42edo: Difference between revisions
→Intervals of 42edo: Removed 7mus column |
updated ups/down notation |
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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|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|84edo]]. On this subgroup 42 has the same [[Comma|comma]]s as 84. | ||
42edo is a | 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: | ||
D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D | D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D | ||
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 | 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<span style="font-size: 90%; vertical-align: super;">3</span>G# - v<span style="font-size: 90%; vertical-align: super;">5</span>B# - v<span style="font-size: 90%; vertical-align: super;">3</span>D# - vF# or as ^<span style="font-size: 90%; vertical-align: super;">3</span>Ab - ^C - ^<span style="font-size: 90%; vertical-align: super;">3</span>Eb - ^<span style="font-size: 90%; vertical-align: super;">5</span>Gb. This is a double-down double-up-seven chord, written either as v<span style="font-size: 90%; vertical-align: super;">3</span>G#vv,^^7 or as ^<span style="font-size: 90%; vertical-align: super;">3</span>Abvv,^^7. | ||
==Intervals of 42edo== | ==Intervals of 42edo== | ||
| Line 28: | Line 28: | ||
| style="text-align:center;" | up 1sn, minor 2nd | | style="text-align:center;" | up 1sn, minor 2nd | ||
| style="text-align:center;" | ^1, m2 | | style="text-align:center;" | ^1, m2 | ||
| style="text-align:center;" | D | | style="text-align:center;" | ^D, Eb | ||
|- | |- | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
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| style="text-align:center;" | double-up 1sn, upminor 2nd | | style="text-align:center;" | double-up 1sn, upminor 2nd | ||
| style="text-align:center;" | ^^1, ^m2 | | style="text-align:center;" | ^^1, ^m2 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^D, ^Eb | ||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
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| style="text-align:center;" | double-up minor 2nd | | style="text-align:center;" | double-up minor 2nd | ||
| style="text-align:center;" | ^^m2 | | style="text-align:center;" | ^^m2 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^Eb | ||
|- | |- | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
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| style="text-align:center;" | downmid 2nd | | style="text-align:center;" | downmid 2nd | ||
| style="text-align:center;" | v~2 | | style="text-align:center;" | v~2 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Eb | ||
|- | |- | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| Line 52: | Line 52: | ||
| style="text-align:center;" | upmid 2nd | | style="text-align:center;" | upmid 2nd | ||
| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E | ||
|- | |- | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
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| style="text-align:center;" | double-down major 2nd | | style="text-align:center;" | double-down major 2nd | ||
| style="text-align:center;" | vvM2 | | style="text-align:center;" | vvM2 | ||
| style="text-align:center;" | | | style="text-align:center;" | vvE | ||
|- | |- | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
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| style="text-align:center;" | downmajor 2nd | | style="text-align:center;" | downmajor 2nd | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| style="text-align:center;" | | | style="text-align:center;" | vE | ||
|- | |- | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| Line 82: | Line 82: | ||
| style="text-align:center;" | upminor 3rd | | style="text-align:center;" | upminor 3rd | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| style="text-align:center;" | F | | style="text-align:center;" | ^F | ||
|- | |- | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| Line 88: | Line 88: | ||
| style="text-align:center;" | double-up minor 3rd | | style="text-align:center;" | double-up minor 3rd | ||
| style="text-align:center;" | ^^m3 | | style="text-align:center;" | ^^m3 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^F | ||
|- | |- | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| Line 94: | Line 94: | ||
| style="text-align:center;" | downmid 3rd | | style="text-align:center;" | downmid 3rd | ||
| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F | ||
|- | |- | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
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| style="text-align:center;" | upmid 3rd | | style="text-align:center;" | upmid 3rd | ||
| style="text-align:center;" | ^~3 | | style="text-align:center;" | ^~3 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>F# | ||
|- | |- | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
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| style="text-align:center;" | double-down major 3rd | | style="text-align:center;" | double-down major 3rd | ||
| style="text-align:center;" | vvM3 | | style="text-align:center;" | vvM3 | ||
| style="text-align:center;" | | | style="text-align:center;" | vvF# | ||
|- | |- | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
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| style="text-align:center;" | downmajor 3rd | | style="text-align:center;" | downmajor 3rd | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| style="text-align:center;" | | | style="text-align:center;" | vF# | ||
|- | |- | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| Line 118: | Line 118: | ||
| style="text-align:center;" | major 3rd, down 4th | | style="text-align:center;" | major 3rd, down 4th | ||
| style="text-align:center;" | M3, v4 | | style="text-align:center;" | M3, v4 | ||
| style="text-align:center;" | F#, | | style="text-align:center;" | F#, vG | ||
|- | |- | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| Line 130: | Line 130: | ||
| style="text-align:center;" | up 4th | | style="text-align:center;" | up 4th | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| style="text-align:center;" | G | | style="text-align:center;" | ^G | ||
|- | |- | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
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| style="text-align:center;" | double-up 4th | | style="text-align:center;" | double-up 4th | ||
| style="text-align:center;" | ^^4 | | style="text-align:center;" | ^^4 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^G | ||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| style="text-align:center;" |571.429 | | style="text-align:center;" |571.429 | ||
