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Title1

Octave stretch or compression

54edo’s approximations of 3/1, 5/1, 7/1, 11/1, 13/1, 17/1, 19/1 and 23/1 are all improved by 139ed6, a stretched-octave version of 54edo. The trade-off is a slightly worse 2/1 and 19/1.

If one prefers a compressed-octave tuning instead, 86edt, 126ed5 and 152ed7 are possible choices. They improve upon 54edo’s 3/1, 5/1, 7/1 and 17/1, at the cost of its 2/1, 11/1 and 13/1.

40ed5/3 is another compressed octave option. It improves upon 54edo’s 3/1, 5/1, 11/1, 13/1, 17/1 and 19/1, at slight cost to the 2/1 and 7/1. Its 2/1 is the least accurate of all the tunings mentioned in this section, though still accurate enough that it has low harmonic entropy.

What follows is a comparison of stretched- and compressed-octave 54edo tunings.

139ed6
  • Octave size: 1205.08 ¢

Stretching the octave of 54edo by around 5 ¢ results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 10.15 ¢. The tuning 139ed6 does this. So does the tuning 262zpi whose octave is identical to 139ed6 within 0.2 ¢.

Approximation of harmonics in 139ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +5.08 -5.08 +10.15 +3.21 +0.00 +0.92 -7.09 -10.15 +8.29 -0.50 +5.08
Relative (%) +22.7 -22.7 +45.5 +14.4 +0.0 +4.1 -31.8 -45.5 +37.1 -2.2 +22.7
Steps
(reduced) 		54
(54) 		85
(85) 		108
(108) 		125
(125) 		139
(0) 		151
(12) 		161
(22) 		170
(31) 		179
(40) 		186
(47) 		193
(54)
Approximation of harmonics in 139ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.40 +6.00 -1.86 -2.01 +4.61 -5.08 -9.41 -8.95 -4.15 +4.58 -5.43 +10.15
Relative (%) +1.8 +26.9 -8.3 -9.0 +20.7 -22.7 -42.2 -40.1 -18.6 +20.5 -24.3 +45.5
Steps
(reduced) 		199
(60) 		205
(66) 		210
(71) 		215
(76) 		220
(81) 		224
(85) 		228
(89) 		232
(93) 		236
(97) 		240
(101) 		243
(104) 		247
(108)
151ed7
  • Octave size: 1204.75 ¢

Stretching the octave of 54edo by around 4.5 ¢ results in improved primes 3, 5, 7, 11, 13 and 17, but a worse prime 2. This approximates all harmonics up to 16 within 11.12 ¢. The tuning 151ed7 does this.

Approximation of harmonics in 151ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.75 -5.60 +9.49 +2.45 -0.85 +0.00 -8.07 +11.12 +7.20 -1.64 +3.90
Relative (%) +21.3 -25.1 +42.5 +11.0 -3.8 +0.0 -36.2 +49.8 +32.3 -7.3 +17.5
Steps
(reduced) 		54
(54) 		85
(85) 		108
(108) 		125
(125) 		139
(139) 		151
(0) 		161
(10) 		171
(20) 		179
(28) 		186
(35) 		193
(42)
Approximation of harmonics in 151ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.82 +4.75 -3.15 -3.33 +3.27 -6.45 -10.81 -10.37 -5.60 +3.11 -6.92 +8.64
Relative (%) -3.7 +21.3 -14.1 -14.9 +14.6 -28.9 -48.4 -46.5 -25.1 +13.9 -31.0 +38.7
Steps
(reduced) 		199
(48) 		205
(54) 		210
(59) 		215
(64) 		220
(69) 		224
(73) 		228
(77) 		232
(81) 		236
(85) 		240
(89) 		243
(92) 		247
(96)
193ed12
  • Octave size: 1203.66 ¢

Stretching the octave of 54edo by around 3.5 ¢ results in improved primes 3, 5, 7, 11 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 10.97 ¢. The tuning 193ed12 does this.

