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 compression

What follows is a comparison of compressed-octave 17edo tunings.

17edo
  • Step size: 70.588 ¢, octave size: 1200.0 ¢

Pure-octaves 17edo approximates the 2.3.11 subgroup well, it arguably might approximate 7, but not well, and it doesn't really approximate 5. It might make tuning for exploring new harmonies with the 7th, 11th and 13th harmonics not found in 12edo, but its very sharp 7th harmonic might arguably hamper that use case.

Approximation of harmonics in 17edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 +3.9 +0.0 -33.4 +3.9 +19.4 +0.0 +7.9 -33.4 +13.4 +3.9
Relative (%) +0.0 +5.6 +0.0 -47.3 +5.6 +27.5 +0.0 +11.1 -47.3 +19.0 +5.6
Steps
(reduced) 		17
(0) 		27
(10) 		34
(0) 		39
(5) 		44
(10) 		48
(14) 		51
(0) 		54
(3) 		56
(5) 		59
(8) 		61
(10)
Approximation of harmonics in 17edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +6.5 +19.4 -29.4 +0.0 -34.4 +7.9 -15.2 -33.4 +23.3 +13.4 +7.0 +3.9
Relative (%) +9.3 +27.5 -41.7 +0.0 -48.7 +11.1 -21.5 -47.3 +33.1 +19.0 +9.9 +5.6
Steps
(reduced) 		63
(12) 		65
(14) 		66
(15) 		68
(0) 		69
(1) 		71
(3) 		72
(4) 		73
(5) 		75
(7) 		76
(8) 		77
(9) 		78
(10)
44ed6
  • Step size: NNN ¢, octave size: 1198.5 ¢

Compressing the octave of 17edo by around 1.5 ¢ results in greatly improved primes 3, 7, 11 and 13, but a slightly worse prime 2. The tuning 44ed6 does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s 13-limit tuning for its size.

Approximation of harmonics in 44ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.5 +1.5 -3.0 +33.6 +0.0 +15.1 -4.6 +3.0 +32.1 +8.1 -1.5
Relative (%) -2.2 +2.2 -4.3 +47.7 +0.0 +21.5 -6.5 +4.3 +45.6 +11.5 -2.2
Steps
(reduced) 		17
(17) 		27
(27) 		34
(34) 		40
(40) 		44
(0) 		48
(4) 		51
(7) 		54
(10) 		57
(13) 		59
(15) 		61
(17)
Approximation of harmonics in 44ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.9 +13.6 +35.2 -6.1 +30.0 +1.5 -21.6 +30.6 +16.6 +6.6 +0.1 -3.0
Relative (%) +1.3 +19.3 +49.9 -8.6 +42.5 +2.2 -30.6 +43.4 +23.6 +9.4 +0.2 -4.3
Steps
(reduced) 		63
(19) 		65
(21) 		67
(23) 		68
(24) 		70
(26) 		71
(27) 		72
(28) 		74
(30) 		75
(31) 		76
(32) 		77
(33) 		78
(34)
27edt
  • Step size: NNN ¢, octave size: 1197.5 ¢

Compressing the octave of 17edo by around 2.5 ¢ results in improved primes NNN, but worse primes NNN. The tuning 27edt does this. Its primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s 13-limit tuning for its size.

Approximation of harmonics in 27edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.5 +0.0 -4.9 +31.4 -2.5 +12.4 -7.4 +0.0 +28.9 +4.8 -4.9
Relative (%) -3.5 +0.0 -7.0 +44.6 -3.5 +17.6 -10.5 +0.0 +41.1 +6.8 -7.0
Steps
(reduced) 		17
(17) 		27
(0) 		34
(7) 		40
(13) 		44
(17) 		48
(21) 		51
(24) 		54
(0) 		57
(3) 		59
(5) 		61
(7)
Approximation of harmonics in 27edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 +10.0 +31.4 -9.9 +26.0 -2.5 -25.6 +26.5 +12.4 +2.3 -4.2 -7.4
Relative (%) -3.7 +14.1 +44.6 -14.0 +37.0 -3.5 -36.4 +37.6 +17.6 +3.3 -5.9 -10.5
Steps
(reduced) 		63
(9) 		65
(11) 		67
(13) 		68
(14) 		70
(16) 		71
(17) 		72
(18) 		74
(20) 		75
(21) 		76
(22) 		77
(23) 		78
(24)
56zpi / 17et, 2.3.7.11.13 WE tuning
  • Step size: 70.403 ¢, octave size: 1296.9 ¢

Compressing the octave of 17edo by around 3 ¢ results in improved primes NNN, but worse primes NNN. The tunings: 56zpi, 17et, 2.3.7.11.13 TE and 17et, 2.3.7.11.13 WE all do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s 13-limit tuning for its size.

Approximation of harmonics in 56zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.1 -1.1 -6.3 +29.8 -4.2 +10.5 -9.4 -2.1 +26.7 +2.5 -7.4
Relative (%) -4.5 -1.5 -8.9 +42.3 -6.0 +14.9 -13.4 -3.1 +37.9 +3.5 -10.5
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 56zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.1 +7.4 +28.7 -12.6 +23.3 -5.3 -28.5 +23.5 +9.4 -0.7 -7.2 -10.5
Relative (%) -7.3 +10.5 +40.8 -17.9 +33.0 -7.5 -40.5 +33.4 +13.4 -1.0 -10.3 -14.9
Step 63 65 67 68 70 71 72 74 75 76 77 78
17et, 2.3.7.11 WE tuning
  • Step size: 70.392 ¢, octave size: 1296.7 ¢

Compressing the octave of 17edo by just over 3 ¢ results in improved primes NNN, but worse primes NNN. Its 2.3.7.11 WE tuning and 2.3.7.11 TE tuning both do this. Their primary purpose is to greatly improve 17edo's approximation of harmonic 7, to make it into an exceptional no-5s 13-limit tuning for its size.

Approximation of harmonics in 17et, 2.3.7.11 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.3 -1.4 -6.7 +29.4 -4.7 +10.0 -10.0 -2.7 +26.0 +1.8 -8.0
Relative (%) -4.7 -1.9 -9.5 +41.7 -6.7 +14.2 -14.2 -3.9 +37.0 +2.6 -11.4
Step 17 27 34 40 44 48 51 54 57 59 61
Approximation of harmonics in 17et, 2.3.7.11 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.8 +6.7 +28.0 -13.3 +22.5 -6.1 -29.3 +22.7 +8.6 -1.5 -8.1 -11.4
Relative (%) -8.3 +9.5 +39.8 -19.0 +31.9 -8.6 -41.6 +32.2 +12.2 -2.2 -11.5 -16.2
Step 63 65 67 68 70 71 72 74 75 76 77 78
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