42edo: Difference between revisions
m →Scales |
→Octave stretch: Additions requested by a composer in the Monthly Tuning Facebook Group when I shared this table |
||
| Line 20: | Line 20: | ||
42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[octave stretching]] or [[octave shrinking]]. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with. | 42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[octave stretching]] or [[octave shrinking]]. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with. | ||
Click “expand” for a table comparing the properties of three stretched and two compressed tunings of 42edo: | |||
{| class="wikitable mw-collapsible" | {| class="wikitable mw-collapsible mw-collapsed" | ||
|+ | |+ | ||
!Tuning | !Tuning | ||
![[42ed257/128]] | ! [[42ed257/128]] | ||
![[1ed28.7c|APS720jot]] | ! [[1ed28.7c|APS720jot]] | ||
![[zpi|189zpi]] | ! [[zpi|189zpi]] | ||
![[42edo]] | ! [[42edo]] | ||
![[191zpi]] | ! [[191zpi]] | ||
![[ed255/128#42ed255/128|42ed255/128]] | ! [[ed255/128#42ed255/128|42ed255/128]] | ||
|- | |- | ||
! Steps / octave | |||
| | | ~41.77 | ||
| | | ~41.81 | ||
| | | ~41.83 | ||
| | | 42.00 | ||
| | | ~42.19 | ||
| | | ~42.24 | ||
|- | |- | ||
! Step size | |||
| | | ~28.73¢ | ||
| | | ~28.70¢ | ||
| | | ~28.69¢ | ||
| | | ~28.57¢ | ||
| | | ~28.44¢ | ||
| | | ~28.41¢ | ||
|- | |- | ||
|3 | ! Overview | ||
| | | great: 5 <br>good: 2, 3, 7 <br>okay: <br>bad: 11, 13 | ||
| | | great: 5 <br>good: 2, 3 <br>okay: 7, 11, 13 <br>bad: | ||
| great: 5 <br>good: 2, 13 <br>okay: 3, 11 <br>bad: 7 | |||
| | | great: 2, 7 <br>good: <br>okay: 11 <br>bad: 3, 5, 13 | ||
|13 | | great: 5, 11, 13 <br>good: 2, 3 <br>okay: <br>bad: 7 | ||
|5 | | great: 3, 5, 11 <br>good: 2 <br>okay: 13 <br>bad: 7 | ||
|- | |- | ||
! 2/1 relative error | |||
| | | +23.5% (6.7¢) | ||
| | | +19.0% (5.5¢) | ||
| | | +17.2% (4.9¢) | ||
| | | 0.0% (0.0¢) | ||
| | | −18.8% (5.3¢) | ||
| | | −23.9% (6.8¢) | ||
|- | |- | ||
! 3/1 rel. err. | |||
| | | −19.6% (5.6¢) | ||
| | | −26.7% (7.7¢) | ||
| | | −29.6% (8.5¢) | ||
| | | +43.2% (12.3¢) | ||
| | | +13.4% (3.8¢) | ||
| | | +5.4% (1.5¢) | ||
|- | |- | ||
! 5/1 rel. err. | |||
| | | +2.4% (0.7¢) | ||
| | | −7.9% (2.3¢) | ||
| | | −12.2% (3.5¢) | ||
| | | +47.9% (13.7¢) | ||
| | | +4.3% (1.2¢) | ||
| | | −7.5% (2.1¢) | ||
|- | |- | ||
| 13/1 rel. err. | ! 7/1 rel. err. | ||
|45.1% | | −24.9% (7.2¢) | ||
|28.6% | | −37.5% (10.8¢) | ||
|21.7% | | −42.6% (12.2¢) | ||
| | | +9.1% (2.6¢) | ||
| | | −43.6% (12.4¢) | ||
| | | +42.2% (12.0¢) | ||
|- | |||
! 11/1 rel. err. | |||
| −48.3% (13.9¢) | |||
| +36.2% (10.4¢) | |||
| +29.8% (8.6¢) | |||
| −29.6% (8.5¢) | |||
| +5.5% (1.6¢) | |||
| −12.1% (3.4¢) | |||
|- | |||
! 13/1 rel. err. | |||
| +45.1% (13.0¢) | |||
| +28.6% (8.2¢) | |||
| +21.7% (6.2¢) | |||
| −41.8% (12.0¢) | |||
| −11.3% (3.2¢) | |||
| −30.1% (8.6¢) | |||
|} | |} | ||