42edo: Difference between revisions
m Moving from Category:Edo to Category:Equal divisions of the octave using Cat-a-lot |
m Proper sharp/flat enharmonicity gets even more important up here, where it can cut the amount of ups and downs you need to notate things correctly in half. |
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The ''42 equal division'' divides the octave into 42 equal parts of 28.571 [[ | The ''42 equal division'' divides the octave into 42 equal parts of 28.571 [[cent]]s each. It has a 3 (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a 5 both over 12 cents sharp, using the same 400 cent interval to represent [[5/4]] as does [[12edo|12]], which means it [[tempering_out|tempers out]] 128/125. In the [[7-limit|7-limit]], it tempers out 64/63 and [[126/125]], making it a tuning supporting [[Augmented_family|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|84edo]]. On this subgroup 42 has the same [[ | 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|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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| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| style="text-align:center;" | upmid 2nd | | style="text-align:center;" | upmid 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vvD# | ||
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| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
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| style="text-align:center;" | vvM2 | | style="text-align:center;" | vvM2 | ||
| style="text-align:center;" | double-down major 2nd | | style="text-align:center;" | double-down major 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | vD# | ||
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| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
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| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | downmajor 2nd | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | | | style="text-align:center;" | D# | ||
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| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
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| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | upminor 3rd | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | | | style="text-align:center;" | Gb | ||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
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| style="text-align:center;" | ^^m3 | | style="text-align:center;" | ^^m3 | ||
| style="text-align:center;" | double-up minor 3rd | | style="text-align:center;" | double-up minor 3rd | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^Gb | ||
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| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
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| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| style="text-align:center;" | downmid 3rd | | style="text-align:center;" | downmid 3rd | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^^Gb | ||
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| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
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| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | | | style="text-align:center;" | A'''b''' | ||
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| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
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| style="text-align:center;" | ^^4 | | style="text-align:center;" | ^^4 | ||
| style="text-align:center;" | double-up 4th | | style="text-align:center;" | double-up 4th | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^Ab | ||
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| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
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| style="text-align:center;" | v~4 | | style="text-align:center;" | v~4 | ||
| style="text-align:center;" | downmid 4th | | style="text-align:center;" | downmid 4th | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^^Ab | ||
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| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
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| style="text-align:center;" | ^~5 | | style="text-align:center;" | ^~5 | ||
| style="text-align:center;" | upmid 5th | | style="text-align:center;" | upmid 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vvG# | ||
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| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
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| style="text-align:center;" | vv5 | | style="text-align:center;" | vv5 | ||
| style="text-align:center;" | double-down 5th | | style="text-align:center;" | double-down 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | vG# | ||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
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| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | G# | ||
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| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
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| style="text-align:center;" | ^~6 | | style="text-align:center;" | ^~6 | ||
| style="text-align:center;" | upmid 6th | | style="text-align:center;" | upmid 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^A# | ||
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| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
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| style="text-align:center;" | vvM6 | | style="text-align:center;" | vvM6 | ||
| style="text-align:center;" | double-down major 6th | | style="text-align:center;" | double-down major 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | ^A# | ||
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| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
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| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | | | style="text-align:center;" | A# | ||
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| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
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| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | | | style="text-align:center;" | D# | ||
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| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
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| style="text-align:center;" | ^^m7 | | style="text-align:center;" | ^^m7 | ||
| style="text-align:center;" | double-up minor 7th | | style="text-align:center;" | double-up minor 7th | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^D# | ||
|- | |- | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
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| style="text-align:center;" | v~7 | | style="text-align:center;" | v~7 | ||
| style="text-align:center;" | downmid 7th | | style="text-align:center;" | downmid 7th | ||
| style="text-align:center;" | ^ | | style="text-align:center;" | ^^D# | ||
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| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||