User:BudjarnLambeth/Sandbox2: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,453 edits
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,453 edits
Line 79: Line 79:


54edo (possibly narrow down edonoi)
54edo (possibly narrow down edonoi)
* 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
{{harmonics in equal | 38 | 5 | 3 | intervals=prime}}
* 85edt (octave is identical to 261zpi within 0.2{{c}})
{{harmonics in equal | 85 | 3 | 1 | intervals=prime}}
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
* 139ed6 (octave is identical to 262zpi within 0.2{{c}})
{{harmonics in equal | 139 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 139 | 6 | 1 | intervals=prime}}
Line 91: Line 87:
* 263zpi (22.243c)
* 263zpi (22.243c)
{{harmonics in cet | 22.243 | intervals=prime}}
{{harmonics in cet | 22.243 | intervals=prime}}
* pure octave 54edo
{{harmonics in equal | 54 | 2 | 1 | intervals=integer | columns=12}}
* 13-limit WE (22.198c)  (octave is identical to 187ed11 within 0.1{{c}})
* 13-limit WE (22.198c)  (octave is identical to 187ed11 within 0.1{{c}})
{{harmonics in cet | 22.198 | intervals=prime}}
{{harmonics in cet | 22.198 | intervals=prime}}
Line 111: Line 105:


59edo (narrow down ZPIs) (Nothing special abt these choices)
59edo (narrow down ZPIs) (Nothing special abt these choices)
* 93edt (octave is identical to 293zpi and 203ed11 within 0.2{{c}})
{{harmonics in equal | 93 | 3 | 1 | intervals=prime}}
* 152ed6
* 152ed6
{{harmonics in equal | 152 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 152 | 6 | 1 | intervals=prime}}
Line 122: Line 114:
{{harmonics in cet | 20.342 | intervals=prime}}
{{harmonics in cet | 20.342 | intervals=prime}}
* pure octaves 59edo  (octave is identical to 137ed5 within 0.05{{c}})
* pure octaves 59edo  (octave is identical to 137ed5 within 0.05{{c}})
{{harmonics in equal | 59 | 2 | 1 | intervals=integer | columns=12}}
* 13-limit WE (20.320c)
* 13-limit WE (20.320c)
{{harmonics in cet | 20.320 | intervals=prime}}
{{harmonics in cet | 20.320 | intervals=prime}}
Line 139: Line 130:
* 205ed11
* 205ed11
{{harmonics in equal | 205 | 11 | 1 | intervals=prime}}
{{harmonics in equal | 205 | 11 | 1 | intervals=prime}}
* 94edt (octave is identical to 297zpi within 0.3{{c}})
{{harmonics in equal | 94 | 3 | 1 | intervals=prime}}


<br><br><br>
<br><br><br>


64edo (narrow down ZPIs)
64edo (narrow down ZPIs)
* 325zpi (18.868c) (octave is identical to 220ed11 within 0.1{{c}})
{{harmonics in cet | 18.868 | intervals=prime}}
{{harmonics in cet | 18.868 | intervals=prime}}
* 101edt
* 101edt
Line 158: Line 146:
{{harmonics in cet | 18.767 | intervals=prime}}
{{harmonics in cet | 18.767 | intervals=prime}}
* 11-limit WE (18.755c)
* 11-limit WE (18.755c)
{{harmonics in cet | 18.755 | intervals=prime}
{{harmonics in cet | 18.755 | intervals=prime}}
* pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}})
* pure octaves 64edo (octave is identical to 13-limit WE within 0.13{{c}})
{{harmonics in equal | 64 | 2 | 1 | intervals=integer | columns=12}}
* 328zpi (18.721c)
* 328zpi (18.721c)
{{harmonics in cet | 18.721 | intervals=prime}}
{{harmonics in cet | 18.721 | intervals=prime}}
Line 171: Line 158:
* 329zpi (18.672c)
* 329zpi (18.672c)
{{harmonics in cet | 18.672 | intervals=prime}}
{{harmonics in cet | 18.672 | intervals=prime}}
* 102edt
{{harmonics in equal | 102 | 3 | 1 | intervals=prime}}
* 330zpi (18.630c)
{{harmonics in cet | 18.630 | intervals=prime}}




Revision as of 09:27, 28 August 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Title1

Octave stretch or compression

Main article: 23edo and octave stretching

23edo is not typically taken seriously as a tuning except by those interested in extreme xenharmony. Its fifths are significantly flat, and is neighbors 22edo and 24edo generally get more attention.

However, when using a slightly stretched octave of around 1216 cents, 23edo looks much better, and it approximates the perfect fifth (and various other intervals involving the 5th, 7th, 11th, and 13th harmonics) to within 18 cents or so. If we can tolerate errors around this size in 12edo, we can probably tolerate them in stretched-23 as well.

Stretched 23edo is one of the best tunings to use for exploring the antidiatonic scale since its fifth is more consonant and less "wolfish" than fifths in other pelogic family temperaments.

What follows is a comparison of stretched- and compressed-octave 23edo tunings.

