User:BudjarnLambeth/Sandbox2: Difference between revisions

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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.
58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[92edt]] or [[150ed6]].


What follows is a comparison of stretched-octave 72edo tunings.
What follows is a comparison of stretched- and compressed-octave 58edo tunings.


; 72edo
; [[zpi|288zpi]]
* Step size: 16.667{{c}}, octave size: 1200.00{{c}}  
* Step size: 20.736{{c}}, octave size: NNN{{c}}
Pure-octaves 72edo approximates all harmonics up to 16 within NNN{{c}}.
_ing the octave of EDONAME 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 ZPINAME does this.
{{Harmonics in equal|72|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72edo}}
{{Harmonics in cet|20.736|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|72|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72edo (continued)}}
{{Harmonics in cet|20.736|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}


; [[249ed11]]
; 58edo
* Step size: NNN{{c}}, octave size: 1200.38{{c}}
* Step size: 20.690{{c}}, octave size: NNN{{c}}  
Stretching the octave of 72edo by around 0.4{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 249ed11 does this.
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|249|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 249ed11}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}}
{{Harmonics in equal|249|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 249ed11 (continued)}}
{{Harmonics in equal|58|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}}


; [[258ed12]]  
; [[WE|58et, 7-limit WE tuning]]  
* Step size: NNN{{c}}, octave size: 1200.55{{c}}
* Step size: 20.667{{c}}, octave size: NNN{{c}}
Stretching the octave of 72edo by around 0.5{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 258ed12 does this.
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in equal|258|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 258ed12}}
{{Harmonics in cet|20.667|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in equal|258|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 258ed12 (continued)}}
{{Harmonics in cet|20.667|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[186ed6]] / [[WE|72et, 11-limit WE tuning]] / [[ed7|202ed7]]
; [[zpi|289zpi]]  
* Step size: NNN{{c}}, octave size: 1200.76{{c}}
* Step size: 20.666{{c}}, octave size: NNN{{c}}
Stretching the octave of 72edo by around 0.75{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 186ed6 does this. 72et's 11-limit WE tuning and 11-limit [[TE]] tuning both do this, their octave differing from 186ed6's by only 0.02{{c}}. The tuning 202ed7 does this also, it's octave differing from 186ed6 by less than a hundredth of a cent.
_ing the octave of EDONAME 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 ZPINAME does this.
{{Harmonics in equal|186|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 186ed6}}
{{Harmonics in cet|20.666|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}}
{{Harmonics in equal|186|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 186ed6 (continued)}}
{{Harmonics in cet|20.666|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}}


; [[zpi|380zpi]]  
; [[WE|58et, 13-limit WE tuning]]  
* Step size: 16.678{{c}}, octave size: 1200.82{{c}}
* Step size: 20.663{{c}}, octave size: NNN{{c}}
Stretching the octave of 72edo by around 0.8{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 380zpi does this.
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this.
{{Harmonics in cet|16.678|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 380zpi}}
{{Harmonics in cet|20.663|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}}
{{Harmonics in cet|16.678|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 380zpi (continued)}}
{{Harmonics in cet|20.663|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}}


; [[WE|72et, 13-limit WE tuning]]  
; [[Ned12]]  
* Step size: 16.680{{c}}, octave size: 1200.96{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 72edo by around NNN{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
_ing the octave of EDONAME 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 EDONOI does this.
{{Harmonics in cet|16.680|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in cet|16.680|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 72et, 13-limit WE tuning (continued)}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}


; [[114edt]] / [[167ed5]]
; [[150ed6]]  
* Step size: NNN{{c}}, octave size: 1201.23{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Stretching the octave of 72edo by around NNN{{c}} results in [[JND|unnoticeably]] better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 144edt does this. The tuning 167ed5 does this also, its octave differing from 114edt by only 0.05{{c}}.
_ing the octave of EDONAME 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 EDONOI does this.
{{Harmonics in equal|114|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 114edt}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|114|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 114edt (continued)}}
{{Harmonics in equal|150|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}
 
; [[92edt]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of EDONAME 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 EDONOI does this.
{{Harmonics in equal|92|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}}
{{Harmonics in equal|92|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}}

