User:BudjarnLambeth/Sandbox2: Difference between revisions

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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
What follows is a comparison of compressed-octave 27edo tunings.
What follows is a comparison of stretched- and compressed-octave 41edo tunings.


; 27edo
; [[184zpi]] / [[WE|41et, 11-limit WE tuning]]
* Step size: 44.444{{c}}, octave size: 1200.0{{c}}  
* Step size: 29.277{{c}}, octave size: NNN{{c}}
Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.
Stretching 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 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, which is identical to WE within 1/1000 of a cent.
{{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}
{{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}}
{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning (continued)}}


; [[WE|27et, 13-limit WE tuning]]
; 41edo
* Step size: 44.375{{c}}, octave size: 1198.9{{c}}
* Step size: 29.268{{c}}, octave size: 1200.0{{c}}  
Compressing the octave of 27edo by around 2{{c}} results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}


; [[97ed12]]  
; [[WE|41et, 13-limit WE tuning]]  
* Step size: 44.350{{c}}, octave size: 1197.5{{c}}
* Step size: 29.267{{c}}, octave size: NNN{{c}}
Compressing the octave of 27edo by around 2.5{{c}} has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6{{c}}. The tuning 97ed12 does this.
Compressing the octave of 41edo 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 equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}}
{{Harmonics in cet|29.267|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning}}
{{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}}
{{Harmonics in cet|29.267|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning (continued)}}


; [[zpi|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]]
; [[147ed12]]  
* Step size (106zpi): 44.306{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Octave size (70ed6): 1196.5{{c}}
Compressing the octave of 41edo 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 147ed12 does this.
* Octave size (7-lim WE): 1196.4{{c}}
{{Harmonics in equal|147|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 147ed12}}
* Octave size (106zpi): 1196.2{{c}}
{{Harmonics in equal|147|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 147ed12 (continued)}}
Compressing the octave of 27edo by around 3.5{{c}} results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So do the tunings 106zpi and 70ed6.
{{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}}
{{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}}


; [[90ed10]]  
; [[106ed6]]  
* Step size: 44.292{{c}}, octave size: 1195.9{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 27edo by around 4{{c}} results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4{{c}}. The tuning 90ed10 does this.
Compressing the octave of 41edo 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 106ed6 does this.
{{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}


; [[43edt]]  
; [[65edt]]  
* Step size: 44.232{{c}}, octave size: 1194.3{{c}}
* Step size: NNN{{c}}, octave size: NNN{{c}}
Compressing the octave of 27edo by around 5.5{{c}} results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2{{c}}. The tuning 43edt does this.
Compressing the octave of 41edo 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 65edt does this.
{{Harmonics in equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}}
{{Harmonics in equal|65|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65edt}}
{{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}}
{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65edt (continued)}}

Revision as of 21:10, 25 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

What follows is a comparison of stretched- and compressed-octave 41edo tunings.

184zpi / 41et, 11-limit WE tuning
  • Step size: 29.277 ¢, octave size: NNN ¢

Stretching 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 11-limit WE tuning and 11-limit TE tuning both do this. So does 184zpi, which is identical to WE within 1/1000 of a cent.

Approximation of harmonics in 41et, 11-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.4 +1.0 +0.7 -5.0 +1.4 -2.0 +1.1 +2.1 -4.6 +6.0 +1.8
Relative (%) +1.2 +3.6 +2.4 -17.1 +4.8 -6.7 +3.7 +7.2 -15.9 +20.5 +6.0
Step 41 65 82 95 106 115 123 130 136 142 147
Approximation of harmonics in 41et, 11-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.6 -1.6 -3.9 +1.4 +13.6 +2.5 -3.3 -4.3 -0.9 +6.4 -12.0 +2.1
Relative (%) +32.7 -5.5 -13.5 +4.9 +46.4 +8.4 -11.3 -14.6 -3.1 +21.8 -41.1 +7.2
Step 152 156 160 164 168 171 174 177 180 183 185 188
41edo
  • Step size: 29.268 ¢, octave size: 1200.0 ¢

Pure-octaves 41edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 41edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +0.5 +0.0 -5.8 +0.5 -3.0 +0.0 +1.0 -5.8 +4.8 +0.5
Relative (%) +0.0 +1.7 +0.0 -19.9 +1.7 -10.2 +0.0 +3.3 -19.9 +16.3 +1.7
Steps
(reduced) 		41
(0) 		65
(24) 		82
(0) 		95
(13) 		106
(24) 		115
(33) 		123
(0) 		130
(7) 		136
(13) 		142
(19) 		147
(24)
Approximation of harmonics in 41edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +8.3 -3.0 -5.3 +0.0 +12.1 +1.0 -4.8 -5.8 -2.5 +4.8 -13.6 +0.5
Relative (%) +28.2 -10.2 -18.3 +0.0 +41.4 +3.3 -16.5 -19.9 -8.5 +16.3 -46.6 +1.7
Steps
(reduced) 		152
(29) 		156
(33) 		160
(37) 		164
(0) 		168
(4) 		171
(7) 		174
(10) 		177
(13) 		180
(16) 		183
(19) 		185
(21) 		188
(24)
41et, 13-limit WE tuning
  • Step size: 29.267 ¢, octave size: NNN ¢