Approximation of harmonics in 193ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +3.66 -7.31 +7.31 -0.07 -3.66 -3.05 +10.97 +7.67 +3.58 -5.39 +0.00
Relative (%) +16.4 -32.8 +32.8 -0.3 -16.4 -13.7 +49.2 +34.4 +16.1 -24.2 +0.0
Steps
(reduced) 		54
(54) 		85
(85) 		108
(108) 		125
(125) 		139
(139) 		151
(151) 		162
(162) 		171
(171) 		179
(179) 		186
(186) 		193
(0)
Approximation of harmonics in 193ed12 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -4.83 +0.61 -7.39 -7.67 -1.17 -10.97 +6.88 +7.24 -10.36 -1.74 +10.47 +3.66
Relative (%) -21.7 +2.7 -33.1 -34.4 -5.3 -49.2 +30.9 +32.5 -46.5 -7.8 +47.0 +16.4
Steps
(reduced) 		199
(6) 		205
(12) 		210
(17) 		215
(22) 		220
(27) 		224
(31) 		229
(36) 		233
(40) 		236
(43) 		240
(47) 		244
(51) 		247
(54)
263zpi
  • Step size: 22.243 ¢, octave size: 1201.12 ¢

Stretching the octave of 54edo by around 1 ¢ results in an improved prime 5, but worse primes 2, 3, 7, 11 and 13. This approximates all harmonics up to 16 within 10.94 ¢. The tuning 263zpi does this.

Approximation of harmonics in 263zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +1.12 +10.94 +2.24 -5.94 -10.18 -10.13 +3.37 -0.36 -4.82 +8.12 -9.06
Relative (%) +5.0 +49.2 +10.1 -26.7 -45.8 -45.6 +15.1 -1.6 -21.7 +36.5 -40.7
Step 54 86 108 125 139 151 162 171 179 187 193
Approximation of harmonics in 263zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +8.07 -9.01 +5.00 +4.49 +10.75 +0.76 -3.87 -3.69 +0.81 +9.25 -0.98 -7.93
Relative (%) +36.3 -40.5 +22.5 +20.2 +48.3 +3.4 -17.4 -16.6 +3.6 +41.6 -4.4 -35.7
Step 200 205 211 216 221 225 229 233 237 241 244 247
54edo
  • Step size: 22.222 ¢, octave size: 1200.00 ¢

Pure-octaves 54edo approximates all harmonics up to 16 within 9.16 ¢.

Approximation of harmonics in 54edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.00 +9.16 +0.00 -8.54 +9.16 +8.95 +0.00 -3.91 -8.54 +4.24 +9.16
Relative (%) +0.0 +41.2 +0.0 -38.4 +41.2 +40.3 +0.0 -17.6 -38.4 +19.1 +41.2
Steps
(reduced) 		54
(0) 		86
(32) 		108
(0) 		125
(17) 		140
(32) 		152
(44) 		162
(0) 		171
(9) 		179
(17) 		187
(25) 		194
(32)
Approximation of harmonics in 54edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.92 +8.95 +0.62 +0.00 +6.16 -3.91 -8.62 -8.54 -4.11 +4.24 -6.05 +9.16
Relative (%) +17.6 +40.3 +2.8 +0.0 +27.7 -17.6 -38.8 -38.4 -18.5 +19.1 -27.2 +41.2
Steps
(reduced) 		200
(38) 		206
(44) 		211
(49) 		216
(0) 		221
(5) 		225
(9) 		229
(13) 		233
(17) 		237
(21) 		241
(25) 		244
(28) 		248
(32)
54et, 13-limit WE tuning
  • Step size: 22.198 ¢, octave size: 1198.69 ¢

Compressing the octave of 54edo by around 1.5 ¢ results in improved primes 3, 7, 11, 13, 17 and 19, but worse primes 2 and 5. This approximates all harmonics up to 16 within 10.63 ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this. So does the tuning 187ed11 whose octave is identical to 13lim WE within 0.1 ¢.

