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

72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly stretching the octave, using tunings such as 114edt or 186ed6. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.

What follows is a comparison of stretched-octave 72edo tunings.

72edo
  • Step size: 16.667 ¢, octave size: 1200.00 ¢

Pure-octaves 72edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 72edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 -1.96 +0.00 -2.98 -1.96 -2.16 +0.00 -3.91 -2.98 -1.32 -1.96
Relative (%) +0.0 -11.7 +0.0 -17.9 -11.7 -13.0 +0.0 -23.5 -17.9 -7.9 -11.7
Steps
(reduced) 		72
(0) 		114
(42) 		144
(0) 		167
(23) 		186
(42) 		202
(58) 		216
(0) 		228
(12) 		239
(23) 		249
(33) 		258
(42)
Approximation of harmonics in 72edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.19 -2.16 -4.94 +0.00 -4.96 -3.91 +2.49 -2.98 -4.11 -1.32 +5.06 -1.96
Relative (%) -43.2 -13.0 -29.6 +0.0 -29.7 -23.5 +14.9 -17.9 -24.7 -7.9 +30.4 -11.7
Steps
(reduced) 		266
(50) 		274
(58) 		281
(65) 		288
(0) 		294
(6) 		300
(12) 		306
(18) 		311
(23) 		316
(28) 		321
(33) 		326
(38) 		330
(42)
249ed11
  • Step size: NNN ¢, octave size: 1200.38 ¢

Stretching the octave of 72edo by around 0.4 ¢ results in unnoticeably better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN ¢. The tuning 249ed11 does this.

Approximation of harmonics in 249ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.38 -1.35 +0.76 -2.10 -0.97 -1.09 +1.14 -2.70 -1.72 +0.00 -0.59
Relative (%) +2.3 -8.1 +4.6 -12.6 -5.8 -6.5 +6.9 -16.2 -10.3 +0.0 -3.5
Steps
(reduced) 		72
(72) 		114
(114) 		144
(144) 		167
(167) 		186
(186) 		202
(202) 		216
(216) 		228
(228) 		239
(239) 		249
(0) 		258
(9)
Approximation of harmonics in 249ed11 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.79 -0.71 -3.45 +1.52 -3.40 -2.32 +4.11 -1.33 -2.44 +0.38 +6.78 -0.21
Relative (%) -34.7 -4.3 -20.7 +9.1 -20.4 -13.9 +24.6 -8.0 -14.6 +2.3 +40.7 -1.2
Steps
(reduced) 		266
(17) 		274
(25) 		281
(32) 		288
(39) 		294
(45) 		300
(51) 		306
(57) 		311
(62) 		316
(67) 		321
(72) 		326
(77) 		330
(81)
258ed12
  • Step size: NNN ¢, octave size: 1200.55 ¢

Stretching the octave of 72edo by around 0.5 ¢ results in unnoticeably better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN ¢. The tuning 258ed12 does this.

Approximation of harmonics in 258ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.55 -1.09 +1.09 -1.71 -0.55 -0.63 +1.64 -2.18 -1.17 +0.57 +0.00
Relative (%) +3.3 -6.5 +6.5 -10.3 -3.3 -3.8 +9.8 -13.1 -7.0 +3.4 +0.0
Steps
(reduced) 		72
(72) 		114
(114) 		144
(144) 		167
(167) 		186
(186) 		202
(202) 		216
(216) 		228
(228) 		239
(239) 		249
(249) 		258
(0)
Approximation of harmonics in 258ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.18 -0.08 -2.81 +2.18 -2.73 -1.64 +4.81 -0.62 -1.72 +1.11 +7.53 +0.55
Relative (%) -31.1 -0.5 -16.8 +13.1 -16.4 -9.8 +28.8 -3.7 -10.3 +6.7 +45.2 +3.3
Steps
(reduced) 		266
(8) 		274
(16) 		281
(23) 		288
(30) 		294
(36) 		300
(42) 		306
(48) 		311
(53) 		316
(58) 		321
(63) 		326
(68) 		330
(72)
186ed6 / 72et, 11-limit WE tuning / 202ed7
  • Step size: NNN ¢, octave size: 1200.76 ¢

Stretching the octave of 72edo by around 0.75 ¢ results in unnoticeably better primes 3, 5, 7, 11 and 13, but an unnoticeably worse prime 2. This approximates all harmonics up to 16 within NNN ¢. The tuning 186ed6 does this. 72et's 11-limit WE tuning and 11-limit TE tuning both do this, their octave differing from 186ed6's by only 0.02 ¢. The tuning 202ed7 does this also, it's octave differing from 186ed6 by less than a hundredth of a cent.

