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User:BudjarnLambeth/Draft related tunings section

Title1

Octave stretch or compression

What follows is a comparison of stretched- and compressed-octave 33edo tunings.

76ed5
  • Octave size: 1209.8 ¢

Stretching the octave of 33edo by around 10 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 76ed5 does this.

Approximation of harmonics in 76ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +9.8 +4.5 -17.0 +0.0 +14.3 +4.1 -7.1 +8.9 +9.8 -8.5 -12.5
Relative (%) +26.9 +12.2 -46.3 +0.0 +39.1 +11.1 -19.4 +24.4 +26.9 -23.2 -34.1
Steps
(reduced) 		33
(33) 		52
(52) 		65
(65) 		76
(0) 		85
(9) 		92
(16) 		98
(22) 		104
(28) 		109
(33) 		113
(37) 		117
(41)
Approximation of harmonics in 76ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.4 +13.9 +4.5 +2.7 +7.8 -17.9 -1.5 -17.0 +8.6 +1.3 -2.3 -2.7
Relative (%) -12.1 +38.0 +12.2 +7.4 +21.2 -48.8 -4.1 -46.3 +23.3 +3.6 -6.3 -7.2
Steps
(reduced) 		121
(45) 		125
(49) 		128
(52) 		131
(55) 		134
(58) 		136
(60) 		139
(63) 		141
(65) 		144
(68) 		146
(70) 		148
(72) 		150
(74)
92ed7
  • Octave size: 1208.4 ¢

Stretching the octave of 33edo by around 8.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 92ed7 does this. So does the tuning 137zpi whose octave differs by only 0.3 ¢.

Approximation of harmonics in 92ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.4 +2.2 +16.8 -3.4 +10.5 +0.0 -11.5 +4.3 +5.0 -13.5 -17.7
Relative (%) +22.9 +5.9 +45.8 -9.2 +28.8 +0.0 -31.3 +11.8 +13.7 -36.9 -48.3
Steps
(reduced) 		33
(33) 		52
(52) 		66
(66) 		76
(76) 		85
(85) 		92
(0) 		98
(6) 		104
(12) 		109
(17) 		113
(21) 		117
(25)
Approximation of harmonics in 92ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.8 +8.4 -1.2 -3.1 +1.8 +12.7 -7.7 +13.4 +2.2 -5.1 -8.9 -9.3
Relative (%) -26.7 +22.9 -3.3 -8.4 +5.0 +34.7 -20.9 +36.6 +5.9 -14.0 -24.2 -25.4
Steps
(reduced) 		121
(29) 		125
(33) 		128
(36) 		131
(39) 		134
(42) 		137
(45) 		139
(47) 		142
(50) 		144
(52) 		146
(54) 		148
(56) 		150
(58)
114ed11
  • Octave size: 1201.7 ¢

Stretching the octave of 33edo by around 2 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 114ed11 does this.

Approximation of harmonics in 114ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.7 -8.4 +3.4 +17.6 -6.7 +17.8 +5.1 -16.7 -17.1 +0.0 -5.0
Relative (%) +4.7 -23.0 +9.3 +48.5 -18.3 +48.8 +14.0 -46.0 -46.9 +0.0 -13.7
Steps
(reduced) 		33
(33) 		52
(52) 		66
(66) 		77
(77) 		85
(85) 		93
(93) 		99
(99) 		104
(104) 		109
(109) 		114
(0) 		118
(4)
Approximation of harmonics in 114ed11 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +2.1 -16.9 +9.3 +6.8 +11.1 -15.0 +0.6 -15.4 +9.4 +1.7 -2.4 -3.3
Relative (%) +5.8 -46.5 +25.5 +18.6 +30.4 -41.3 +1.6 -42.2 +25.8 +4.7 -6.7 -9.0
Steps
(reduced) 		122
(8) 		125
(11) 		129
(15) 		132
(18) 		135
(21) 		137
(23) 		140
(26) 		142
(28) 		145
(31) 		147
(33) 		149
(35) 		151
(37)
138zpi
  • Step size: 36.394 ¢, octave size: 1201.0 ¢

Stretching the octave of 33edo by around 1 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 138zpi does this. So does the tuning 122ed13 whose octave differs by only 0.1 ¢.

