User:BudjarnLambeth/Sandbox2: Difference between revisions

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18edo
18edo
* 42ed5
* 42ed5
* 47ed6
* 13lim WE (66.291)
* 61zpi (66.228)
* 65ed12
* 65ed12
* 7lim WE (66.148)
* 7lim WE (66.148)
* 13lim WE (66.291)
* 47ed6
* 61zpi (66.228)
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].
18edo's [[prime]]s 3, 5, 7 and 13 are all tuned sharp, so it can benefit from [[octave shrinking]].


25edo
25edo
* 95zpi (48.067)
* 13lim WE (47.946)
* 90ed12
* 65ed6
* 65ed6
* 90ed12
* 13lim WE (47.946)
* 95zpi (48.067)
* 96zpi (47.642)
* 96zpi (47.642)
25edo's [[prime]] 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from [[octave shrinking]].
25edo's [[prime]] 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from [[octave shrinking]].
{{harmonics in equal | 25 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 65 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 90 | 12 | 1 | intervals=prime}}
{{harmonics in cet | 47.946 | intervals=prime}}
{{harmonics in cet | 48.067 | intervals=prime}}
{{harmonics in cet | 47.642 | intervals=prime}}


26edo
26edo
* 41edt
* 13lim WE (46.249) (octave identical to 11lim within 1/20th of a cent)
* 67ed6
* 86ed10
* 93ed12
* 93ed12
* 96ed14
* 13lim WE (46.249) (octave identical to 11lim within 1/20th of a cent)
* 100zpi (46.268)
* 100zpi (46.268)
26edo's simple [[prime]]s with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from [[octave stretching]].
26edo's simple [[prime]]s with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from [[octave stretching]].
{{harmonics in equal | 26 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 41 | 3 | 1 | intervals=prime}}
{{harmonics in equal | 67 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 86 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 93 | 12 | 1 | intervals=prime}}
{{harmonics in equal | 96 | 14 | 1 | intervals=prime}}
{{harmonics in cet | 46.249 | intervals=prime}}
{{harmonics in cet | 46.268 | intervals=prime}}


29edo
29edo
* 46edt
* 46edt
* 105ed12
* [[116zpi]] (41.465)
* 13lim WE (41.484)
* 107ed13
* 100ed11
* 96ed10
* 96ed10
* 100ed11
* 107ed13
* 16edf
* 11lim WE (41.482)
* 13lim WE (41.484)
* [[116zpi]] (41.465)
29edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]].
29edo's [[prime]]s 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from [[octave stretching]].
{{harmonics in equal | 29 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 46 | 3 | 1 | intervals=prime}}
{{harmonics in equal | 96 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 100 | 11 | 1 | intervals=prime}}
{{harmonics in equal | 105 | 12 | 1 | intervals=prime}}
{{harmonics in equal | 107 | 13 | 1 | intervals=prime}}
{{harmonics in equal | 16 | 3 | 2 | intervals=prime}}
{{harmonics in cet | 41.482 | intervals=prime}}
{{harmonics in cet | 41.484 | intervals=prime}}
{{harmonics in cet | 41.465 | intervals=prime}}


30edo
30edo
* 78ed6
* 39.918zpi (39.918) (octave identical to 104ed11 within 0.1{{c}})
* 13lim WE (39.904)
* 11lim WE (79.770)
* 100ed10
* 100ed10
* 104ed11
* 108ed12
* 108ed12
* 11lim WE (79.770)
* 78ed6
* 13lim WE (39.904)
* 39.918zpi (39.918)
30edo's simple [[prime]]s with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from [[octave shrinking]].
30edo's simple [[prime]]s with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from [[octave shrinking]].
{{harmonics in equal | 30 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 78 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 100 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 104 | 11 | 1 | intervals=prime}}
{{harmonics in equal | 108 | 12 | 1 | intervals=prime}}
{{harmonics in cet | 79.770 | intervals=prime}}
{{harmonics in cet | 39.904 | intervals=prime}}
{{harmonics in cet | 39.918 | intervals=prime}}


34edo
34edo
* 54edt
* 11lim WE (35.284)
* 13lim WE (35.276) (identical to 113ed10)
* 79ed5
* 79ed5
* 122ed12
* 88ed6
* 88ed6
* 108ed9
* 144zpi (35.248)
* 113ed10
* 122ed12
* 126ed13
* 126ed13
* 11lim WE (35.284)
* 54edt
* 13lim WE (35.276)
* 144zpi (35.248)
34edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from [[octave shrinking]].
34edo's [[prime]]s 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from [[octave shrinking]].
{{harmonics in equal | 34 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 54 | 3 | 1 | intervals=prime}}
{{harmonics in equal | 79 | 5 | 1 | intervals=prime}}
{{harmonics in equal | 88 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 108 | 9 | 1 | intervals=prime}}
{{harmonics in equal | 113 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 122 | 12 | 1 | intervals=prime}}
{{harmonics in equal | 126 | 13 | 1 | intervals=prime}}
{{harmonics in cet | 35.284 | intervals=prime}}
{{harmonics in cet | 35.276 | intervals=prime}}
{{harmonics in cet | 35.248 | intervals=prime}}


35edo
35edo
* 81ed5
* 90ed6
* 98ed7
* 116ed10
* 121ed11
* 125ed12
* 11lim WE (35.284)
* 11lim WE (35.284)
* 13lim WE (35.276)
* 13lim WE (35.276)
* 121ed11
* [[149zpi]] (34.359)
* [[149zpi]] (34.359)
* 116ed10
* 98ed7
* 81ed5
* 125ed12
* 90ed6
35edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]].
35edo's [[prime]]s 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from [[octave stretching]].
{{harmonics in equal | 35 | 2 | 1 | intervals=prime}}
{{harmonics in equal | 81 | 5 | 1 | intervals=prime}}
{{harmonics in equal | 90 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 98 | 7 | 1 | intervals=prime}}
{{harmonics in equal | 116 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 121 | 11 | 1 | intervals=prime}}
{{harmonics in equal | 125 | 12 | 1 | intervals=prime}}
{{harmonics in cet | 35.284 | intervals=prime}}
{{harmonics in cet | 35.276 | intervals=prime}}
{{harmonics in cet | 34.359 | intervals=prime}}


37edo
37edo
Line 157: Line 94:
{{harmonics in equal | 86 | 5 | 1 | intervals=prime}}
{{harmonics in equal | 86 | 5 | 1 | intervals=prime}}
{{harmonics in equal | 96 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 96 | 6 | 1 | intervals=prime}}
{{harmonics in equal | 104 | 7 | 1 | intervals=prime}
{{harmonics in equal | 104 | 7 | 1 | intervals=prime}}
{{harmonics in equal | 123 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 123 | 10 | 1 | intervals=prime}}
{{harmonics in equal | 128 | 11 | 1 | intervals=prime}}
{{harmonics in equal | 128 | 11 | 1 | intervals=prime}}

Revision as of 09:08, 14 September 2025

Quick link

User:BudjarnLambeth/Draft related tunings section

Lab

15edo

  • 52ed11
  • 11lim WE (79.770)
  • 50ed10
  • 47zpi (79.715)
  • 54ed12

15edo's primes 3, 5, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking.

