User:BudjarnLambeth/Sandbox2

From Xenharmonic Wiki
Jump to navigation Jump to search

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 compression

What follows is a comparison of compressed-octave 17edo tunings.

17edo
  • Step size: NNN ¢, octave size: NNN ¢

Pure-octaves 17edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 17edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +3.9 +0.0 -33.4 +3.9 +19.4 +0.0 +7.9 -33.4 +13.4 +3.9
Relative (%) +0.0 +5.6 +0.0 -47.3 +5.6 +27.5 +0.0 +11.1 -47.3 +19.0 +5.6
Steps
(reduced) 		17
(0) 		27
(10) 		34
(0) 		39
(5) 		44
(10) 		48
(14) 		51
(0) 		54
(3) 		56
(5) 		59
(8) 		61
(10)
Approximation of harmonics in 17edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.5 +19.4 -29.4 +0.0 -34.4 +7.9 -15.2 -33.4 +23.3 +13.4 +7.0 +3.9
Relative (%) +9.3 +27.5 -41.7 +0.0 -48.7 +11.1 -21.5 -47.3 +33.1 +19.0 +9.9 +5.6
Steps
(reduced) 		63
(12) 		65
(14) 		66
(15) 		68
(0) 		69
(1) 		71
(3) 		72
(4) 		73
(5) 		75
(7) 		76
(8) 		77
(9) 		78
(10)
44ed6
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 44ed6 does this.

Approximation of harmonics in 44ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.5 +1.5 -3.0 +33.6 +0.0 +15.1 -4.6 +3.0 +32.1 +8.1 -1.5
Relative (%) -2.2 +2.2 -4.3 +47.7 +0.0 +21.5 -6.5 +4.3 +45.6 +11.5 -2.2
Steps
(reduced) 		17
(17) 		27
(27) 		34
(34) 		40
(40) 		44
(0) 		48
(4) 		51
(7) 		54
(10) 		57
(13) 		59
(15) 		61
(17)
Approximation of harmonics in 44ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.9 +13.6 +35.2 -6.1 +30.0 +1.5 -21.6 +30.6 +16.6 +6.6 +0.1 -3.0
Relative (%) +1.3 +19.3 +49.9 -8.6 +42.5 +2.2 -30.6 +43.4 +23.6 +9.4 +0.2 -4.3
Steps
(reduced) 		63
(19) 		65
(21) 		67
(23) 		68
(24) 		70
(26) 		71
(27) 		72
(28) 		74
(30) 		75
(31) 		76
(32) 		77
(33) 		78
(34)
27edt
  • Step size: NNN ¢, octave size: NNN ¢

_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 27edt does this.

Approximation of harmonics in 27edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.5 +0.0 -4.9 +31.4 -2.5 +12.4 -7.4 +0.0 +28.9 +4.8 -4.9
Relative (%) -3.5 +0.0 -7.0 +44.6 -3.5 +17.6 -10.5 +0.0 +41.1 +6.8 -7.0
Steps
(reduced) 		17
(17) 		27
(0) 		34
(7) 		40
(13) 		44
(17) 		48
(21) 		51
(24) 		54
(0) 		57
(3) 		59
(5) 		61
(7)
Approximation of harmonics in 27edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 +10.0 +31.4 -9.9 +26.0 -2.5 -25.6 +26.5 +12.4 +2.3 -4.2 -7.4
Relative (%) -3.7 +14.1 +44.6 -14.0 +37.0 -3.5 -36.4 +37.6 +17.6 +3.3 -5.9 -10.5
Steps
(reduced) 		63
(9) 		65
(11) 		67
(13) 		68
(14) 		70
(16) 		71
(17) 		72
(18) 		74
(20) 		75
(21) 		76
(22) 		77
(23) 		78
(24)
56zpi
  • Step size: 70.403 ¢, octave size: NNN ¢

_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 56zpi does this.

Approximation of harmonics in 56zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.1 -1.1 -6.3 +29.8 -4.2 +10.5 -9.4 -2.1 +26.7 +2.5 -7.4
Relative (%) -4.5 -1.5 -8.9 +42.3 -6.0 +14.9 -13.4 -3.1 +37.9 +3.5 -10.5
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 56zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.1 +7.4 +28.7 -12.6 +23.3 -5.3 -28.5 +23.5 +9.4 -0.7 -7.2 -10.5
Relative (%) -7.3 +10.5 +40.8 -17.9 +33.0 -7.5 -40.5 +33.4 +13.4 -1.0 -10.3 -14.9
Step 63 65 67 68 70 71 72 74 75 76 77 78
17et, 2.3.7.11.13 WE tuning
  • Step size: 70.410 ¢, octave size: NNN ¢

_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.3.7.11.13 WE tuning and 2.3.7.11.13 TE tuning both do this.

Approximation of harmonics in 17et, 2.3.7.11.13 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.0 -0.9 -6.1 +30.1 -3.9 +10.9 -9.1 -1.8 +27.1 +2.9 -6.9
Relative (%) -4.3 -1.3 -8.6 +42.7 -5.6 +15.4 -12.9 -2.5 +38.4 +4.1 -9.9
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 17et, 2.3.7.11.13 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.7 +7.8 +29.2 -12.1 +23.7 -4.8 -28.0 +24.0 +10.0 -0.2 -6.7 -10.0
Relative (%) -6.7 +11.1 +41.5 -17.2 +33.7 -6.8 -39.8 +34.1 +14.2 -0.2 -9.5 -14.2
Step 63 65 67 68 70 71 72 74 75 76 77 78
17et, 2.3.7.11 WE tuning
  • Step size: 70.392 ¢, octave size: NNN ¢

_ing the octave of 17edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 2.3.7.11 WE tuning and 2.3.7.11 TE tuning both do this.

Approximation of harmonics in 17et, 2.3.7.11 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.3 -1.4 -6.7 +29.4 -4.7 +10.0 -10.0 -2.7 +26.0 +1.8 -8.0
Relative (%) -4.7 -1.9 -9.5 +41.7 -6.7 +14.2 -14.2 -3.9 +37.0 +2.6 -11.4
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.8 +6.7 +28.0 -13.3 +22.5 -6.1 -29.3 +22.7 +8.6 -1.5 -8.1 -11.4
Relative (%) -8.3 +9.5 +39.8 -19.0 +31.9 -8.6 -41.6 +32.2 +12.2 -2.2 -11.5 -16.2
Step 63 65 67 68 70 71 72 74 75 76 77 78
Retrieved from "https://en.xen.wiki/index.php?title=User:BudjarnLambeth/Sandbox2&oldid=207997"