86zpi
  • Step size: 51.653 ¢, octave size: 1188.0 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. 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 (¢) -12.0 +9.2 -24.0 +2.9 -2.8 -11.4 +15.7 +18.4 -9.0 -19.1 -14.8
Relative (%) -23.2 +17.8 -46.4 +5.7 -5.4 -22.0 +30.4 +35.6 -17.5 -36.9 -28.6
Step 23 37 46 54 60 65 70 74 77 80 83
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.6 -23.4 +12.2 +3.7 +2.1 +6.4 +16.1 -21.0 -2.2 +20.6 -4.7 +24.9
Relative (%) +3.2 -45.2 +23.5 +7.2 +4.0 +12.5 +31.2 -40.7 -4.2 +39.9 -9.1 +48.2
Step 86 88 91 93 95 97 99 100 102 104 105 107
60ed6
  • Step size: 51.700 ¢, octave size: 1189.1 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 60ed6 does this. So does the tuning 105ed23 whose octave is identical within 0.01 ¢.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -10.9 +10.9 -21.8 +5.4 +0.0 -8.4 +18.9 +21.8 -5.5 -15.4 -10.9
Relative (%) -21.1 +21.1 -42.2 +10.5 +0.0 -16.2 +36.6 +42.2 -10.6 -29.7 -21.1
Steps
(reduced) 		23
(23) 		37
(37) 		46
(46) 		54
(54) 		60
(0) 		65
(5) 		70
(10) 		74
(14) 		77
(17) 		80
(20) 		83
(23)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.6 -19.3 +16.4 +8.0 +6.5 +10.9 +20.7 -16.4 +2.5 +25.4 +0.1 -21.8
Relative (%) +10.8 -37.3 +31.7 +15.5 +12.5 +21.1 +40.1 -31.7 +4.9 +49.1 +0.3 -42.2
Steps
(reduced) 		86
(26) 		88
(28) 		91
(31) 		93
(33) 		95
(35) 		97
(37) 		99
(39) 		100
(40) 		102
(42) 		104
(44) 		105
(45) 		106
(46)
85zpi
  • Step size: 52.114 ¢, octave size: 1198.6 ¢

Compressing the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 85zpi does this. So does the tuning 73ed9 whose octave is identical within 0.02 ¢.

Approximation of harmonics in ZPINAME
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.4 -25.9 -2.8 -24.3 +24.9 +18.6 -4.1 +0.4 -25.6 +17.8 +23.5
Relative (%) -2.6 -49.6 -5.3 -46.6 +47.8 +35.7 -7.9 +0.8 -49.2 +34.2 +45.1
Step 23 36 46 53 60 65 69 73 76 80 83
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -10.8 +17.2 +2.0 -5.5 -6.2 -1.0 +9.7 +25.1 -7.3 +16.4 -8.4 +22.1
Relative (%) -20.8 +33.0 +3.8 -10.6 -12.0 -1.9 +18.5 +48.1 -13.9 +31.5 -16.2 +42.5
Step 85 88 90 92 94 96 98 100 101 103 104 106
23edo
  • Step size: NNN ¢, octave size: 1200.0 ¢

Pure-octaves EDONAME 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 (¢) +0.0 -23.7 +0.0 -21.1 -23.7 +22.5 +0.0 +4.8 -21.1 +22.6 -23.7
Relative (%) +0.0 -45.4 +0.0 -40.4 -45.4 +43.1 +0.0 +9.2 -40.4 +43.3 -45.4
Steps
(reduced) 		23
(0) 		36
(13) 		46
(0) 		53
(7) 		59
(13) 		65
(19) 		69
(0) 		73
(4) 		76
(7) 		80
(11) 		82
(13)
Approximation of harmonics in EDONAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -5.7 +22.5 +7.4 +0.0 -0.6 +4.8 +15.5 -21.1 -1.2 +22.6 -2.2 -23.7
Relative (%) -11.0 +43.1 +14.2 +0.0 -1.2 +9.2 +29.8 -40.4 -2.3 +43.3 -4.2 -45.4
Steps
(reduced) 		85
(16) 		88
(19) 		90
(21) 		92
(0) 		94
(2) 		96
(4) 		98
(6) 		99
(7) 		101
(9) 		103
(11) 		104
(12) 		105
(13)
23et, 13-limit WE tuning
  • Step size: 52.237 ¢, octave size: 1201.5 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. 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 (¢) +1.5 -21.4 +2.9 -17.8 -20.0 -25.7 +4.4 +9.4 -16.3 -24.6 -18.5
Relative (%) +2.8 -41.0 +5.6 -34.0 -38.2 -49.1 +8.3 +18.0 -31.2 -47.1 -35.5
Step 23 36 46 53 59 64 69 73 76 79 82
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -0.4 -24.2 +13.1 +5.8 +5.3 +10.8 +21.7 -14.9 +5.2 -23.1 +4.4 -17.1
Relative (%) -0.7 -46.3 +25.0 +11.1 +10.2 +20.8 +41.6 -28.4 +9.9 -44.3 +8.4 -32.7
Step 85 87 90 92 94 96 98 99 101 102 104 105
23et, 2.3.5.13 WE tuning
  • Step size: 52.447 ¢, octave size: 1206.3 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its SUBGROUP WE tuning and SUBGROUP TE tuning both do this. So does the tuning 76ed10 whose octave is identical within 0.01 ¢.