18edo

  • 42ed5
  • 13lim WE (66.291)
  • 61zpi (66.228)
  • 65ed12
  • 7lim WE (66.148)
  • 47ed6

18edo's primes 3, 5, 7 and 13 are all tuned sharp, so it can benefit from octave shrinking.

25edo

  • 95zpi (48.067)
  • 13lim WE (47.946)
  • 90ed12
  • 65ed6
  • 96zpi (47.642)

25edo's prime 3 is very sharp, and its sharp and flat mapping of 11 and 13 are about equally bad, it can benefit from octave shrinking.

26edo

  • 13lim WE (46.249) (octave identical to 11lim within 1/20th of a cent)
  • 93ed12
  • 100zpi (46.268)

26edo's simple primes with the most error - 3, 5 and 13 - are all tuned flat, so it can benefit from octave stretching.

29edo

  • 46edt
  • 116zpi (41.465)
  • 13lim WE (41.484)
  • 107ed13
  • 100ed11
  • 96ed10

29edo's primes 5, 7, 11 and 13 are all tuned flat and the 3 has relatively little error, so 29edo can benefit from octave stretching.

30edo

  • 39.918zpi (39.918) (octave identical to 104ed11 within 0.1 ¢)
  • 13lim WE (39.904)
  • 11lim WE (79.770)
  • 100ed10
  • 108ed12
  • 78ed6

30edo's simple primes with the most error - 3, 5 and 11 - are all tuned sharp, so it can benefit from octave shrinking.

34edo

  • 11lim WE (35.284)
  • 13lim WE (35.276) (identical to 113ed10)
  • 79ed5
  • 122ed12
  • 88ed6
  • 144zpi (35.248)
  • 126ed13
  • 54edt

34edo's primes 3, 5, 11 and 13 are all tuned sharp, and it has two about equally bad mappings of 7, so 34edo can benefit from octave shrinking.

35edo

  • 11lim WE (35.284)
  • 13lim WE (35.276)
  • 121ed11
  • 149zpi (34.359)
  • 116ed10
  • 98ed7
  • 81ed5
  • 125ed12
  • 90ed6

35edo's primes 3, 5, 7 and 11 are all tuned flat, and it has two about equally bad mappings of 13, so 35edo can benefit from octave stretching.

37edo

  • 59edt
  • 86ed5
  • 96ed6
  • 104ed7
  • 123ed10
  • 128ed11
  • 133ed12
  • 137ed13
  • 11lim WE (32.377)
  • 13lim WE (32.383)
  • 161zpi (32.408)

37edo's primes 3, 5, 7, 11 and 13 are all tuned sharp, so it can benefit from octave shrinking.

