42edo: Difference between revisions
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→Intervals: made the notes match the interval names |
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| Line 66: | Line 66: | ||
| ^~2 | | ^~2 | ||
| upmid 2nd | | upmid 2nd | ||
| | | v<sup>3</sup>E | ||
|- | |- | ||
| 6 | | 6 | ||
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| vvM2 | | vvM2 | ||
| double-down major 2nd | | double-down major 2nd | ||
| | | vvE | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 78: | Line 78: | ||
| vM2 | | vM2 | ||
| downmajor 2nd | | downmajor 2nd | ||
| | | vE | ||
|- | |- | ||
| 8 | | 8 | ||
| Line 96: | Line 96: | ||
| ^m3 | | ^m3 | ||
| upminor 3rd | | upminor 3rd | ||
| | | ^F | ||
|- | |- | ||
| 11 | | 11 | ||
| Line 102: | Line 102: | ||
| ^^m3 | | ^^m3 | ||
| double-up minor 3rd | | double-up minor 3rd | ||
| ^ | | ^^F | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 108: | Line 108: | ||
| v~3 | | v~3 | ||
| downmid 3rd | | downmid 3rd | ||
| ^ | | ^<sup>3</sup>F | ||
|- | |- | ||
| 13 | | 13 | ||
| Line 144: | Line 144: | ||
| ^4 | | ^4 | ||
| up 4th | | up 4th | ||
| | | ^G | ||
|- | |- | ||
| 19 | | 19 | ||
| Line 150: | Line 150: | ||
| ^^4 | | ^^4 | ||
| double-up 4th | | double-up 4th | ||
| ^ | | ^^G | ||
|- | |- | ||
| 20 | | 20 | ||
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| v~4 | | v~4 | ||
| downmid 4th | | downmid 4th | ||
| ^ | | ^<sup>3</sup>G | ||
|- | |- | ||
| 21 | | 21 | ||
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| ^~5 | | ^~5 | ||
| upmid 5th | | upmid 5th | ||
| | | v<sup>3</sup>A | ||
|- | |- | ||
| 23 | | 23 | ||
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| vv5 | | vv5 | ||
| double-down 5th | | double-down 5th | ||
| | | vvA | ||
|- | |- | ||
| 24 | | 24 | ||
| Line 180: | Line 180: | ||
| v5 | | v5 | ||
| down 5th | | down 5th | ||
| | | vA | ||
|- | |- | ||
| 25 | | 25 | ||
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| ^~6 | | ^~6 | ||
| upmid 6th | | upmid 6th | ||
| | | v<sup>3</sup>B | ||
|- | |- | ||
| 31 | | 31 | ||
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| vvM6 | | vvM6 | ||
| double-down major 6th | | double-down major 6th | ||
| | | vvB | ||
|- | |- | ||
| 32 | | 32 | ||
| Line 228: | Line 228: | ||
| vM6 | | vM6 | ||
| downmajor 6th | | downmajor 6th | ||
| | | vB | ||
|- | |- | ||
| 33 | | 33 | ||
| Line 246: | Line 246: | ||
| ^m7 | | ^m7 | ||
| upminor 7th | | upminor 7th | ||
| | | ^C | ||
|- | |- | ||
| 36 | | 36 | ||
| Line 252: | Line 252: | ||
| ^^m7 | | ^^m7 | ||
| double-up minor 7th | | double-up minor 7th | ||
| ^ | | ^^C | ||
|- | |- | ||
| 37 | | 37 | ||
| Line 258: | Line 258: | ||
| v~7 | | v~7 | ||
| downmid 7th | | downmid 7th | ||
| ^ | | ^<sup>3</sup>C | ||
|- | |- | ||
| 38 | | 38 | ||
Revision as of 07:27, 25 June 2022
| ← 41edo | 42edo | 43edo → |
The 42 equal divisions of the octave (42edo), or 42(-tone) equal temperament when viewed from a regular temperament perspective, is the equal division of the octave into 42 equal parts of 28.6 cents 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 7n) and a third both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12edo, which means it 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 2*42 subgroup, having the same tuning on it as does 84edo. On this subgroup 42 has the same commas 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:
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 v3G# - v5B# - v3D# - vF# or as ^3Ab - ^C - ^3Eb - ^5Gb. This is a double-down double-up-seven chord, written either as v3G#vv,^^7 or as ^3Abvv,^^7.
