User:BudjarnLambeth/Sandbox2

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Title1

Octave stretch or compression

Having a flat tendency, 16et is best tuned with stretched octaves, which improve the accuracy of wide-voiced JI chords and rooted harmonics especially on inharmonic timbres such as bells and gamelan, with 37ed5 and 57ed12 being good options.

What follows is a comparison of stretched- and compressed-octave 16edo tunings.

16edo
  • Step size: 75.000 ¢, octave size: 1200.0 ¢

Pure-octaves 16edo approximates all harmonics up to 16 within 36.7 ¢.

Approximation of harmonics in 16edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -27.0 +0.0 -11.3 -27.0 +6.2 +0.0 +21.1 -11.3 -26.3 -27.0
Relative (%) +0.0 -35.9 +0.0 -15.1 -35.9 +8.2 +0.0 +28.1 -15.1 -35.1 -35.9
Steps
(reduced) 		16
(0) 		25
(9) 		32
(0) 		37
(5) 		41
(9) 		45
(13) 		48
(0) 		51
(3) 		53
(5) 		55
(7) 		57
(9)
Approximation of harmonics in 16edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -15.5 +6.2 +36.7 +0.0 -30.0 +21.1 +2.5 -11.3 -20.8 -26.3 -28.3 -27.0
Relative (%) -20.7 +8.2 +49.0 +0.0 -39.9 +28.1 +3.3 -15.1 -27.7 -35.1 -37.7 -35.9
Steps
(reduced) 		59
(11) 		61
(13) 		63
(15) 		64
(0) 		65
(1) 		67
(3) 		68
(4) 		69
(5) 		70
(6) 		71
(7) 		72
(8) 		73
(9)
16et, 2.5.7.13 WE tuning
  • Step size: 75.105 ¢, octave size: 1201.7 ¢

Stretching the octave of 16edo by around 2 ¢ results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 31.8 ¢. Its 2.5.7.13 WE tuning and 2.5.7.13 TE tuning both do this.

Approximation of harmonics in 16et, 2.5.7.13 WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.7 -24.3 +3.4 -7.4 -22.7 +10.9 +5.0 +26.4 -5.7 -20.5 -21.0
Relative (%) +2.2 -32.4 +4.5 -9.9 -30.2 +14.5 +6.7 +35.2 -7.7 -27.4 -27.9
Step 16 25 32 37 41 45 48 51 53 55 57
Approximation of harmonics in 16et, 2.5.7.13 WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.3 +12.6 -31.8 +6.7 -23.1 +28.1 +9.6 -4.1 -13.4 -18.9 -20.7 -19.3
Relative (%) -12.4 +16.7 -42.3 +8.9 -30.8 +37.4 +12.8 -5.4 -17.9 -25.1 -27.6 -25.7
Step 59 61 62 64 65 67 68 69 70 71 72 73
15zpi / 59ed13
  • Step size: 75.262 ¢, octave size: 1204.2 ¢

Stretching the octave of 16edo by around 4 ¢ results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 34.5 ¢. The tunings 15zpi and 59ed13 do this.

Approximation of harmonics in 15zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.2 -20.4 +8.4 -1.6 -16.2 +18.0 +12.6 +34.5 +2.6 -11.9 -12.0
Relative (%) +5.6 -27.1 +11.1 -2.2 -21.5 +23.9 +16.7 +45.8 +3.4 -15.8 -16.0
Step 16 25 32 37 41 45 48 51 53 55 57
Approximation of harmonics in 15zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.1 +22.2 -22.0 +16.8 -12.9 -36.6 +20.3 +6.8 -2.4 -7.7 -9.4 -7.8
Relative (%) -0.1 +29.4 -29.3 +22.3 -17.2 -48.7 +27.0 +9.0 -3.2 -10.3 -12.5 -10.4
Step 59 61 62 64 65 66 68 69 70 71 72 73
16et, 13-limit WE tuning / 37ed5
  • Step size (WE 16et): 75.315 ¢, octave size (WE 16et): 1205.0 ¢

Stretching the octave of 16edo by around 5 ¢ results in very improved primes 3, 5, 11 and 13, but much worse primes 2 and 7. This approximates all harmonics up to 16 within 37.2 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this, so does the tuning 37ed5.