Approximation of prime harmonics in 37edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +11.6 +2.9 +4.1 +0.0 +2.7 -7.7 -5.6 -12.1 +8.3 -9.9
Relative (%) +0.0 +35.6 +8.9 +12.8 +0.1 +8.4 -23.6 -17.3 -37.2 +25.5 -30.5
Steps
(reduced) 		37
(0) 		59
(22) 		86
(12) 		104
(30) 		128
(17) 		137
(26) 		151
(3) 		157
(9) 		167
(19) 		180
(32) 		183
(35)
Approximation of prime harmonics in 59edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -7.2 +0.0 -14.0 +16.0 +7.2 +8.1 -5.0 -4.1 -12.5 +5.2 -13.5
Relative (%) -22.5 +0.0 -43.3 +49.7 +22.3 +25.2 -15.5 -12.8 -38.9 +16.2 -41.9
Steps
(reduced) 		37
(37) 		59
(0) 		86
(27) 		105
(46) 		129
(11) 		138
(20) 		152
(34) 		158
(40) 		168
(50) 		181
(4) 		184
(7)
Approximation of prime harmonics in 86ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.2 +9.6 +0.0 +0.7 -4.2 -1.9 -12.7 -10.9 +14.8 +2.2 -16.0
Relative (%) -3.8 +29.6 +0.0 +2.1 -13.1 -5.8 -39.2 -33.6 +45.5 +6.9 -49.4
Steps
(reduced) 		37
(37) 		59
(59) 		86
(0) 		104
(18) 		128
(42) 		137
(51) 		151
(65) 		157
(71) 		168
(82) 		180
(8) 		183
(11)
Approximation of prime harmonics in 96ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.5 +4.5 -7.5 -8.4 -15.4 -13.8 +6.5 +7.8 +0.1 -13.4 +0.4
Relative (%) -13.8 +13.8 -23.1 -25.9 -47.6 -42.6 +20.0 +24.1 +0.5 -41.5 +1.2
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(8) 		128
(32) 		137
(41) 		152
(56) 		158
(62) 		168
(72) 		180
(84) 		184
(88)
Approximation of prime harmonics in 104ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.5 +9.2 -0.6 +0.0 -5.1 -2.7 -13.7 -11.9 +13.7 +1.1 +15.2
Relative (%) -4.6 +28.4 -1.7 +0.0 -15.7 -8.5 -42.2 -36.7 +42.2 +3.3 +46.9
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(0) 		128
(24) 		137
(33) 		151
(47) 		157
(53) 		168
(64) 		180
(76) 		184
(80)
Approximation of prime harmonics in 123ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.9 +10.2 +0.9 +1.7 -3.0 -0.5 -11.2 -9.3 -16.0 +4.1 -14.2
Relative (%) -2.7 +31.4 +2.7 +5.3 -9.1 -1.5 -34.5 -28.7 -49.3 +12.5 -43.7
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(104) 		128
(5) 		137
(14) 		151
(28) 		157
(34) 		167
(44) 		180
(57) 		183
(60)
Approximation of prime harmonics in 128ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.0 +11.5 +2.9 +4.1 +0.0 +2.7 -7.7 -5.7 -12.1 +8.2 -9.9
Relative (%) -0.0 +35.6 +8.8 +12.7 +0.0 +8.3 -23.7 -17.5 -37.3 +25.3 -30.7
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(104) 		128
(0) 		137
(9) 		151
(23) 		157
(29) 		167
(39) 		180
(52) 		183
(55)
Approximation of prime harmonics in 133ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.2 +6.4 -4.6 -4.9 -11.1 -9.2 +11.6 +13.1 +5.8 -7.4 +6.5
Relative (%) -9.9 +19.9 -14.2 -15.1 -34.3 -28.4 +35.8 +40.4 +17.9 -22.8 +20.2
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(104) 		128
(128) 		137
(4) 		152
(19) 		158
(25) 		168
(35) 		180
(47) 		184
(51)
Approximation of prime harmonics in 137ed13
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.7 +10.4 +1.2 +2.1 -2.5 +0.0 -10.7 -8.7 -15.4 +4.7 -13.5
Relative (%) -2.3 +32.1 +3.6 +6.4 -7.7 +0.0 -32.9 -26.9 -47.4 +14.5 -41.7
Steps
(reduced) 		37
(37) 		59
(59) 		86
(86) 		104
(104) 		128
(128) 		137
(0) 		151
(14) 		157
(20) 		167
(30) 		180
(43) 		183
(46)
Approximation of prime harmonics in 1ed32.377c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.1 +8.3 -1.9 -1.6 -7.1 -4.9 -16.0 -14.3 +11.1 -1.7 +12.3
Relative (%) -6.3 +25.6 -5.8 -5.0 -21.8 -15.1 -49.5 -44.2 +34.2 -5.3 +38.1
Step 37 59 86 104 128 137 151 157 168 180 184
Approximation of prime harmonics in 1ed32.383c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.8 +8.6 -1.4 -1.0 -6.3 -4.1 -15.1 -13.4 +12.1 -0.6 +13.4
Relative (%) -5.6 +26.7 -4.2 -3.1 -19.4 -12.5 -46.7 -41.3 +37.3 -2.0 +41.5
Step 37 59 86 104 128 137 151 157 168 180 184
Approximation of prime harmonics in 1ed32.408c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.9 +10.1 +0.8 +1.6 -3.1 -0.6 -11.3 -9.5 -16.1 +3.9 -14.4
Relative (%) -2.8 +31.2 +2.4 +5.0 -9.5 -1.9 -35.0 -29.2 -49.8 +11.9 -44.3
