User:BudjarnLambeth/Sandbox2: Difference between revisions
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| Line 52: | Line 52: | ||
* Main: "23edo and octave stretching" | * Main: "23edo and octave stretching" | ||
* 36edt | * 36edt | ||
{{harmonics in equal|23|3|1|intervals=prime}} | |||
* 59ed6 | * 59ed6 | ||
* 60ed6 | * 60ed6 | ||
| Line 58: | Line 59: | ||
* 1ed33/32 | * 1ed33/32 | ||
* 2.3.5.13 WE (52.447c) | * 2.3.5.13 WE (52.447c) | ||
{{harmonics in cet|52.447|intervals=prime}} | |||
* 2.7.11 WE (51.962c) | * 2.7.11 WE (51.962c) | ||
* 13-limit WE (52.237c) | * 13-limit WE (52.237c) | ||
Revision as of 02:31, 28 August 2025
Title1
Octave stretch or compression
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly compressing the octave is acceptable, using tunings such as 157edt or 256ed6. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
What follows is a comparison of stretched- and compressed-octave 99edo tunings.
- Step size: 12.138 ¢, octave size: 1201.66 ¢
Stretching the octave of 99edo by around 1.5 ¢ results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54 ¢. The tuning 567zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.66 | +3.71 | +3.32 | +5.43 | +5.37 | +5.54 | +4.99 | -4.72 | -5.05 | -0.12 | -5.10 |
| Relative (%) | +13.7 | +30.6 | +27.4 | +44.7 | +44.3 | +45.6 | +41.1 | -38.9 | -41.6 | -1.0 | -42.0 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 313 | 328 | 342 | 354 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.98 | -4.94 | -3.00 | -5.49 | -1.20 | -3.05 | +0.45 | -3.39 | -2.89 | +1.54 | -2.59 | -3.44 |
| Relative (%) | +16.3 | -40.7 | -24.7 | -45.2 | -9.9 | -25.2 | +3.7 | -27.9 | -23.8 | +12.7 | -21.3 | -28.3 | |
| Step | 366 | 376 | 386 | 395 | 404 | 412 | 420 | 427 | 434 | 441 | 447 | 453 | |
- Step size: 12.123 ¢, octave size: 1200.18 ¢
Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.18 | +1.36 | +0.35 | +1.98 | +1.53 | +1.37 | +0.53 | +2.71 | +2.15 | -5.25 | +1.71 |
| Relative (%) | +1.5 | +11.2 | +2.9 | +16.3 | +12.6 | +11.3 | +4.4 | +22.4 | +17.8 | -43.3 | +14.1 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 342 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.51 | +1.55 | +3.33 | +0.71 | +4.86 | +2.89 | -5.85 | +2.33 | +2.72 | -5.07 | +2.83 | +1.89 |
| Relative (%) | -29.0 | +12.7 | +27.5 | +5.8 | +40.1 | +23.8 | -48.3 | +19.2 | +22.5 | -41.9 | +23.3 | +15.6 | |
| Step | 366 | 377 | 387 | 396 | 405 | 413 | 420 | 428 | 435 | 441 | 448 | 454 | |
- 99edo
- Step size: 12.121 ¢, octave size: 1200.00 ¢
Pure-octaves 99edo approximates all harmonics up to 16 within 5.86 ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +1.08 | +0.00 | +1.57 | +1.08 | +0.87 | +0.00 | +2.15 | +1.57 | -5.86 | +1.08 |
| Relative (%) | +0.0 | +8.9 | +0.0 | +12.9 | +8.9 | +7.2 | +0.0 | +17.7 | +12.9 | -48.4 | +8.9 | |
| Steps (reduced) |
99 (0) |
157 (58) |
198 (0) |
230 (32) |
256 (58) |
278 (80) |
297 (0) |
314 (17) |
329 (32) |
342 (45) |
355 (58) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -4.16 | +0.87 | +2.64 | +0.00 | +4.14 | +2.15 | +5.52 | +1.57 | +1.95 | -5.86 | +2.03 | +1.08 |
| Relative (%) | -34.4 | +7.2 | +21.8 | +0.0 | +34.1 | +17.7 | +45.5 | +12.9 | +16.1 | -48.4 | +16.7 | +8.9 | |
| Steps (reduced) |
366 (69) |
377 (80) |
387 (90) |
396 (0) |
405 (9) |
413 (17) |
421 (25) |
428 (32) |
435 (39) |
441 (45) |
448 (52) |
454 (58) | |
- Step size: 12.117 ¢, octave size: 1199.58 ¢
Compressing the octave of 99edo by around 0.6 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.42 | +0.41 | -0.83 | +0.60 | -0.00 | -0.30 | -1.25 | +0.83 | +0.18 | +4.81 | -0.42 |
| Relative (%) | -3.4 | +3.4 | -6.9 | +4.9 | -0.0 | -2.5 | -10.3 | +6.8 | +1.5 | +39.7 | -3.5 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 343 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.71 | -0.72 | +1.01 | -1.67 | +2.43 | +0.41 | +3.74 | -0.24 | +0.11 | +4.40 | +0.14 | -0.84 |
| Relative (%) | -47.1 | -5.9 | +8.3 | -13.8 | +20.1 | +3.4 | +30.9 | -2.0 | +0.9 | +36.3 | +1.2 | -6.9 | |
| Step | 366 | 377 | 387 | 396 | 405 | 413 | 421 | 428 | 435 | 442 | 448 | 454 | |
- Step size: 12.115 ¢, octave size: 1199.39 ¢
Compressing the octave of 99edo by around 0.4 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68 ¢. The tuning 568zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.61 | +0.10 | -1.23 | +0.14 | -0.52 | -0.86 | -1.84 | +0.20 | -0.48 | +4.13 | -1.13 |
| Relative (%) | -5.1 | +0.8 | -10.2 | +1.1 | -4.3 | -7.1 | -15.2 | +1.7 | -4.0 | +34.1 | -9.3 | |
| Step | 99 | 157 | 198 | 230 | 256 | 278 | 297 | 314 | 329 | 343 | 355 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.68 | -1.47 | +0.24 | -2.46 | +1.62 | -0.42 | +2.90 | -1.09 | -0.76 | +3.51 | -0.75 | -1.75 |
| Relative (%) | +46.9 | -12.1 | +2.0 | -20.3 | +13.4 | -3.4 | +24.0 | -9.0 | -6.2 | +29.0 | -6.2 | -14.4 | |
| Step | 367 | 377 | 387 | 396 | 405 | 413 | 421 | 428 | 435 | 442 | 448 | 454 | |
- Step size: 12.114 ¢, octave size: 1199.32 ¢