Approximation of harmonics in 138zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.0 -9.5 +2.0 +16.0 -8.5 +15.8 +3.0 +17.5 +17.0 -2.4 -7.5
Relative (%) +2.8 -26.0 +5.5 +44.0 -23.3 +43.5 +8.3 +48.0 +46.8 -6.6 -20.5
Step 33 52 66 77 85 93 99 105 110 114 118
Approximation of harmonics in 138zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.5 +16.8 +6.6 +4.0 +8.2 -17.9 -2.4 +18.0 +6.3 -1.4 -5.6 -6.5
Relative (%) -1.3 +46.2 +18.0 +11.0 +22.6 -49.3 -6.5 +49.5 +17.4 -3.8 -15.3 -17.8
Step 122 126 129 132 135 137 140 143 145 147 149 151
33edo
  • Step size: 36.363 ¢, octave size: 1200.0 ¢

Pure-octaves 33edo approximates all harmonics up to 16 within NNN ¢.

Approximation of harmonics in 33edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -11.0 +0.0 +13.7 -11.0 +13.0 +0.0 +14.3 +13.7 -5.9 -11.0
Relative (%) +0.0 -30.4 +0.0 +37.6 -30.4 +35.7 +0.0 +39.2 +37.6 -16.1 -30.4
Steps
(reduced) 		33
(0) 		52
(19) 		66
(0) 		77
(11) 		85
(19) 		93
(27) 		99
(0) 		105
(6) 		110
(11) 		114
(15) 		118
(19)
Approximation of harmonics in 33edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.2 +13.0 +2.6 +0.0 +4.1 +14.3 -6.6 +13.7 +1.9 -5.9 -10.1 -11.0
Relative (%) -11.5 +35.7 +7.3 +0.0 +11.4 +39.2 -18.2 +37.6 +5.4 -16.1 -27.8 -30.4
Steps
(reduced) 		122
(23) 		126
(27) 		129
(30) 		132
(0) 		135
(3) 		138
(6) 		140
(8) 		143
(11) 		145
(13) 		147
(15) 		149
(17) 		151
(19)
33et, 13-limit WE tuning
  • Step size: 36.357 ¢, octave size: 1199.8 ¢

Compressing the octave of 33edo by a fifth of a cent results in improved primes NNN, but worse primes NNN. 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 33et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -0.2 -11.4 -0.4 +13.2 -11.6 +12.4 -0.7 +13.6 +13.0 -6.6 -11.8
Relative (%) -0.6 -31.3 -1.2 +36.2 -31.9 +34.0 -1.8 +37.3 +35.6 -18.2 -32.5
Step 33 52 66 77 85 93 99 105 110 114 118
Approximation of harmonics in 33et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.0 +12.2 +1.8 -0.9 +3.2 +13.4 -7.5 +12.7 +1.0 -6.8 -11.1 -12.0
Relative (%) -13.7 +33.4 +4.9 -2.4 +8.9 +36.7 -20.7 +35.0 +2.7 -18.8 -30.5 -33.1
Step 122 126 129 132 135 138 140 143 145 147 149 151
93ed7
  • Octave size: 1196.4 ¢

Compressing the octave of 33edo by around 4.5 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. If one wishes to use both 33edo's sharp and flat fifths simultaneously (see dual-fifth tuning), then this amount of stretch is ideal, because it evenly shares error between the two fifths. The tuning 93ed7 does this. So does the tuning 52ed13 whose octave differs by only 0.1 ¢.

Approximation of harmonics in 93ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.6 +17.9 -9.2 +2.9 +13.3 +0.0 -13.8 -0.4 -1.7 +14.4 +8.7
Relative (%) -12.7 +49.5 -25.5 +8.1 +36.7 +0.0 -38.2 -1.1 -4.6 +39.8 +24.0
Steps
(reduced) 		33
(33) 		53
(53) 		66
(66) 		77
(77) 		86
(86) 		93
(0) 		99
(6) 		105
(12) 		110
(17) 		115
(22) 		119
(26)
Approximation of harmonics in 93ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +15.0 -4.6 -15.4 +17.8 -14.7 -5.0 +10.1 -6.3 +17.9 +9.8 +5.3 +4.1
Relative (%) +41.5 -12.7 -42.5 +49.1 -40.6 -13.8 +27.8 -17.4 +49.5 +27.1 +14.7 +11.3
Steps
(reduced) 		123
(30) 		126
(33) 		129
(36) 		133
(40) 		135
(42) 		138
(45) 		141
(48) 		143
(50) 		146
(53) 		148
(55) 		150
(57) 		152
(59)
77ed5
  • Octave size: 1204.1 ¢

Compressing the octave of 33edo by around 6 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 77ed5 does this. So does the tuning 139zpi whose octave differs by only 0.2 ¢.