Revision as of 23:29, 26 August 2025

Title1

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 -8.5 -8.2 +4.1 -12.6 +19.5 -12.3 -16.9 +0.0 +34.3 -16.7
Relative (%) -4.1 -8.5 -8.2 +4.1 -12.6 +19.6 -12.4 -17.0 +0.0 +34.4 -16.7
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(31) 		34
(34) 		36
(36) 		38
(38) 		40
(0) 		42
(2) 		43
(3)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.4 +3.4 +6.7 +21.5 +6.7 +40.7 +10.1 +6.7 +24.9 -39.9 +10.1
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 +0.0 +2.5 +16.6 +1.2 +34.7 +3.7 +0.0 +17.8 -47.1 +2.5
Relative (%) +1.2 +0.0 +2.5 +16.6 +1.2 +34.6 +3.7 +0.0 +17.8 -47.1 +2.5
Steps
(reduced) 		12
(12) 		19
(0) 		24
(5) 		28
(9) 		31
(12) 		34
(15) 		36
(17) 		38
(0) 		40
(2) 		41
(3) 		43
(5)
Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -0.8 +1.5 +15.5 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Relative (%) +0.8 -0.8 +1.5 +15.4 +0.0 +33.3 +2.3 -1.5 +16.2 -48.7 +0.8
Steps
(reduced) 		12
(12) 		19
(19) 		24
(24) 		28
(28) 		31
(0) 		34
(3) 		36
(5) 		38
(7) 		40
(9) 		41
(10) 		43
(12)

Title2

Octave stretch or compression

58edo's approximations of harmonics 3, 5, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as 92edt or 150ed6.

What follows is a comparison of stretched- and compressed-octave 58edo tunings.

288zpi
  • Step size: 20.736 ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.69 +5.76 +5.38 -7.69 +8.44 -9.59 +8.06 -9.22 -5.00 -4.12 -9.60
Relative (%) +13.0 +27.8 +25.9 -37.1 +40.7 -46.3 +38.9 -44.5 -24.1 -19.9 -46.3
Step 58 92 116 134 150 162 174 183 192 200 207
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.02 -6.91 -1.93 -9.98 +9.48 -6.53 +3.54 -2.31 -3.84 -1.43 +4.56 -6.92
Relative (%) -14.6 -33.3 -9.3 -48.1 +45.7 -31.5 +17.1 -11.2 -18.5 -6.9 +22.0 -33.3
Step 214 220 226 231 237 241 246 250 254 258 262 265
58edo
  • Step size: 20.690 ¢, octave size: NNN ¢