Step 37 59 86 104 128 137 151 157 167 180 183

48edo

  • 76edt
  • 124ed6
  • 152ed9
  • 159ed10
  • 166ed11
  • 172ed12
  • 28edf
  • 11lim WE (25.017)
  • 13lim WE (25.005)
  • 226zpi (25.006)

Most of 48edo's simple primes have low error, but its 5 is substantially flat, so 48edo can benefit from slight octave stretching.

Approximation of prime harmonics in 48edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -2.0 -11.3 +6.2 -1.3 +9.5 -5.0 +2.5 -3.3 -4.6 +5.0
Relative (%) +0.0 -7.8 -45.3 +24.7 -5.3 +37.9 -19.8 +9.9 -13.1 -18.3 +19.9
Steps
(reduced) 		48
(0) 		76
(28) 		111
(15) 		135
(39) 		166
(22) 		178
(34) 		196
(4) 		204
(12) 		217
(25) 		233
(41) 		238
(46)
Approximation of prime harmonics in 76edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.2 +0.0 -8.5 +9.6 +3.0 -11.0 +0.1 +7.7 +2.3 +1.4 +11.1
Relative (%) +4.9 +0.0 -33.8 +38.5 +11.8 -43.9 +0.3 +30.9 +9.2 +5.7 +44.3
Steps
(reduced) 		48
(48) 		76
(0) 		111
(35) 		135
(59) 		166
(14) 		177
(25) 		196
(44) 		204
(52) 		217
(65) 		233
(5) 		238
(10)
Approximation of prime harmonics in 124ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.8 -0.8 -9.6 +8.3 +1.3 +12.3 -1.9 +5.7 +0.1 -0.9 +8.7
Relative (%) +3.0 -3.0 -38.2 +33.2 +5.2 +49.1 -7.5 +22.8 +0.6 -3.6 +34.8
Steps
(reduced) 		48
(48) 		76
(76) 		111
(111) 		135
(11) 		166
(42) 		178
(54) 		196
(72) 		204
(80) 		217
(93) 		233
(109) 		238
(114)
Approximation of prime harmonics in 152ed9
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.2 +0.0 -8.5 +9.6 +3.0 -11.0 +0.1 +7.7 +2.3 +1.4 +11.1
Relative (%) +4.9 +0.0 -33.8 +38.5 +11.8 -43.9 +0.3 +30.9 +9.2 +5.7 +44.3
Steps
(reduced) 		48
(48) 		76
(76) 		111
(111) 		135
(135) 		166
(14) 		177
(25) 		196
(44) 		204
(52) 		217
(65) 		233
(81) 		238
(86)
Approximation of prime harmonics in 159ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.4 +3.5 -3.4 -9.3 +10.5 -2.9 +9.0 -8.1 +12.2 +12.0 -3.2
Relative (%) +13.6 +13.8 -13.6 -37.1 +41.9 -11.7 +35.9 -32.2 +48.5 +47.9 -12.7
Steps
(reduced) 		48
(48) 		76
(76) 		111
(111) 		134
(134) 		166
(7) 		177
(18) 		196
(37) 		203
(44) 		217
(58) 		233
(74) 		237
(78)
Approximation of prime harmonics in 166ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.4 -1.4 -10.4 +7.2 +0.0 +10.9 -3.4 +4.1 -1.6 -2.7 +6.9
Relative (%) +1.5 -5.4 -41.7 +29.0 +0.0 +43.5 -13.6 +16.4 -6.2 -10.9 +27.4
Steps
(reduced) 		48
(48) 		76
(76) 		111
(111) 		135
(135) 		166
(0) 		178
(12) 		196
(30) 		204
(38) 		217
(51) 		233
(67) 		238
(72)
Approximation of prime harmonics in 172ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.5 -1.1 -10.1 +7.7 +0.6 +11.5 -2.7 +4.8 -0.8 -1.9 +7.7
Relative (%) +2.2 -4.4 -40.2 +30.8 +2.3 +46.0 -10.9 +19.2 -3.2 -7.7 +30.7
Steps
(reduced) 		48
(48) 		76
(76) 		111
(111) 		135
(135) 		166
(166) 		178
(6) 		196
(24) 		204
(32) 		217
(45) 		233
(61) 		238
(66)
Approximation of prime harmonics in 28edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.4 +3.4 -3.6 -9.5 +10.3 -3.2 +8.7 -8.3 +11.9 +11.7 -3.5
Relative (%) +13.4 +13.4 -14.2 -37.8 +41.0 -12.6 +34.8 -33.3 +47.4 +46.6 -13.9
Steps
(reduced) 		48
(20) 		76
(20) 		111
(27) 		134
(22) 		166
(26) 		177
(9) 		196
(0) 		203
(7) 		217
(21) 		233
(9) 		237
(13)
Approximation of prime harmonics in 1ed25.017c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.8 -0.7 -9.4 +8.5 +1.5 +12.5 -1.6 +6.0 +0.4 -0.6 +9.0
Relative (%) +3.3 -2.7 -37.7 +33.9 +6.0 +50.0 -6.5 +23.8 +1.7 -2.5 +36.0
Step 48 76 111 135 166 178 196 204 217 233 238
Approximation of prime harmonics in 1ed25.005c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.2 -1.6 -10.8 +6.8 -0.5 +10.4 -4.0 +3.5 -2.2 -3.4 +6.2
Relative (%) +1.0 -6.3 -43.0 +27.4 -2.0 +41.4 -15.9 +14.0 -8.8 -13.6 +24.6
Step 48 76 111 135 166 178 196 204 217 233 238
Approximation of prime harmonics in 1ed25.006c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.3 -1.5 -10.6 +7.0 -0.3 +10.5 -3.8 +3.7 -2.0 -3.2 +6.4
Relative (%) +1.2 -6.0 -42.6 +27.9 -1.3 +42.2 -15.1 +14.8 -7.9 -12.7 +25.6
Step 48 76 111 135 166 178 196 204 217 233 238
Medium-low priority

