User:BudjarnLambeth/Sandbox2: Difference between revisions

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; 18edo
; 18edo
* Step size: NNN{{c}}, octave size: NNN{{c}}  
* Step size: 66.667{{c}}, octave size: 1200.0{{c}}  
Pure-octaves 18edo approximates all harmonics up to 16 within NNN{{c}}.
Pure-octaves 18edo approximates all harmonics up to 15 within 31.4{{c}}.
{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}
{{Harmonics in equal|18|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18edo}}
{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}
{{Harmonics in equal|18|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edo (continued)}}
Line 14: Line 14:
; [[WE|18et, 13-limit WE tuning]]  
; [[WE|18et, 13-limit WE tuning]]  
* Step size: 66.291{{c}}, octave size: 1193.2{{c}}
* Step size: 66.291{{c}}, octave size: 1193.2{{c}}
Compressing the octave of 18edo by around 7{{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.
Compressing the octave of 18edo by around 7{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 25.3{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}
{{Harmonics in cet|66.291|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning}}
{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}
{{Harmonics in cet|66.291|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 13-limit WE tuning (continued)}}
Line 20: Line 20:
; [[zpi|61zpi]]  
; [[zpi|61zpi]]  
* Step size: 66.228{{c}}, octave size: 1192.1{{c}}
* Step size: 66.228{{c}}, octave size: 1192.1{{c}}
Compressing the octave of 18edo by around 8{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 61zpi does this.
Compressing the octave of 18edo by around 8{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 28.9{{c}}. The tuning 61zpi does this.
{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}
{{Harmonics in cet| 66.228 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 61zpi}}
{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}
{{Harmonics in cet| 66.228 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 61zpi (continued)}}


; [[65ed12]]  
; [[65ed12]]  
* Step size: NNN{{c}}, octave size: 1191.3{{c}}
* Octave size: 1191.3{{c}}
Compressing the octave of 18edo by around 9{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 65ed12 does this.
Compressing the octave of 18edo by around 9{{c}} results in improved primes 3, 5, 7 and 13, but a worse primes 2. This approximates all harmonics up to 15 within 31.4{{c}}. The tuning 65ed12 does this.
{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}
{{Harmonics in equal|65|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 65ed12}}
{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}
{{Harmonics in equal|65|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 65ed12 (continued)}}
; [[WE|18et, 7-limit WE tuning]]
* Step size: 66.148{{c}}, octave size: 1190.7{{c}}
Compressing the octave of 18edo by around 9.5{{c}} results in improved primes NNN, but worse primes NNN. 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| 66.148 |intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 18et, 7-limit WE tuning}}
{{Harmonics in cet| 66.148 |intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18et, 7-limit WE tuning (continued)}}


; [[47ed6]]  
; [[47ed6]]  
* Step size: NNN{{c}}, octave size: 1188.0{{c}}
* Step size: NNN{{c}}, octave size: 1188.0{{c}}
Compressing the octave of 18edo by around 12{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 47ed6 does this.
Compressing the octave of 18edo by around 12{{c}} results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 15 within 29.9{{c}}. The tuning 47ed6 does this.
{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}
{{Harmonics in equal|47|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 47ed6}}
{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}
{{Harmonics in equal|47|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 47ed6 (continued)}}

Revision as of 07:30, 16 September 2025

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User:BudjarnLambeth/Draft related tunings section

Octave stretch or compression

18edo's primes 3, 5, 7 and 13 are all tuned sharp, so it can benefit from octave shrinking.

18edo
  • Step size: 66.667 ¢, octave size: 1200.0 ¢

Pure-octaves 18edo approximates all harmonics up to 15 within 31.4 ¢.

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4
Relative (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11)
Approximation of harmonics in 18edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +26.1 +31.2 -21.6 +0.0 +28.4 -3.9 -30.8 +13.7 -4.1 -18.0 -28.3 +31.4
Relative (%) +39.2 +46.8 -32.4 +0.0 +42.6 -5.9 -46.3 +20.5 -6.2 -27.0 -42.4 +47.1
Steps
(reduced) 		67
(13) 		69
(15) 		70
(16) 		72
(0) 		74
(2) 		75
(3) 		76
(4) 		78
(6) 		79
(7) 		80
(8) 		81
(9) 		83
(11)
18et, 13-limit WE tuning
  • Step size: 66.291 ¢, octave size: 1193.2 ¢