Compressing the octave of 99edo by around 0.3 ¢ results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44 ¢. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.68 | +0.00 | -1.36 | -0.01 | -0.68 | -1.03 | -2.03 | +0.00 | -0.69 | +3.91 | -1.36 |
| Relative (%) | -5.6 | +0.0 | -11.2 | -0.1 | -5.6 | -8.5 | -16.8 | +0.0 | -5.7 | +32.3 | -11.2 | |
| Steps (reduced) |
99 (99) |
157 (0) |
198 (41) |
230 (73) |
256 (99) |
278 (121) |
297 (140) |
314 (0) |
329 (15) |
343 (29) |
355 (41) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.44 | -1.71 | -0.01 | -2.71 | +1.36 | -0.68 | +2.63 | -1.37 | -1.03 | +3.23 | -1.04 | -2.03 |
| Relative (%) | +44.9 | -14.1 | -0.1 | -22.4 | +11.2 | -5.6 | +21.7 | -11.3 | -8.5 | +26.7 | -8.6 | -16.8 | |
| Steps (reduced) |
367 (53) |
377 (63) |
387 (73) |
396 (82) |
405 (91) |
413 (99) |
421 (107) |
428 (114) |
435 (121) |
442 (128) |
448 (134) |
454 (140) | |
Title2
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
23edo (narrow down edonoi & ZPIs)
- Main: "23edo and octave stretching"
- 36edt
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +40.4 | +0.0 | +25.3 | +21.6 | -16.6 | +24.9 | -26.0 | +29.5 | +29.5 | -41.0 | +8.9 |
| Relative (%) | +48.9 | +0.0 | +30.6 | +26.1 | -20.1 | +30.1 | -31.5 | +35.7 | +35.7 | -49.6 | +10.8 | |
| Steps (reduced) |
15 (15) |
23 (0) |
34 (11) |
41 (18) |
50 (4) |
54 (8) |
59 (13) |
62 (16) |
66 (20) |
70 (1) |
72 (3) | |
- 59ed6
- 60ed6
- 68ed8
- 11ed7/5
- 1ed33/32
- 2.3.5.13 WE (52.447c)
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.3 | -13.9 | -6.6 | -12.2 | -8.0 | +17.5 | +25.1 | -10.2 | +26.2 | -8.0 | -18.5 |
| Relative (%) | +12.0 | -26.4 | -12.6 | -23.3 | -15.3 | +33.3 | +47.8 | -19.4 | +50.0 | -15.2 | -35.3 | |
| Step | 23 | 36 | 53 | 64 | 79 | 85 | 94 | 97 | 104 | 111 | 113 | |
- 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 (narrow down edonoi & ZPIs)
- 95edt
- 35edf
- 139ed5
- 155ed6
- 208ed11
- (???)ed12
- 255ed19
- 272ed23 (great for catnip temperament, maybe there's a similar but simpler tuning w similar benefits?)
- 13-limit WE (20.013c)
- 299zpi (20.128c)
- 300zpi (20.093c)
- 301zpi (20.027c)
- 302zpi (19.962c)
- 303zpi (19.913c)
- 304zpi (19.869c)
- Medium priority
13edo
- Main: "13edo and optimal octave stretching"
- 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)
32edo (narrow down 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 (narrow down 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 (narrow down slightly)
- 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 (narrow down slightly)
- 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 (narrow down slightly)
- 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 (narrow down 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)
64edo (narrow down ZPIs)
- 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)
103edo (narrow down edonoi, choose ZPIS)
- 163edt
- 239ed5
- (???)ed6
- 289ed7
- 356ed11
- (???)ed12
- 381ed13
- 421ed17
- 466ed23
- 13-limit WE (11.658c)
- Best nearby ZPI(s)
111edo (choose ZPIS)
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
118edo (choose ZPIS)
- 187edt
- 69edf
- 13-limit WE (10.171c)
- Best nearby ZPI(s)
- Low priority
104edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
125edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
145edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
152edo
- 241edt
- 13-limit WE (7.894c)
- Best nearby ZPI(s)
159edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
166edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
182edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
198edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
212edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
243edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
247edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
- Optional
25edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
26edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
29edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
30edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
34edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
35edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
36edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
37edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
5edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
6edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
9edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
10edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
11edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
15edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
18edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
48edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
20edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
24edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
28edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)