Approximation of harmonics in 186ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.76 -0.76 +1.51 -1.23 +0.00 -0.04 +2.27 -1.51 -0.47 +1.30 +0.76
Relative (%) +4.5 -4.5 +9.1 -7.3 +0.0 -0.2 +13.6 -9.1 -2.8 +7.8 +4.5
Steps
(reduced) 		72
(72) 		114
(114) 		144
(144) 		167
(167) 		186
(0) 		202
(16) 		216
(30) 		228
(42) 		239
(53) 		249
(63) 		258
(72)
Approximation of harmonics in 186ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.40 +0.72 -1.98 +3.03 -1.87 -0.76 +5.70 +0.29 -0.79 +2.06 -8.19 +1.51
Relative (%) -26.4 +4.3 -11.9 +18.2 -11.2 -4.5 +34.2 +1.7 -4.8 +12.3 -49.1 +9.1
Steps
(reduced) 		266
(80) 		274
(88) 		281
(95) 		288
(102) 		294
(108) 		300
(114) 		306
(120) 		311
(125) 		316
(130) 		321
(135) 		325
(139) 		330
(144)
380zpi
  • Step size: 16.678 ¢, octave size: 1200.82 ¢

Stretching the octave of 72edo by around 0.8 ¢ results in unnoticeably better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN ¢. The tuning 380zpi does this.

Approximation of harmonics in 380zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.82 -0.66 +1.63 -1.09 +0.15 +0.13 +2.45 -1.33 -0.27 +1.50 +0.97
Relative (%) +4.9 -4.0 +9.8 -6.5 +0.9 +0.8 +14.7 -8.0 -1.6 +9.0 +5.8
Step 72 114 144 167 186 202 216 228 239 249 258
Approximation of harmonics in 380zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.18 +0.95 -1.75 +3.26 -1.62 -0.51 +5.95 +0.54 -0.53 +2.32 -7.92 +1.78
Relative (%) -25.1 +5.7 -10.5 +19.6 -9.7 -3.1 +35.7 +3.3 -3.2 +13.9 -47.5 +10.7
Step 266 274 281 288 294 300 306 311 316 321 325 330
72et, 13-limit WE tuning
  • Step size: 16.680 ¢, octave size: 1200.96 ¢

Stretching the octave of 72edo by around NNN ¢ results in unnoticeably better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.

Approximation of harmonics in 72et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.96 -0.44 +1.92 -0.75 +0.52 +0.53 +2.88 -0.87 +0.21 +2.00 +1.48
Relative (%) +5.8 -2.6 +11.5 -4.5 +3.1 +3.2 +17.3 -5.2 +1.2 +12.0 +8.9
Step 72 114 144 167 186 202 216 228 239 249 258
Approximation of harmonics in 72et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -3.65 +1.49 -1.19 +3.84 -1.04 +0.09 +6.57 +1.17 +0.10 +2.96 -7.27 +2.44
Relative (%) -21.9 +9.0 -7.1 +23.0 -6.2 +0.5 +39.4 +7.0 +0.6 +17.8 -43.6 +14.7
Step 266 274 281 288 294 300 306 311 316 321 325 330
114edt / 167ed5
  • Step size: NNN ¢, octave size: 1201.23 ¢

Stretching the octave of 72edo by around NNN ¢ results in unnoticeably better primes 3, 5, 7 and 13, but unnoticeably worse primes 2 and 11. This approximates all harmonics up to 16 within NNN ¢. The tuning 144edt does this. The tuning 167ed5 does this also, its octave differing from 114edt by only 0.05 ¢.

Approximation of harmonics in 114edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.23 +0.00 +2.47 -0.12 +1.23 +1.30 +3.70 +0.00 +1.12 +2.95 +2.47
Relative (%) +7.4 +0.0 +14.8 -0.7 +7.4 +7.8 +22.2 +0.0 +6.7 +17.7 +14.8
Steps
(reduced) 		72
(72) 		114
(0) 		144
(30) 		167
(53) 		186
(72) 		202
(88) 		216
(102) 		228
(0) 		239
(11) 		249
(21) 		258
(30)
Approximation of harmonics in 114edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.63 +2.54 -0.12 +4.94 +0.09 +1.23 +7.73 +2.35 +1.30 +4.19 -6.03 +3.70
Relative (%) -15.8 +15.2 -0.7 +29.6 +0.5 +7.4 +46.4 +14.1 +7.8 +25.1 -36.2 +22.2
Steps
(reduced) 		266
(38) 		274
(46) 		281
(53) 		288
(60) 		294
(66) 		300
(72) 		306
(78) 		311
(83) 		316
(88) 		321
(93) 		325
(97) 		330
(102)
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