User:BudjarnLambeth/Sandbox2: Difference between revisions

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


; [[184zpi]] / [[WE|41et, 11-limit WE tuning]]  
; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]  
* Step size: 29.277{{c}}, octave size: NNN{{c}}
* Step size: 29.277{{c}}, octave size: NNN{{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.
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, whose octave is identical to WE within 0.02{{c}}.
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning}}
{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 11-limit WE tuning (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)}}
Line 17: Line 17:
; 41edo
; 41edo
* Step size: 29.268{{c}}, octave size: 1200.0{{c}}  
* Step size: 29.268{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}.
Pure-octaves 41edo approximates all harmonics up to 16 within NNN{{c}}. The octaves of its compressed tuning [[147ed12]] differ by only 0.1{{c}} from pure. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}


; [[WE|41et, 13-limit WE tuning]]  
; [[147ed12]] / [[106ed6]] / [[65edt]]
* Step size: 29.267{{c}}, octave size: NNN{{c}}
* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{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}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
{{Harmonics in cet|29.267|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning}}
* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
{{Harmonics in cet|29.267|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 13-limit WE tuning (continued)}}
Compressing the octave of 41edo by around 0.2{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tunings 147ed12, 106ed6 and 65edt do this.
 
; [[147ed12]]
* Step size: NNN{{c}}, octave size: NNN{{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.
{{Harmonics in equal|147|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 147ed12}}
{{Harmonics in equal|147|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 147ed12 (continued)}}
 
; [[106ed6]]
* Step size: NNN{{c}}, octave size: NNN{{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 106ed6 does this.
{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}
; [[65edt]]
* Step size: NNN{{c}}, octave size: NNN{{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 65edt does this.
{{Harmonics in equal|65|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65edt}}
{{Harmonics in equal|65|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65edt (continued)}}

Revision as of 21:23, 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, whose octave is identical to WE within 0.02 ¢.

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 ¢. The octaves of its compressed tuning 147ed12 differ by only 0.1 ¢ from pure. The octaves of its 13-limit WE and TE tuning differ by less than 0.1 ¢ from pure.

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)
147ed12 / 106ed6 / 65edt
  • 147ed12 — step size: 29.265 ¢, octave size: 1199.87 ¢
  • 106ed6 — step size: 29.264 ¢, octave size: 1199.69 ¢
  • 65edt — step size: 29.261 ¢, octave size: 1199.81 ¢

Compressing the octave of 41edo by around 0.2 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tunings 147ed12, 106ed6 and 65edt do 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)
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