Approximation of harmonics in ETNAME, SUBGROUP WE tuning
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.3 -13.9 +12.6 -6.6 -7.6 -12.2 +18.8 +24.7 -0.3 -8.0 -1.3
Relative (%) +12.0 -26.4 +24.0 -12.6 -14.5 -23.3 +35.9 +47.1 -0.7 -15.3 -2.5
Step 23 36 46 53 59 64 69 73 76 79 82
Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +17.5 -5.9 -20.5 +25.1 +25.1 -21.4 -10.2 +5.9 -26.1 -1.7 +26.2 +5.0
Relative (%) +33.3 -11.3 -39.1 +47.9 +47.8 -40.9 -19.4 +11.3 -49.7 -3.3 +50.0 +9.5
Step 85 87 89 92 94 95 97 99 100 102 104 105
59ed6
  • Step size: 52.575 ¢, octave size: 1209.2 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 59ed6 does this. So does the tuning 53ed5 whose octave is identical within 0.01 ¢.

Approximation of harmonics in EDONOI
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +9.2 -9.2 +18.5 +0.2 +0.0 -4.0 -24.9 -18.5 +9.4 +2.1 +9.2
Relative (%) +17.6 -17.6 +35.1 +0.4 +0.0 -7.6 -47.3 -35.1 +17.9 +4.1 +17.6
Steps
(reduced) 		23
(23) 		36
(36) 		46
(46) 		53
(53) 		59
(0) 		64
(5) 		68
(9) 		72
(13) 		76
(17) 		79
(20) 		82
(23)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -24.2 +5.2 -9.0 -15.6 -15.4 -9.2 +2.3 +18.7 -13.2 +11.4 -13.0 +18.5
Relative (%) -46.0 +10.0 -17.2 -29.7 -29.4 -17.6 +4.4 +35.5 -25.2 +21.7 -24.7 +35.1
Steps
(reduced) 		84
(25) 		87
(28) 		89
(30) 		91
(32) 		93
(34) 		95
(36) 		97
(38) 		99
(40) 		100
(41) 		102
(43) 		103
(44) 		105
(46)
84zpi
  • Step size: 52.615 ¢, octave size: 1210.1 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. 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 (¢) +10.1 -7.8 +20.3 +2.3 +2.3 -1.5 -22.2 -15.6 +12.4 +5.3 +12.5
Relative (%) +19.3 -14.9 +38.6 +4.3 +4.4 -2.8 -42.2 -29.7 +23.6 +10.0 +23.7
Step 23 36 46 53 59 64 68 72 76 79 82
Approximation of harmonics in ZPINAME (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -20.9 +8.7 -5.5 -12.0 -11.8 -5.5 +6.1 +22.6 -9.3 +15.4 -8.9 +22.6
Relative (%) -39.7 +16.5 -10.5 -22.9 -22.4 -10.4 +11.7 +42.9 -17.6 +29.3 -17.0 +43.0
Step 84 87 89 91 93 95 97 99 100 102 103 105
36edt
  • Step size: 52.832 ¢, octave size: 1215.1 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. 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 (¢) +15.1 +0.0 -22.6 +13.8 +15.1 +12.4 -7.4 +0.0 -23.9 +22.4 -22.6
Relative (%) +28.7 +0.0 -42.7 +26.1 +28.7 +23.5 -14.0 +0.0 -45.3 +42.4 -42.7
Steps
(reduced) 		23
(23) 		36
(0) 		45
(9) 		53
(17) 		59
(23) 		64
(28) 		68
(32) 		72
(0) 		75
(3) 		79
(7) 		81
(9)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -2.6 -25.3 +13.8 +7.7 +8.4 +15.1 -25.6 -8.8 +12.4 -15.3 +13.4 -7.4
Relative (%) -5.0 -47.8 +26.1 +14.6 +16.0 +28.7 -48.5 -16.6 +23.5 -28.9 +25.4 -14.0
Steps
(reduced) 		84
(12) 		86
(14) 		89
(17) 		91
(19) 		93
(21) 		95
(23) 		96
(24) 		98
(26) 		100
(28) 		101
(29) 		103
(31) 		104
(32)
84ed13
  • Step size: 52.863 ¢, octave size: 1215.9 ¢