Approximation of harmonics in 77ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.9 +15.9 -11.7 +0.0 +10.0 -3.5 -17.6 -4.4 -5.9 +10.1 +4.2
Relative (%) -16.2 +43.9 -32.4 +0.0 +27.7 -9.8 -48.6 -12.1 -16.2 +27.8 +11.5
Steps
(reduced) 		33
(33) 		53
(53) 		66
(66) 		77
(0) 		86
(9) 		93
(16) 		99
(22) 		105
(28) 		110
(33) 		115
(38) 		119
(42)
Approximation of harmonics in 77ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +10.3 -9.4 +15.9 +12.7 +16.3 -10.3 +4.7 -11.7 +12.4 +4.2 -0.4 -1.7
Relative (%) +28.6 -26.0 +43.9 +35.2 +45.1 -28.3 +13.0 -32.4 +34.2 +11.6 -1.1 -4.7
Steps
(reduced) 		123
(46) 		126
(49) 		130
(53) 		133
(56) 		136
(59) 		138
(61) 		141
(64) 		143
(66) 		146
(69) 		148
(71) 		150
(73) 		152
(75)
115ed11
  • Octave size: 1191.2 ¢

Compressing the octave of 33edo by around 9 ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 115ed11 does this. So do the tunings 123ed13 and 1ed47/46 whose octaves are within 0.3 ¢ of 115ed11.

Approximation of harmonics in 115ed11
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -8.8 +11.3 -17.5 -6.7 +2.5 -11.7 +9.8 -13.6 -15.5 +0.0 -6.2
Relative (%) -24.2 +31.2 -48.5 -18.7 +7.0 -32.3 +27.3 -37.6 -42.9 +0.0 -17.3
Steps
(reduced) 		33
(33) 		53
(53) 		66
(66) 		77
(77) 		86
(86) 		93
(93) 		100
(100) 		105
(105) 		110
(110) 		115
(0) 		119
(4)
Approximation of harmonics in 115ed11 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.4 +15.7 +4.5 +1.1 +4.4 +13.8 -7.6 +11.9 -0.4 -8.8 -13.5 -15.0
Relative (%) -1.2 +43.4 +12.5 +3.0 +12.3 +38.1 -21.2 +32.8 -1.1 -24.2 -37.4 -41.5
Steps
(reduced) 		123
(8) 		127
(12) 		130
(15) 		133
(18) 		136
(21) 		139
(24) 		141
(26) 		144
(29) 		146
(31) 		148
(33) 		150
(35) 		152
(37)

Title2

Lab

Place holder








Approximation of prime harmonics in 1ed300c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0 -102 -86 -69 +49 +59 -105 +2 -28 -130 +55
Relative (%) +0.0 -34.0 -28.8 -22.9 +16.2 +19.8 -35.0 +0.8 -9.4 -43.2 +18.3
Step 4 6 9 11 14 15 16 17 18 19 20


Approximation of prime harmonics in 140ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.6 +3.2 +10.0 +11.3 -3.0 +15.1 +11.6 +3.4 +10.6 +8.8 -14.5
Relative (%) -5.2 +10.4 +32.4 +36.7 -9.8 +49.0 +37.6 +11.0 +34.6 +28.6 -47.1
Steps
(reduced) 		39
(39) 		62
(62) 		91
(91) 		110
(110) 		135
(135) 		145
(5) 		160
(20) 		166
(26) 		177
(37) 		190
(50) 		193
(53)

Possible tunings to be used on each page

You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.

