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User:BudjarnLambeth/Sandbox2

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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: NNN ¢, octave size: NNN ¢

Pure-octaves 18edo approximates all harmonics up to 16 within NNN ¢.

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 NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. 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 NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. 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
  • Step size: NNN ¢, octave size: 1191.3 ¢

Compressing the octave of 18edo by around 9 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. 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)
18et, 7-limit WE tuning
  • Step size: 66.148 ¢, octave size: 1190.7 ¢

Compressing the octave of 18edo by around 9.5 ¢ results in improved primes NNN, but worse primes NNN. 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 18et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -9.3 +16.3 -18.7 -8.1 +7.0 +4.7 -28.0 +32.7 -17.4 +16.0 -2.3
Relative (%) -14.1 +24.7 -28.2 -12.2 +10.6 +7.1 -42.3 +49.4 -26.4 +24.2 -3.5
Step 18 29 36 42 47 51 54 58 60 63 65
Approximation of harmonics in 18et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -8.6 -4.6 +8.2 +28.8 -10.0 +23.3 -4.1 -26.8 +21.1 +6.7 -4.1 -11.7
Relative (%) -13.0 -7.0 +12.5 +43.5 -15.1 +35.3 -6.2 -40.5 +31.8 +10.1 -6.3 -17.6
Step 67 69 71 73 74 76 77 78 80 81 82 83
47ed6
  • Step size: NNN ¢, octave size: 1188.0 ¢

Compressing the octave of 18edo by around 12 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. 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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