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User:BudjarnLambeth/Sandbox2

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Revision as of 05:38, 24 August 2025 by BudjarnLambeth (talk | contribs) (→Octave stretch or compression)

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

31edo can benefit from slightly stretching the octave, especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13.

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: 1200.8 ¢

Stretching the octave of 31edo by slightly less than 1 ¢ results in slightly improved primes 3 and 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 14.2 ¢. 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: 1201.2 ¢

Stretching the octave of 31edo by slightly more than 1 ¢ results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This approximates all harmonics up to 16 within 15.5 ¢ Its 11-limit WE tuning and 11-limit TE tuning both do this, so does the tuning 111ed12.

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
80ed6
  • Step size: 38.774 ¢, octave size: 1202.0 ¢

Stretching the octave of 31edo by about 2 ¢ results in slightly improved primes 3 and 11, but slightly worse primes 2, 5 and 7, and much worse 13. This is approaching 2.239 ¢ - the most octave stretch 31edo can tolerate before the mapping of the 13th harmonic changes. This approximates all harmonics up to 16 within 18.5 ¢. 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)

{{Harmonics in equal|80|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 80ed6 (continued)}

49edt
  • Step size: 38.815 ¢, octave size: 1203.3 ¢

Stretching the octave of 31edo by about 3.5 ¢ results in improved primes 3 and 11, especially 11, but slightly worse primes 2, 5, 7 and 13. This approximates all harmonics up to 16 within 15.6 ¢. The tuning 49edt does this.

Approximation of harmonics in 49edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.3 +0.0 +6.6 +8.4 +3.3 +8.1 +9.8 +0.0 +11.7 +1.9 +6.6
Relative (%) +8.4 +0.0 +16.9 +21.6 +8.4 +20.9 +25.3 +0.0 +30.1 +5.0 +16.9
Steps
(reduced) 		31
(31) 		49
(0) 		62
(13) 		72
(23) 		80
(31) 		87
(38) 		93
(44) 		98
(0) 		103
(5) 		107
(9) 		111
(13)
Approximation of harmonics in 49edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -15.6 +11.4 +8.4 +13.1 -14.2 +3.3 -12.7 +15.0 +8.1 +5.2 +5.9 +9.8
Relative (%) -40.1 +29.3 +21.6 +33.8 -36.6 +8.4 -32.7 +38.5 +20.9 +13.4 +15.2 +25.3
Steps
(reduced) 		114
(16) 		118
(20) 		121
(23) 		124
(26) 		126
(28) 		129
(31) 		131
(33) 		134
(36) 		136
(38) 		138
(40) 		140
(42) 		142
(44)
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