Stretching the octave of EDONAME by around NNN ¢ results in improved primes NNN, but worse primes NNN. 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 (¢) +15.9 +1.1 -21.1 +15.4 +17.0 +14.4 -5.3 +2.3 -21.6 +24.9 -20.0
Relative (%) +30.0 +2.1 -40.0 +29.2 +32.1 +27.3 -10.0 +4.3 -40.8 +47.1 -37.9
Steps
(reduced) 		23
(23) 		36
(36) 		45
(45) 		53
(53) 		59
(59) 		64
(64) 		68
(68) 		72
(72) 		75
(75) 		79
(79) 		81
(81)
Approximation of harmonics in EDONOI (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +0.0 -22.6 +16.6 +10.6 +11.3 +18.1 -22.6 -5.7 +15.6 -12.1 +16.7 -4.2
Relative (%) +0.0 -42.7 +31.4 +20.0 +21.5 +34.3 -42.8 -10.8 +29.4 -22.9 +31.5 -7.9
Steps
(reduced) 		84
(0) 		86
(2) 		89
(5) 		91
(7) 		93
(9) 		95
(11) 		96
(12) 		98
(14) 		100
(16) 		101
(17) 		103
(19) 		104
(20)

Title2

Lab

54edo (possibly narrow down edonoi)

  • 139ed6 (octave is identical to 262zpi within 0.2 ¢)
Approximation of prime harmonics in 139ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +5.08 -5.08 +3.21 +0.92 -0.50 +0.40 +4.61 -9.41 -5.43 -5.04 -8.92
Relative (%) +22.7 -22.7 +14.4 +4.1 -2.2 +1.8 +20.7 -42.2 -24.3 -22.6 -40.0
Steps
(reduced) 		54
(54) 		85
(85) 		125
(125) 		151
(12) 		186
(47) 		199
(60) 		220
(81) 		228
(89) 		243
(104) 		261
(122) 		266
(127)
  • 151ed7
Approximation of prime harmonics in 151ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +4.75 -5.60 +2.45 +0.00 -1.64 -0.82 +3.27 -10.81 -6.92 -6.64 -10.55
Relative (%) +21.3 -25.1 +11.0 +0.0 -7.3 -3.7 +14.6 -48.4 -31.0 -29.8 -47.3
Steps
(reduced) 		54
(54) 		85
(85) 		125
(125) 		151
(0) 		186
(35) 		199
(48) 		220
(69) 		228
(77) 		243
(92) 		261
(110) 		266
(115)
  • 193ed12
Approximation of prime harmonics in 193ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.66 -7.31 -0.07 -3.05 -5.39 -4.83 -1.17 +6.88 +10.47 +10.38 +6.37
Relative (%) +16.4 -32.8 -0.3 -13.7 -24.2 -21.7 -5.3 +30.9 +47.0 +46.6 +28.6
Steps
(reduced) 		54
(54) 		85
(85) 		125
(125) 		151
(151) 		186
(186) 		199
(6) 		220
(27) 		229
(36) 		244
(51) 		262
(69) 		267
(74)
  • 263zpi (22.243c)
Approximation of prime harmonics in 1ed22.243c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.12 +10.94 -5.94 -10.13 +8.12 +8.07 +10.75 -3.87 -0.98 -1.91 -6.15
Relative (%) +5.0 +49.2 -26.7 -45.6 +36.5 +36.3 +48.3 -17.4 -4.4 -8.6 -27.7
Step 54 86 125 151 187 200 221 229 244 262 267
  • 13-limit WE (22.198c) (octave is identical to 187ed11 within 0.1 ¢)