(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)

High-priority

39edo

  • 171zpi (30.973c) (optimised for dual-fifths use)
  • 13-limit WE (30.757c) (octave of 135ed11 differs by only 0.2 ¢)
  • 101ed6 (octave of 172zpi differs by only 0.4 ¢)
  • 173zpi (30.672c) (octave of 62edt differs by only 0.2 ¢)
  • 110ed7 (octave of 145ed13 differs by only 0.1 ¢)
  • 91ed5

45edo

  • 209zpi (26.550)
  • 13-limit WE (26.695c)
  • 161ed12
  • 116ed6 (octave identical to 126ed7 within 0.1 ¢)
  • 7-limit WE (26.745c)
  • 207zpi (26.762)
  • 71edt (octave identical to 155ed11 within 0.3 ¢)

54edo

  • 139ed6 (octave is identical to 262zpi within 0.2 ¢)
  • 151ed7
  • 193ed12
  • 263zpi (22.243c)
  • 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1 ¢)
  • 264zpi (22.175c) (octave is identical to 194ed12 within 0.01 ¢)
  • 152ed7
  • 140ed6
  • 126ed5 (octave is identical to 86edt within 0.1 ¢)

64edo

  • 179ed7 (octave is identical to 326zpi within 0.3 ¢)
  • 165ed6
  • 229ed12 (octave is identical to 221ed11 within 0.1 ¢)
  • 327zpi (18.767c)
  • 11-limit WE (18.755c)

pure octaves 64edo (octave is identical to 13-limit WE within 0.13 ¢

  • 328zpi (18.721c)
  • 180ed7
  • 230ed12
  • 149ed5

33edo (reduce # of edonoi)

  • 76ed5
  • 92ed7 (137zpi's octave differs by only 0.3 ¢)
  • 52ed13
  • 114ed11
  • 138zpi (36.394c) (122ed13's octave differs by only 0.1 ¢)
  • 13-limit WE (36.357c)
  • 93ed7 (optimised for dual-fifths)
  • 77ed5 (139zpi's octave differs by only 0.2 ¢)
  • 123ed13 / 1ed47/46 (identical within <0.1 ¢)
  • 115ed11

42edo (reduce # of edonoi)

  • 108ed6 (octave is identical to 97ed5 within 0.1 ¢)
  • 189zpi (28.689c)
  • 150ed12
  • 145ed11

190zpi's octave is within 0.05 ¢ of pure-octaves 42edo

  • 118ed7
  • 13-limit WE (28.534c)
  • 151ed12 (octave is identical to 7-limit WE within 0.3 ¢)
  • 109ed6
  • 191zpi (28.444c)
  • 67edt

59edo (reduce # of edonoi or zpi)

  • 152ed6
  • 294zpi (20.399c)
  • 211ed12
  • 295zpi (20.342c)

pure octaves 59edo octave is identical to 137ed5 within 0.05 ¢

  • 13-limit WE (20.320c)
  • 7-limit WE (20.301c)
  • 166ed7
  • 212ed12
  • 296zpi (20.282c)
  • 153ed6
Medium priority

118edo (choose ZPIS)

Approximation of harmonics in 118edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -0.26 +0.00 +0.13 -0.26 -2.72 +0.00 -0.52 +0.13 -2.17 -0.26 +3.54
Relative (%) +0.0 -2.6 +0.0 +1.2 -2.6 -26.8 +0.0 -5.1 +1.2 -21.3 -2.6 +34.8
Steps
(reduced) 		118
(0) 		187
(69) 		236
(0) 		274
(38) 		305
(69) 		331
(95) 		354
(0) 		374
(20) 		392
(38) 		408
(54) 		423
(69) 		437
(83)
  • 187edt
  • 69edf
  • 13-limit WE (10.171c)
  • Best nearby ZPI(s)

13edo

Approximation of harmonics in 13edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +36.5 +0.0 -17.1 +36.5 -45.7 +0.0 -19.3 -17.1 +2.5 +36.5 -9.8
Relative (%) +0.0 +39.5 +0.0 -18.5 +39.5 -49.6 +0.0 -20.9 -18.5 +2.7 +39.5 -10.6
Steps
(reduced) 		13
(0) 		21
(8) 		26
(0) 		30
(4) 		34
(8) 		36
(10) 		39
(0) 		41
(2) 		43
(4) 		45
(6) 		47
(8) 		48
(9)
  • Main: "13edo and optimal octave stretching"
  • 2.5.11.13 WE (92.483c)
  • 2.5.7.13 WE (92.804c)
  • 2.3 WE (91.405c) (good for opposite 7 mapping)
  • 38zpi (92.531c)

103edo (narrow down edonoi, choose ZPIS)

