42edo

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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) pions 7mus relative notation absolute notation
0 0 perfect unison P1 D
1 29.571 30.286 36.571 (24.92516) up 1sn, minor 2nd ^1, m2 D^, Eb
2 57.143 60.571 73.143 (49.24916) double-up 1sn, upminor 2nd ^^1, ^m2 D^^, Eb^
3 85.714 90.857 109.714 (6D.B6E16) double-up minor 2nd ^^m2 Eb^^
4 114.286 121.143 146.286 (92.49216) downmid 2nd v~2 Eb^3
5 143.857 151.429 182.857 (B6.DB716) upmid 2nd ^~2 Ev3
6 171.429 181.714 219.429 (DB.6DB16) double-down major 2nd vvM2 Evv
7 200 212 256 (10016) downmajor 2nd vM2 Ev
8 228.571 242.286 292.571 (124.92516) major 2nd M2 E
9 257.143 282.571 329.143 (149.24916) minor 3rd m3 F
10 285.714 302.857 365.714 (16D.B6E16) upminor 3rd ^m3 F^
11 314.286 333.143 402.286 (192.49216) double-up minor 3rd ^^m3 F^^
12 342.857 363.429 438.857 (1B6.DB716) downmid 3rd v~3 F^3
13 371.429 393.714 475.429 (1DB.6DB16) upmid 3rd ^~3 F#v3
14 400 424 512 (20016) double-down major 3rd vvM3 F#vv
15 428.571 454.286 548.571 (224.92516) downmajor 3rd vM3 F#v
16 457.143 484.571 585.143 (249.24916) major 3rd, down 4th M3, v4 F#, Gv
17 485.714 514.857 621.714 (26D.B6E16) perfect 4th P4 G
18 514.286 545.143 658.286 (292.49216) up 4th ^4 G^
19 543.857 575.429 694.857 (2B6.DB716) double-up 4th ^^4 G^^
20 571.429 605.714 731.429 (2DB.6DB16) triple-up 4th ^34 G^3
21 600 6 768 (30016) triple-down aug 4th, triple-up dim 5th v3A4, ^3d5 G#v3, Ab^3
22 628.571 666.286 804.571 (324.92516) triple-down 5th v35 Av3
23 657.143 696.571 841.143 (349.24916) double-down 5th vv5 Avv
24 685.714 726.857 877.714 (36D.B6E16) down 5th v5 Av
25 714.286 757.143 950.286 (392.49216) perfect 5th P5 A
26 742.857 787.429 950.857 (3B6.DB716) up 5th, minor 6th ^5, m6 A^, Bb
27 771.429 817.714 987.429 (3DB.6DB16) upminor 6th ^m6 Bb^
28 800 848 1024 (40016) double-up minor 6th ^^m6 Bb^^
29 829.571 878.286 1060.571 (424.92516) downmid 6th v~6 Bb^3
30 857.143 908.571 1097.143 (449.24916) upmid 6th ^~6 Bv3
31 885.714 938.857 1133.714 (46D.B6E16) double-down major 6th vvM6 Bvv
32 914.286 969.143 1170.286 (492.49216) downmajor 6th vM6 Bv
33 942.857 999.429 1206.857 (4B6.DB716) major 6th M6 B
34 971.429 1029.714 1243.429 (4DB.6DB16) minor 7th m7 C
35 1000 1060 1280 (50016) upminor 7th ^m7 C^
36 1028.571 1090.286 1316.571 (524.92516) double-up minor 7th ^^m7 C^^
37 1057.143 1120.571 1353.143 (549.24916) downmid 7th v~7 C^3
38 1085.714 1150.857 1389.714 (56D.B6E16) upmid 7th ^~7 C#v3
39 1114.286 1181.143 1426.286 (592.49216) double-down major 7th vvM7 C#vv
40 1142.857 1211.429 1462.857 (5B6.DB716) downmajor 7th vM7 C#v
41 1171.429 1241.714 1499.429 (5DB.6DB16) major 7th, down 8ve M7, v8 C#, Dv
42 1200 1272 1536 (60016) perfect 8ve P8 D