Approximation of prime harmonics in 1ed22.198c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.31 +7.07 +10.63 +5.27 -0.29 -0.93 +0.80 +8.03 +10.24 +8.50 +4.03
Relative (%) -5.9 +31.9 +47.9 +23.7 -1.3 -4.2 +3.6 +36.2 +46.1 +38.3 +18.1
Step 54 86 126 152 187 200 221 230 245 263 268
  • 2.3.7.11.13 WE (22.180c)
Approximation of prime harmonics in 1ed22.18c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.28 +5.52 +8.37 +2.53 -3.66 -4.53 -3.18 +3.89 +5.83 +3.76 -0.80
Relative (%) -10.3 +24.9 +37.7 +11.4 -16.5 -20.4 -14.3 +17.5 +26.3 +17.0 -3.6
Step 54 86 126 152 187 200 221 230 245 263 268
  • 264zpi (22.175c) (octave is identical to 194ed12 within 0.01 ¢)
Approximation of prime harmonics in 1ed22.175c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.55 +5.09 +7.74 +1.77 -4.59 -5.53 -4.28 +2.74 +4.60 +2.45 -2.14
Relative (%) -11.5 +23.0 +34.9 +8.0 -20.7 -24.9 -19.3 +12.3 +20.7 +11.0 -9.6
Step 54 86 126 152 187 200 221 230 245 263 268
  • 152ed7
Approximation of prime harmonics in 152ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.18 +4.09 +6.27 +0.00 -6.78 -7.86 -6.86 +0.05 +1.74 -0.62 -5.26
Relative (%) -14.3 +18.5 +28.3 +0.0 -30.6 -35.5 -31.0 +0.2 +7.9 -2.8 -23.7
Steps
(reduced) 		54
(54) 		86
(86) 		126
(126) 		152
(0) 		187
(35) 		200
(48) 		221
(69) 		230
(78) 		245
(93) 		263
(111) 		268
(116)
  • 140ed6
Approximation of prime harmonics in 140ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.53 +3.53 +5.45 -0.99 -7.99 -9.16 -8.30 -1.44 +0.15 -2.33 -7.01
Relative (%) -15.9 +15.9 +24.6 -4.5 -36.1 -41.4 -37.5 -6.5 +0.7 -10.5 -31.6
Steps
(reduced) 		54
(54) 		86
(86) 		126
(126) 		152
(12) 		187
(47) 		200
(60) 		221
(81) 		230
(90) 		245
(105) 		263
(123) 		268
(128)
  • 126ed5 (octave is identical to 86edt within 0.1 ¢)
Approximation of prime harmonics in 126ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5.87 -0.19 +0.00 -7.56 +6.04 +4.31 +4.26 +10.73 -10.44 +8.41 +3.52
Relative (%) -26.5 -0.8 +0.0 -34.2 +27.3 +19.5 +19.3 +48.5 -47.2 +38.0 +15.9
Steps
(reduced) 		54
(54) 		86
(86) 		126
(0) 		152
(26) 		188
(62) 		201
(75) 		222
(96) 		231
(105) 		245
(119) 		264
(12) 		269
(17)
  • 265zpi (22.100c)
Approximation of prime harmonics in 1ed22.1c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.60 -1.36 -1.71 -9.63 +3.48 +1.57 +1.24 +7.59 +8.33 +4.82 -0.14
Relative (%) -29.9 -6.1 -7.8 -43.6 +15.8 +7.1 +5.6 +34.3 +37.7 +21.8 -0.6
Step 54 86 126 152 188 201 222 231 246 264 269