10edo

  • 16edt
  • 23ed5
  • 26ed6
  • 28ed7
  • 32ed8
  • 33ed10
  • 36ed12
  • 37ed13
  • 6edf
  • 2.3.7.13 WE (119.785)
  • 2.5.7.13 WE (120.358)
  • 13lim WE (119.776)
  • 26zpi (119.899)

If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from octave shrinking. If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from octave stretching.

Approximation of prime harmonics in 10edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 -26.3 -8.8 +48.7 -0.5 +15.0 -57.5 -28.3 +50.4 +55.0
Relative (%) +0.0 +15.0 -21.9 -7.4 +40.6 -0.4 +12.5 -47.9 -23.6 +42.0 +45.8
Steps
(reduced) 		10
(0) 		16
(6) 		23
(3) 		28
(8) 		35
(5) 		37
(7) 		41
(1) 		42
(2) 		45
(5) 		49
(9) 		50
(0)
Approximation of prime harmonics in 23ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +11.4 +36.4 +0.0 +23.2 -32.4 +41.8 -59.2 -9.5 +23.2 -14.7 -9.0
Relative (%) +9.4 +30.0 +0.0 +19.2 -26.8 +34.5 -48.9 -7.8 +19.2 -12.1 -7.4
Steps
(reduced) 		10
(10) 		16
(16) 		23
(0) 		28
(5) 		34
(11) 		37
(14) 		40
(17) 		42
(19) 		45
(22) 		48
(2) 		49
(3)
Approximation of prime harmonics in 26ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.9 +6.9 -42.3 -28.3 +24.4 -26.2 -13.4 +32.6 -59.5 +16.4 +20.3
Relative (%) -5.8 +5.8 -35.4 -23.7 +20.4 -22.0 -11.2 +27.4 -49.9 +13.8 +17.0
Steps
(reduced) 		10
(10) 		16
(16) 		23
(23) 		28
(2) 		35
(9) 		37
(11) 		41
(15) 		43
(17) 		45
(19) 		49
(23) 		50
(24)
Approximation of prime harmonics in 28ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.2 +23.1 -19.1 +0.0 +59.7 +11.1 +28.0 -44.3 -14.1 -54.4 -49.6
Relative (%) +2.6 +19.2 -15.8 +0.0 +49.6 +9.3 +23.2 -36.8 -11.7 -45.3 -41.2
Steps
(reduced) 		10
(10) 		16
(16) 		23
(23) 		28
(0) 		35
(7) 		37
(9) 		41
(13) 		42
(14) 		45
(17) 		48
(20) 		49
(21)
Approximation of prime harmonics in 32ed8
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +37.5 +10.5 +26.2 +6.2 +11.2 -53.0 +45.0 -35.0 -28.3 +20.4 +17.5
Relative (%) +33.3 +9.4 +23.3 +5.5 +9.9 -47.1 +40.0 -31.1 -25.1 +18.2 +15.5
Steps
(reduced) 		11
(11) 		17
(17) 		25
(25) 		30
(30) 		37
(5) 		39
(7) 		44
(12) 		45
(13) 		48
(16) 		52
(20) 		53
(21)
Approximation of prime harmonics in 33ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +8.0 +30.8 -8.0 +13.5 -44.2 +29.0 +47.7 -24.0 +7.6 -31.3 -26.0
Relative (%) +6.6 +25.5 -6.6 +11.2 -36.6 +24.0 +39.5 -19.9 +6.3 -25.9 -21.5
Steps
(reduced) 		10
(10) 		16
(16) 		23
(23) 		28
(28) 		34
(1) 		37
(4) 		41
(8) 		42
(9) 		45
(12) 		48
(15) 		49
(16)
Approximation of prime harmonics in 36ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5.0 +10.0 -37.8 -22.9 +31.1 -19.1 -5.5 +40.9 -50.8 +25.9 +29.9
Relative (%) -4.2 +8.4 -31.7 -19.1 +26.1 -16.0 -4.6 +34.3 -42.5 +21.6 +25.0
Steps
(reduced) 		10
(10) 		16
(16) 		23
(23) 		28
(28) 		35
(35) 		37
(1) 		41
(5) 		43
(7) 		45
(9) 		49
(13) 		50
(14)
Approximation of prime harmonics in 37ed13
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.1 +18.3 -26.0 -8.4 +49.2 +0.0 +15.6 -56.9 -27.6 +51.1 +55.7
Relative (%) +0.1 +15.2 -21.7 -7.0 +41.0 +0.0 +13.0 -47.4 -23.0 +42.6 +46.4
Steps
(reduced) 		10
(10) 		16
(16) 		23
(23) 		28
(28) 		35
(35) 		37
(0) 		41
(4) 		42
(5) 		45
(8) 		49
(12) 		50
(13)
Approximation of prime harmonics in 6edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -30.1 -30.1 +21.5 +24.0 -56.6 +5.2 +8.7 +50.2 -46.6 +20.0 +21.6
Relative (%) -25.7 -25.7 +18.4 +20.5 -48.4 +4.4 +7.5 +42.9 -39.8 +17.1 +18.4
Steps
(reduced) 		10
(4) 		16
(4) 		24
(0) 		29
(5) 		35
(5) 		38
(2) 		42
(0) 		44
(2) 		46
(4) 		50
(2) 		51
(3)
Approximation of prime harmonics in 1ed119.785c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.2 +14.6 -31.3 -14.8 +41.2 -8.5 +6.2 +53.2 -37.9 +39.9 +44.2
Relative (%) -1.8 +12.2 -26.1 -12.4 +34.4 -7.1 +5.2 +44.4 -31.7 +33.3 +36.9
Step 10 16 23 28 35 37 41 43 45 49 50
Approximation of prime harmonics in 1ed120.358c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.6 +23.8 -18.1 +1.2 -59.1 +12.7 +29.7 -42.5 -12.2 -52.4 -47.5
Relative (%) +3.0 +19.8 -15.0 +1.0 -49.1 +10.6 +24.7 -35.3 -10.1 -43.5 -39.5
Step 10 16 23 28 34 37 41 42 45 48 49
Approximation of prime harmonics in 1ed119.776c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.2 +14.5 -31.5 -15.1 +40.8 -8.8 +5.9 +52.9 -38.4 +39.4 +43.8
Relative (%) -1.9 +12.1 -26.3 -12.6 +34.1 -7.4 +4.9 +44.1 -32.0 +32.9 +36.5
Step 10 16 23 28 35 37 41 43 45 49 50
Approximation of prime harmonics in 1ed119.899c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.0 +16.4 -28.6 -11.7 +45.1 -4.3 +10.9 +58.1 -32.8 +45.5 +49.9
Relative (%) -0.8 +13.7 -23.9 -9.7 +37.7 -3.6 +9.1 +48.5 -27.4 +37.9 +41.6
Step 10 16 23 28 35 37 41 43 45 49 50