Compressing the octave of 41edo 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 41et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.1 +0.4 -0.1 -5.9 +0.3 -3.1 -0.2 +0.8 -6.0 +4.6 +0.3
Relative (%) -0.2 +1.4 -0.4 -20.3 +1.2 -10.7 -0.5 +2.7 -20.5 +15.7 +1.0
Step 41 65 82 95 106 115 123 130 136 142 147
Approximation of harmonics in 41et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +8.1 -3.2 -5.5 -0.2 +11.9 +0.7 -5.1 -6.1 -2.7 +4.5 -13.9 +0.2
Relative (%) +27.5 -10.8 -19.0 -0.7 +40.7 +2.6 -17.3 -20.7 -9.3 +15.5 -47.4 +0.8
Step 152 156 160 164 168 171 174 177 180 183 185 188
147ed12
  • Step size: NNN ¢, octave size: NNN ¢

Compressing the octave of 41edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 147ed12 does this.

Approximation of harmonics in 147ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.1 +0.3 -0.3 -6.1 +0.1 -3.4 -0.4 +0.5 -6.3 +4.3 +0.0
Relative (%) -0.5 +0.9 -0.9 -21.0 +0.5 -11.5 -1.4 +1.8 -21.4 +14.7 +0.0
Steps
(reduced) 		41
(41) 		65
(65) 		82
(82) 		95
(95) 		106
(106) 		115
(115) 		123
(123) 		130
(130) 		136
(136) 		142
(142) 		147
(0)
Approximation of harmonics in 147ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.8 -3.5 -5.9 -0.5 +11.6 +0.4 -5.4 -6.4 -3.1 +4.2 -14.2 -0.1
Relative (%) +26.5 -11.9 -20.1 -1.8 +39.5 +1.4 -18.5 -21.9 -10.5 +14.3 -48.7 -0.5
Steps
(reduced) 		152
(5) 		156
(9) 		160
(13) 		164
(17) 		168
(21) 		171
(24) 		174
(27) 		177
(30) 		180
(33) 		183
(36) 		185
(38) 		188
(41)
106ed6
  • Step size: NNN ¢, octave size: NNN ¢

Compressing the octave of 41edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 106ed6 does this.

Approximation of harmonics in 106ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.2 +0.2 -0.4 -6.3 +0.0 -3.5 -0.6 +0.4 -6.4 +4.1 -0.2
Relative (%) -0.6 +0.6 -1.3 -21.4 +0.0 -12.0 -1.9 +1.3 -22.0 +14.1 -0.6
Steps
(reduced) 		41
(41) 		65
(65) 		82
(82) 		95
(95) 		106
(0) 		115
(9) 		123
(17) 		130
(24) 		136
(30) 		142
(36) 		147
(41)
Approximation of harmonics in 106ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.6 -3.7 -6.1 -0.7 +11.4 +0.2 -5.6 -6.6 -3.3 +3.9 -14.5 -0.4
Relative (%) +25.8 -12.6 -20.8 -2.6 +38.8 +0.6 -19.2 -22.7 -11.3 +13.5 -49.5 -1.3
Steps
(reduced) 		152
(46) 		156
(50) 		160
(54) 		164
(58) 		168
(62) 		171
(65) 		174
(68) 		177
(71) 		180
(74) 		183
(77) 		185
(79) 		188
(82)
65edt
  • Step size: NNN ¢, octave size: NNN ¢

Compressing the octave of 41edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 65edt does this.

Approximation of harmonics in 65edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.3 +0.0 -0.6 -6.5 -0.3 -3.8 -0.9 +0.0 -6.8 +3.7 -0.6
Relative (%) -1.0 +0.0 -2.1 -22.3 -1.0 -13.1 -3.1 +0.0 -23.4 +12.7 -2.1
Steps
(reduced) 		41
(41) 		65
(0) 		82
(17) 		95
(30) 		106
(41) 		115
(50) 		123
(58) 		130
(0) 		136
(6) 		142
(12) 		147
(17)
Approximation of harmonics in 65edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +7.1 -4.1 -6.5 -1.2 +10.9 -0.3 -6.1 -7.1 -3.8 +3.4 +14.2 -0.9
Relative (%) +24.3 -14.1 -22.3 -4.2 +37.1 -1.0 -20.9 -24.4 -13.1 +11.7 +48.7 -3.1
Steps
(reduced) 		152
(22) 		156
(26) 		160
(30) 		164
(34) 		168
(38) 		171
(41) 		174
(44) 		177
(47) 		180
(50) 		183
(53) 		186
(56) 		188
(58)
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