Approximation of harmonics in 103edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -2.93 +0.00 -1.85 -2.93 -1.84 +0.00 +5.80 -1.85 -3.75 -2.93 -1.69
Relative (%) +0.0 -25.1 +0.0 -15.9 -25.1 -15.8 +0.0 +49.8 -15.9 -32.1 -25.1 -14.5
Steps
(reduced) 		103
(0) 		163
(60) 		206
(0) 		239
(33) 		266
(60) 		289
(83) 		309
(0) 		327
(18) 		342
(33) 		356
(47) 		369
(60) 		381
(72)
  • 163edt
  • 239ed5
  • 266ed6
  • 289ed7
  • 356ed11
  • 369ed12
  • 381ed13
  • 421ed17
  • 466ed23
  • 13-limit WE (11.658c)
  • Best nearby ZPI(s)

111edo (choose ZPIS)

Approximation of harmonics in 111edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 +0.75 +0.00 +2.88 +0.75 +4.15 +0.00 +1.50 +2.88 +0.03 +0.75 +2.72
Relative (%) +0.0 +6.9 +0.0 +26.6 +6.9 +38.4 +0.0 +13.8 +26.6 +0.3 +6.9 +25.1
Steps
(reduced) 		111
(0) 		176
(65) 		222
(0) 		258
(36) 		287
(65) 		312
(90) 		333
(0) 		352
(19) 		369
(36) 		384
(51) 		398
(65) 		411
(78)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
Low priority

104edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

125edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

145edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

152edo

  • 241edt
  • 13-limit WE (7.894c)
  • Best nearby ZPI(s)

159edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

166edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

182edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

198edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

212edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

243edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

247edo

  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
Optional

25edo

Approximation of harmonics in 25edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -2.3 +18.0 -8.8 +0.0 -11.9 -2.3 -23.3 +18.0 +23.5
Relative (%) +0.0 +37.6 +0.0 -4.8 +37.6 -18.4 +0.0 -24.8 -4.8 -48.6 +37.6 +48.9
Steps
(reduced) 		25
(0) 		40
(15) 		50
(0) 		58
(8) 		65
(15) 		70
(20) 		75
(0) 		79
(4) 		83
(8) 		86
(11) 		90
(15) 		93
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

26edo

Approximation of harmonics in 26edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -9.6 +0.0 -17.1 -9.6 +0.4 +0.0 -19.3 -17.1 +2.5 -9.6 -9.8
Relative (%) +0.0 -20.9 +0.0 -37.0 -20.9 +0.9 +0.0 -41.8 -37.0 +5.5 -20.9 -21.1
Steps
(reduced) 		26
(0) 		41
(15) 		52
(0) 		60
(8) 		67
(15) 		73
(21) 		78
(0) 		82
(4) 		86
(8) 		90
(12) 		93
(15) 		96
(18)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

29edo

Approximation of harmonics in 29edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +1.5 +0.0 -13.9 +1.5 -17.1 +0.0 +3.0 -13.9 -13.4 +1.5 -12.9
Relative (%) +0.0 +3.6 +0.0 -33.6 +3.6 -41.3 +0.0 +7.2 -33.6 -32.4 +3.6 -31.3
Steps
(reduced) 		29
(0) 		46
(17) 		58
(0) 		67
(9) 		75
(17) 		81
(23) 		87
(0) 		92
(5) 		96
(9) 		100
(13) 		104
(17) 		107
(20)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

30edo

Approximation of harmonics in 30edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 -3.9 +13.7 +8.7 +18.0 -0.5
Relative (%) +0.0 +45.1 +0.0 +34.2 +45.1 -22.1 +0.0 -9.8 +34.2 +21.7 +45.1 -1.3
Steps
(reduced) 		30
(0) 		48
(18) 		60
(0) 		70
(10) 		78
(18) 		84
(24) 		90
(0) 		95
(5) 		100
(10) 		104
(14) 		108
(18) 		111
(21)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

34edo

Approximation of harmonics in 34edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +3.9 +0.0 +1.9 +3.9 -15.9 +0.0 +7.9 +1.9 +13.4 +3.9 +6.5
Relative (%) +0.0 +11.1 +0.0 +5.4 +11.1 -45.0 +0.0 +22.3 +5.4 +37.9 +11.1 +18.5
Steps
(reduced) 		34
(0) 		54
(20) 		68
(0) 		79
(11) 		88
(20) 		95
(27) 		102
(0) 		108
(6) 		113
(11) 		118
(16) 		122
(20) 		126
(24)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

