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 stretched- and compressed-octave 12edo tunings.

40ed10
  • Step size: 99.658 ¢, octave size: 1195.9 ¢

Compressing the octave of EDONAME by around 4 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 40ed10 does this.

Approximation of harmonics in 40ed10
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 40ed10 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.1 +15.4 -4.4 -16.4 -21.7 -21.0 -15.0 -4.1 +11.1 +30.2 -46.8 -20.8
Relative (%) +44.2 +15.5 -4.4 -16.5 -21.8 -21.1 -15.0 -4.1 +11.1 +30.3 -46.9 -20.8
Steps
(reduced) 		45
(5) 		46
(6) 		47
(7) 		48
(8) 		49
(9) 		50
(10) 		51
(11) 		52
(12) 		53
(13) 		54
(14) 		54
(14) 		55
(15)
12et, 7-limit WE tuning
  • Step size: 99.664 ¢, octave size: NNN ¢

Compressing 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 7-limit WE tuning and 7-limit TE tuning both do this.

Approximation of harmonics in 12et, 7-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.0 -8.3 -8.1 +4.3 -12.4 +19.8 -12.1 -16.7 +0.2 +34.6 -16.4
Relative (%) -4.0 -8.4 -8.1 +4.3 -12.4 +19.8 -12.1 -16.7 +0.2 +34.7 -16.5
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 12et, 7-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.4 +15.7 -4.1 -16.1 -21.4 -20.7 -14.6 -3.8 +11.4 +30.5 -46.4 -20.4
Relative (%) +44.5 +15.8 -4.1 -16.2 -21.5 -20.8 -14.7 -3.8 +11.4 +30.6 -46.6 -20.5
Step 45 46 47 48 49 50 51 52 53 54 54 55
34zpi
  • Step size: 99.807 ¢, octave size: NNN ¢

Compressing the octave of 12edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 34zpi does this.

Approximation of harmonics in 34zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.3 -5.6 -4.6 +8.3 -7.9 +24.6 -6.9 -11.2 +6.0 +40.6 -10.3
Relative (%) -2.3 -5.6 -4.6 +8.3 -8.0 +24.7 -7.0 -11.3 +6.0 +40.7 -10.3
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 34zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -49.0 +22.3 +2.7 -9.3 -14.4 -13.6 -7.4 +3.7 +19.0 +38.3 -38.7 -12.6
Relative (%) -49.1 +22.3 +2.7 -9.3 -14.4 -13.6 -7.4 +3.7 +19.0 +38.3 -38.8 -12.6
Step 44 46 47 48 49 50 51 52 53 54 54 55
12et, 5-limit WE tuning
  • Step size: 99.868 ¢, octave size: NNN ¢

Compressing 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 5-limit WE tuning and 5-limit TE tuning both do this.

Approximation of harmonics in 12et, 5-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.6 -4.5 -3.2 +10.0 -6.0 +26.7 -4.8 -8.9 +8.4 +43.1 -7.6
Relative (%) -1.6 -4.5 -3.2 +10.0 -6.1 +26.7 -4.8 -8.9 +8.4 +43.2 -7.6
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 12et, 5-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -46.3 +25.1 +5.5 -6.3 -11.4 -10.5 -4.2 +6.8 +22.2 +41.6 -35.4 -9.2
Relative (%) -46.4 +25.1 +5.5 -6.3 -11.4 -10.5 -4.3 +6.8 +22.3 +41.6 -35.4 -9.2
Step 44 46 47 48 49 50 51 52 53 54 54 55
12et, 2.3.5.17.19 WE tuning
  • Step size: 99.930 ¢, octave size: NNN ¢

Compressing the octave of 12edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The 2.3.5.17.19 WE tuning and 2.3.5.17.19 TE tuning both do this.

Approximation of harmonics in 12et, 2.3.5.17.19 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.8 -3.3 -1.7 +11.7 -4.1 +28.8 -2.5 -6.6 +10.9 +45.7 -5.0
Relative (%) -0.8 -3.3 -1.7 +11.7 -4.1 +28.8 -2.5 -6.6 +10.9 +45.8 -5.0
Step 12 19 24 28 31 34 36 38 40 42 43
Approximation of harmonics in 12et, 2.3.5.17.19 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -43.6 +28.0 +8.4 -3.4 -8.4 -7.4 -1.1 +10.0 +25.5 +44.9 -32.1 -5.8
Relative (%) -43.6 +28.0 +8.4 -3.4 -8.4 -7.4 -1.1 +10.1 +25.5 +44.9 -32.1 -5.8
Step 44 46 47 48 49 50 51 52 53 54 54 55
12edo
  • Step size: 100.000 ¢, octave size: 1200.0 ¢

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

Approximation of harmonics in EDONAME
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
Steps
(reduced) 		12
(0) 		19
(7) 		24
(0) 		28
(4) 		31
(7) 		34
(10) 		36
(0) 		38
(2) 		40
(4) 		42
(6) 		43
(7)
Approximation of harmonics in EDONAME (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
Steps
(reduced) 		44
(8) 		46
(10) 		47
(11) 		48
(0) 		49
(1) 		50
(2) 		51
(3) 		52
(4) 		53
(5) 		54
(6) 		54
(6) 		55
(7)
31ed6
  • Step size: 100.063 ¢, octave size: 1200.8 ¢

Stretching the octave of 12edo by a little less than 1 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 31ed6 does this.

Approximation of harmonics in 31ed6
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)
Approximation of harmonics in 31ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -37.8 +34.1 +14.7 +3.0 -1.9 -0.8 +5.7 +17.0 +32.6 -48.0 -24.9 +1.5
Relative (%) -37.7 +34.1 +14.7 +3.0 -1.9 -0.8 +5.7 +17.0 +32.5 -47.9 -24.9 +1.5
Steps
(reduced) 		44
(13) 		46
(15) 		47
(16) 		48
(17) 		49
(18) 		50
(19) 		51
(20) 		52
(21) 		53
(22) 		53
(22) 		54
(23) 		55
(24)
19edt
  • Step size: 101.103 ¢, octave size: 1201.2 ¢

Stretching the octave of 12edo by a little more than 1 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 19edt does this.

Approximation of harmonics in 19edt
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 19edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -36.0 +35.9 +16.6 +4.9 +0.1 +1.2 +7.7 +19.0 +34.7 -45.9 -22.7 +3.7
Relative (%) -36.0 +35.9 +16.6 +4.9 +0.1 +1.2 +7.7 +19.0 +34.6 -45.8 -22.7 +3.7
Steps
(reduced) 		44
(6) 		46
(8) 		47
(9) 		48
(10) 		49
(11) 		50
(12) 		51
(13) 		52
(14) 		53
(15) 		53
(15) 		54
(16) 		55
(17)
7edf
  • Step size: 100.3 ¢, octave size: 1203.35 ¢

Stretching the octave of 12edo by around 3 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 7edf does this.

Approximation of harmonics in 7edf
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 7edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -28.2 +44.0 +24.9 +13.4 +8.7 +10.1 +16.7 +28.2 +44.0 -36.5 -13.2 +13.4
Relative (%) -28.2 +43.9 +24.8 +13.4 +8.7 +10.0 +16.7 +28.1 +43.9 -36.4 -13.2 +13.4
Steps
(reduced) 		44
(2) 		46
(4) 		47
(5) 		48
(6) 		49
(0) 		50
(1) 		51
(2) 		52
(3) 		53
(4) 		53
(4) 		54
(5) 		55
(6)
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