Compressing the octave of 18edo by around 7 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 25.3 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 18et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.8 +20.5 -13.5 -2.1 +13.7 +12.0 -20.3 -25.3 -8.9 +25.0 +7.0
Relative (%) -10.2 +30.9 -20.4 -3.2 +20.7 +18.1 -30.6 -38.2 -13.4 +37.7 +10.5
Step 18 29 36 42 47 51 54 57 60 63 65
Approximation of harmonics in 18et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.0 +5.3 +18.4 -27.0 +0.6 -32.1 +6.9 -15.6 +32.5 +18.3 +7.6 +0.2
Relative (%) +1.5 +7.9 +27.7 -40.8 +0.9 -48.4 +10.4 -23.6 +49.0 +27.5 +11.4 +0.3
Step 67 69 71 72 74 75 77 78 80 81 82 83
61zpi
  • Step size: 66.228 ¢, octave size: 1192.1 ¢

Compressing the octave of 18edo by around 8 ¢ results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 15 within 28.9 ¢. The tuning 61zpi does this.

Approximation of harmonics in 61zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -7.9 +18.7 -15.8 -4.7 +10.8 +8.8 -23.7 -28.9 -12.6 +21.0 +2.9
Relative (%) -11.9 +28.2 -23.8 -7.2 +16.2 +13.3 -35.8 -43.7 -19.1 +31.8 +4.3
Step 18 29 36 42 47 51 54 57 60 63 65
Approximation of harmonics in 61zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.3 +0.9 +13.9 -31.6 -4.1 +29.4 +2.0 -20.5 +27.5 +13.2 +2.4 -5.0
Relative (%) -4.9 +1.4 +21.0 -47.7 -6.2 +44.4 +3.1 -31.0 +41.5 +19.9 +3.7 -7.6
Step 67 69 71 72 74 76 77 78 80 81 82 83
65ed12
  • Octave size: 1191.3 ¢

Compressing the octave of 18edo by around 9 ¢ results in improved primes 3, 5, 7 and 13, but a worse primes 2. This approximates all harmonics up to 15 within 31.4 ¢. The tuning 65ed12 does this.

Approximation of harmonics in 65ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -8.7 +17.4 -17.4 -6.6 +8.7 +6.6 -26.1 -31.4 -15.3 +18.3 +0.0
Relative (%) -13.1 +26.3 -26.3 -10.0 +13.1 +9.9 -39.4 -47.5 -23.1 +27.6 +0.0
Steps
(reduced) 		18
(18) 		29
(29) 		36
(36) 		42
(42) 		47
(47) 		51
(51) 		54
(54) 		57
(57) 		60
(60) 		63
(63) 		65
(0)
Approximation of harmonics in 65ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -6.2 -2.1 +10.8 +31.4 -7.3 +26.1 -1.4 -24.0 +23.9 +9.6 -1.2 -8.7
Relative (%) -9.4 -3.2 +16.3 +47.5 -11.1 +39.4 -2.0 -36.2 +36.2 +14.5 -1.8 -13.1
Steps
(reduced) 		67
(2) 		69
(4) 		71
(6) 		73
(8) 		74
(9) 		76
(11) 		77
(12) 		78
(13) 		80
(15) 		81
(16) 		82
(17) 		83
(18)
47ed6
  • Step size: NNN ¢, octave size: 1188.0 ¢

Compressing the octave of 18edo by around 12 ¢ results in improved primes 3, 7, 11 and 13, but worse primes 2 and 5. This approximates all harmonics up to 15 within 29.9 ¢. The tuning 47ed6 does this.

Approximation of harmonics in 47ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -12.0 +12.0 -24.0 -14.4 +0.0 -2.9 +29.9 +24.0 -26.4 +6.6 -12.0
Relative (%) -18.2 +18.2 -36.4 -21.7 +0.0 -4.4 +45.4 +36.4 -40.0 +10.0 -18.2
Steps
(reduced) 		18
(18) 		29
(29) 		36
(36) 		42
(42) 		47
(0) 		51
(4) 		55
(8) 		58
(11) 		60
(13) 		63
(16) 		65
(18)
Approximation of harmonics in 47ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -18.6 -14.9 -2.3 +17.9 -21.0 +12.0 -15.6 +27.6 +9.1 -5.4 -16.4 -24.0
Relative (%) -28.2 -22.6 -3.5 +27.2 -31.9 +18.2 -23.6 +41.8 +13.9 -8.2 -24.8 -36.4
Steps
(reduced) 		67
(20) 		69
(22) 		71
(24) 		73
(26) 		74
(27) 		76
(29) 		77
(30) 		79
(32) 		80
(33) 		81
(34) 		82
(35) 		83
(36)
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