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 22edo tunings.

22edo
  • Step size: 54.545 ¢, octave size: 1200.0 ¢

Pure-octaves 22edo approximates all harmonics up to 16 within 22.3 ¢. The optimal 13-limit WE tuning has octaves only 0.01 ¢ different from pure-octaves 22edo, and the 13-limit TE tuning has octaves only 0.08 ¢ different, so by those metrics pure-octaves 22edo might be considered already optimal.

Approximation of harmonics in 22edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +7.1 +0.0 -4.5 +7.1 +13.0 +0.0 +14.3 -4.5 -5.9 +7.1
Relative (%) +0.0 +13.1 +0.0 -8.2 +13.1 +23.8 +0.0 +26.2 -8.2 -10.7 +13.1
Steps
(reduced) 		22
(0) 		35
(13) 		44
(0) 		51
(7) 		57
(13) 		62
(18) 		66
(0) 		70
(4) 		73
(7) 		76
(10) 		79
(13)
Approximation of harmonics in 22edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -22.3 +13.0 +2.6 +0.0 +4.1 +14.3 -24.8 -4.5 +20.1 -5.9 +26.3 +7.1
Relative (%) -41.0 +23.8 +4.8 +0.0 +7.6 +26.2 -45.4 -8.2 +36.9 -10.7 +48.2 +13.1
Steps
(reduced) 		81
(15) 		84
(18) 		86
(20) 		88
(0) 		90
(2) 		92
(4) 		93
(5) 		95
(7) 		97
(9) 		98
(10) 		100
(12) 		101
(13)
22et, 11-limit WE tuning
  • Step size: 54.494 ¢, octave size: NNN ¢

Compressing the octave of 22edo by around half a cent results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this.

Approximation of harmonics in 22et, 11-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.1 +5.3 -2.3 -7.1 +4.2 +9.8 -3.4 +10.7 -8.3 -9.8 +3.1
Relative (%) -2.1 +9.8 -4.2 -13.1 +7.7 +18.0 -6.2 +19.6 -15.1 -17.9 +5.6
Step 22 35 44 51 57 62 66 70 73 76 79
Approximation of harmonics in 22et, 11-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -26.5 +8.7 -1.8 -4.5 -0.5 +9.5 +24.9 -9.4 +15.1 -10.9 +21.1 +1.9
Relative (%) -48.7 +15.9 -3.3 -8.3 -0.9 +17.5 +45.7 -17.2 +27.8 -20.0 +38.8 +3.6
Step 81 84 86 88 90 92 94 95 97 98 100 101
80zpi
  • Step size: 54.483 ¢, octave size: NNN ¢

Compressing the octave of 22edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 80zpi does this.

Approximation of harmonics in 80zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.4 +4.9 -2.7 -7.7 +3.6 +9.1 -4.1 +9.9 -9.1 -10.6 +2.2
Relative (%) -2.5 +9.1 -5.0 -14.1 +6.6 +16.7 -7.6 +18.2 -16.6 -19.5 +4.0
Step 22 35 44 51 57 62 66 70 73 76 79
Approximation of harmonics in 80zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +27.1 +7.7 -2.7 -5.5 -1.5 +8.5 +23.9 -10.4 +14.1 -12.0 +20.0 +0.8
Relative (%) +49.7 +14.2 -5.0 -10.1 -2.7 +15.6 +43.8 -19.1 +25.8 -22.0 +36.8 +1.5
Step 82 84 86 88 90 92 94 95 97 98 100 101
35edt
  • 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 EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.8 +2.8 -5.5 -10.9 +0.0 +5.2 -8.3 +5.5 -13.6 -15.4 -2.8
Relative (%) -5.1 +5.1 -10.1 -20.0 +0.0 +9.6 -15.2 +10.1 -25.1 -28.3 -5.1
Steps
(reduced) 		22
(22) 		35
(35) 		44
(44) 		51
(51) 		57
(0) 		62
(5) 		66
(9) 		70
(13) 		73
(16) 		76
(19) 		79
(22)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +21.9 +2.5 -8.1 -11.0 -7.1 +2.8 +18.0 -16.4 +8.0 -18.1 +13.8 -5.5
Relative (%) +40.3 +4.6 -14.9 -20.2 -13.1 +5.1 +33.1 -30.1 +14.7 -33.3 +25.3 -10.1
Steps
(reduced) 		82
(25) 		84
(27) 		86
(29) 		88
(31) 		90
(33) 		92
(35) 		94
(37) 		95
(38) 		97
(40) 		98
(41) 		100
(43) 		101
(44)
13edf
  • 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 EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -12.1 -12.1 -24.2 +21.5 -24.2 -21.0 +17.8 -24.2 +9.4 +6.4 +17.8
Relative (%) -22.4 -22.4 -44.7 +39.8 -44.7 -39.0 +32.9 -44.7 +17.5 +11.9 +32.9
Steps
(reduced) 		22
(9) 		35
(9) 		44
(5) 		52
(0) 		57
(5) 		62
(10) 		67
(2) 		70
(5) 		74
(9) 		77
(12) 		80
(2)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -12.8 +20.9 +9.4 +5.7 +8.7 +17.8 -21.8 -2.6 +20.9 -5.7 +25.4 +5.7
Relative (%) -23.7 +38.7 +17.5 +10.5 +16.2 +32.9 -40.4 -4.9 +38.7 -10.5 +47.0 +10.5
Steps
(reduced) 		82
(4) 		85
(7) 		87
(9) 		89
(11) 		91
(0) 		93
(2) 		94
(3) 		96
(5) 		98
(7) 		99
(8) 		101
(10) 		102
(11)
35edt
  • 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 EDONOI does this.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.5 +0.0 -9.0 -14.9 -4.5 +0.4 -13.5 +0.0 -19.4 -21.4 -9.0
Relative (%) -8.3 +0.0 -16.5 -27.4 -8.3 +0.6 -24.8 +0.0 -35.7 -39.3 -16.5
Steps
(reduced) 		22
(22) 		35
(0) 		44
(9) 		51
(16) 		57
(22) 		62
(27) 		66
(31) 		70
(0) 		73
(3) 		76
(6) 		79
(9)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.5 -4.1 -14.9 -17.9 -14.2 -4.5 +10.6 -23.9 +0.4 -25.8 +5.9 -13.5
Relative (%) +28.5 -7.6 -27.4 -33.0 -26.2 -8.3 +19.5 -43.9 +0.6 -47.6 +10.8 -24.8
Steps
(reduced) 		82
(12) 		84
(14) 		86
(16) 		88
(18) 		90
(20) 		92
(22) 		94
(24) 		95
(25) 		97
(27) 		98
(28) 		100
(30) 		101
(31)


13edf 35edt

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