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 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
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 | +48 | +9 | -14 | -30 | -42 | -9 | +33 | -41 | -48 | +1 | |
Steps (reduced) |
67 (25) |
98 (14) |
118 (34) |
133 (7) |
145 (19) |
155 (29) |
164 (38) |
172 (4) |
178 (10) |
184 (16) |
190 (22) |
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.
Instruments
Lumatone