User:BudjarnLambeth/Draft related tunings section
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The guidelines
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
- Temperament Calculator by Sintel (calculates WE & TE)
- x31eq Temperament Finder by Graham Breed (calculates TE)
- 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 subgroup TE- or WE-optimal tunings, based on the best choice(s) of subgroup for the edo
- 1 other equal tuning of any kind at all (optional)
Additional guidelines for selecting tunings:
- In total, 3 to 8 tunings should be listed.
- The selection of tunings should cover a range of meaningfully different tunings (eg with a range of different mappings).
- Further instructions
- Adding the comparison table at the end is optional.
- 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.
- Where this section should be placed on an edo page
- 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.
Plan for roll-out
Edo pages which currently have an "octave stretch", "related tunings", "zeta properties", etc. section:
- High priority pages: 7, 12, 17, 19, 22, 27, 31, 36, 41, 58 & 72 edos.
- Medium-high priority pages: 8, 13, 14, 16, 23, 60, 99, 103, 118, & 152 edos.
- Low-medium priority pages: 32, 33, 39, 42, 45, 54, 59 & 64 edos.
- Low priority pages: 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: "{{See also|36edo and octave stretch}}".
Example (36edo)
Octave stretch or compression
What follows is a comparison of stretched- and compressed-octave 36edo tunings.
- Step size: 33.426 ¢
- Octave size: 1203.3 ¢
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.4 | +3.4 | +6.7 | -11.9 | +6.7 | +7.2 | +10.1 | +6.7 | -8.6 | -6.4 | +10.1 | +5.2 |
| Relative (%) | +10.0 | +10.0 | +20.1 | -35.7 | +20.1 | +21.7 | +30.1 | +20.1 | -25.6 | -19.3 | +30.1 | +15.5 | |
| Steps (reduced) |
36 (15) |
57 (15) |
72 (9) |
83 (20) |
93 (9) |
101 (17) |
108 (3) |
114 (9) |
119 (14) |
124 (19) |
129 (3) |
133 (7) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.2 | +10.6 | -8.6 | +13.4 | +8.7 | +10.1 | -16.7 | -5.2 | +10.6 | -3.1 | -13.2 | +13.4 |
| Relative (%) | +15.5 | +31.7 | -25.6 | +40.1 | +26.1 | +30.1 | -49.9 | -15.6 | +31.7 | -9.2 | -39.5 | +40.1 | |
| Steps (reduced) |
133 (7) |
137 (11) |
140 (14) |
144 (18) |
147 (0) |
150 (3) |
152 (5) |
155 (8) |
158 (11) |
160 (13) |
162 (15) |
165 (18) | |
Stretching the octave of 36edo by a little over 3 ¢ 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 ¢. The tuning 21edf does this.
- Step size: 33.368 ¢
- Octave size: 1201.2 ¢
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.2 | +0.0 | +2.5 | +16.6 | +1.2 | +1.3 | +3.7 | +0.0 | -15.6 | -13.7 | +2.5 | -2.6 |
| Relative (%) | +3.7 | +0.0 | +7.4 | +49.7 | +3.7 | +3.9 | +11.1 | +0.0 | -46.6 | -41.2 | +7.4 | -7.9 | |
| Steps (reduced) |
36 (36) |
57 (0) |
72 (15) |
84 (27) |
93 (36) |
101 (44) |
108 (51) |
114 (0) |
119 (5) |
124 (10) |
129 (15) |
133 (19) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.6 | +2.5 | +16.6 | +4.9 | +0.1 | +1.2 | +7.7 | -14.3 | +1.3 | -12.5 | +10.6 | +3.7 |
| Relative (%) | -7.9 | +7.6 | +49.7 | +14.8 | +0.3 | +3.7 | +23.2 | -42.9 | +3.9 | -37.5 | +31.9 | +11.1 | |
| Steps (reduced) |
133 (19) |
137 (23) |
141 (27) |
144 (30) |
147 (33) |
150 (36) |
153 (39) |
155 (41) |
158 (44) |
160 (46) |
163 (49) |
165 (51) | |
If one intends to use both 36edo's vals for 5/1 at once, stretching the octave of 36edo by about 1 ¢ 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 ¢. Five almost-identical tunings do this: 57edt, 101ed7, 155zpi, and the TE and WE 2.3.7.13 subgroup WE tunings of 36edo.
- Pure-octaves 36edo
- Step size: 33.333 ¢
- Octave size: 1200.0 ¢
Pure-octaves 36edo approximates all primes up to 11 within 15.3 ¢.
- Step size: 33.287 ¢
- Octave size: 1198.3 ¢
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.7 | -4.6 | -3.3 | +9.8 | -6.3 | -6.8 | -5.0 | -9.2 | +8.1 | +9.6 | -7.9 | -13.4 |
| Relative (%) | -5.0 | -13.8 | -10.0 | +29.4 | -18.8 | -20.5 | -15.0 | -27.6 | +24.4 | +28.7 | -23.8 | -40.1 | |
| Step | 36 | 57 | 72 | 84 | 93 | 101 | 108 | 114 | 120 | 125 | 129 | 133 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -13.4 | -8.5 | +5.2 | -6.7 | -11.8 | -10.9 | -4.6 | +6.5 | -11.4 | +7.9 | -2.5 | -9.6 |
| Relative (%) | -40.1 | -25.6 | +15.6 | -20.0 | -35.3 | -32.6 | -13.8 | +19.4 | -34.4 | +23.7 | -7.5 | -28.8 | |
| Step | 133 | 137 | 141 | 144 | 147 | 150 | 153 | 156 | 158 | 161 | 163 | 165 | |
Compressing the octave of 36edo by about 2 ¢ 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 ¢. The 11- and 13-limit TE tunings of 36edo both do this, as do their respective WE tunings.
| Tuning | Step size (cents) |
Prime error (cents) | Mapping of primes 2–13 (steps) | Octave stretch | |||||
|---|---|---|---|---|---|---|---|---|---|
| 2 | 3 | 5 | 7 | 11 | 13 | ||||
| 21edf | 33.426 | +3.3 | +3.3 | −12.0 | +7.2 | −6.5 | +5.1 | 36, 57, 83, 101, 124, 133 | +0.275% |
| 57edt | 33.368 | +1.2 | 0.0 | +16.6 | +1.3 | −13.7 | −2.6 | 36, 57, 84, 101, 124, 133 | +0.001% |
| 155zpi | 33.346 | +0.6 | −1.0 | +15.1 | −0.5 | −16.0 | −5.0 | 36, 57, 83, 101, 124, 133 | +0.0005% |
| 36edo | 33.333 | 0.0 | −2.0 | +13.7 | −2.2 | +15.3 | −7.2 | 36, 57, 84, 101, 125, 133 | 0% |
| 13-limit WE | 33.302 | −1.1 | −3.7 | +11.1 | −5.3 | +11.4 | −11.4 | 36, 57, 84, 101, 125, 133 | -0.0009% |
| 11-limit WE | 33.286 | −1.7 | −4.7 | +9.7 | −6.9 | +9.4 | −13.5 | 36, 57, 84, 101, 125, 133 | -0.00142% |