42edo: Difference between revisions
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| Step size = 28.57143¢ | | Step size = 28.57143¢ | ||
| Fifth = 25\42 (714¢) | | Fifth = 25\42 (714¢) | ||
| Major 2nd = 8\42 (229¢ | | Major 2nd = 8\42 (229¢) | ||
| | | Semitones = 7:1 (200¢ : 29¢) | ||
| | | Consistency = 7 | ||
}} | }} | ||
The '''42edo''' is the equal division of the octave into 42 equal parts of 28.6 [[cent]]s each. | |||
== Theory == | |||
42edo has a fifth (the step of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7''n'') and a third both over 12 cents sharp, using the same 400 cent interval to represent [[5/4]] as does [[12edo]], which means it [[tempering out|tempers out]] 128/125. In the [[7-limit]], it tempers out 64/63 and [[126/125]], making it a tuning supporting [[augene]] temperament. | |||
While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 [[k*N subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo]]. On this subgroup 42 has the same [[comma]]s as 84. | While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 [[k*N subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo]]. On this subgroup 42 has the same [[comma]]s as 84. | ||
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D# is next to E. The notation requires triple ups and downs, even more if chords are to be spelled correctly. For example, a 1/1 - 5/4 - 3/2 - 9/5 chord with a root on the key or fret midway between G and A would be written either as v<sup>3</sup>G# - v<sup>5</sup>B# - v<sup>3</sup>D# - vF# or as ^<sup>3</sup>Ab - ^C - ^<sup>3</sup>Eb - ^<sup>5</sup>Gb. This is a double-down double-up-seven chord, written either as v<sup>3</sup>G#vv,^^7 or as ^<sup>3</sup>Abvv,^^7. | D# is next to E. The notation requires triple ups and downs, even more if chords are to be spelled correctly. For example, a 1/1 - 5/4 - 3/2 - 9/5 chord with a root on the key or fret midway between G and A would be written either as v<sup>3</sup>G# - v<sup>5</sup>B# - v<sup>3</sup>D# - vF# or as ^<sup>3</sup>Ab - ^C - ^<sup>3</sup>Eb - ^<sup>5</sup>Gb. This is a double-down double-up-seven chord, written either as v<sup>3</sup>G#vv,^^7 or as ^<sup>3</sup>Abvv,^^7. | ||
=== Odd harmonics === | |||
{{Odd harmonics in edo|42}} | |||
== Intervals == | == Intervals == | ||