42edo: Difference between revisions
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Swap "third" and "five" as it originally meant; clarify the coprime stuff; minor cleanup |
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2 | | Prime factorization = 2 × 3 × 7 | ||
| Step size = 29.571¢ | | Step size = 29.571¢ | ||
| Fifth = 25\42 = 714.286¢ | | Fifth = 25\42 = 714.286¢ | ||
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}} | }} | ||
The '''42edo''' is the equal division of the octave into 42 equal parts of 28.571 [[cent]]s each. It has a | The '''42edo''' is the equal division of the octave into 42 equal parts of 28.571 [[cent]]s each. It 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* | 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. | ||
42edo is a diatonic edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult diatonic edos to notate, because no other diatonic edo's 5th is as sharp (see [[47edo]] for the opposite extreme). Assuming the natural notes form a chain of fifths, the major 2nd is 8 edosteps and the minor 2nd is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep: | 42edo is a diatonic edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult diatonic edos to notate, because no other diatonic edo's 5th is as sharp (see [[47edo]] for the opposite extreme). Assuming the natural notes form a chain of fifths, the major 2nd is 8 edosteps and the minor 2nd is only one. The naturals create a 5edo-like scale, with two of the notes inflected by a comma-sized edostep: | ||
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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. | ||
==Intervals | == Intervals == | ||
{| class="wikitable center-all right-2 left-4" | {| class="wikitable center-all right-2 left-4" | ||