Approximation of harmonics in 16et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +5.0 -19.1 +10.1 +0.3 -14.0 +20.3 +15.1 +37.2 +5.4 -9.0 -9.0
Relative (%) +6.7 -25.3 +13.4 +0.5 -18.6 +27.0 +20.1 +49.3 +7.1 -11.9 -11.9
Step 16 25 32 37 41 45 48 51 53 55 57
Approximation of harmonics in 16et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.1 +25.4 -18.7 +20.2 -9.5 -33.1 +23.9 +10.4 +1.3 -4.0 -5.6 -4.0
Relative (%) +4.1 +33.7 -24.9 +26.8 -12.6 -44.0 +31.7 +13.8 +1.7 -5.2 -7.4 -5.3
Step 59 61 62 64 65 66 68 69 70 71 72 73
Approximation of harmonics in 37ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.9 -19.3 +9.8 +0.0 -14.4 +19.9 +14.7 +36.7 +4.9 -9.5 -9.5
Relative (%) +6.5 -25.6 +13.0 +0.0 -19.1 +26.5 +19.5 +48.7 +6.5 -12.6 -12.6
Steps
(reduced) 		16
(16) 		25
(25) 		32
(32) 		37
(0) 		41
(4) 		45
(8) 		48
(11) 		51
(14) 		53
(16) 		55
(18) 		57
(20)
Approximation of harmonics in 37ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.5 +24.8 -19.3 +19.6 -10.1 -33.7 +23.3 +9.8 +0.6 -4.6 -6.3 -4.6
Relative (%) +3.3 +33.0 -25.6 +26.0 -13.4 -44.8 +30.9 +13.0 +0.8 -6.1 -8.3 -6.2
Steps
(reduced) 		59
(22) 		61
(24) 		62
(25) 		64
(27) 		65
(28) 		66
(29) 		68
(31) 		69
(32) 		70
(33) 		71
(34) 		72
(35) 		73
(36)
57ed12 / 55ed11
  • Step size (57ed12): 75.473 ¢, octave size (57ed12): 1207.6 ¢

Stretching the octave of 16edo by around 7.5 ¢ results in especially improved primes 3, 5 and 11, but far worse primes 2 and 7. This approximates all harmonics up to 16 within NNN ¢. The tunings 57ed12 and 55ed11 do this.

Approximation of harmonics in 57ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +7.6 -15.1 +15.1 +6.2 -7.6 +27.5 +22.7 -30.3 +13.7 -0.3 +0.0
Relative (%) +10.0 -20.1 +20.1 +8.2 -10.0 +36.4 +30.1 -40.1 +18.2 -0.4 +0.0
Steps
(reduced) 		16
(16) 		25
(25) 		32
(32) 		37
(37) 		41
(41) 		45
(45) 		48
(48) 		50
(50) 		53
(53) 		55
(55) 		57
(0)
Approximation of harmonics in 57ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.4 +35.0 -8.9 +30.3 +0.8 -22.7 +34.6 +21.3 +12.3 +7.3 +5.8 +7.6
Relative (%) +16.4 +46.4 -11.9 +40.1 +1.0 -30.1 +45.9 +28.2 +16.3 +9.6 +7.7 +10.0
Steps
(reduced) 		59
(2) 		61
(4) 		62
(5) 		64
(7) 		65
(8) 		66
(9) 		68
(11) 		69
(12) 		70
(13) 		71
(14) 		72
(15) 		73
(16)
Approximation of harmonics in 55ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +7.7 -15.0 +15.3 +6.4 -7.3 +27.7 +23.0 -30.0 +14.0 +0.0 +0.3
Relative (%) +10.1 -19.9 +20.3 +8.5 -9.7 +36.7 +30.4 -39.7 +18.6 +0.0 +0.4
Steps
(reduced) 		16
(16) 		25
(25) 		32
(32) 		37
(37) 		41
(41) 		45
(45) 		48
(48) 		50
(50) 		53
(53) 		55
(0) 		57
(2)
Approximation of harmonics in 55ed11 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +12.7 +35.4 -8.6 +30.6 +1.1 -22.3 +35.0 +21.7 +12.7 +7.7 +6.2 +8.0
Relative (%) +16.8 +46.9 -11.4 +40.6 +1.5 -29.6 +46.4 +28.8 +16.8 +10.1 +8.2 +10.6
Steps
(reduced) 		59
(4) 		61
(6) 		62
(7) 		64
(9) 		65
(10) 		66
(11) 		68
(13) 		69
(14) 		70
(15) 		71
(16) 		72
(17) 		73
(18)

Title2

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