
| +11.4
| -11.4
| 36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| -3.3%
| −3.3%
|-
|-
! 11-limit WE
! 11-limit WE
| 33.286
| 33.286
| -1.7
| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5
| -4.7
| +9.7
| -6.9
| +9.4
| -13.5
| 36, 57, 84, 101, 125, 133
| 36, 57, 84, 101, 125, 133
| -5.1%
| −5.1%
|-
|-
! 156zpi
! 156zpi
| 33.152 || -6.5 || -12.3 || -1.5 || +12.7 || -7.3 || +1.8
| 33.152
| −6.5 || −12.3 || −1.5 || +12.7 || −7.3 || +1.8
| 36, 57, 84, 102, 125, 134
| 36, 57, 84, 102, 125, 134
| -19.5%
| −19.5%
|}
|}


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{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}
{{Harmonics in equal|21|3|2|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 21edf (continued)}}


Stretching the octave of 36edo by about 3.5 cents 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.
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.


; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]
; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]
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{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}
{{Harmonics in equal|57|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57edt (continued)}}


If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 cent 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 cents''. Four almost identical tunings do this: 57edt, 101ed7, 155zpi, and the 2.3.7.13 subgroup WE tuning of 36edo.
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.


; Pure-octaves 36edo
; Pure-octaves 36edo
Pure-octaves 36edo approximates all primes up to 11 within ''15.3 cents''.
Pure-octaves 36edo approximates all primes up to 11 within ''15.3{{c}}''.


; [[WE|11-limit WE 36edo / 13-limit WE 36edo]]
; [[WE|11-limit WE 36edo / 13-limit WE 36edo]]
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{{Harmonics in cet|33.302|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed33.302c (continued)}}
{{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 cents 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 cents''. The 11- and 13-limit WE tunings of 36edo both do this, as do their respective [[TE]] tunings.
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.
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