User:BudjarnLambeth/Sandbox2: Difference between revisions

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What follows is a comparison of stretched- and compressed-octave 7edo tunings.
What follows is a comparison of stretched- and compressed-octave 7edo tunings.


; [[zpi|15zpi]]
; 7edo
* Step size: 172.495{{c}}, octave size: NNN{{c}}
* Step size: 171.429{{c}}, octave size: 1200.0{{c}}  
_ing the octave of 7edo 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 15zpi does this.
Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}
{{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}}
 
; [[11edt]]
* Step size: NNN{{c}}, octave size: NNN{{c}}
_ing the octave of 7edo 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 11edt does this.
{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}


; [[WE|7et, 2.3.11.13 WE]]  
; [[WE|7et, 2.3.11.13 WE]]  
* Step size: 171.993{{c}}, octave size: NNN{{c}}
* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
_ing the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
{{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
{{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}


; 7edo
; [[18ed6]]
* Step size: NNN{{c}}, octave size: NNN{{c}}  
* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
Stretching the octave of 7edo 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 18ed6 does this.
{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
{{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}


; [[WE|7et, 2.3.5.11.13 WE]]  
; [[WE|7et, 2.3.5.11.13 WE]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 172.390{{c}}, octave size: 1206.7{{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 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
Stretching the octave of 7edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in TETUNING}}
{{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
{{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in TETUNING (continued)}}
{{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
 
; [[zpi|15zpi]]
* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
Stretching the octave of 7edo 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 15zpi does this.
{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}


; [[18ed6]]  
; [[11edt]]  
* Step size: NNN{{c}}, octave size: NNN{{c}}
* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
_ing the octave of 7edo 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 18ed6 does this.
Stretching the octave of 7edo 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 11edt does this.
{{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
{{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}
{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}

Revision as of 21:47, 21 August 2025

7edo

Octave stretch or compression

What follows is a comparison of stretched- and compressed-octave 7edo tunings.

7edo
  • Step size: 171.429 ¢, octave size: 1200.0 ¢

Pure-octaves 7edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 7edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 -43.5 -16.2 +59.7 +0.0 -32.5 -43.5 -37.0 -16.2
Relative (%) +0.0 -9.5 +0.0 -25.3 -9.5 +34.9 +0.0 -18.9 -25.3 -21.6 -9.5
Steps
(reduced) 		7
(0) 		11
(4) 		14
(0) 		16
(2) 		18
(4) 		20
(6) 		21
(0) 		22
(1) 		23
(2) 		24
(3) 		25
(4)
Approximation of harmonics in 7edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.6 +59.7 -59.7 +0.0 +66.5 -32.5 +45.3 -43.5 +43.5 -37.0 +57.4 -16.2
Relative (%) +9.7 +34.9 -34.8 +0.0 +38.8 -18.9 +26.5 -25.3 +25.4 -21.6 +33.5 -9.5
Steps
(reduced) 		26
(5) 		27
(6) 		27
(6) 		28
(0) 		29
(1) 		29
(1) 		30
(2) 		30
(2) 		31
(3) 		31
(3) 		32
(4) 		32
(4)
7et, 2.3.11.13 WE
  • Step size: 171.993 ¢, octave size: 1204.0 ¢

Stretching the octave of 7edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The 2.3.11.13 WE tuning and 2.3.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.0 -10.0 +7.9 -34.4 -6.1 +71.0 +11.9 -20.1 -30.5 -23.5 -2.1
Relative (%) +2.3 -5.8 +4.6 -20.0 -3.5 +41.3 +6.9 -11.7 -17.7 -13.7 -1.2
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +31.3 +75.0 -44.5 +15.8 +82.8 -16.1 +62.3 -26.5 +61.0 -19.5 +75.5 +1.8
Relative (%) +18.2 +43.6 -25.8 +9.2 +48.2 -9.4 +36.2 -15.4 +35.5 -11.4 +43.9 +1.1
Step 26 27 27 28 29 29 30 30 31 31 32 32
18ed6
  • Step size: 172.331 ¢, octave size: 1206.3 ¢

Stretching the octave of 7edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 18ed6 does this.

Approximation of harmonics in 18ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Relative (%) +0.0 -2.0 +0.0 +13.7 -2.0 +31.2 +0.0 -3.9 +13.7 +48.7 -2.0
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in 18ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Relative (%) -40.5 +31.2 +11.7 +0.0 -5.0 -3.9 +2.5 +13.7 +29.2 +48.7 -28.3 -2.0
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
7et, 2.3.5.11.13 WE
  • Step size: 172.390 ¢, octave size: 1206.7 ¢

Stretching the octave of 7edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.5.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -5.7 +13.5 -28.1 +1.1 +79.0 +20.2 -11.3 -21.3 -14.0 +7.8
Relative (%) +3.9 -3.3 +7.8 -16.3 +0.6 +45.8 +11.7 -6.6 -12.4 -8.1 +4.5
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +41.6 +85.7 -33.7 +26.9 -78.0 -4.6 +74.2 -14.6 +73.3 -7.2 -84.2 +14.5
Relative (%) +24.1 +49.7 -19.6 +15.6 -45.3 -2.7 +43.0 -8.5 +42.5 -4.2 -48.8 +8.4
Step 26 27 27 28 28 29 30 30 31 31 31 32
15zpi
  • Step size: 172.495 ¢, octave size: 1207.5 ¢

Stretching the octave of 7edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 15zpi does this.

Approximation of harmonics in 15zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +7.5 -4.5 +14.9 -26.4 +3.0 +81.1 +22.4 -9.0 -18.9 -11.4 +10.4
Relative (%) +4.3 -2.6 +8.7 -15.3 +1.7 +47.0 +13.0 -5.2 -11.0 -6.6 +6.0
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 15zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.3 -84.0 -30.9 +29.9 -75.1 -1.6 +77.3 -11.5 +76.6 -4.0 -80.9 +17.9
Relative (%) +25.7 -48.7 -17.9 +17.3 -43.5 -0.9 +44.8 -6.6 +44.4 -2.3 -46.9 +10.4
Step 26 26 27 28 28 29 30 30 31 31 31 32
11edt
  • Step size: 172.905 ¢, octave size: 1210.3 ¢

Stretching the octave of 7edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 11edt does this.

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(0) 		14
(3) 		16
(5) 		18
(7) 		19
(8) 		21
(10) 		22
(0) 		23
(1) 		24
(2) 		25
(3)
Approximation of harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced) 		26
(4) 		26
(4) 		27
(5) 		28
(6) 		28
(6) 		29
(7) 		29
(7) 		30
(8) 		30
(8) 		31
(9) 		31
(9) 		32
(10)
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