59edo (narrow down ZPIs) (Nothing special abt these choices)

  • 152ed6
Approximation of prime harmonics in 152ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +4.05 -4.05 +9.53 -1.57 -8.58 +8.33 -7.13 +4.39 +0.15 +7.00 -6.42
Relative (%) +19.8 -19.8 +46.7 -7.7 -42.0 +40.8 -34.9 +21.5 +0.7 +34.3 -31.5
Steps
(reduced) 		59
(59) 		93
(93) 		137
(137) 		165
(13) 		203
(51) 		218
(66) 		240
(88) 		250
(98) 		266
(114) 		286
(134) 		291
(139)
  • 294zpi (20.399c)
Approximation of prime harmonics in 1ed20.399c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.54 -4.85 +8.35 -2.99 +10.08 +6.45 -9.20 +2.24 -2.14 +4.54 -8.93
Relative (%) +17.4 -23.8 +40.9 -14.7 +49.4 +31.6 -45.1 +11.0 -10.5 +22.2 -43.8
Step 59 93 137 165 204 218 240 250 266 286 291
  • 211ed12
Approximation of prime harmonics in 211ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +2.92 -5.83 +6.90 -4.74 +7.92 +4.15 +8.65 -0.41 -4.96 +1.51 +8.38
Relative (%) +14.3 -28.6 +33.8 -23.2 +38.8 +20.3 +42.4 -2.0 -24.3 +7.4 +41.1
Steps
(reduced) 		59
(59) 		93
(93) 		137
(137) 		165
(165) 		204
(204) 		218
(7) 		241
(30) 		250
(39) 		266
(55) 		286
(75) 		292
(81)
  • 295zpi (20.342c)
Approximation of prime harmonics in 1ed20.342c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.18 -10.15 +0.54 +7.95 -1.55 -5.97 -2.53 +8.33 +3.04 +8.58 -5.17
Relative (%) +0.9 -49.9 +2.7 +39.1 -7.6 -29.4 -12.5 +40.9 +14.9 +42.2 -25.4
Step 59 93 137 166 204 218 241 251 267 287 292
  • pure octaves 59edo (octave is identical to 137ed5 within 0.05 ¢)
  • 13-limit WE (20.320c)
Approximation of prime harmonics in 1ed20.32c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.12 +8.12 -2.47 +4.29 -6.04 +9.55 -7.84 +2.81 -2.83 +2.26 +8.72
Relative (%) -5.5 +40.0 -12.2 +21.1 -29.7 +47.0 -38.6 +13.8 -13.9 +11.1 +42.9
Step 59 94 137 166 204 219 241 251 267 287 293
  • 11-limit WE (20.310c)
Approximation of prime harmonics in 1ed20.31c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.71 +7.18 -3.84 +2.63 -8.08 +7.36 +10.06 +0.30 -5.50 -0.61 +5.79
Relative (%) -8.4 +35.4 -18.9 +13.0 -39.8 +36.2 +49.6 +1.5 -27.1 -3.0 +28.5
Step 59 94 137 166 204 219 242 251 267 287 293
  • 7-limit WE (20.301c)
Approximation of prime harmonics in 1ed20.301c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.24 +6.34 -5.08 +1.14 -9.91 +5.39 +7.89 -1.96 -7.91 -3.19 +3.16
Relative (%) -11.0 +31.2 -25.0 +5.6 -48.8 +26.6 +38.8 -9.7 -39.0 -15.7 +15.6
Step 59 94 137 166 204 219 242 251 267 287 293
  • 166ed7
Approximation of prime harmonics in 166ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.65 +5.69 -6.02 +0.00 +8.98 +3.89 +6.22 -3.69 -9.74 -5.16 +1.15
Relative (%) -13.0 +28.1 -29.7 +0.0 +44.2 +19.2 +30.7 -18.2 -48.0 -25.4 +5.6
Steps
(reduced) 		59
(59) 		94
(94) 		137
(137) 		166
(0) 		205
(39) 		219
(53) 		242
(76) 		251
(85) 		267
(101) 		287
(121) 		293
(127)
  • 212ed12
Approximation of prime harmonics in 212ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.76 +5.52 -6.28 -0.31 +8.59 +3.47 +5.77 -4.16 +10.05 -5.70 +0.59
Relative (%) -13.6 +27.2 -30.9 -1.5 +42.3 +17.1 +28.4 -20.5 +49.5 -28.1 +2.9
Steps
(reduced) 		59
(59) 		94
(94) 		137
(137) 		166
(166) 		205
(205) 		219
(7) 		242
(30) 		251
(39) 		268
(56) 		287
(75) 		293
(81)
  • 296zpi (20.282c)
Approximation of prime harmonics in 1ed20.282c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.36 +4.55 -7.68 -2.01 +6.49 +1.23 +3.29 -6.73 +7.30 -8.64 -2.41
Relative (%) -16.6 +22.4 -37.9 -9.9 +32.0 +6.1 +16.2 -33.2 +36.0 -42.6 -11.9
Step 59 94 137 166 205 219 242 251 268 287 293
  • 153ed6
Approximation of prime harmonics in 153ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.82 +3.82 -8.75 -3.31 +4.90 -0.47 +1.40 -8.68 +5.22 +9.40 -4.69
Relative (%) -18.8 +18.8 -43.1 -16.3 +24.2 -2.3 +6.9 -42.8 +25.7 +46.3 -23.1
Steps
(reduced) 		59
(59) 		94
(94) 		137
(137) 		166
(13) 		205
(52) 		219
(66) 		242
(89) 		251
(98) 		268
(115) 		288
(135) 		293
(140)
  • 205ed11
Approximation of prime harmonics in 205ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5.23 +1.58 +8.23 -7.27 +0.00 -5.71 -4.38 +5.57 -1.19 +2.52 +8.56
Relative (%) -25.8 +7.8 +40.7 -35.9 +0.0 -28.2 -21.6 +27.5 -5.9 +12.4 +42.3
Steps
(reduced) 		59
(59) 		94
(94) 		138
(138) 		166
(166) 		205
(0) 		219
(14) 		242
(37) 		252
(47) 		268
(63) 		288
(83) 		294
(89)