Odd harmonics
| Odd harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +12.3 | +13.7 | +2.6 | -3.9 | -8.5 | -12.0 | -2.6 | +9.3 | -11.8 | -13.6 | +0.3 | -1.2 | +8.4 | -1.0 | -2.2 |
| relative (%) | +43 | +48 | +9 | -14 | -30 | -42 | -9 | +33 | -41 | -48 | +1 | -4 | +29 | -4 | -8 | |
| Steps (reduced) | 67 (25) | 98 (14) | 118 (34) | 133 (7) | 145 (19) | 155 (29) | 164 (38) | 172 (4) | 178 (10) | 184 (16) | 190 (22) | 195 (27) | 200 (32) | 204 (36) | 208 (40) | |
Intervals
| # | Cents | Ups and Downs Notation | ||
|---|---|---|---|---|
| 0 | 0.000 | P1 | perfect unison | D |
| 1 | 28.571 | ^1, m2 | up unison, minor 2nd | ^D, Eb |
| 2 | 57.143 | ^^1, ^m2 | double-up 1sn, upminor 2nd | ^^D, ^Eb |
| 3 | 85.714 | ^^m2 | double-up minor 2nd | ^^Eb |
| 4 | 114.286 | v~2 | downmid 2nd | ^3Eb |
| 5 | 143.857 | ^~2 | upmid 2nd | v3E |
| 6 | 171.429 | vvM2 | double-down major 2nd | vvE |
| 7 | 200.000 | vM2 | downmajor 2nd | vE |
| 8 | 228.571 | M2 | major 2nd | E |
| 9 | 257.143 | m3 | minor 3rd | F |
| 10 | 285.714 | ^m3 | upminor 3rd | ^F |
| 11 | 314.286 | ^^m3 | double-up minor 3rd | ^^F |
| 12 | 342.857 | v~3 | downmid 3rd | ^3F |
| 13 | 371.429 | ^~3 | upmid 3rd | v3F# |
| 14 | 400.000 | vvM3 | double-down major 3rd | vvF# |
| 15 | 428.571 | vM3 | downmajor 3rd | vF# |
| 16 | 457.143 | M3, v4 | major 3rd, down 4th | F#, vG |
| 17 | 485.714 | P4 | perfect 4th | G |
| 18 | 514.286 | ^4 | up 4th | ^G |
| 19 | 543.857 | ^^4 | double-up 4th | ^^G |
| 20 | 571.429 | v~4 | downmid 4th | ^3G |
| 21 | 600.000 | ^~4, v~5 | upmid 4th, downmid 5th | v3G#, ^3Ab |
| 22 | 628.571 | ^~5 | upmid 5th | v3A |
| 23 | 657.143 | vv5 | double-down 5th | vvA |
| 24 | 685.714 | v5 | down 5th | vA |
| 25 | 714.286 | P5 | perfect 5th | A |
| 26 | 742.857 | ^5, m6 | up 5th, minor 6th | ^A, Bb |
| 27 | 771.429 | ^m6 | upminor 6th | ^Bb |
| 28 | 800.000 | ^^m6 | double-up minor 6th | ^^Bb |
| 29 | 828.571 | v~6 | downmid 6th | ^3Bb |
| 30 | 857.143 | ^~6 | upmid 6th | v3B |
| 31 | 885.714 | vvM6 | double-down major 6th | vvB |
| 32 | 914.286 | vM6 | downmajor 6th | vB |
| 33 | 942.857 | M6 | major 6th | B |
| 34 | 971.429 | m7 | minor 7th | C |
| 35 | 1000.000 | ^m7 | upminor 7th | ^C |
| 36 | 1028.571 | ^^m7 | double-up minor 7th | ^^C |
| 37 | 1057.143 | v~7 | downmid 7th | ^3C |
| 38 | 1085.714 | ^~7 | upmid 7th | v3C# |
| 39 | 1114.286 | vvM7 | double-down major 7th | vvC# |
| 40 | 1142.857 | vM7 | downmajor 7th | vC# |
| 41 | 1171.429 | M7, v8 | major 7th, down 8ve | C#, vD |
| 42 | 1200.000 | P8 | perfect 8ve | 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.