User:BudjarnLambeth/Sandbox2
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) | |