42edo: Difference between revisions
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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 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 | 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 dud dup-seven chord, written either as v<sup>3</sup>G#vv,^^7 or as ^<sup>3</sup>Abvv,^^7. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 46: | Line 46: | ||
| 57.143 | | 57.143 | ||
| ^^1, ^m2 | | ^^1, ^m2 | ||
| | | dup 1sn, upminor 2nd | ||
| ^^D, ^Eb | | ^^D, ^Eb | ||
|- | |- | ||
| Line 52: | Line 52: | ||
| 85.714 | | 85.714 | ||
| ^^m2 | | ^^m2 | ||
| | | dupminor 2nd | ||
| ^^Eb | | ^^Eb | ||
|- | |- | ||
| Line 70: | Line 70: | ||
| 171.429 | | 171.429 | ||
| vvM2 | | vvM2 | ||
| | | dudmajor 2nd | ||
| vvE | | vvE | ||
|- | |- | ||
| Line 100: | Line 100: | ||
| 314.286 | | 314.286 | ||
| ^^m3 | | ^^m3 | ||
| | | dupminor 3rd | ||
| ^^F | | ^^F | ||
|- | |- | ||
| Line 118: | Line 118: | ||
| 400.000 | | 400.000 | ||
| vvM3 | | vvM3 | ||
| | | dudmajor 3rd | ||
| vvF# | | vvF# | ||
|- | |- | ||
| Line 148: | Line 148: | ||
| 543.857 | | 543.857 | ||
| ^^4 | | ^^4 | ||
| | | dup 4th | ||
| ^^G | | ^^G | ||
|- | |- | ||
| Line 172: | Line 172: | ||
| 657.143 | | 657.143 | ||
| vv5 | | vv5 | ||
| | | dud 5th | ||
| vvA | | vvA | ||
|- | |- | ||
| Line 202: | Line 202: | ||
| 800.000 | | 800.000 | ||
| ^^m6 | | ^^m6 | ||
| | | dupminor 6th | ||
| ^^Bb | | ^^Bb | ||
|- | |- | ||
| Line 220: | Line 220: | ||
| 885.714 | | 885.714 | ||
| vvM6 | | vvM6 | ||
| | | dudmajor 6th | ||
| vvB | | vvB | ||
|- | |- | ||
| Line 250: | Line 250: | ||
| 1028.571 | | 1028.571 | ||
| ^^m7 | | ^^m7 | ||
| | | dupminor 7th | ||
| ^^C | | ^^C | ||
|- | |- | ||
| Line 268: | Line 268: | ||
| 1114.286 | | 1114.286 | ||
| vvM7 | | vvM7 | ||
| | | dudmajor 7th | ||
| vvC# | | vvC# | ||
|- | |- | ||
Revision as of 15:00, 13 September 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 dud dup-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 | dup 1sn, upminor 2nd | ^^D, ^Eb |
| 3 | 85.714 | ^^m2 | dupminor 2nd | ^^Eb |
| 4 | 114.286 | v~2 | downmid 2nd | ^3Eb |
| 5 | 143.857 | ^~2 | upmid 2nd | v3E |
| 6 | 171.429 | vvM2 | dudmajor 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 | dupminor 3rd | ^^F |
| 12 | 342.857 | v~3 | downmid 3rd | ^3F |
| 13 | 371.429 | ^~3 | upmid 3rd | v3F# |
| 14 | 400.000 | vvM3 | dudmajor 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 | dup 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 | dud 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 | dupminor 6th | ^^Bb |
| 29 | 828.571 | v~6 | downmid 6th | ^3Bb |
| 30 | 857.143 | ^~6 | upmid 6th | v3B |
| 31 | 885.714 | vvM6 | dudmajor 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 | dupminor 7th | ^^C |
| 37 | 1057.143 | v~7 | downmid 7th | ^3C |
| 38 | 1085.714 | ^~7 | upmid 7th | v3C# |
| 39 | 1114.286 | vvM7 | dudmajor 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.