User:BudjarnLambeth/Table of stretched 42edo tunings: Difference between revisions

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{{Editable user page}}
[[42edo]] has almost 50% relative error on harmonics 3/1 and 5/1. This means that whether one [[Octave stretch|stretches]] or [[Octave shrinking|compresses]] the octave, either way it will improve 42edo’s approximations of [[JI]], but in opposite directions.
[[42edo]] has almost 50% relative error on harmonics 3/1 and 5/1. This means that whether one [[Octave stretch|stretches]] or [[Octave shrinking|compresses]] the octave, either way it will improve 42edo’s approximations of [[JI]], but in opposite directions.


This table provides a detailed comparison of three stretched- and two compressed-octave tunings for 42edo.
This table provides a detailed comparison of three stretched- and three compressed-octave tunings for 42edo.


{| class="wikitable"
{| class="wikitable"
|+
|+
!Tuning
!Tuning
! [[42ed257/128]]
! '''[[42ed257/128]]'''
! [[1ed28.7c|APS720jot]]
! [[1ed28.7c|'''APS720jot''']]
! [[zpi|189zpi]]
! [[zpi|'''189zpi''']]
! [[42edo]]
! [[42edo]]
! [[1ed28.5c|APS715jot]]
! [[1ed28.5c|APS715jot]]
Line 18: Line 19:
| ~41.81
| ~41.81
| ~41.83
| ~41.83
| 42.00
| '''42.00'''
| ~42.10
| ~42.10
| ~42.19
| ~42.19
Line 27: Line 28:
| ~28.70¢
| ~28.70¢
| ~28.69¢
| ~28.69¢
| ~28.57¢
| '''~28.57¢'''
| ~28.50¢
| ~28.50¢
| ~28.44¢
| ~28.44¢
Line 33: Line 34:
|-
|-
! Overview
! Overview
| great: 5 <br>good: 2, 3, 7 <br>okay: <br>bad: 11, 13
| <small>ex: 5 <br>gd: 2, 3, 7 <br>ok: <br>ba: 11, 13</small>
| great: 5 <br>good: 2, 3 <br>okay: 7, 11, 13 <br>bad:  
| <small>ex: 5 <br>gd: 2, 3 <br>ok: 7, 11, 13 <br>ba:</small>
| great: 5 <br>good: 2, 13 <br>okay: 3, 11 <br>bad: 7
| <small>ex: 5 <br>gd: 2, 13 <br>ok: 3, 11 <br>ba: 7</small>
| great: 2, 7 <br>good: <br>okay: 11 <br>bad: 3, 5, 13
| '''<small>ex: 2, 7 <br>gd: <br>ok: 11 <br>ba: 3, 5, 13</small>'''
| great: 2 <br>good: 3, 5, 7, 13 <br>okay: 11 <br>bad:  
| <small>ex: 2 <br>gd: 5, 7, 13 <br>ok: 3, 11 <br>ba:</small>
| great: 5, 11, 13 <br>good: 2, 3 <br>okay: <br>bad: 7
| <small>ex: 5, 11, 13 <br>gd: 2, 3 <br>ok: <br>ba: 7</small>
| great: 3, 5, 11 <br>good: 2 <br>okay: 13 <br>bad: 7
| <small>ex: 3, 5, 11 <br>gd: 2 <br>ok: 13 <br>ba: 7</small>
|-
|-
! 2/1 error
! 2/1 error
Line 45: Line 46:
| +19.0% <br>(5.5¢)
| +19.0% <br>(5.5¢)
| +17.2% <br>(4.9¢)
| +17.2% <br>(4.9¢)
| 0.0% <br>(0.0¢)
| '''0.0% <br>(0.0¢)'''
| -10.2% <br>(2.9¢)
| -10.2% <br>(2.9¢)
| −18.8% <br>(5.3¢)
| −18.8% <br>(5.3¢)
Line 54: Line 55:
| −26.7% <br>(7.7¢)
| −26.7% <br>(7.7¢)
| −29.6% <br>(8.5¢)
| −29.6% <br>(8.5¢)
| +43.2% <br>(12.3¢)
| '''+43.2% <br>(12.3¢)'''
| +27.0% <br>(7.7¢)
| +27.0% <br>(7.7¢)
| +13.4% <br>(3.8¢)
| +13.4% <br>(3.8¢)
Line 63: Line 64:
| −7.9% <br>(2.3¢)
| −7.9% <br>(2.3¢)
| −12.2% <br>(3.5¢)
| −12.2% <br>(3.5¢)
| +47.9% <br>(13.7¢)
| '''+47.9% <br>(13.7¢)'''
| +24.2% <br>(6.9¢)
| +24.2% <br>(6.9¢)
