User:BudjarnLambeth/Draft related tunings section
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Introduction
This is a draft of what a standard "related tunings" section might look like on edo pages, using 36edo as an example.
Useful links for working on this:
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 n-limit WE or n-limit TE tunings, valid examples include:
- 13-limit WE and 13-limit TE
- 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
- 0 to 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
- 1 other equal tuning of any kind at all (optional)
Additional guidelines for selecting tunings:
- 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.
Plan for roll-out
Edo pages which currently have an "octave stretch", "related tunings", "zeta properties", etc. section:
- 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:
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.
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.
Octave stretch or compression
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
| Tuning | Step size (cents) |
Prime error (cents) | Mapping of primes 2-13 (steps) | Stretch | |||||
|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 5 | 7 | 11 | 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 | 33.426 | +3.3 | +3.3 | -12.0 | +7.2 | -6.5 | +5.1 | 36, 57, 83, 101, 124, 133 | +10.2% |
| 57edt | 33.368 | +1.2 | 0 | +16.6 | +1.3 | -13.7 | -2.6 | 36, 57, 84, 101, 124, 133 | +3.6% |
| 36edo | 33.333 | 0 | -2.0 | +13.7 | -2.2 | +15.3 | -7.2 | 36, 57, 84, 101, 125, 133 | 0% |
| 13lim WE | 33.302 | -1.1 | -3.7 | +11.1 | -5.3 | +11.4 | -11.4 | 36, 57, 84, 101, 125, 133 | -3.3% |
| 11lim WE | 33.286 | -1.7 | -4.7 | +9.7 | -6.9 | +9.4 | -13.5 | 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% |
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.
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.
- Pure-octaves 36edo
Pure-octaves 36edo approximates all primes up to 11 within 15.3 cents.
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.