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

What follows is a comparison of stretched-octave 19edo tunings.

19edo
  • Step size: 63.158 ¢, octave size: 1200.0 ¢

Pure-octaves 19edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 19edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 -21.5 +0.0 -14.4 -7.4 +17.1 -7.2
Relative (%) +0.0 -11.4 +0.0 -11.7 -11.4 -34.0 +0.0 -22.9 -11.7 +27.1 -11.4
Steps
(reduced) 		19
(0) 		30
(11) 		38
(0) 		44
(6) 		49
(11) 		53
(15) 		57
(0) 		60
(3) 		63
(6) 		66
(9) 		68
(11)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.53 -2.51 -1.95 +0.00 +2.41 -1.80 -0.67 -1.05 +2.90 -1.84 -3.01 -0.90
Relative (%) -8.4 -39.7 -30.9 +0.0 +38.2 -28.6 -10.6 -16.6 +46.0 -29.2 -47.7 -14.3
Steps
(reduced) 		703
(133) 		723
(153) 		742
(172) 		760
(0) 		777
(17) 		792
(32) 		807
(47) 		821
(61) 		835
(75) 		847
(87) 		859
(99) 		871
(111)
19et, 2.3.5.11 WE tuning
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of EDONAME 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 WE tuning and 2.3.5.11 TE tuning both do this.

Approximation of harmonics in 19et, 2.3.5.11 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.6 -6.2 +1.3 -5.9 -5.5 -19.6 +1.9 -12.4 -5.2 +19.4 -4.9
Relative (%) +1.0 -9.8 +2.1 -9.3 -8.8 -31.1 +3.1 -19.6 -8.3 +30.6 -7.8
Step 19 30 38 44 49 53 57 60 63 66 68
Approximation of harmonics in 19et, 2.3.5.11 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -17.1 -19.0 -12.1 +2.6 +24.0 -11.7 +21.0 -4.6 -25.8 +20.0 +6.2 -4.3
Relative (%) -27.0 -30.1 -19.1 +4.1 +38.0 -18.6 +33.3 -7.2 -40.9 +31.7 +9.9 -6.7
Step 70 72 74 76 78 79 81 82 83 85 86 87
19et, 13-limit WE tuning
  • Step size: 63.291 ¢, octave size: NNN ¢

_ing the octave of EDONAME 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 19et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.5 -3.2 +5.1 -1.5 -0.7 -14.4 +7.6 -6.5 +1.0 +25.9 +1.8
Relative (%) +4.0 -5.1 +8.0 -2.4 -1.1 -22.8 +12.0 -10.2 +1.6 +40.9 +2.9
Step 19 30 38 44 49 53 57 60 63 66 68
Approximation of harmonics in 19et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -10.2 -11.9 -4.7 +10.1 -31.5 -3.9 +29.1 +3.5 -17.6 +28.4 +14.8 +4.4
Relative (%) -16.0 -18.8 -7.5 +16.0 -49.8 -6.2 +45.9 +5.6 -27.9 +44.9 +23.3 +6.9
Step 70 72 74 76 77 79 81 82 83 85 86 87
ZPINAME
  • Step size: NNN ¢, octave size: NNN ¢

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

Approximation of harmonics in ZPINAME
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
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in ZPINAME (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
Step 44 46 47 48 49 50 51 52 53 54 54 55
49ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 49ed6 does this.

Approximation of harmonics in 49ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.8 -2.8 +5.6 -0.9 +0.0 -13.7 +8.4 -5.6 +1.9 +26.8 +2.8
Relative (%) +4.4 -4.4 +8.8 -1.4 +0.0 -21.6 +13.3 -8.8 +3.0 +42.4 +4.4
Steps
(reduced) 		19
(19) 		30
(30) 		38
(38) 		44
(44) 		49
(0) 		53
(4) 		57
(8) 		60
(11) 		63
(14) 		66
(17) 		68
(19)
Approximation of harmonics in 49ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.2 -10.9 -3.7 +11.2 -30.5 -2.8 +30.2 +4.7 -16.4 +29.6 +16.0 +5.6
Relative (%) -14.5 -17.1 -5.8 +17.7 -48.1 -4.4 +47.7 +7.4 -26.0 +46.8 +25.2 +8.8
Steps
(reduced) 		70
(21) 		72
(23) 		74
(25) 		76
(27) 		77
(28) 		79
(30) 		81
(32) 		82
(33) 		83
(34) 		85
(36) 		86
(37) 		87
(38)
30edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 30edt does this.

Approximation of harmonics in 30edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.6 +0.0 +9.1 +3.2 +4.6 -8.7 +13.7 +0.0 +7.8 -30.4 +9.1
Relative (%) +7.2 +0.0 +14.4 +5.1 +7.2 -13.7 +21.6 +0.0 +12.3 -48.0 +14.4
Steps
(reduced) 		19
(19) 		30
(0) 		38
(8) 		44
(14) 		49
(19) 		53
(23) 		57
(27) 		60
(0) 		63
(3) 		65
(5) 		68
(8)
Approximation of harmonics in 30edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -4.1 +3.2 +18.3 -23.3 +4.6 -25.6 +12.4 -8.7 -25.8 +24.0 +13.7
Relative (%) -4.2 -6.5 +5.1 +28.8 -36.7 +7.2 -40.4 +19.5 -13.7 -40.8 +37.9 +21.6
Steps
(reduced) 		70
(10) 		72
(12) 		74
(14) 		76
(16) 		77
(17) 		79
(19) 		80
(20) 		82
(22) 		83
(23) 		84
(24) 		86
(26) 		87
(27)
11edf
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 19edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 11edf does this.

Approximation of harmonics in 11edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +12.5 +12.5 +24.9 +21.5 +24.9 +13.3 -26.4 +24.9 -29.8 -3.4 -26.4
Relative (%) +19.5 +19.5 +39.1 +33.7 +39.1 +20.9 -41.4 +39.1 -46.8 -5.3 -41.4
Steps
(reduced) 		19
(8) 		30
(8) 		38
(5) 		44
(0) 		49
(5) 		53
(9) 		56
(1) 		60
(5) 		62
(7) 		65
(10) 		67
(1)
Approximation of harmonics in 11edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +26.5 +25.8 -29.8 -13.9 +8.7 -26.4 +7.6 -17.4 +25.8 +9.1 -4.1 -13.9
Relative (%) +41.5 +40.4 -46.8 -21.8 +13.7 -41.4 +11.9 -27.2 +40.4 +14.2 -6.4 -21.8
Steps
(reduced) 		70
(4) 		72
(6) 		73
(7) 		75
(9) 		77
(0) 		78
(1) 		80
(3) 		81
(4) 		83
(6) 		84
(7) 		85
(8) 		86
(9)
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