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-octave 31edo tunings.

31edo
  • Step size: 38.710 ¢, octave size: 1200.0 ¢

Pure-octaves 31edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 31edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -5.2 +0.0 +0.8 -5.2 -1.1 +0.0 -10.4 +0.8 -9.4 -5.2
Relative (%) +0.0 -13.4 +0.0 +2.0 -13.4 -2.8 +0.0 -26.8 +2.0 -24.2 -13.4
Steps
(reduced) 		31
(0) 		49
(18) 		62
(0) 		72
(10) 		80
(18) 		87
(25) 		93
(0) 		98
(5) 		103
(10) 		107
(14) 		111
(18)
Approximation of harmonics in 31edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +11.1 -1.1 -4.4 +0.0 +11.2 -10.4 +12.2 +0.8 -6.3 -9.4 -8.9 -5.2
Relative (%) +28.6 -2.8 -11.4 +0.0 +28.9 -26.8 +31.4 +2.0 -16.2 -24.2 -23.0 -13.4
Steps
(reduced) 		115
(22) 		118
(25) 		121
(28) 		124
(0) 		127
(3) 		129
(5) 		132
(8) 		134
(10) 		136
(12) 		138
(14) 		140
(16) 		142
(18)
31et, 13-limit WE tuning
  • Step size: 38.725 ¢, octave size: 1200.5 ¢

Stretching the octave of 31edo by around 0.5 ¢ results in slightly improved primes 3, 7 and 11, but slightly worse primes 2, 5 and 13. This approximates all harmonics up to 16 within 12.8 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 31et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.5 -4.4 +0.9 +1.9 -4.0 +0.2 +1.4 -8.9 +2.4 -7.7 -3.5
Relative (%) +1.2 -11.4 +2.5 +4.9 -10.2 +0.6 +3.7 -22.9 +6.1 -20.0 -9.0
Step 31 49 62 72 80 87 93 98 103 107 111
Approximation of harmonics in 31et, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.8 +0.7 -2.5 +1.9 +13.1 -8.4 +14.2 +2.8 -4.2 -7.3 -6.8 -3.0
Relative (%) +33.2 +1.9 -6.6 +4.9 +33.9 -21.7 +36.6 +7.3 -10.8 -18.8 -17.5 -7.8
Step 115 118 121 124 127 129 132 134 136 138 140 142
127zpi
  • Step size: 38.737 ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 127zpi does this.

Approximation of harmonics in 127zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.8 -3.8 +1.7 +2.8 -3.0 +1.3 +2.5 -7.7 +3.6 -6.5 -2.1
Relative (%) +2.2 -9.9 +4.4 +7.1 -7.7 +3.3 +6.6 -19.8 +9.3 -16.7 -5.5
Step 31 49 62 72 80 87 93 98 103 107 111
Approximation of harmonics in 127zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +14.2 +2.1 -1.1 +3.4 +14.6 -6.8 +15.8 +4.4 -2.5 -5.6 -5.1 -1.3
Relative (%) +36.7 +5.5 -2.8 +8.7 +37.8 -17.6 +40.7 +11.5 -6.6 -14.5 -13.2 -3.4
Step 115 118 121 124 127 129 132 134 136 138 140 142
31et, 11-limit WE tuning
  • Step size: 38.748 ¢, octave size: NNN ¢

_Stretching the octave of 31edo by around NNN ¢ 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 31et, 11-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.2 -3.3 +2.4 +3.5 -2.1 +2.3 +3.6 -6.6 +4.7 -5.3 -0.9
Relative (%) +3.1 -8.5 +6.1 +9.1 -5.5 +5.8 +9.2 -17.0 +12.2 -13.6 -2.4
Step 31 49 62 72 80 87 93 98 103 107 111
Approximation of harmonics in 31et, 11-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.5 +3.4 +0.2 +4.8 +16.0 -5.4 +17.2 +5.9 -1.1 -4.1 -3.6 +0.3
Relative (%) +40.0 +8.9 +0.6 +12.3 +41.4 -14.0 +44.4 +15.3 -2.7 -10.6 -9.2 +0.7
Step 115 118 121 124 127 129 132 134 136 138 140 142
111ed12
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 111ed12 does this.

Approximation of harmonics in 111ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.4 -2.9 +2.9 +4.1 -1.4 +3.0 +4.3 -5.8 +5.6 -4.4 +0.0
Relative (%) +3.7 -7.5 +7.5 +10.7 -3.7 +7.7 +11.2 -14.9 +14.4 -11.3 +0.0
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(80) 		87
(87) 		93
(93) 		98
(98) 		103
(103) 		107
(107) 		111
(0)
Approximation of harmonics in 111ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.5 +4.4 +1.2 +5.8 +17.1 -4.3 +18.3 +7.0 +0.1 -2.9 -2.4 +1.4
Relative (%) +42.5 +11.4 +3.2 +14.9 +44.1 -11.2 +47.3 +18.2 +0.2 -7.6 -6.2 +3.7
Steps
(reduced) 		115
(4) 		118
(7) 		121
(10) 		124
(13) 		127
(16) 		129
(18) 		132
(21) 		134
(23) 		136
(25) 		138
(27) 		140
(29) 		142
(31)
80ed6
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 80ed6 does this.

Approximation of harmonics in 80ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.0 -2.0 +4.0 +5.4 +0.0 +4.6 +6.0 -4.0 +7.5 -2.5 +2.0
Relative (%) +5.2 -5.2 +10.4 +14.0 +0.0 +11.7 +15.5 -10.4 +19.2 -6.3 +5.2
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(0) 		87
(7) 		93
(13) 		98
(18) 		103
(23) 		107
(27) 		111
(31)
Approximation of harmonics in 80ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +18.5 +6.6 +3.4 +8.0 -19.4 -2.0 -18.1 +9.5 +2.5 -0.4 +0.1 +4.0
Relative (%) +47.8 +16.9 +8.9 +20.7 -50.0 -5.2 -46.6 +24.4 +6.6 -1.1 +0.4 +10.4
Steps
(reduced) 		115
(35) 		118
(38) 		121
(41) 		124
(44) 		126
(46) 		129
(49) 		131
(51) 		134
(54) 		136
(56) 		138
(58) 		140
(60) 		142
(62)
25ed7/4
  • Step size: NNN ¢, octave size: NNN ¢

Stretching the octave of 31edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 25ed7/4 does this.

Approximation of harmonics in 25ed7/4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.3 -3.1 +2.7 +3.9 -1.7 +2.7 +4.0 -6.1 +5.2 -4.7 -0.4
Relative (%) +3.5 -7.9 +6.9 +10.1 -4.4 +6.9 +10.4 -15.8 +13.5 -12.2 -0.9
Steps
(reduced) 		31
(6) 		49
(24) 		62
(12) 		72
(22) 		80
(5) 		87
(12) 		93
(18) 		98
(23) 		103
(3) 		107
(7) 		111
(11)
Approximation of harmonics in 25ed7/4 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.1 +4.0 +0.8 +5.4 +16.7 -4.8 +17.9 +6.6 -0.4 -3.4 -2.8 +1.0
Relative (%) +41.5 +10.4 +2.2 +13.9 +43.0 -12.3 +46.2 +17.0 -0.9 -8.8 -7.4 +2.5
Steps
(reduced) 		115
(15) 		118
(18) 		121
(21) 		124
(24) 		127
(2) 		129
(4) 		132
(7) 		134
(9) 		136
(11) 		138
(13) 		140
(15) 		142
(17)
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