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.
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.  


; 27edo
For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].
* Step size: 44.444{{c}}, octave size: 1200.0{{c}}
Pure-octaves 27edo approximates all harmonics up to 16 within 18.3{{c}}.
{{Harmonics in equal|27|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo}}
{{Harmonics in equal|27|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27edo (continued)}}


; [[WE|27et, 13-limit WE tuning]]
Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.
* Step size: 44.375{{c}}, octave size: 1198.9{{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.
{{Harmonics in cet|44.375|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning}}
{{Harmonics in cet|44.375|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 27et, 13-limit WE tuning (continued)}}


; [[97ed12]]
What follows is a comparison of stretched- and compressed-octave 41edo tunings.
* Step size: 44.350{{c}}, octave size: 1197.5{{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.
{{Harmonics in equal|97|12|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12}}
{{Harmonics in equal|97|12|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 97ed12 (continued)}}


; [[zpi|106zpi]] / [[WE|27et, 7-limit WE tuning]] / [[70ed6]]
; [[147ed12]] / [[106ed6]] / [[65edt]]
* Step size (106zpi): 44.306{{c}}
* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
* Octave size (70ed6): 1196.5{{c}}
* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
* Octave size (7-lim WE): 1196.4{{c}}
* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
* Octave size (106zpi): 1196.2{{c}}
Compressing the octave of 41edo by around 0.2{{c}} results in just slightly improved primes 3, 11 and 13, but just slightly worse primes , 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this.
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 equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
{{Harmonics in cet|44.306|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi}}
{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}
{{Harmonics in cet|44.306|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 106zpi (continued)}}


; [[90ed10]]
; 41edo
* Step size: 44.292{{c}}, octave size: 1195.9{{c}}
* Step size: 29.268{{c}}, octave size: 1200.00{{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.
Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
{{Harmonics in equal|90|10|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in equal|90|10|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 90ed10 (continued)}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}


; [[43edt]]  
; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]  
* Step size: 44.232{{c}}, octave size: 1204.3{{c}}
* Step size: 29.277{{c}}, octave size: 1200.35{{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.
Stretching the octave of 41edo by around 0.5{{c}} results in just slightly improved primes 5 and 7, but just slightly worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{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 equal|43|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}
{{Harmonics in equal|43|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 43edt (continued)}}
{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}
 
; [[WE|41et, 7-limit WE tuning]]
* Step size: 29.288{{c}}, octave size: 1200.81{{c}}
Stretching the octave of 41edo by just under 1{{c}} results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. 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|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}
{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}

Latest revision as of 21:33, 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

Whether there is intonational improvement from octave stretch or compression depends on which subgroup we are focusing on.

For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as 65edt, 106ed6, and 147ed12.

Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.

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

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

Compressing the octave of 41edo by around 0.2 ¢ results in just slightly improved primes 3, 11 and 13, but just slightly worse primes , 5 and 7. This approximates all harmonics up to 16 within 7.6 ¢. The tunings 147ed12, 106ed6 and 65edt each 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)
41edo
  • Step size: 29.268 ¢, octave size: 1200.00 ¢

Pure-octaves 41edo approximates all harmonics up to 16 within 8.3 ¢. 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)
184zpi / 41et, 11-limit WE tuning
  • Step size: 29.277 ¢, octave size: 1200.35 ¢

Stretching the octave of 41edo by around 0.5 ¢ results in just slightly improved primes 5 and 7, but just slightly worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6 ¢. 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 184zpi 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 184zpi (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
41et, 7-limit WE tuning
  • Step size: 29.288 ¢, octave size: 1200.81 ¢

Stretching the octave of 41edo by just under 1 ¢ results in improved primes 5 and 7, but worse primes 2, 3, 11 and 13. 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 41et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 +1.8 +1.6 -4.0 +2.6 -0.7 +2.4 +3.5 -3.1 +7.6 +3.4
Relative (%) +2.8 +6.0 +5.5 -13.5 +8.8 -2.4 +8.3 +12.1 -10.7 +25.9 +11.5
Step 41 65 82 95 106 115 123 130 136 142 147
Approximation of harmonics in 41et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +11.2 +0.1 -2.2 +3.2 -13.9 +4.3 -1.4 -2.3 +1.1 +8.4 -10.0 +4.2
Relative (%) +38.4 +0.3 -7.5 +11.0 -47.3 +14.8 -4.8 -8.0 +3.6 +28.6 -34.1 +14.3
Step 152 156 160 164 167 171 174 177 180 183 185 188
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