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: NNN ¢, octave size: 1200.00 ¢

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

Approximation of harmonics in EDONAME
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 EDONAME (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)
258ed12
  • Step size: NNN ¢, octave size: 1200.55 ¢

Stretching the octave of 72edo by around NNN ¢ 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 EDONOI does this.

Approximation of harmonics in EDONOI
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 EDONOI (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 NNN ¢ 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 ¢. Its 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 EDONOI
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 EDONOI (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 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 ZPINAME does this.

Approximation of harmonics in ZPINAME
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 ZPINAME (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 SUBGROUP WE tuning and SUBGROUP TE tuning both do this.

Approximation of harmonics in ETNAME, SUBGROUP 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 ETNAME, SUBGROUP 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
  • 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 EDONOI does this.

Approximation of harmonics in EDONOI
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 EDONOI (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)



Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.28 +0.08 +2.57 +0.00 +1.36 +1.45 +3.85 +0.16 +1.28 +3.13 +2.65
Relative (%) +7.7 +0.5 +15.4 +0.0 +8.2 +8.7 +23.1 +1.0 +7.7 +18.7 +15.9
Steps
(reduced) 		72
(72) 		114
(114) 		144
(144) 		167
(0) 		186
(19) 		202
(35) 		216
(49) 		228
(61) 		239
(72) 		249
(82) 		258
(91)


Approximation of harmonics in EDONOI
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)
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