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User:BudjarnLambeth/Sandbox2

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Revision as of 01:32, 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

What follows is a comparison of stretched- and compressed-octave 22edo tunings.

51ed5
  • Step size: NNN ¢, octave size: nnn ¢

Stretching the octave of 22edo by around 3 ¢ results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a just noticeably worse 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9 ¢. With its worse 5 and 11, it only really makes sense as a 2.3.7 tuning (eg for archy temperament). The tuning 57ed6 does this.

Approximation of harmonics in 57ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.9 +10.2 +3.9 +0.0 +12.2 +18.5 +5.8 +20.4 +1.9 +0.8 +14.1
Relative (%) +3.5 +18.7 +7.1 +0.0 +22.3 +33.8 +10.6 +37.4 +3.5 +1.5 +25.8
Steps
(reduced) 		22
(22) 		35
(35) 		44
(44) 		51
(0) 		57
(6) 		62
(11) 		66
(15) 		70
(19) 		73
(22) 		76
(25) 		79
(28)
Approximation of harmonics in 57ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -15.2 +20.4 +10.2 +7.8 +12.1 +22.4 -16.6 +3.9 -26.0 +2.8 -19.5 +16.0
Relative (%) -27.8 +37.3 +18.7 +14.2 +22.1 +41.0 -30.4 +7.1 -47.5 +5.1 -35.8 +29.4
Steps
(reduced) 		81
(30) 		84
(33) 		86
(35) 		88
(37) 		90
(39) 		92
(41) 		93
(42) 		95
(44) 		96
(45) 		98
(47) 		99
(48) 		101
(50)
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. It is a good 13-limit tuning for its size.

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: 1198.9 ¢

Compressing the octave of 22edo by around 1 ¢ results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 26.5 ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this. It is a good 11-limit tuning for its size.

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: 1198.6 ¢

Compressing the octave of 22edo by around 1 ¢ results in slightly improved primes 3 and 7, but slightly worse primes 5 and 11, and a much worse 13. This approximates all harmonics up to 16 within 27.1 ¢. The tuning 80zpi does this. It is a good 11-limit tuning for its size.

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
57ed6
  • Step size: NNN ¢, octave size: 1197.2 ¢

Compressing the octave of 22edo by around 3 ¢ results in greatly improved primes 3 and 7, but far worse primes 5 and 11 and a just noticeably worse 2. The mapping of 13 differs from 22edo but has about the same amount of error. This approximates all harmonics up to 16 within 21.9 ¢. With its worse 5 and 11, it only really makes sense as a 2.3.7 tuning (eg for archy temperament). The tuning 57ed6 does this.

Approximation of harmonics in 57ed6
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 57ed6 (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)
35edt
  • Step size: NNN ¢, octave size: 1195.5 ¢

Compressing the octave of 22edo by around 4.5 ¢ results in greatly improved primes 3, 7 and 13, but far worse primes 5 and 11 and a just noticeably worse 2. This approximates all harmonics up to 16 within 21.4 ¢. The tunings 35edt and 62ed7 both do this. This extends 57ed6's 2.3.7 tuning into a 2.3.7.13 subgroup tuning.

Approximation of harmonics in 35edt
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 35edt (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)
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