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= Title2 =
= Title2 =
== Octave stretch or compression ==
== Octave stretch or compression ==
31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13.
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.  


What follows is a comparison of stretched-octave 31edo tunings.
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]].  


; 31edo
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: 38.710{{c}}, octave size: 1200.0{{c}}
Pure-octaves 31edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in equal|31|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo}}
{{Harmonics in equal|31|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31edo (continued)}}


; [[WE|31et, 13-limit WE tuning]]
What follows is a comparison of stretched- and compressed-octave 41edo tunings.
* Step size: 38.725{{c}}, octave size: 1200.5{{c}}
Stretching the octave of 31edo by around 0.5{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|38.725|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 13-limit WE tuning}}
{{Harmonics in cet|38.725|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)}}


; [[zpi|127zpi]]  
; [[147ed12]] / [[106ed6]] / [[65edt]]
* Step size: 38.737{{c}}, octave size: 1200.8{{c}}
* 65edt — step size: 29.261{{c}}, octave size: 1199.81{{c}}
Stretching the octave of 31edo by slightly less than 1{{c}} results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 127zpi does this.
* 106ed6 — step size: 29.264{{c}}, octave size: 1199.69{{c}}
{{Harmonics in cet|38.737|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi}}
* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
{{Harmonics in cet|38.737|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 127zpi (continued)}}
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.
{{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)}}


; [[WE|31et, 11-limit WE tuning]]
; 41edo
* Step size: 38.748{{c}}, octave size: 1201.2{{c}}
* Step size: 29.268{{c}}, octave size: 1200.00{{c}}  
Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3, 7 and 11, but moderately worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]].
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 cet|38.748|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-limit WE tuning}}
{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
{{Harmonics in cet|38.748|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31et, 11-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)}}


; [[80ed6]]  
; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]  
* Step size: 38.774{{c}}, octave size: 1202.0{{c}}
* Step size: 29.277{{c}}, octave size: 1200.35{{c}}
Stretching the octave of 31edo by slightly more than 1{{c}} results in moderately improved primes 3, 7 and 11, but moderately worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 15.5{{c}} Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this, so does the tuning [[111ed12]]... Slightly more than this - a stretch of 2.239{{c}} - is the absolute maximum amount of octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes.
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|80|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6}}
{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}
{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (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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