Approximation of harmonics in 54et, 13-limit WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.31 +7.07 -2.62 +10.63 +5.76 +5.27 -3.92 -8.05 +9.33 -0.29 +4.46
Relative (%) -5.9 +31.9 -11.8 +47.9 +26.0 +23.7 -17.7 -36.3 +42.0 -1.3 +20.1
Step 54 86 108 126 140 152 162 171 180 187 194
Approximation of harmonics in 54et, 13-limit WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.93 +3.96 -4.49 -5.23 +0.80 -9.36 +8.03 +8.02 -9.85 -1.60 +10.24 +3.15
Relative (%) -4.2 +17.8 -20.2 -23.6 +3.6 -42.2 +36.2 +36.1 -44.4 -7.2 +46.1 +14.2
Step 200 206 211 216 221 225 230 234 237 241 245 248
264zpi
  • Step size: 22.175 ¢, octave size: 1197.45 ¢

Compressing the octave of 54edo by around 2.5 ¢ results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.19 ¢. The tuning 264zpi does this. So does the tuning 194ed12 whose octave is identical to 264zpi within 0.01 ¢.

Approximation of harmonics in 264zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -2.55 +5.09 -5.10 +7.74 +2.54 +1.77 -7.65 +10.19 +5.19 -4.59 -0.01
Relative (%) -11.5 +23.0 -23.0 +34.9 +11.5 +8.0 -34.5 +46.0 +23.4 -20.7 -0.0
Step 54 86 108 126 140 152 162 172 180 187 194
Approximation of harmonics in 264zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.53 -0.78 -9.34 -10.20 -4.28 +7.64 +2.74 +2.64 +6.87 -7.14 +4.60 -2.56
Relative (%) -24.9 -3.5 -42.1 -46.0 -19.3 +34.5 +12.3 +11.9 +31.0 -32.2 +20.7 -11.5
Step 200 206 211 216 221 226 230 234 238 241 245 248
152ed7
  • Octave size: 1196.82 ¢

Compressing the octave of 54edo by around 3 ¢ results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.36 ¢. The tuning 152ed7 does this.

Approximation of harmonics in 152ed7
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.18 +4.09 -6.36 +6.27 +0.91 +0.00 -9.54 +8.18 +3.09 -6.78 -2.27
Relative (%) -14.3 +18.5 -28.7 +28.3 +4.1 +0.0 -43.0 +36.9 +13.9 -30.6 -10.2
Steps
(reduced) 		54
(54) 		86
(86) 		108
(108) 		126
(126) 		140
(140) 		152
(0) 		162
(10) 		172
(20) 		180
(28) 		187
(35) 		194
(42)
Approximation of harmonics in 152ed7 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -7.86 -3.18 +10.36 +9.44 -6.86 +5.00 +0.05 -0.09 +4.09 -9.96 +1.74 -5.45
Relative (%) -35.5 -14.3 +46.7 +42.6 -31.0 +22.6 +0.2 -0.4 +18.5 -44.9 +7.9 -24.6
Steps
(reduced) 		200
(48) 		206
(54) 		212
(60) 		217
(65) 		221
(69) 		226
(74) 		230
(78) 		234
(82) 		238
(86) 		241
(89) 		245
(93) 		248
(96)
140ed6
  • Octave size: 1196.47 ¢

Compressing the octave of 54edo by around 3.5 ¢ results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.59 ¢. The tuning 140ed6 does this.

