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

== Introduction ==

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

Which tunings should be listed for any given edo:

* 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 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 editor can choose how many decimal places to use as long as it's self-consistent.

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}}.

High-priority pages:

* 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!!)

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.

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.

What follows is a comparison of stretched- and compressed-octave 36edo tunings.

{| class="wikitable sortable center-all"

! rowspan="2" | Tuning !! rowspan="2" | Step size<br>(cents) !! colspan="6" | Prime error (cents)  

! 154zpi

| +7.7 || +10.2 || −1.9 || −14.1 || +8.5 || −12.3

| 33.426

| 33.368 || +1.2 || 0.0 || +16.6 || +1.3 || −13.7 || −2.6

| 33.346

| '''33.333'''

! 13-limit WE

| 33.302

| −1.1 || −3.7 || +11.1 || −5.3 || +11.4 || −11.4

! 11-limit WE

| 33.286

| −1.7 || −4.7 || +9.7 || −6.9 || +9.4 || −13.5

; [[21edf]]

{{Harmonics in equal|21|3|2|columns=12|collapsed=true}}

{{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{{c}} results in improved primes 5, 11, and 13, but worse primes 2, 3, and 7. This approximates all primes up to 11 within ''12.0{{c}}''. The tuning 21edf does this.

; [[57edt]] / [[ed7|101ed7]] / [[zpi|155zpi]] / [[WE|2.3.7.13 WE-tuned 36edo]]

{{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{{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

{{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.