User:BudjarnLambeth/Sandbox2: Difference between revisions

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= Title1 =
= Title1 =
== Octave stretch or compression ==
== Octave stretch or compression ==
Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and [[gamelan]], with [[37ed5]] and [[57ed12]] being good options.
99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|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 16edo tunings.
What follows is a comparison of stretched- and compressed-octave 99edo tunings.


; 16edo
; [[zpi|567zpi]]
* Step size: 75.000{{c}}, octave size: 1200.0{{c}}  
* Step size: 12.138{{c}}, octave size: NNN{{c}}
Pure-octaves 16edo approximates all harmonics up to 16 within 36.7{{c}}.
Stretching the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 567zpi does this.
{{Harmonics in equal|16|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16edo}}
{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}
{{Harmonics in equal|16|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16edo (continued)}}
{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}


; [[WE|16et, 2.5.7.13 WE tuning]]  
; [[WE|99et, 13-limit WE tuning]]  
* Step size: 75.105{{c}}, octave size: 1201.7{{c}}
* Step size: 12.123{{c}}, octave size: NNN{{c}}
Stretching the octave of 16edo by around 2{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8{{c}}. Its 2.5.7.13 WE tuning and 2.5.7.13 [[TE]] tuning both do this.
Stretching the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|75.105|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning}}
{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}
{{Harmonics in cet|75.105|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)}}
{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}


; [[zpi|15zpi]] / [[equal tuning|59ed13]]
; 99edo
* Step size: 75.262{{c}}, octave size: 1204.2{{c}}
* Step size: 12.121{{c}}, octave size: 1200.00{{c}}  
Stretching the octave of 16edo by around 4{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5{{c}}. The tunings 15zpi and 59ed13 do this.
Pure-octaves 99edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in cet|75.262|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 15zpi}}
{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}
{{Harmonics in cet|75.262|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 15zpi (continued)}}
{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}


; [[WE|16et, 13-limit WE tuning]] / [[37ed5]]
; [[WE|99et, 7-limit WE tuning]]  
* Step size (WE 16et): 75.315{{c}}, octave size (WE 16et): 1205.0{{c}}
* Step size: 12.117{{c}}, octave size: NNN{{c}}
Stretching the octave of 16edo by around 5{{c}} results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this, so does the tuning 37ed5.
Compressing the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
{{Harmonics in cet|75.315|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning}}
{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}
{{Harmonics in cet|75.315|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 16et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}
{{Harmonics in equal|37|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 37ed5}}
{{Harmonics in equal|37|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 37ed5 (continued)}}


; [[57ed12]] / [[equal tuning|55ed11]]
; [[zpi|568zpi]]  
* Step size (57ed12): 75.473{{c}}, octave size (57ed12): 1207.6{{c}}
* Step size: 12.115{{c}}, octave size: NNN{{c}}
Stretching the octave of 16edo by around 7.5{{c}} results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 57ed12 and 55ed11 do this.
Compressing the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 568zpi does this.
{{Harmonics in equal|57|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 57ed12}}
{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}
{{Harmonics in equal|57|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 57ed12 (continued)}}
{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}
{{Harmonics in equal|55|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 55ed11}}
 
{{Harmonics in equal|55|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 55ed11 (continued)}}
; [[256ed6]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 256ed6 does this.
{{Harmonics in equal|256|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 256ed6}}
{{Harmonics in equal|256|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 256ed6 (continued)}}
 
; [[157edt]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 99edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 157edt does this.
{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}
{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}


= Title2 =
= Title2 =
=== Placeholder ===
=== Placeholder ===

Revision as of 01:46, 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.

567zpi
  • Step size: 12.138 ¢, octave size: NNN ¢

Stretching the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 567zpi does this.

Approximation of harmonics in 567zpi
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
Approximation of harmonics in 567zpi (continued)
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
99et, 13-limit WE tuning
  • Step size: 12.123 ¢, octave size: NNN ¢

Stretching the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 99et, 13-limit WE tuning
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
Approximation of harmonics in 99et, 13-limit WE tuning (continued)
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 NNN ¢.

Approximation of harmonics in 99edo
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)
Approximation of harmonics in 99edo (continued)
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)
99et, 7-limit WE tuning
  • Step size: 12.117 ¢, octave size: NNN ¢

Compressing the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this.

Approximation of harmonics in 99et, 7-limit WE tuning
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
Approximation of harmonics in 99et, 7-limit WE tuning (continued)
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
568zpi
  • Step size: 12.115 ¢, octave size: NNN ¢

Compressing the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 568zpi does this.

Approximation of harmonics in 568zpi
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
Approximation of harmonics in 568zpi (continued)
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
256ed6
  • Step size: NNN ¢, octave size: NNN ¢

Compressing the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 256ed6 does this.

Approximation of harmonics in 256ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.42 +0.42 -0.83 +0.60 +0.00 -0.30 -1.25 +0.83 +0.18 +4.82 -0.42
Relative (%) -3.4 +3.4 -6.9 +4.9 +0.0 -2.4 -10.3 +6.9 +1.5 +39.8 -3.4
Steps
(reduced) 		99
(99) 		157
(157) 		198
(198) 		230
(230) 		256
(0) 		278
(22) 		297
(41) 		314
(58) 		329
(73) 		343
(87) 		355
(99)
Approximation of harmonics in 256ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.70 -0.71 +1.01 -1.66 +2.43 +0.42 +3.75 -0.23 +0.12 +4.40 +0.15 -0.83
Relative (%) -47.1 -5.9 +8.4 -13.7 +20.1 +3.4 +30.9 -1.9 +1.0 +36.3 +1.2 -6.9
Steps
(reduced) 		366
(110) 		377
(121) 		387
(131) 		396
(140) 		405
(149) 		413
(157) 		421
(165) 		428
(172) 		435
(179) 		442
(186) 		448
(192) 		454
(198)
157edt
  • Step size: NNN ¢, octave size: NNN ¢

Compressing the octave of 99edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 157edt does this.

Approximation of harmonics in 157edt
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)
Approximation of harmonics in 157edt (continued)
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

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