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.
42edo is on the optimal ET sequence of the eugene, joan, lemba, neutron, qeema, seville, sevond, skateboard, tritikleismic and vines temperaments.
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
While not an accurate tuning on the full 7-limit, 42edo 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.
| 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) | |
Octave stretch
42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through octave stretching or octave shrinking. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.
The following table compares three stretched tunings of 42edo:
| Tuning | 42ed257/128 | APS720jot | 189zpi | 42edo |
|---|---|---|---|---|
| Steps / octave | ~41.77 | ~41.81 | ~41.83 | 42.00 |
| Approximation of harmonics |
great: 5 good: 2, 3, 7 okay: bad: 11, 13 |
great: 5 good: 2, 3 okay: 7, 11, 13 bad: |
great: 5 good: 2, 13 okay: 3, 11 bad: 7 |
great: 2, 7 good: okay: 11 bad: 3, 5, 13 |
“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%
| ||||
The following table compares three compressed tunings of 42edo:
| Tuning | 42edo | APS715jot | 191zpi | 42ed255/128 |
|---|---|---|---|---|
| Steps / octave | 42.00 | ~42.10 | ~42.19 | ~42.24 |
| Approximation of harmonics |
great: 2, 7 good: okay: 11 bad: 3, 5, 13 |
great: 2 good: 5, 7, 13 okay: 3, 11 bad: |
great: 5, 11, 13 good: 2, 3 okay: bad: 7 |
great: 3, 5, 11 good: 2 okay: 13 bad: 7 |
“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%
| ||||
For a more detailed comparison see Table of stretched 42edo tunings.
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.
Scales
- Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
- Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
- Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
- Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
- Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2
- Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5
- Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1
- Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4
- Subsets of MOS scales
(Names used are idiosyncratic.)
- Eugene/Tritikleismic[9]
- Groovy aeolian pentatonic: 11 6 8 3 14
- Otonal mixolydian pentatonic: 14 3 8 11 6
- Pseudo-equipentatonic: 11 6 8 6 11
- Septimal melodic minor pentatonic: 8 3 14 14 3
- Septimal Picardy pentatonic: 8 6 11 3 14
- Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
- Yokai pentatonic: 3 14 8 3 14
Instruments
- Lumatone
See Lumatone mapping for 42edo
Music
- Improvisation in 42edo - composed and played by Bryan Deister (May 2023), transcribed by Stephen Weigel (Sept 2024)
- Four Short Experiments in Octave Stretched 42edo - Budjarn Lambeth (Dec 2024)
- Glory of Them - Mundoworld (July 2024)