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User:BudjarnLambeth/Sandbox2

< User:BudjarnLambeth
Revision as of 23:49, 24 August 2025 by BudjarnLambeth (talk | contribs) (→Octave stretch or compression)

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

What follows is a comparison of stretched- and compressed-octave 27edo tunings.

105zpi
  • Step size: 44.674 ¢, octave size: NNN ¢

Stretching the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 105zpi does this.

Approximation of harmonics in 105zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.2 +19.0 +12.4 -16.5 -19.4 -18.3 +18.6 -6.6 -10.3 +3.4 -13.3
Relative (%) +13.9 +42.6 +27.7 -37.0 -43.5 -40.9 +41.6 -14.8 -23.1 +7.5 -29.7
Step 27 43 54 62 69 75 81 85 89 93 96
Approximation of harmonics in 105zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.8 -12.1 +2.5 -19.9 +9.2 -0.4 -4.7 -4.1 +0.8 +9.6 +22.0 -7.1
Relative (%) -39.8 -27.0 +5.6 -44.5 +20.6 -0.9 -10.5 -9.2 +1.7 +21.4 +49.1 -15.8
Step 99 102 105 107 110 112 114 116 118 120 122 123
27edo
  • Step size: 44.444 ¢, octave size: 1200.0 ¢

Pure-octaves 27edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 27edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +9.2 +0.0 +13.7 +9.2 +9.0 +0.0 +18.3 +13.7 -18.0 +9.2
Relative (%) +0.0 +20.6 +0.0 +30.8 +20.6 +20.1 +0.0 +41.2 +30.8 -40.5 +20.6
Steps
(reduced) 		27
(0) 		43
(16) 		54
(0) 		63
(9) 		70
(16) 		76
(22) 		81
(0) 		86
(5) 		90
(9) 		93
(12) 		97
(16)
Approximation of harmonics in 27edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.9 +9.0 -21.6 +0.0 -16.1 +18.3 +13.6 +13.7 +18.1 -18.0 -6.1 +9.2
Relative (%) +8.8 +20.1 -48.6 +0.0 -36.1 +41.2 +30.6 +30.8 +40.7 -40.5 -13.6 +20.6
Steps
(reduced) 		100
(19) 		103
(22) 		105
(24) 		108
(0) 		110
(2) 		113
(5) 		115
(7) 		117
(9) 		119
(11) 		120
(12) 		122
(14) 		124
(16)
27et, 13-limit WE tuning
  • Step size: 44.375 ¢, octave size: NNN ¢

Compressing the octave of 27edo by around NNN ¢ 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 27et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.9 +6.2 -3.7 +9.3 +4.3 +3.7 -5.6 +12.3 +7.4 +19.9 +2.4
Relative (%) -4.2 +13.9 -8.5 +21.0 +9.7 +8.3 -12.7 +27.8 +16.8 +44.9 +5.5
Step 27 43 54 63 70 76 81 86 90 94 97
Approximation of harmonics in 27et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.0 +1.8 +15.5 -7.5 +20.7 +10.5 +5.6 +5.6 +9.8 +18.1 -14.5 +0.5
Relative (%) -6.8 +4.1 +34.9 -16.9 +46.6 +23.6 +12.6 +12.5 +22.2 +40.7 -32.7 +1.2
Step 100 103 106 108 111 113 115 117 119 121 122 124
27et, 7-limit WE tuning
  • Step size: 44.306 ¢, octave size: NNN ¢

Compressing the octave of 27edo by around NNN ¢ 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 27et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.7 +3.2 -7.5 +5.0 -0.5 -1.6 -11.2 +6.4 +1.2 +13.4 -4.3
Relative (%) -8.4 +7.2 -16.9 +11.2 -1.2 -3.5 -25.3 +14.5 +2.8 +30.3 -9.6
Step 27 43 54 63 70 76 81 86 90 94 97
Approximation of harmonics in 27et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.9 -5.3 +8.2 -15.0 +13.0 +2.7 -2.3 -2.5 +1.6 +9.7 +21.4 -8.0
Relative (%) -22.4 -12.0 +18.4 -33.7 +29.4 +6.0 -5.2 -5.7 +3.7 +21.9 +48.2 -18.1
Step 100 103 106 108 111 113 115 117 119 121 123 124
106zpi
  • Step size: 44.302 ¢, octave size: NNN ¢

Compressing the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 106zpi does this.

Approximation of harmonics in 106zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.8 +3.0 -7.7 +4.7 -0.8 -1.9 -11.5 +6.1 +0.9 +13.1 -4.7
Relative (%) -8.7 +6.8 -17.4 +10.6 -1.8 -4.2 -26.0 +13.7 +2.0 +29.5 -10.5
Step 27 43 54 63 70 76 81 86 90 94 97
Approximation of harmonics in 106zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -10.3 -5.7 +7.7 -15.4 +12.6 +2.2 -2.8 -3.0 +1.2 +9.2 +20.9 -8.5
Relative (%) -23.3 -12.9 +17.5 -34.7 +28.4 +5.0 -6.3 -6.7 +2.6 +20.8 +47.1 -19.2
Step 100 103 106 108 111 113 115 117 119 121 123 124
97ed12
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 97ed12 does this.

