42edo
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 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 regular edo because its 5th falls between 4\7 = 686¢ and 3\5 = 720¢. 42edo is one of the most difficult regular edos to notate, because no other regular 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 roughly 5edo-ish 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 G#v3 - B#v5 - D#v3 - F#v or as Ab^3 - C^ - Eb^3 - Gb^5. This is a dot-double-down double-up-seven chord, written either as G#v3.vv,^^7 or as Ab^3.vv,^^7.
Intervals of 42edo
| Degree | Size (Cents) | 7mus | relative notation | absolute notation | |
|---|---|---|---|---|---|
| 0 | 0 | 0 | perfect unison | P1 | D |
| 1 | 29.571 | 36.571 (24.92516) | up 1sn, minor 2nd | ^1, m2 | D^, Eb |
| 2 | 57.143 | 73.143 (49.24916) | double-up 1sn, upminor 2nd | ^^1, ^m2 | D^^, Eb^ |
| 3 | 85.714 | 109.714 (6D.B6E16) | double-up minor 2nd | ^^m2 | Eb^^ |
| 4 | 114.286 | 146.286 (92.49216) | downmid 2nd | v~2 | Eb^3 |
| 5 | 143.857 | 182.857 (B6.DB716) | upmid 2nd | ^~2 | Ev3 |
| 6 | 171.429 | 219.429 (DB.6DB16) | double-down major 2nd | vvM2 | Evv |
| 7 | 200 | 256 (10016) | downmajor 2nd | vM2 | Ev |
| 8 | 228.571 | 292.571 (124.92516) | major 2nd | M2 | E |
| 9 | 257.143 | 329.143 (149.24916) | minor 3rd | m3 | F |
| 10 | 285.714 | 365.714 (16D.B6E16) | upminor 3rd | ^m3 | F^ |
| 11 | 314.286 | 402.286 (192.49216) | double-up minor 3rd | ^^m3 | F^^ |
| 12 | 342.857 | 438.857 (1B6.DB716) | downmid 3rd | v~3 | F^3 |
| 13 | 371.429 | 475.429 (1DB.6DB16) | upmid 3rd | ^~3 | F#v3 |
| 14 | 400 | 512 (20016) | double-down major 3rd | vvM3 | F#vv |
| 15 | 428.571 | 548.571 (224.92516) | downmajor 3rd | vM3 | F#v |
| 16 | 457.143 | 585.143 (249.24916) | major 3rd, down 4th | M3, v4 | F#, Gv |
| 17 | 485.714 | 621.714 (26D.B6E16) | perfect 4th | P4 | G |
| 18 | 514.286 | 658.286 (292.49216) | up 4th | ^4 | G^ |
| 19 | 543.857 | 694.857 (2B6.DB716) | double-up 4th | ^^4 | G^^ |
| 20 | 571.429 | 731.429 (2DB.6DB16) | triple-up 4th | ^34 | G^3 |
| 21 | 600 | 768 (30016) | triple-down aug 4th, triple-up dim 5th | v3A4, ^3d5 | G#v3, Ab^3 |
| 22 | 628.571 | 804.571 (324.92516) | triple-down 5th | v35 | Av3 |
| 23 | 657.143 | 841.143 (349.24916) | double-down 5th | vv5 | Avv |
| 24 | 685.714 | 877.714 (36D.B6E16) | down 5th | v5 | Av |
| 25 | 714.286 | 950.286 (392.49216) | perfect 5th | P5 | A |
| 26 | 742.857 | 950.857 (3B6.DB716) | up 5th, minor 6th | ^5, m6 | A^, Bb |
| 27 | 771.429 | 987.429 (3DB.6DB16) | upminor 6th | ^m6 | Bb^ |
| 28 | 800 | 1024 (40016) | double-up minor 6th | ^^m6 | Bb^^ |
| 29 | 829.571 | 1060.571 (424.92516) | downmid 6th | v~6 | Bb^3 |
| 30 | 857.143 | 1097.143 (449.24916) | upmid 6th | ^~6 | Bv3 |
| 31 | 885.714 | 1133.714 (46D.B6E16) | double-down major 6th | vvM6 | Bvv |
| 32 | 914.286 | 1170.286 (492.49216) | downmajor 6th | vM6 | Bv |
| 33 | 942.857 | 1206.857 (4B6.DB716) | major 6th | M6 | B |
| 34 | 971.429 | 1243.429 (4DB.6DB16) | minor 7th | m7 | C |
| 35 | 1000 | 1280 (50016) | upminor 7th | ^m7 | C^ |
| 36 | 1028.571 | 1316.571 (524.92516) | double-up minor 7th | ^^m7 | C^^ |
| 37 | 1057.143 | 1353.143 (549.24916) | downmid 7th | v~7 | C^3 |
| 38 | 1085.714 | 1389.714 (56D.B6E16) | upmid 7th | ^~7 | C#v3 |
| 39 | 1114.286 | 1426.286 (592.49216) | double-down major 7th | vvM7 | C#vv |
| 40 | 1142.857 | 1462.857 (5B6.DB716) | downmajor 7th | vM7 | C#v |
| 41 | 1171.429 | 1499.429 (5DB.6DB16) | major 7th, down 8ve | M7, v8 | C#, Dv |
| 42 | 1200 | 1536 (60016) | perfect 8ve | P8 | D |