42edo
| ← 41edo | 42edo | 43edo → |
42 equal divisions of the octave (abbreviated 42edo or 42ed2), also called 42-tone equal temperament (42tet) or 42 equal temperament (42et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 42 equal parts of about 28.6 ¢ each. Each step represents a frequency ratio of 21/42, or the 42nd root of 2.
Theory
42edo has a patent val fifth (the step of which is not from 7edo, this being a first for edos of the form 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 the 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).
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.3 | +13.7 | +2.6 | -3.9 | -8.5 | -12.0 | -2.6 | +9.3 | -11.8 | -13.6 | +0.3 |
| Relative (%) | +43.2 | +47.9 | +9.1 | -13.7 | -29.6 | -41.8 | -8.9 | +32.7 | -41.3 | -47.7 | +1.0 | |
| Steps (reduced) |
67 (25) |
98 (14) |
118 (34) |
133 (7) |
145 (19) |
155 (29) |
164 (38) |
172 (4) |
178 (10) |
184 (16) |
190 (22) | |
Subsets and supersets
Since 42 factors into 2 × 3 × 7, 42edo contains subset edos 2, 3, 6, 7, 14, and 21.
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 | ^3m | trupminor 2nd | ^3Eb |
| 5 | 143.857 | v3M | trudmajor 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 | ^3m3 | trupminor 3rd | ^3F |
| 13 | 371.429 | v3M3 | trudmajor 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 | ^34, ^^d5 | trup 4th, dupdim 5th | ^3G, ^^Ab |
| 21 | 600.000 | v3A4, ^3d5 | trudaug 4th, trupdim 5th | v3G#, ^3Ab |
| 22 | 628.571 | vvA4, v35 | dudaug 4th, trud 5th | vvG#, 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 | ^3m6 | trupminor 6th | ^3Bb |
| 30 | 857.143 | v3M6 | trudmajor 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 | ^3m7 | trupminor 7th | ^3C |
| 38 | 1085.714 | v3M7 | trudmajor 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.
Notation
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.
Instruments
- Lumatone