35edo

Approximation of harmonics in 35edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -16.2 +0.0 -9.2 -16.2 -8.8 +0.0 +1.8 -9.2 -2.7 -16.2 +16.6
Relative (%) +0.0 -47.4 +0.0 -26.7 -47.4 -25.7 +0.0 +5.3 -26.7 -8.0 -47.4 +48.5
Steps
(reduced) 		35
(0) 		55
(20) 		70
(0) 		81
(11) 		90
(20) 		98
(28) 		105
(0) 		111
(6) 		116
(11) 		121
(16) 		125
(20) 		130
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

36edo

Approximation of harmonics in 36edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -2.2 +0.0 -3.9 +13.7 +15.3 -2.0 -7.2
Relative (%) +0.0 -5.9 +0.0 +41.1 -5.9 -6.5 +0.0 -11.7 +41.1 +46.0 -5.9 -21.6
Steps
(reduced) 		36
(0) 		57
(21) 		72
(0) 		84
(12) 		93
(21) 		101
(29) 		108
(0) 		114
(6) 		120
(12) 		125
(17) 		129
(21) 		133
(25)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

37edo

Approximation of harmonics in 37edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +11.6 +0.0 +2.9 +11.6 +4.1 +0.0 -9.3 +2.9 +0.0 +11.6 +2.7
Relative (%) +0.0 +35.6 +0.0 +8.9 +35.6 +12.8 +0.0 -28.7 +8.9 +0.1 +35.6 +8.4
Steps
(reduced) 		37
(0) 		59
(22) 		74
(0) 		86
(12) 		96
(22) 		104
(30) 		111
(0) 		117
(6) 		123
(12) 		128
(17) 		133
(22) 		137
(26)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

9edo

Approximation of harmonics in 9edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -35.3 +0.0 +13.7 -35.3 -35.5 +0.0 +62.8 +13.7 -18.0 -35.3 -40.5
Relative (%) +0.0 -26.5 +0.0 +10.3 -26.5 -26.6 +0.0 +47.1 +10.3 -13.5 -26.5 -30.4
Steps
(reduced) 		9
(0) 		14
(5) 		18
(0) 		21
(3) 		23
(5) 		25
(7) 		27
(0) 		29
(2) 		30
(3) 		31
(4) 		32
(5) 		33
(6)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

10edo

Approximation of harmonics in 10edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -26.3 +18.0 -8.8 +0.0 +36.1 -26.3 +48.7 +18.0 -0.5
Relative (%) +0.0 +15.0 +0.0 -21.9 +15.0 -7.4 +0.0 +30.1 -21.9 +40.6 +15.0 -0.4
Steps
(reduced) 		10
(0) 		16
(6) 		20
(0) 		23
(3) 		26
(6) 		28
(8) 		30
(0) 		32
(2) 		33
(3) 		35
(5) 		36
(6) 		37
(7)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

11edo

Approximation of harmonics in 11edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -47.4 +0.0 +50.0 -47.4 +13.0 +0.0 +14.3 +50.0 -5.9 -47.4 +32.2
Relative (%) +0.0 -43.5 +0.0 +45.9 -43.5 +11.9 +0.0 +13.1 +45.9 -5.4 -43.5 +29.5
Steps
(reduced) 		11
(0) 		17
(6) 		22
(0) 		26
(4) 		28
(6) 		31
(9) 		33
(0) 		35
(2) 		37
(4) 		38
(5) 		39
(6) 		41
(8)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

15edo

Approximation of harmonics in 15edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 +13.7 +18.0 -8.8 +0.0 +36.1 +13.7 +8.7 +18.0 +39.5
Relative (%) +0.0 +22.6 +0.0 +17.1 +22.6 -11.0 +0.0 +45.1 +17.1 +10.9 +22.6 +49.3
Steps
(reduced) 		15
(0) 		24
(9) 		30
(0) 		35
(5) 		39
(9) 		42
(12) 		45
(0) 		48
(3) 		50
(5) 		52
(7) 		54
(9) 		56
(11)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