11edo

  • 27ed6
  • 28ed6
  • 31ed7
  • 35ed9
  • 37ed10
  • 38ed10
  • 38ed12
  • 39ed12
  • 41ed13
  • 2.7.11.13 WE (108.821)
  • 30zpi (108.722)

11edo has about equally bad sharp and flat mappings of primes 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight octave shrinking is advisable. To use its primes 3 or 5, extreme octave shrinking or octave stretching can be used, at the cost of making the octaves sound significantly weaker.

Approximation of prime harmonics in 11edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -47.4 +50.0 +13.0 -5.9 +32.2 +4.1 +29.8 +26.3 -47.8 -54.1
Relative (%) +0.0 -43.5 +45.9 +11.9 -5.4 +29.5 +3.8 +27.3 +24.1 -43.8 -49.6
Steps
(reduced) 		11
(0) 		17
(6) 		26
(4) 		31
(9) 		38
(5) 		41
(8) 		45
(1) 		47
(3) 		50
(6) 		53
(9) 		54
(10)
Approximation of prime harmonics in 27ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -51.1 +51.1 -29.0 -37.1 -15.4 +40.1 +35.2 -42.5 -28.6 +29.7 +29.1
Relative (%) -44.5 +44.5 -25.3 -32.3 -13.4 +34.9 +30.6 -37.0 -24.9 +25.8 +25.3
Steps
(reduced) 		10
(10) 		17
(17) 		24
(24) 		29
(2) 		36
(9) 		39
(12) 		43
(16) 		44
(17) 		47
(20) 		51
(24) 		52
(25)
Approximation of prime harmonics in 28ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +18.6 -18.6 -16.7 -45.3 -52.3 -9.2 -30.5 -1.4 +0.1 +42.0 +37.3
Relative (%) +16.8 -16.8 -15.1 -40.9 -47.2 -8.3 -27.5 -1.3 +0.1 +37.9 +33.7
Steps
(reduced) 		11
(11) 		17
(17) 		25
(25) 		30
(2) 		37
(9) 		40
(12) 		44
(16) 		46
(18) 		49
(21) 		53
(25) 		54
(26)
Approximation of prime harmonics in 31ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.6 +54.1 +39.2 +0.0 -21.8 +15.0 -14.7 +10.1 +5.3 +38.7 +31.9
Relative (%) -4.2 +49.8 +36.0 +0.0 -20.1 +13.8 -13.5 +9.3 +4.9 +35.6 +29.4
Steps
(reduced) 		11
(11) 		18
(18) 		26
(26) 		31
(0) 		38
(7) 		41
(10) 		45
(14) 		47
(16) 		50
(19) 		54
(23) 		55
(24)
Approximation of prime harmonics in 35ed9
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.5 +54.3 +39.4 +0.4 -21.4 +15.5 -14.2 +10.6 +5.9 +39.3 +32.5
Relative (%) -4.1 +50.0 +36.3 +0.3 -19.7 +14.2 -13.1 +9.7 +5.4 +36.2 +29.9
Steps
(reduced) 		11
(11) 		18
(18) 		26
(26) 		31
(31) 		38
(3) 		41
(6) 		45
(10) 		47
(12) 		50
(15) 		54
(19) 		55
(20)
Approximation of prime harmonics in 37ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -14.9 +37.3 +14.9 -28.9 +50.5 -23.3 +51.0 -33.8 -41.4 -11.7 -19.4
Relative (%) -13.8 +34.7 +13.8 -26.9 +46.8 -21.6 +47.3 -31.4 -38.4 -10.9 -18.0
Steps
(reduced) 		11
(11) 		18
(18) 		26
(26) 		31
(31) 		39
(2) 		41
(4) 		46
(9) 		47
(10) 		50
(13) 		54
(17) 		55
(18)
Approximation of prime harmonics in 38ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -46.1 -13.7 +46.1 -11.9 +44.8 -34.6 +25.5 +42.7 +26.7 +45.0 +34.4
Relative (%) -43.9 -13.1 +43.9 -11.4 +42.7 -33.0 +24.3 +40.7 +25.4 +42.9 +32.8
Steps
(reduced) 		11
(11) 		18
(18) 		27
(27) 		32
(32) 		40
(2) 		42
(4) 		47
(9) 		49
(11) 		52
(14) 		56
(18) 		57
(19)
Approximation of prime harmonics in 38ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +45.3 +22.6 +43.9 +27.5 +37.4 -25.4 -37.0 -3.1 +5.8 -55.9 +55.1
Relative (%) +40.0 +20.0 +38.8 +24.3 +33.1 -22.4 -32.6 -2.7 +5.1 -49.4 +48.6
Steps
(reduced) 		11
(11) 		17
(17) 		25
(25) 		30
(30) 		37
(37) 		39
(1) 		43
(5) 		45
(7) 		48
(10) 		51
(13) 		53
(15)
Approximation of prime harmonics in 39ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +13.4 -26.7 -28.7 +50.7 +40.3 -28.3 -51.5 -23.4 -23.3 +16.7 +11.5
Relative (%) +12.1 -24.2 -26.0 +45.9 +36.6 -25.6 -46.7 -21.2 -21.1 +15.1 +10.4
Steps
(reduced) 		11
(11) 		17
(17) 		25
(25) 		31
(31) 		38
(38) 		40
(1) 		44
(5) 		46
(7) 		49
(10) 		53
(14) 		54
(15)
Approximation of prime harmonics in 41ed13
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -8.6 +47.5 +29.6 -11.4 -35.7 +0.0 -31.2 -7.2 -13.0 +18.9 +11.8
Relative (%) -8.0 +43.9 +27.4 -10.5 -33.0 +0.0 -28.8 -6.6 -12.0 +17.5 +10.9
Steps
(reduced) 		11
(11) 		18
(18) 		26
(26) 		31
(31) 		38
(38) 		41
(0) 		45
(4) 		47
(6) 		50
(9) 		54
(13) 		55
(14)
Approximation of prime harmonics in 1ed108.821c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.0 -52.0 +43.0 +4.6 -16.1 +21.1 -8.0 +17.1 +12.8 +46.8 +40.1
Relative (%) -2.7 -47.8 +39.5 +4.3 -14.8 +19.4 -7.4 +15.7 +11.7 +43.0 +36.9
Step 11 17 26 31 38 41 45 47 50 54 55
Approximation of prime harmonics in 1ed108.722c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.1 -53.7 +40.5 +1.6 -19.9 +17.1 -12.5 +12.4 +7.8 +41.4 +34.7
Relative (%) -3.7 -49.4 +37.2 +1.4 -18.3 +15.7 -11.5 +11.4 +7.2 +38.1 +31.9
Step 11 17 26 31 38 41 45 47 50 54 55

24edo ((13lim WE's octave is only 1/10th of a cent different from 24edo))

  • 38edt
  • 56ed5
  • 62ed6
  • 67ed7
  • 9ed7/6
  • 80ed10
  • 83ed11
  • 86ed12
  • 89ed13
  • 14edf
  • 2.3.5.11.13 WE (49.942)
  • 11lim WE (50.017)
  • 90zpi (49.988)

If one wishes to use 24edo as a full 19-or-lower-limit tuning, then it benefits from slight octave stretching, mostly to improve its prime 7. If one wishes to use 24edo as a no-7s 19-or-lower-limit tuning, then it benefits from slight octave shrinking, mostly to improve its primes 5 and 13.