| +4.3% <br>(1.2¢)
| +4.3% <br>(1.2¢)
Line 72: Line 73:
| −37.5% <br>(10.8¢)
| −37.5% <br>(10.8¢)
| −42.6% <br>(12.2¢)
| −42.6% <br>(12.2¢)
| +9.1% <br>(2.6¢)
| '''+9.1% <br>(2.6¢)'''
| -19.6% <br>(5.6¢)
| -19.6% <br>(5.6¢)
| −43.6% <br>(12.4¢)
| −43.6% <br>(12.4¢)
Line 81: Line 82:
| +36.2% <br>(10.4¢)
| +36.2% <br>(10.4¢)
| +29.8% <br>(8.6¢)
| +29.8% <br>(8.6¢)
| −29.6% <br>(8.5¢)
| '''−29.6% <br>(8.5¢)'''
| +35.1% <br>(10.0¢)
| +35.1% <br>(10.0¢)
| +5.5% <br>(1.6¢)
| +5.5% <br>(1.6¢)
| −12.1% <br>3.4¢)
| −12.1% <br>(3.4¢)
|-
|-
! 13/1 error
! 13/1 error
Line 90: Line 91:
| +28.6% <br>(8.2¢)
| +28.6% <br>(8.2¢)
| +21.7% <br>(6.2¢)
| +21.7% <br>(6.2¢)
| −41.8% <br>(12.0¢)
| '''−41.8% <br>(12.0¢)'''
| +20.4% <br>(5.8¢)
| +20.4% <br>(5.8¢)
| −11.3% <br>(3.2¢)
| −11.3% <br>(3.2¢)
| −30.1% <br>(8.6¢)
| −30.1% <br>(8.6¢)
|-
! 17/1 error
| +28.7% <br>(8.2¢)
| +10.4% <br>(3.0¢)
| +2.9% <br>(0.8¢)
| '''+32.7% <br>(9.3¢)'''
| -9.1% <br>(2.5¢)
| -44.0% <br>(12.5¢)
| +35.2% <br>(10.0¢)
|-
|-
! 2/1 val <br>(& reduced)
! 2/1 val <br>(& reduced)
Line 99: Line 109:
| 42 <br>(0)
| 42 <br>(0)
| 42 <br>(0)
| 42 <br>(0)
| 42 <br>(0)
| '''42 <br>(0)'''
| 42 <br>(0)
| 42 <br>(0)
| 42 <br>(0)
| 42 <br>(0)
Line 108: Line 118:
| 66 <br>(24)
| 66 <br>(24)
| 66 <br>(24)
| 66 <br>(24)
| 67 <br>(25)
| '''67 <br>(25)'''
| 67 <br>(25)
| 67 <br>(25)
| 67 <br>(25)
| 67 <br>(25)
Line 117: Line 127:
| 97 <br>(13)
| 97 <br>(13)
| 97 <br>(13)
| 97 <br>(13)
| 98 <br>(14)
| '''98 <br>(14)'''
| 98 <br>(14)
| 98 <br>(14)
| 98 <br>(14)
| 98 <br>(14)
Line 126: Line 136:
| 117 <br>(33)
| 117 <br>(33)
| 117 <br>(33)
| 117 <br>(33)
| 118 <br>(34)
| '''118 <br>(34)'''
| 118 <br>(34)
| 118 <br>(34)
| 118 <br>(34)
| 118 <br>(34)
Line 135: Line 145:
| 145 <br>(19)
| 145 <br>(19)
| 145 <br>(19)
| 145 <br>(19)
| 145 <br>(19)
| '''145 <br>(19)'''
| 146 <br>(20)
| 146 <br>(20)
| 146 <br>(20)
| 146 <br>(20)
Line 144: Line 154:
| 155 <br>(29)
| 155 <br>(29)
| 155 <br>(29)
| 155 <br>(29)
| 155 <br>(29)
| '''155 <br>(29)'''
| 156 <br>(30)
| 156 <br>(30)
| 156 <br>(30)
| 156 <br>(30)
| 156 <br>(30)
| 156 <br>(30)
|-
|-
| colspan="8" | <small><div style="text-align: center;">''“great” = 0-12.5% relative error • “good” = 12.5-27.5% • “okay” = 27.5-40% • “bad” = 40-50%''</div></small>
! 17/1 val
| 171 <br>(3)
| 171 <br>(3)
| 171 <br>(3)
| '''172 <br>(4)'''
| 172 <br>(4)
| 172 <br>(4)
| 173 <br>(5)
|-
| colspan="8" | <small><div style="text-align: center;">''ex = “excellent”, 0-13% relative error • gd = “good”, 13-27% • ok = “okay”, 27-40% • bd = “bad”, 40-50%''</div></small>
|}
|}


[[Category:42edo]][[Category:Tables]]
[[Category:42edo]][[Category:Tables]]

Latest revision as of 00:13, 2 August 2025

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42edo has almost 50% relative error on harmonics 3/1 and 5/1. This means that whether one stretches or compresses the octave, either way it will improve 42edo’s approximations of JI, but in opposite directions.

