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User:BudjarnLambeth/Sandbox2

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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 compressed-octave 27edo tunings.

27edo
  • Step size: 44.444 ¢, octave size: 1200.0 ¢

Pure-octaves 27edo approximates all harmonics up to 16 within 18.3 ¢.

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: 1198.9 ¢

Compressing the octave of 27edo by around 2 ¢ results in substantially improved primes 3, 5 and 7 at little cost. This approximates all harmonics up to 16 within 19.9 ¢. 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
97ed12
  • Step size: 44.350 ¢, octave size: 1197.5 ¢

Compressing the octave of 27edo by around 2.5 ¢ has the same benefits and drawbacks as the 13-limit tuning, but both are slightly amplified. This approximates all harmonics up to 16 within 17.6 ¢. 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)
106zpi / 27et, 7-limit WE tuning / 70ed6
  • Step size (106zpi): 44.306 ¢
  • Octave size (70ed6): 1196.5 ¢
  • Octave size (7-lim WE): 1196.4 ¢
  • Octave size (106zpi): 1196.2 ¢

Compressing the octave of 27edo by around 3.5 ¢ results in even more improvement to primes 3, 5 and 7 than the 13-limit tuning, but now at the cost of moderate damage to 2, 11 and 13. This approximates all harmonics up to 16 within 15.4 ¢. Its 7-limit WE tuning and 7-limit TE tuning both do this. So do the tunings 106zpi and 70ed6.

Approximation of harmonics in 106zpi
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 106zpi (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
90ed10
  • Step size: 44.292 ¢, octave size: 1195.9 ¢

Compressing the octave of 27edo by around 4 ¢ results in improved primes 3, 5, 7 and 11, but a worse prime 2 and much worse 13. This approximates all harmonics up to 16 within 16.4 ¢. 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)
43edt
  • Step size: 44.232 ¢, octave size: 1194.3 ¢

Compressing the octave of 27edo by around 5.5 ¢ results in the same benefits and drawbacks as 90ed10, but amplified. This approximates all harmonics up to 16 within 21.2 ¢. 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)
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