Approximation of prime harmonics in 24edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -2.0 +13.7 -18.8 -1.3 +9.5 -5.0 +2.5 +21.7 +20.4 +5.0
Relative (%) +0.0 -3.9 +27.4 -37.7 -2.6 +18.9 -9.9 +5.0 +43.5 +40.8 +9.9
Steps
(reduced) 		24
(0) 		38
(14) 		56
(8) 		67
(19) 		83
(11) 		89
(17) 		98
(2) 		102
(6) 		109
(13) 		117
(21) 		119
(23)
Approximation of prime harmonics in 38edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.2 +0.0 +16.6 -15.4 +3.0 +14.1 +0.1 +7.7 -22.7 -23.6 +11.1
Relative (%) +2.5 +0.0 +33.1 -30.7 +5.9 +28.1 +0.2 +15.5 -45.4 -47.2 +22.2
Steps
(reduced) 		24
(24) 		38
(0) 		56
(18) 		67
(29) 		83
(7) 		89
(13) 		98
(22) 		102
(26) 		108
(32) 		116
(2) 		119
(5)
Approximation of prime harmonics in 56ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5.9 -11.2 +0.0 +14.6 -21.6 -12.3 +20.8 -22.4 -4.9 -8.2 -24.1
Relative (%) -11.8 -22.6 +0.0 +29.3 -43.4 -24.7 +41.9 -45.1 -9.9 -16.4 -48.5
Steps
(reduced) 		24
(24) 		38
(38) 		56
(0) 		68
(12) 		83
(27) 		89
(33) 		99
(43) 		102
(46) 		109
(53) 		117
(5) 		119
(7)
Approximation of prime harmonics in 62ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.8 -0.8 +15.5 -16.7 +1.3 +12.3 -1.9 +5.7 -24.9 +24.1 +8.7
Relative (%) +1.5 -1.5 +30.9 -33.4 +2.6 +24.5 -3.7 +11.4 -49.7 +48.2 +17.4
Steps
(reduced) 		24
(24) 		38
(38) 		56
(56) 		67
(5) 		83
(21) 		89
(27) 		98
(36) 		102
(40) 		108
(46) 		117
(55) 		119
(57)
Approximation of prime harmonics in 67ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.7 +8.7 -20.9 +0.0 +22.0 -15.8 +22.6 -19.1 +2.1 +3.0 -11.9
Relative (%) +13.4 +17.3 -41.5 +0.0 +43.8 -31.4 +44.9 -38.1 +4.1 +6.0 -23.6
Steps
(reduced) 		24
(24) 		38
(38) 		55
(55) 		67
(0) 		83
(16) 		88
(21) 		98
(31) 		101
(34) 		108
(41) 		116
(49) 		118
(51)
Approximation of prime harmonics in 9ed7/6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -13.9 -4.2 +1.0 +11.5 +0.0 +7.3 -12.3 +2.7 -1.9 +11.9 -14.6
Relative (%) -46.9 -14.2 +3.4 +38.9 +0.0 +24.7 -41.6 +9.1 -6.4 +40.2 -49.1
Steps
(reduced) 		40
(4) 		64
(1) 		94
(4) 		114
(6) 		140
(5) 		150
(6) 		165
(3) 		172
(1) 		183
(3) 		197
(8) 		200
(2)
Approximation of prime harmonics in 80ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.1 -8.5 +4.1 +19.5 -15.5 -5.8 -21.7 -15.0 +3.1 +0.4 -15.4
Relative (%) -8.2 -17.0 +8.2 +39.2 -31.1 -11.5 -43.6 -30.0 +6.2 +0.8 -30.9
Steps
(reduced) 		24
(24) 		38
(38) 		56
(56) 		68
(68) 		83
(3) 		89
(9) 		98
(18) 		102
(22) 		109
(29) 		117
(37) 		119
(39)
Approximation of prime harmonics in 83ed11
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.4 -1.4 +14.6 -17.8 +0.0 +10.9 -3.4 +4.1 +23.5 +22.3 +6.9
Relative (%) +0.8 -2.7 +29.1 -35.5 +0.0 +21.8 -6.8 +8.2 +46.9 +44.5 +13.7
Steps
(reduced) 		24
(24) 		38
(38) 		56
(56) 		67
(67) 		83
(0) 		89
(6) 		98
(15) 		102
(19) 		109
(26) 		117
(34) 		119
(36)
Approximation of prime harmonics in 86ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.5 -1.1 +15.0 -17.3 +0.6 +11.5 -2.7 +4.8 +24.2 +23.1 +7.7
Relative (%) +1.1 -2.2 +29.9 -34.6 +1.1 +23.0 -5.5 +9.6 +48.4 +46.1 +15.3
Steps
(reduced) 		24
(24) 		38
(38) 		56
(56) 		67
(67) 		83
(83) 		89
(3) 		98
(12) 		102
(16) 		109
(23) 		117
(31) 		119
(33)
Approximation of prime harmonics in 89ed13
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.6 -6.0 +7.7 +23.9 -10.2 +0.0 -15.4 -8.4 +10.1 +8.0 -7.7
Relative (%) -5.1 -12.0 +15.5 +48.0 -20.3 +0.0 -30.8 -16.8 +20.3 +16.0 -15.4
Steps
(reduced) 		24
(24) 		38
(38) 		56
(56) 		68
(68) 		83
(83) 		89
(0) 		98
(9) 		102
(13) 		109
(20) 		117
(28) 		119
(30)
Approximation of prime harmonics in 14edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3.4 +3.4 +21.5 -9.5 +10.3 +21.9 +8.7 +16.7 -13.2 -13.4 +21.6
Relative (%) +6.7 +6.7 +42.9 -18.9 +20.5 +43.7 +17.4 +33.4 -26.3 -26.7 +43.0
Steps
(reduced) 		24
(10) 		38
(10) 		56
(0) 		67
(11) 		83
(13) 		89
(5) 		98
(0) 		102
(4) 		108
(10) 		116
(4) 		119
(7)
Approximation of prime harmonics in 1ed49.942c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.4 -4.2 +10.4 -22.7 -6.1 +4.3 -10.6 -3.4 +15.4 +13.6 -1.9
Relative (%) -2.8 -8.3 +20.9 -45.5 -12.3 +8.6 -21.3 -6.9 +30.8 +27.3 -3.9
Step 24 38 56 67 83 89 98 102 109 117 119
Approximation of prime harmonics in 1ed50.017c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.4 -1.3 +14.6 -17.7 +0.1 +11.0 -3.3 +4.2 +23.6 +22.4 +7.0
Relative (%) +0.8 -2.6 +29.3 -35.4 +0.2 +22.0 -6.6 +8.4 +47.1 +44.8 +14.0
Step 24 38 56 67 83 89 98 102 109 117 119
Approximation of prime harmonics in 1ed49.988c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -0.3 -2.4 +13.0 -19.6 -2.3 +8.4 -6.1 +1.3 +20.4 +19.0 +3.5
Relative (%) -0.6 -4.8 +26.0 -39.3 -4.6 +16.8 -12.3 +2.5 +40.8 +38.0 +7.1
Step 24 38 56 67 83 89 98 102 109 117 119

5edo

  • 8edt
  • 13ed6
  • 14ed7
  • 18ed12
  • 3edf
  • 2.3.7 WE (239.426)
  • 9zpi (238.357)

If one wishes to use 5edo as a 2.3.7 subgroup tuning, then it benefits from slight octave shrinking to improve its prime 3.