64edo (narrow down ZPIs)

Approximation of prime harmonics in 1ed18.868c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +7.55 +3.71 +6.15 +8.55 -0.36 -6.55 +0.72 -3.15 +5.71 +0.63 -1.62
Relative (%) +40.0 +19.7 +32.6 +45.3 -1.9 -34.7 +3.8 -16.7 +30.3 +3.4 -8.6
Step 64 101 148 179 220 235 260 270 288 309 315
  • 101edt
Approximation of prime harmonics in 101edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +5.20 +0.00 +0.71 +1.97 -8.45 +3.64 -8.83 +5.75 -4.88 +8.11 +5.64
Relative (%) +27.6 +0.0 +3.8 +10.4 -44.8 +19.4 -46.9 +30.5 -25.9 +43.0 +29.9
Steps
(reduced) 		64
(64) 		101
(0) 		148
(47) 		179
(78) 		220
(18) 		236
(34) 		260
(58) 		271
(69) 		288
(86) 		310
(7) 		316
(13)
  • 179ed7 (octave is identical to 326zpi within 0.3 ¢)
Approximation of prime harmonics in 179ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +4.50 -1.11 -0.92 +0.00 +7.96 +1.05 +7.13 +2.78 -8.04 +4.70 +2.17
Relative (%) +23.9 -5.9 -4.9 +0.0 +42.3 +5.6 +37.9 +14.8 -42.7 +25.0 +11.5
Steps
(reduced) 		64
(64) 		101
(101) 		148
(148) 		179
(0) 		221
(42) 		236
(57) 		261
(82) 		271
(92) 		288
(109) 		310
(131) 		316
(137)
  • 165ed6
Approximation of prime harmonics in 165ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.18 -3.18 -3.95 -3.67 +3.42 -3.79 +1.77 -2.79 +4.85 -1.66 -4.32
Relative (%) +16.9 -16.9 -21.0 -19.5 +18.2 -20.2 +9.4 -14.8 +25.8 -8.8 -23.0
Steps
(reduced) 		64
(64) 		101
(101) 		148
(148) 		179
(14) 		221
(56) 		236
(71) 		261
(96) 		271
(106) 		289
(124) 		310
(145) 		316
(151)
  • 229ed12 (octave is identical to 221ed11 within 0.1 ¢)
Approximation of prime harmonics in 229ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +2.29 -4.59 -6.01 -6.16 +0.35 -7.07 -1.85 -6.55 +0.83 -5.97 -8.71
Relative (%) +12.2 -24.4 -32.0 -32.8 +1.9 -37.6 -9.9 -34.9 +4.4 -31.8 -46.4
Steps
(reduced) 		64
(64) 		101
(101) 		148
(148) 		179
(179) 		221
(221) 		236
(7) 		261
(32) 		271
(42) 		289
(60) 		310
(81) 		316
(87)
  • 327zpi (18.767c)
Approximation of prime harmonics in 1ed18.767c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.09 -6.49 -8.80 +9.23 -3.81 +7.25 -6.77 +7.11 -4.61 +6.96 +4.10
Relative (%) +5.8 -34.6 -46.9 +49.2 -20.3 +38.6 -36.1 +37.9 -24.6 +37.1 +21.9
Step 64 101 148 180 221 237 261 272 289 311 317
  • 11-limit WE (18.755c)
Approximation of prime harmonics in 1ed18.755c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.32 -7.70 +8.18 +7.07 -6.46 +4.41 +8.85 +3.85 -8.08 +3.23 +0.30
Relative (%) +1.7 -41.1 +43.6 +37.7 -34.5 +23.5 +47.2 +20.5 -43.1 +17.2 +1.6
Step 64 101 149 180 221 237 262 272 289 311 317
  • pure octaves 64edo (octave is identical to 13-limit WE within 0.13 ¢)
  • 328zpi (18.721c)
Approximation of prime harmonics in 1ed18.721c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.86 +7.59 +3.12 +0.95 +4.74 -3.65 -0.05 -5.40 +0.82 -7.35 +8.24
Relative (%) -9.9 +40.5 +16.6 +5.1 +25.3 -19.5 -0.3 -28.9 +4.4 -39.2 +44.0
Step 64 102 149 180 222 237 262 272 290 311 318
  • 180ed7
Approximation of prime harmonics in 180ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.20 +7.05 +2.33 +0.00 +3.57 -4.91 -1.44 -6.84 -0.72 -8.99 +6.56
Relative (%) -11.7 +37.6 +12.4 +0.0 +19.1 -26.2 -7.7 -36.6 -3.9 -48.1 +35.0
Steps
(reduced) 		64
(64) 		102
(102) 		149
(149) 		180
(0) 		222
(42) 		237
(57) 		262
(82) 		272
(92) 		290
(110) 		311
(131) 		318
(138)
  • 230ed12
Approximation of prime harmonics in 230ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.93 +5.87 +0.60 -2.08 +1.00 -7.64 -4.47 +8.72 -4.07 +6.12 +2.88
Relative (%) -15.7 +31.4 +3.2 -11.1 +5.4 -40.9 -23.9 +46.6 -21.8 +32.7 +15.4
Steps
(reduced) 		64
(64) 		102
(102) 		149
(149) 		180
(180) 		222
(222) 		237
(7) 		262
(32) 		273
(43) 		290
(60) 		312
(82) 		318
(88)
  • 149ed5 (octave is identical to 222ed11 within 0.03 ¢)
Approximation of prime harmonics in 149ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.19 +5.45 +0.00 -2.81 +0.10 -8.61 -5.53 +7.61 -5.25 +4.85 +1.59
Relative (%) -17.1 +29.2 +0.0 -15.0 +0.5 -46.0 -29.6 +40.7 -28.1 +25.9 +8.5
Steps
(reduced) 		64
(64) 		102
(102) 		149
(0) 		180
(31) 		222
(73) 		237
(88) 		262
(113) 		273
(124) 		290
(141) 		312
(14) 		318
(20)
  • 329zpi (18.672c)
Approximation of prime harmonics in 1ed18.672c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.99 +2.59 -4.19 -7.87 -6.13 +3.41 +5.78 -0.06 +5.28 -3.91 -7.34
Relative (%) -26.7 +13.9 -22.4 -42.1 -32.9 +18.3 +31.0 -0.3 +28.3 -21.0 -39.3
Step 64 102 149 180 222 238 263 273 291 312 318








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

60edo (narrow down edonoi & ZPIs)

  • 35edf
  • 139ed5
  • 301zpi (20.027c)
  • 95edt
  • 13-limit WE (20.013c) (155ed6 has octaves only 0.02 ¢ different)
  • 215ed12
  • 302zpi (19.962c)
  • 208ed11 (ideal for catnip temperament)
  • 303zpi (19.913c)

32edo

  • 13-limit WE (37.481c)
  • 11-limit WE (37.453c)
  • 90ed7 (optimal for dual-5) (133zpi's octave only differs by 0.4 ¢)
  • 51edt
  • 134zpi (37.176c)
  • 75ed5