Pure-octaves EDONAME approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in EDONAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 +1.49 +0.00 +6.79 +1.49 +3.59 +0.00 +2.99 +6.79 +7.30 +1.49
Relative (%) +0.0 +7.2 +0.0 +32.8 +7.2 +17.3 +0.0 +14.4 +32.8 +35.3 +7.2
Steps
(reduced) 		58
(0) 		92
(34) 		116
(0) 		135
(19) 		150
(34) 		163
(47) 		174
(0) 		184
(10) 		193
(19) 		201
(27) 		208
(34)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.75 +3.59 +8.28 +0.00 -1.51 +2.99 -7.86 +6.79 +5.08 +7.30 -7.58 +1.49
Relative (%) +37.4 +17.3 +40.0 +0.0 -7.3 +14.4 -38.0 +32.8 +24.6 +35.3 -36.7 +7.2
Steps
(reduced) 		215
(41) 		221
(47) 		227
(53) 		232
(0) 		237
(5) 		242
(10) 		246
(14) 		251
(19) 		255
(23) 		259
(27) 		262
(30) 		266
(34)
58et, 7-limit WE tuning
  • Step size: 20.667 ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.31 -0.59 -2.63 +3.73 -1.91 -0.10 -3.94 -1.18 +2.42 +2.75 -3.22
Relative (%) -6.4 -2.9 -12.7 +18.1 -9.2 -0.5 -19.1 -5.7 +11.7 +13.3 -15.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.88 -1.42 +3.14 -5.26 -6.88 -2.50 +7.24 +1.10 -0.70 +1.44 +7.15 -4.53
Relative (%) +13.9 -6.9 +15.2 -25.4 -33.3 -12.1 +35.0 +5.3 -3.4 +6.9 +34.6 -21.9
Step 215 221 227 232 237 242 247 251 255 259 263 266
289zpi
  • Step size: 20.666 ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning ZPINAME does this.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.37 -0.68 -2.74 +3.60 -2.06 -0.27 -4.12 -1.37 +2.22 +2.55 -3.43
Relative (%) -6.6 -3.3 -13.3 +17.4 -9.9 -1.3 -19.9 -6.6 +10.8 +12.3 -16.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.66 -1.64 +2.91 -5.49 -7.11 -2.74 +6.99 +0.85 -0.95 +1.18 +6.88 -4.80
Relative (%) +12.9 -7.9 +14.1 -26.6 -34.4 -13.2 +33.8 +4.1 -4.6 +5.7 +33.3 -23.2
Step 215 221 227 232 237 242 247 251 255 259 263 266
58et, 13-limit WE tuning
  • Step size: 20.663 ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.55 -0.96 -3.09 +3.19 -2.51 -0.76 -4.64 -1.92 +1.65 +1.95 -4.05
Relative (%) -7.5 -4.6 -15.0 +15.4 -12.1 -3.7 -22.4 -9.3 +8.0 +9.4 -19.6
Step 58 92 116 135 150 163 174 184 193 201 208
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.02 -2.30 +2.23 -6.18 -7.82 -3.46 +6.25 +0.10 -1.72 +0.40 +6.09 -5.60
Relative (%) +9.8 -11.1 +10.8 -29.9 -37.9 -16.8 +30.2 +0.5 -8.3 +1.9 +29.5 -27.1
Step 215 221 227 232 237 242 247 251 255 259 263 266
Ned12
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.58 +0.58 -1.15 +5.45 +0.00 +1.97 -1.73 +1.15 +4.87 +5.30 -0.58
Relative (%) -2.8 +2.8 -5.6 +26.3 +0.0 +9.5 -8.4 +5.6 +23.5 +25.6 -2.8
Steps
(reduced) 		58
(58) 		92
(92) 		116
(116) 		135
(135) 		150
(0) 		163
(13) 		174
(24) 		184
(34) 		193
(43) 		201
(51) 		208
(58)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.61 +1.39 +6.02 -2.31 -3.87 +0.58 -10.31 +4.29 +2.54 +4.72 -10.19 -1.15
Relative (%) +27.1 +6.7 +29.1 -11.2 -18.7 +2.8 -49.8 +20.7 +12.3 +22.8 -49.3 -5.6
Steps
(reduced) 		215
(65) 		221
(71) 		227
(77) 		232
(82) 		237
(87) 		242
(92) 		246
(96) 		251
(101) 		255
(105) 		259
(109) 		262
(112) 		266
(116)
150ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.58 +0.58 -1.15 +5.45 +0.00 +1.97 -1.73 +1.15 +4.87 +5.30 -0.58
Relative (%) -2.8 +2.8 -5.6 +26.3 +0.0 +9.5 -8.4 +5.6 +23.5 +25.6 -2.8
Steps
(reduced) 		58
(58) 		92
(92) 		116
(116) 		135
(135) 		150
(0) 		163
(13) 		174
(24) 		184
(34) 		193
(43) 		201
(51) 		208
(58)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.61 +1.39 +6.02 -2.31 -3.87 +0.58 -10.31 +4.29 +2.54 +4.72 -10.19 -1.15
Relative (%) +27.1 +6.7 +29.1 -11.2 -18.7 +2.8 -49.8 +20.7 +12.3 +22.8 -49.3 -5.6
Steps
(reduced) 		215
(65) 		221
(71) 		227
(77) 		232
(82) 		237
(87) 		242
(92) 		246
(96) 		251
(101) 		255
(105) 		259
(109) 		262
(112) 		266
(116)
92edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.94 +0.00 -1.88 +4.60 -0.94 +0.94 -2.82 +0.00 +3.66 +4.04 -1.88
Relative (%) -4.6 +0.0 -9.1 +22.2 -4.6 +4.6 -13.7 +0.0 +17.7 +19.5 -9.1
Steps
(reduced) 		58
(58) 		92
(0) 		116
(24) 		135
(43) 		150
(58) 		163
(71) 		174
(82) 		184
(0) 		193
(9) 		201
(17) 		208
(24)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.26 +0.00 +4.60 -3.77 -5.35 -0.94 +8.82 +2.72 +0.94 +3.10 +8.84 -2.82
Relative (%) +20.6 +0.0 +22.2 -18.2 -25.9 -4.6 +42.7 +13.1 +4.6 +15.0 +42.7 -13.7
Steps
(reduced) 		215
(31) 		221
(37) 		227
(43) 		232
(48) 		237
(53) 		242
(58) 		247
(63) 		251
(67) 		255
(71) 		259
(75) 		263
(79) 		266
(82)
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