Approximation of prime harmonics in 5edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0 +18 +94 -9 -71 +119 -105 -58 +92 -70 +55
Relative (%) +0.0 +7.5 +39.0 -3.7 -29.7 +49.8 -43.7 -24.0 +38.2 -29.0 +22.9
Steps
(reduced) 		5
(0) 		8
(3) 		12
(2) 		14
(4) 		17
(2) 		19
(4) 		20
(0) 		21
(1) 		23
(3) 		24
(4) 		25
(0)
Approximation of prime harmonics in 8edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -11 +0 +67 -40 -110 +77 +88 -105 +40 +114 -1
Relative (%) -4.7 +0.0 +28.0 -17.0 -46.1 +32.2 +36.9 -44.1 +16.8 +48.0 -0.6
Steps
(reduced) 		5
(5) 		8
(0) 		12
(4) 		14
(6) 		17
(1) 		19
(3) 		21
(5) 		21
(5) 		23
(7) 		25
(1) 		25
(1)
Approximation of prime harmonics in 13ed6
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -7 +7 +77 -28 -95 +93 +106 -87 +60 -103 +20
Relative (%) -2.9 +2.9 +32.3 -11.8 -39.8 +39.0 +44.4 -36.3 +25.1 -43.1 +8.5
Steps
(reduced) 		5
(5) 		8
(8) 		12
(12) 		14
(1) 		17
(4) 		19
(6) 		21
(8) 		21
(8) 		23
(10) 		24
(11) 		25
(12)
Approximation of prime harmonics in 14ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +3 +23 +101 +0 -61 -109 -92 -44 +106 -54 +71
Relative (%) +1.3 +9.6 +42.1 +0.0 -25.2 -45.4 -38.4 -18.4 +44.1 -22.6 +29.4
Steps
(reduced) 		5
(5) 		8
(8) 		12
(12) 		14
(0) 		17
(3) 		18
(4) 		20
(6) 		21
(7) 		23
(9) 		24
(10) 		25
(11)
Approximation of prime harmonics in 18ed12
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5 +10 +82 -23 -88 +100 +114 -79 +69 -94 +30
Relative (%) -2.1 +4.2 +34.2 -9.6 -37.0 +42.0 +47.7 -32.9 +28.7 -39.2 +12.5
Steps
(reduced) 		5
(5) 		8
(8) 		12
(12) 		14
(14) 		17
(17) 		19
(1) 		21
(3) 		21
(3) 		23
(5) 		24
(6) 		25
(7)
Approximation of prime harmonics in 3edf
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -30 -30 +22 -93 +60 +5 +9 +50 -47 +20 -95
Relative (%) -12.9 -12.9 +9.2 -39.8 +25.8 +2.2 +3.7 +21.4 -19.9 +8.6 -40.8
Steps
(reduced) 		5
(2) 		8
(2) 		12
(0) 		14
(2) 		18
(0) 		19
(1) 		21
(0) 		22
(1) 		23
(2) 		25
(1) 		25
(1)
Approximation of prime harmonics in 1ed239.426c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3 +13 +87 -17 -81 +109 -116 -70 +79 -83 +41
Relative (%) -1.2 +5.6 +36.3 -7.0 -33.9 +45.3 -48.6 -29.1 +32.8 -34.8 +17.0
Step 5 8 12 14 17 19 20 21 23 24 25
Approximation of prime harmonics in 1ed238.357c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -8 +5 +74 -32 -99 +88 +101 -92 +54 -109 +14
Relative (%) -3.4 +2.1 +31.0 -13.4 -41.6 +37.0 +42.2 -38.6 +22.6 -45.7 +5.8
Step 5 8 12 14 17 19 21 21 23 24 25

6edo

  • 14ed5
  • 17ed7
  • 19ed9
  • 20ed10
  • 2.9.5 WE (199.736)
  • 2.9.5.7 WE (199.329)
  • 12zpi (198.843)

If one wishes to use 6edo as a 2.9.5 or 2.9.5.7 subgroup tuning, then it benefits from octave shrinking.

Approximation of prime harmonics in 14ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -5.9 +88.3 +0.0 +14.6 +28.2 -62.0 +70.6 +77.1 -54.7 -57.9 +25.6
Relative (%) -2.9 +44.4 +0.0 +7.3 +14.1 -31.2 +35.5 +38.7 -27.5 -29.1 +12.9
Steps
(reduced) 		6
(6) 		10
(10) 		14
(0) 		17
(3) 		21
(7) 		22
(8) 		25
(11) 		26
(12) 		27
(13) 		29
(1) 		30
(2)
Approximation of prime harmonics in 17ed7
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -11.0 +79.7 -12.0 +0.0 +10.2 -80.9 +49.2 +54.8 -77.8 -82.8 -0.0
Relative (%) -5.6 +40.2 -6.0 +0.0 +5.1 -40.8 +24.8 +27.7 -39.3 -41.8 -0.0
Steps
(reduced) 		6
(6) 		10
(10) 		14
(14) 		17
(0) 		21
(4) 		22
(5) 		25
(8) 		26
(9) 		27
(10) 		29
(12) 		30
(13)
Approximation of prime harmonics in 19ed9
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +1.2 +100.1 +16.6 +34.7 +53.0 -36.0 -100.0 -92.4 -22.7 -23.6 +61.1
Relative (%) +0.6 +50.0 +8.3 +17.3 +26.5 -18.0 -50.0 -46.1 -11.3 -11.8 +30.5
Steps
(reduced) 		6
(6) 		10
(10) 		14
(14) 		17
(17) 		21
(2) 		22
(3) 		24
(5) 		25
(6) 		27
(8) 		29
(10) 		30
(11)
Approximation of prime harmonics in 20ed10
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.1 +91.2 +4.1 +19.5 +34.3 -55.6 +77.9 +84.7 -46.8 -49.4 +34.4
Relative (%) -2.1 +45.8 +2.1 +9.8 +17.2 -27.9 +39.1 +42.5 -23.5 -24.8 +17.3
Steps
(reduced) 		6
(6) 		10
(10) 		14
(14) 		17
(17) 		21
(1) 		22
(2) 		25
(5) 		26
(6) 		27
(7) 		29
(9) 		30
(10)
Approximation of prime harmonics in 1ed199.736c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -1.6 +95.4 +10.0 +26.7 +43.1 -46.3 +88.4 +95.6 -35.4 -37.2 +47.0
Relative (%) -0.8 +47.8 +5.0 +13.4 +21.6 -23.2 +44.3 +47.9 -17.7 -18.6 +23.6
Step 6 10 14 17 21 22 25 26 27 29 30
Approximation of prime harmonics in 1ed199.329c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -4.0 +91.3 +4.3 +19.8 +34.6 -55.3 +78.3 +85.0 -46.4 -49.0 +34.8
Relative (%) -2.0 +45.8 +2.2 +9.9 +17.4 -27.7 +39.3 +42.7 -23.3 -24.6 +17.5
Step 6 10 14 17 21 22 25 26 27 29 30
Approximation of prime harmonics in 1ed198.843c
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -6.9 +86.5 -2.5 +11.5 +24.4 -66.0 +66.1 +72.4 -59.5 -63.1 +20.3
Relative (%) -3.5 +43.5 -1.3 +5.8 +12.3 -33.2 +33.3 +36.4 -29.9 -31.7 +10.2
Step 6 10 14 17 21 22 25 26 27 29 30
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