Approximation of harmonics in 140ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.53 +3.53 -7.06 +5.45 +0.00 -0.99 -10.59 +7.06 +1.91 -7.99 -3.53
Relative (%) -15.9 +15.9 -31.9 +24.6 +0.0 -4.5 -47.8 +31.9 +8.6 -36.1 -15.9
Steps
(reduced) 		54
(54) 		86
(86) 		108
(108) 		126
(126) 		140
(0) 		152
(12) 		162
(22) 		172
(32) 		180
(40) 		187
(47) 		194
(54)
Approximation of harmonics in 140ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -9.16 -4.52 +8.98 +8.03 -8.30 +3.53 -1.44 -1.62 +2.54 +10.63 +0.15 -7.06
Relative (%) -41.4 -20.4 +40.5 +36.2 -37.5 +15.9 -6.5 -7.3 +11.5 +48.0 +0.7 -31.9
Steps
(reduced) 		200
(60) 		206
(66) 		212
(72) 		217
(77) 		221
(81) 		226
(86) 		230
(90) 		234
(94) 		238
(98) 		242
(102) 		245
(105) 		248
(108)
126ed5
  • Octave size: 1194.13 ¢

Compressing the octave of 54edo by around 6 ¢ results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20 ¢. The tuning 126ed5 does this. So does the tuning 86edt whose octave is identical to 126ed5 within 0.1 ¢.

Approximation of harmonics in 126ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.87 -0.19 +10.38 +0.00 -6.05 -7.56 +4.52 -0.37 -5.87 +6.04 +10.20
Relative (%) -26.5 -0.8 +47.0 +0.0 -27.4 -34.2 +20.4 -1.7 -26.5 +27.3 +46.1
Steps
(reduced) 		54
(54) 		86
(86) 		109
(109) 		126
(0) 		140
(14) 		152
(26) 		163
(37) 		172
(46) 		180
(54) 		188
(62) 		195
(69)
Approximation of harmonics in 126ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +4.31 +8.69 -0.19 -1.35 +4.26 -6.24 +10.73 +10.38 -7.74 +0.17 -10.44 +4.33
Relative (%) +19.5 +39.3 -0.8 -6.1 +19.3 -28.2 +48.5 +47.0 -35.0 +0.8 -47.2 +19.6
Steps
(reduced) 		201
(75) 		207
(81) 		212
(86) 		217
(91) 		222
(96) 		226
(100) 		231
(105) 		235
(109) 		238
(112) 		242
(116) 		245
(119) 		249
(123)
40ed5/3
  • Octave size: 1194.13 ¢

Compressing the octave of 54edo by around 6 ¢ results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 10.20 ¢. The tuning 126ed5 does this. So does the tuning 86edt whose octave is identical to 126ed5 within 0.1 ¢.

Approximation of harmonics in 126ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6.12 -0.58 +9.88 -0.58 -6.70 -8.26 +3.76 -1.17 -6.70 +5.17 +9.29
Relative (%) -27.7 -2.6 +44.7 -2.6 -30.3 -37.4 +17.0 -5.3 -30.3 +23.4 +42.0
Steps
(reduced) 		54
(14) 		86
(6) 		109
(29) 		126
(6) 		140
(20) 		152
(32) 		163
(3) 		172
(12) 		180
(20) 		188
(28) 		195
(35)
Approximation of harmonics in 126ed5 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +3.37 +7.73 -1.17 -2.35 +3.24 -7.28 +9.66 +9.29 -8.85 -0.95 +10.53 +3.18
Relative (%) +15.3 +35.0 -5.3 -10.6 +14.6 -32.9 +43.7 +42.0 -40.0 -4.3 +47.6 +14.4
Steps
(reduced) 		201
(1) 		207
(7) 		212
(12) 		217
(17) 		222
(22) 		226
(26) 		231
(31) 		235
(35) 		238
(38) 		242
(2) 		246
(6) 		249
(9)

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

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

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

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)

38edo

Approximation of harmonics in 38edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 +10.1 +0.0 -14.4 -7.4 -14.5 -7.2 +12.1
Relative (%) +0.0 -22.9 +0.0 -23.3 -22.9 +32.1 +0.0 -45.7 -23.3 -45.8 -22.9 +38.3
Steps
(reduced) 		38
(0) 		60
(22) 		76
(0) 		88
(12) 		98
(22) 		107
(31) 		114
(0) 		120
(6) 		126
(12) 		131
(17) 		136
(22) 		141
(27)
  • 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)

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)

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)

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)

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)

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)
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