The 42 equal division divides the octave into 42 equal parts of 28.571 cents each. It has a 3 (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a 5 both over 12 cents sharp, using the same 400 cent interval to represent 5/4 as does 12, 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 188.8.131.52.33.39 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.
Intervals of 42edo
|Degree||Cents||ups and downs notation|
|1||29.571||^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|
|6||171.429||vvM2||double-down major 2nd||vvE|
|11||314.286||^^m3||double-up minor 3rd||^^F|
|14||400||vvM3||double-down major 3rd||vvF#|
|16||457.143||M3, v4||major 3rd, down 4th||F#, vG|
|21||600||^~4, v~5||upmid 4th, downmid 5th||v3G#, ^3Ab|
|26||742.857||^5, m6||up 5th, minor 6th||^A, Bb|
|28||800||^^m6||double-up minor 6th||^^Bb|
|31||885.714||vvM6||double-down major 6th||vvB|
|36||1028.571||^^m7||double-up minor 7th||^^C|
|39||1114.286||vvM7||double-down major 7th||vvC#|
|41||1171.429||M7, v8||major 7th, down 8ve||C#, vD|
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See Ups and Downs Notation - Chords and Chord Progressions.