33edo

  • 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)
  • 11-limit WE (36.349c)
  • 93ed7 (optimised for dual-fifths)
  • 77ed5 (139zpi's octave differs by only 0.2 ¢)
  • 123ed13 / 1ed47/46 (identical within <0.1 ¢)
  • 115ed11

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 ¢)
  • 2.3.5.11 WE (30.703c)
  • 173zpi (30.672c) (octave of 62edt differs by only 0.2 ¢)
  • 110ed7 (octave of 145ed13 differs by only 0.1 ¢)
  • 91ed5

42edo

  • Good <27% rel err
  • Okay <40% rel err
Approximation of harmonics in 42edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.0 +12.3 +0.0 +13.7 +12.3 +2.6 +0.0 -3.9 +13.7 -8.5 +12.3 -12.0
Relative (%) +0.0 +43.2 +0.0 +47.9 +43.2 +9.1 +0.0 -13.7 +47.9 -29.6 +43.2 -41.8
Steps
(reduced) 		42
(0) 		67
(25) 		84
(0) 		98
(14) 		109
(25) 		118
(34) 		126
(0) 		133
(7) 		140
(14) 		145
(19) 		151
(25) 		155
(29)
  • 42ed257/128 (good 2.3.5.7; bad 11.13)
  • 11ed6/5 (good 2.3.5; okay 7.11.13)
  • 189zpi (28.689c) (good 2.5.13; okay 3.11; bad 7)
  • 190zpi (28.572c)
  • 13-limit WE (28.534c)
  • 34ed7/4 (good 2.5.7.13; okay 3.11)
  • 7-limit WE (28.484c) (good 2.3.5.11.13; bad 7)
  • 191zpi (28.444c)
  • 1ed123/121 (good 2.3.5.11; okay 13; bad 7)

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 (possibly narrow down edonoi)

Approximation of harmonics in 54edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 +9.16 +0.00 -8.54 +9.16 +8.95 +0.00 -3.91 -8.54 +4.24 +9.16 +3.92
Relative (%) +0.0 +41.2 +0.0 -38.4 +41.2 +40.3 +0.0 -17.6 -38.4 +19.1 +41.2 +17.6
Steps
(reduced) 		54
(0) 		86
(32) 		108
(0) 		125
(17) 		140
(32) 		152
(44) 		162
(0) 		171
(9) 		179
(17) 		187
(25) 		194
(32) 		200
(38)
  • 126ed5
  • 38ed5/3 (stretch, improves 3.5.7.11.13.17.19.23)
  • 262zpi (22.313c)
  • 263zpi (22.243c)
  • 13-limit WE (22.198c)
  • 2.3.7.11.13 WE (22.180c)
  • 264zpi (22.175c)
  • 40ed5/3 (compress, improves 3.5.11.13.17.19 (not 7))
  • 152ed7
  • 86edt

59edo (narrow down ZPIs)

  • (Nothing special abt these choices)
Approximation of harmonics in 59edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 +9.91 +0.00 +0.13 +9.91 +7.45 +0.00 -0.52 +0.13 -2.17 +9.91 -6.63
Relative (%) +0.0 +48.7 +0.0 +0.6 +48.7 +36.6 +0.0 -2.6 +0.6 -10.6 +48.7 -32.6
Steps
(reduced) 		59
(0) 		94
(35) 		118
(0) 		137
(19) 		153
(35) 		166
(48) 		177
(0) 		187
(10) 		196
(19) 		204
(27) 		212
(35) 		218
(41)
  • 93edt
  • 203ed11
  • 293zpi (20.454c)
  • 294zpi (20.399c)
  • 295zpi (20.342c)
  • 13-limit WE (20.320c)
  • 11-limit WE (20.310c)
  • 7-limit WE (20.301c)
  • 296zpi (20.282c)
  • 297zpi (20.229c)
  • 166ed7

64edo (narrow down ZPIs)

Approximation of harmonics in 64edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13
Error Absolute (¢) +0.00 -8.21 +0.00 +7.44 -8.21 +6.17 +0.00 +2.34 +7.44 -7.57 -8.21 +3.22
Relative (%) +0.0 -43.8 +0.0 +39.7 -43.8 +32.9 +0.0 +12.5 +39.7 -40.4 -43.8 +17.2
Steps
(reduced) 		64
(0) 		101
(37) 		128
(0) 		149
(21) 		165
(37) 		180
(52) 		192
(0) 		203
(11) 		213
(21) 		221
(29) 		229
(37) 		237
(45)
  • 47ed5/3 (like 221ed11 but benefits & drawbacks both amplified)
  • 221ed11
  • 325zpi (18.868c)
  • 326zpi (18.816c)
  • 327zpi (18.767c)
  • 11-limit WE (18.755c)
  • 13-limit WE (18.752c)
  • 328zpi (18.721c)
  • 329zpi (18.672c)
  • 330zpi (18.630c)
  • 180ed7
  • 149ed5
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)

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)

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)

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)
Retrieved from "https://en.xen.wiki/index.php?title=User:BudjarnLambeth/Sandbox2&oldid=208558"