This table provides a detailed comparison of three stretched- and three compressed-octave tunings for 42edo.

Tuning 42ed257/128 APS720jot 189zpi 42edo APS715jot 191zpi 42ed255/128
Steps / octave ~41.77 ~41.81 ~41.83 42.00 ~42.10 ~42.19 ~42.24
Step size ~28.73¢ ~28.70¢ ~28.69¢ ~28.57¢ ~28.50¢ ~28.44¢ ~28.41¢
Overview ex: 5
gd: 2, 3, 7
ok:
ba: 11, 13 		ex: 5
gd: 2, 3
ok: 7, 11, 13
ba: 		ex: 5
gd: 2, 13
ok: 3, 11
ba: 7 		ex: 2, 7
gd:
ok: 11
ba: 3, 5, 13 		ex: 2
gd: 5, 7, 13
ok: 3, 11
ba: 		ex: 5, 11, 13
gd: 2, 3
ok:
ba: 7 		ex: 3, 5, 11
gd: 2
ok: 13
ba: 7
2/1 error +23.5%
(6.7¢) 		+19.0%
(5.5¢) 		+17.2%
(4.9¢) 		0.0%
(0.0¢) 		-10.2%
(2.9¢) 		−18.8%
(5.3¢) 		−23.9%
(6.8¢)
3/1 error −19.6%
(5.6¢) 		−26.7%
(7.7¢) 		−29.6%
(8.5¢) 		+43.2%
(12.3¢) 		+27.0%
(7.7¢) 		+13.4%
(3.8¢) 		+5.4%
(1.5¢)
5/1 error +2.4%
(0.7¢) 		−7.9%
(2.3¢) 		−12.2%
(3.5¢) 		+47.9%
(13.7¢) 		+24.2%
(6.9¢) 		+4.3%
(1.2¢) 		−7.5%
(2.1¢)
7/1 error −24.9%
(7.2¢) 		−37.5%
(10.8¢) 		−42.6%
(12.2¢) 		+9.1%
(2.6¢) 		-19.6%
(5.6¢) 		−43.6%
(12.4¢) 		+42.2%
(12.0¢)
11/1 error −48.3%
(13.9¢) 		+36.2%
(10.4¢) 		+29.8%
(8.6¢) 		−29.6%
(8.5¢) 		+35.1%
(10.0¢) 		+5.5%
(1.6¢) 		−12.1%
(3.4¢)
13/1 error +45.1%
(13.0¢) 		+28.6%
(8.2¢) 		+21.7%
(6.2¢) 		−41.8%
(12.0¢) 		+20.4%
(5.8¢) 		−11.3%
(3.2¢) 		−30.1%
(8.6¢)
17/1 error +28.7%
(8.2¢) 		+10.4%
(3.0¢) 		+2.9%
(0.8¢) 		+32.7%
(9.3¢) 		-9.1%
(2.5¢) 		-44.0%
(12.5¢) 		+35.2%
(10.0¢)
2/1 val
(& reduced) 		42
(0) 		42
(0) 		42
(0) 		42
(0) 		42
(0) 		42
(0) 		42
(0)
3/1 val 66
(24) 		66
(24) 		66
(24) 		67
(25) 		67
(25) 		67
(25) 		67
(25)
5/1 val 97
(13) 		97
(13) 		97
(13) 		98
(14) 		98
(14) 		98
(14) 		98
(14)
7/1 val 117
(33) 		117
(33) 		117
(33) 		118
(34) 		118
(34) 		118
(34) 		118
(34)
11/1 val 144
(18) 		145
(19) 		145
(19) 		145
(19) 		146
(20) 		146
(20) 		146
(20)
13/1 val 155
(29) 		155
(29) 		155
(29) 		155
(29) 		156
(30) 		156
(30) 		156
(30)
17/1 val 171
(3) 		171
(3) 		171
(3) 		172
(4) 		172
(4) 		172
(4) 		173
(5)
ex = “excellent”, 0-13% relative error • gd = “good”, 13-27% • ok = “okay”, 27-40% • bd = “bad”, 40-50%
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