18edo

Approximation of harmonics in 18edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +31.4 +0.0 +13.7 +31.4 +31.2 +0.0 -3.9 +13.7 -18.0 +31.4 +26.1
Relative (%) +0.0 +47.1 +0.0 +20.5 +47.1 +46.8 +0.0 -5.9 +20.5 -27.0 +47.1 +39.2
Steps
(reduced) 		18
(0) 		29
(11) 		36
(0) 		42
(6) 		47
(11) 		51
(15) 		54
(0) 		57
(3) 		60
(6) 		62
(8) 		65
(11) 		67
(13)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

48edo

Approximation of harmonics in 48edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 -11.3 -2.0 +6.2 +0.0 -3.9 -11.3 -1.3 -2.0 +9.5
Relative (%) +0.0 -7.8 +0.0 -45.3 -7.8 +24.7 +0.0 -15.6 -45.3 -5.3 -7.8 +37.9
Steps
(reduced) 		48
(0) 		76
(28) 		96
(0) 		111
(15) 		124
(28) 		135
(39) 		144
(0) 		152
(8) 		159
(15) 		166
(22) 		172
(28) 		178
(34)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

5edo

Approximation of harmonics in 5edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0 +18 +0 +94 +18 -9 +0 +36 +94 -71 +18 +119
Relative (%) +0.0 +7.5 +0.0 +39.0 +7.5 -3.7 +0.0 +15.0 +39.0 -29.7 +7.5 +49.8
Steps
(reduced) 		5
(0) 		8
(3) 		10
(0) 		12
(2) 		13
(3) 		14
(4) 		15
(0) 		16
(1) 		17
(2) 		17
(2) 		18
(3) 		19
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

6edo

Approximation of harmonics in 6edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +98.0 +0.0 +13.7 +98.0 +31.2 +0.0 -3.9 +13.7 +48.7 +98.0 -40.5
Relative (%) +0.0 +49.0 +0.0 +6.8 +49.0 +15.6 +0.0 -2.0 +6.8 +24.3 +49.0 -20.3
Steps
(reduced) 		6
(0) 		10
(4) 		12
(0) 		14
(2) 		16
(4) 		17
(5) 		18
(0) 		19
(1) 		20
(2) 		21
(3) 		22
(4) 		22
(4)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

20edo

Approximation of harmonics in 20edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +18.0 +0.0 -26.3 +18.0 -8.8 +0.0 -23.9 -26.3 -11.3 +18.0 -0.5
Relative (%) +0.0 +30.1 +0.0 -43.9 +30.1 -14.7 +0.0 -39.9 -43.9 -18.9 +30.1 -0.9
Steps
(reduced) 		20
(0) 		32
(12) 		40
(0) 		46
(6) 		52
(12) 		56
(16) 		60
(0) 		63
(3) 		66
(6) 		69
(9) 		72
(12) 		74
(14)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

24edo

Approximation of harmonics in 24edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -2.0 +0.0 +13.7 -2.0 -18.8 +0.0 -3.9 +13.7 -1.3 -2.0 +9.5
Relative (%) +0.0 -3.9 +0.0 +27.4 -3.9 -37.7 +0.0 -7.8 +27.4 -2.6 -3.9 +18.9
Steps
(reduced) 		24
(0) 		38
(14) 		48
(0) 		56
(8) 		62
(14) 		67
(19) 		72
(0) 		76
(4) 		80
(8) 		83
(11) 		86
(14) 		89
(17)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)

28edo

Approximation of harmonics in 28edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -16.2 +0.0 -0.6 -16.2 +16.9 +0.0 +10.4 -0.6 +5.8 -16.2 +16.6
Relative (%) +0.0 -37.9 +0.0 -1.4 -37.9 +39.4 +0.0 +24.2 -1.4 +13.6 -37.9 +38.8
Steps
(reduced) 		28
(0) 		44
(16) 		56
(0) 		65
(9) 		72
(16) 		79
(23) 		84
(0) 		89
(5) 		93
(9) 		97
(13) 		100
(16) 		104
(20)
  • Nearby edt, ed6, ed12 and/or edf
  • Nearby ed5, ed10, ed7 and/or ed11 (optional)
  • 1-2 WE tunings
  • Best nearby ZPI(s)
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