Approximation of harmonics in 97ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.5 +5.1 -5.1 +7.7 +2.5 +1.8 -7.6 +10.2 +5.2 +17.6 +0.0
Relative (%) -5.7 +11.5 -11.5 +17.5 +5.7 +4.0 -17.2 +23.0 +11.7 +39.7 +0.0
Steps
(reduced) 		27
(27) 		43
(43) 		54
(54) 		63
(63) 		70
(70) 		76
(76) 		81
(81) 		86
(86) 		90
(90) 		94
(94) 		97
(0)
Approximation of harmonics in 97ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.5 -0.8 +12.8 -10.2 +17.9 +7.6 +2.7 +2.6 +6.9 +15.0 -17.6 -2.5
Relative (%) -12.5 -1.7 +28.9 -23.0 +40.4 +17.2 +6.2 +6.0 +15.5 +33.9 -39.6 -5.7
Steps
(reduced) 		100
(3) 		103
(6) 		106
(9) 		108
(11) 		111
(14) 		113
(16) 		115
(18) 		117
(20) 		119
(22) 		121
(24) 		122
(25) 		124
(27)
70ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 70ed6 does this.

Approximation of harmonics in 70ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.5 +3.5 -7.1 +5.4 +0.0 -1.0 -10.6 +7.1 +1.9 +14.2 -3.5
Relative (%) -8.0 +8.0 -15.9 +12.3 +0.0 -2.2 -23.9 +15.9 +4.3 +32.0 -8.0
Steps
(reduced) 		27
(27) 		43
(43) 		54
(54) 		63
(63) 		70
(0) 		76
(6) 		81
(11) 		86
(16) 		90
(20) 		94
(24) 		97
(27)
Approximation of harmonics in 70ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.2 -4.5 +9.0 -14.1 +13.9 +3.5 -1.4 -1.6 +2.5 +10.6 -22.0 -7.1
Relative (%) -20.7 -10.2 +20.3 -31.9 +31.3 +8.0 -3.3 -3.7 +5.7 +24.0 -49.7 -15.9
Steps
(reduced) 		100
(30) 		103
(33) 		106
(36) 		108
(38) 		111
(41) 		113
(43) 		115
(45) 		117
(47) 		119
(49) 		121
(51) 		122
(52) 		124
(54)
43edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 43edt does this.

Approximation of harmonics in 43edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.7 +0.0 -11.5 +0.3 -5.7 -7.2 -17.2 +0.0 -5.5 +6.4 -11.5
Relative (%) -13.0 +0.0 -26.0 +0.6 -13.0 -16.3 -39.0 +0.0 -12.4 +14.6 -26.0
Steps
(reduced) 		27
(27) 		43
(0) 		54
(11) 		63
(20) 		70
(27) 		76
(33) 		81
(38) 		86
(0) 		90
(4) 		94
(8) 		97
(11)
Approximation of harmonics in 43edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.4 -13.0 +0.3 +21.2 +4.7 -5.7 -10.9 -11.2 -7.2 +0.7 +12.2 -17.2
Relative (%) -39.3 -29.3 +0.6 +48.0 +10.7 -13.0 -24.6 -25.4 -16.3 +1.6 +27.6 -39.0
Steps
(reduced) 		100
(14) 		103
(17) 		106
(20) 		109
(23) 		111
(25) 		113
(27) 		115
(29) 		117
(31) 		119
(33) 		121
(35) 		123
(37) 		124
(38)
90ed10
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 27edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 90ed10 does this.

Approximation of harmonics in 90ed10
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.1 +2.6 -8.2 +4.1 -1.5 -2.6 -12.3 +5.2 +0.0 +12.2 -5.6
Relative (%) -9.3 +5.9 -18.5 +9.3 -3.4 -5.9 -27.8 +11.8 +0.0 +27.5 -12.6
Steps
(reduced) 		27
(27) 		43
(43) 		54
(54) 		63
(63) 		70
(70) 		76
(76) 		81
(81) 		86
(86) 		90
(0) 		94
(4) 		97
(7)
Approximation of harmonics in 90ed10 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -11.3 -6.7 +6.7 -16.4 +11.5 +1.1 -3.9 -4.1 +0.0 +8.1 +19.7 -9.7
Relative (%) -25.5 -15.2 +15.2 -37.1 +26.0 +2.5 -8.8 -9.3 +0.0 +18.2 +44.4 -21.9
Steps
(reduced) 		100
(10) 		103
(13) 		106
(16) 		108
(18) 		111
(21) 		113
(23) 		115
(25) 		117
(27) 		119
(29) 		121
(31) 		123
(33) 		124
(34)
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