| style="text-align:center;" | | | style="text-align:center;" | downmid 4th | ||
| style="text-align:center;" | | | style="text-align:center;" | v~4 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G | ||
|- | |- | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| style="text-align:center;" | 600 | | style="text-align:center;" | 600 | ||
| style="text-align:center;" | | | style="text-align:center;" | upmid 4th, downmid 5th | ||
| style="text-align:center;" | v | | style="text-align:center;" | ^~4, v~5 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>G#, ^<span style="font-size: 90%; vertical-align: super;">3</span>Ab | ||
|- | |- | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:center;" |628.571 | | style="text-align:center;" |628.571 | ||
| style="text-align:center;" | | | style="text-align:center;" | upmid 5th | ||
| style="text-align:center;" | | | style="text-align:center;" | ^~5 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A | ||
|- | |- | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| Line 160: | Line 160: | ||
| style="text-align:center;" | double-down 5th | | style="text-align:center;" | double-down 5th | ||
| style="text-align:center;" | vv5 | | style="text-align:center;" | vv5 | ||
| style="text-align:center;" | | | style="text-align:center;" | vvA | ||
|- | |- | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| Line 166: | Line 166: | ||
| style="text-align:center;" | down 5th | | style="text-align:center;" | down 5th | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| style="text-align:center;" | | | style="text-align:center;" | vA | ||
|- | |- | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| Line 178: | Line 178: | ||
| style="text-align:center;" | up 5th, minor 6th | | style="text-align:center;" | up 5th, minor 6th | ||
| style="text-align:center;" | ^5, m6 | | style="text-align:center;" | ^5, m6 | ||
| style="text-align:center;" | A | | style="text-align:center;" | ^A, Bb | ||
|- | |- | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| Line 184: | Line 184: | ||
| style="text-align:center;" | upminor 6th | | style="text-align:center;" | upminor 6th | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| style="text-align:center;" | Bb | | style="text-align:center;" | ^Bb | ||
|- | |- | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| Line 190: | Line 190: | ||
| style="text-align:center;" | double-up minor 6th | | style="text-align:center;" | double-up minor 6th | ||
| style="text-align:center;" | ^^m6 | | style="text-align:center;" | ^^m6 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^Bb | ||
|- | |- | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| Line 196: | Line 196: | ||
| style="text-align:center;" | downmid 6th | | style="text-align:center;" | downmid 6th | ||
| style="text-align:center;" | v~6 | | style="text-align:center;" | v~6 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Bb | ||
|- | |- | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| Line 202: | Line 202: | ||
| style="text-align:center;" | upmid 6th | | style="text-align:center;" | upmid 6th | ||
| style="text-align:center;" | ^~6 | | style="text-align:center;" | ^~6 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B | ||
|- | |- | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| Line 208: | Line 208: | ||
| style="text-align:center;" | double-down major 6th | | style="text-align:center;" | double-down major 6th | ||
| style="text-align:center;" | vvM6 | | style="text-align:center;" | vvM6 | ||
| style="text-align:center;" | | | style="text-align:center;" | vvB | ||
|- | |- | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| Line 214: | Line 214: | ||
| style="text-align:center;" | downmajor 6th | | style="text-align:center;" | downmajor 6th | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| style="text-align:center;" | | | style="text-align:center;" | vB | ||
|- | |- | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| Line 232: | Line 232: | ||
| style="text-align:center;" | upminor 7th | | style="text-align:center;" | upminor 7th | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| style="text-align:center;" | C | | style="text-align:center;" | ^C | ||
|- | |- | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| Line 238: | Line 238: | ||
| style="text-align:center;" | double-up minor 7th | | style="text-align:center;" | double-up minor 7th | ||
| style="text-align:center;" | ^^m7 | | style="text-align:center;" | ^^m7 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^^C | ||
|- | |- | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| Line 244: | Line 244: | ||
| style="text-align:center;" | downmid 7th | | style="text-align:center;" | downmid 7th | ||
| style="text-align:center;" | v~7 | | style="text-align:center;" | v~7 | ||
| style="text-align:center;" | | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C | ||
|- | |- | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| Line 250: | Line 250: | ||
| style="text-align:center;" | upmid 7th | | style="text-align:center;" | upmid 7th | ||
| style="text-align:center;" | ^~7 | | style="text-align:center;" | ^~7 | ||
| style="text-align:center;" | | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>C# | ||
|- | |- | ||
| style="text-align:center;" | 39 | | style="text-align:center;" | 39 | ||
| Line 256: | Line 256: | ||
| style="text-align:center;" | double-down major 7th | | style="text-align:center;" | double-down major 7th | ||
| style="text-align:center;" | vvM7 | | style="text-align:center;" | vvM7 | ||
| style="text-align:center;" | | | style="text-align:center;" | vvC# | ||
|- | |- | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
| Line 262: | Line 262: | ||
| style="text-align:center;" | downmajor 7th | | style="text-align:center;" | downmajor 7th | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| style="text-align:center;" | | | style="text-align:center;" | vC# | ||
|- | |- | ||
| style="text-align:center;" | 41 | | style="text-align:center;" | 41 | ||
| Line 268: | Line 268: | ||
| style="text-align:center;" | major 7th, down 8ve | | style="text-align:center;" | major 7th, down 8ve | ||
| style="text-align:center;" | M7, v8 | | style="text-align:center;" | M7, v8 | ||
| style="text-align:center;" | C#, | | style="text-align:center;" | C#, vD | ||
|- | |- | ||
| style="text-align:center;" | 42 | | style="text-align:center;" | 42 | ||
| Line 276: | Line 276: | ||
| style="text-align:center;" | D | | style="text-align:center;" | D | ||
|} | |} | ||
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | |||
[[Category:augene]] | [[Category:augene]] | ||
[[Category:edo]] | [[Category:edo]] | ||
[[Category:subgroup]] | [[Category:subgroup]] | ||