42edo: Difference between revisions

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→Intervals of 42edo: Removed pions column
Line 15: Line 15:
! | Degree
! | Degree
! | Size (Cents)
! | Size (Cents)
!pions
!7mus
!7mus
! colspan="2" | relative notation
! colspan="2" | relative notation
! | absolute notation
! | absolute notation
|-
|-
| colspan="4" style="text-align:center;" | 0
| style="text-align:center;" | 0
|0
|0
| style="text-align:center;" | perfect unison
| style="text-align:center;" | perfect unison
| style="text-align:center;" | P1
| style="text-align:center;" | P1
Line 27: Line 28:
| style="text-align:center;" | 1
| style="text-align:center;" | 1
| style="text-align:center;" | 29.571
| style="text-align:center;" | 29.571
|30.286
|36.571 (24.925<sub>16</sub>)
|36.571 (24.925<sub>16</sub>)
| style="text-align:center;" | up 1sn, minor 2nd
| style="text-align:center;" | up 1sn, minor 2nd
Line 35: Line 35:
| style="text-align:center;" | 2
| style="text-align:center;" | 2
| style="text-align:center;" |57.143
| style="text-align:center;" |57.143
|60.571
|73.143 (49.249<sub>16</sub>)
|73.143 (49.249<sub>16</sub>)
| style="text-align:center;" | double-up 1sn, upminor 2nd
| style="text-align:center;" | double-up 1sn, upminor 2nd
Line 43: Line 42:
| style="text-align:center;" | 3
| style="text-align:center;" | 3
| style="text-align:center;" |85.714
| style="text-align:center;" |85.714
|90.857
|109.714 (6D.B6E<sub>16</sub>)
|109.714 (6D.B6E<sub>16</sub>)
| style="text-align:center;" | double-up minor 2nd
| style="text-align:center;" | double-up minor 2nd
Line 51: Line 49:
| style="text-align:center;" | 4
| style="text-align:center;" | 4
| style="text-align:center;" |114.286
| style="text-align:center;" |114.286
|121.143
|146.286 (92.492<sub>16</sub>)
|146.286 (92.492<sub>16</sub>)
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | downmid 2nd
Line 59: Line 56:
| style="text-align:center;" | 5
| style="text-align:center;" | 5
| style="text-align:center;" | 143.857
| style="text-align:center;" | 143.857
|151.429
|182.857 (B6.DB7<sub>16</sub>)
|182.857 (B6.DB7<sub>16</sub>)
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | upmid 2nd
Line 67: Line 63:
| style="text-align:center;" | 6
| style="text-align:center;" | 6
| style="text-align:center;" |171.429
| style="text-align:center;" |171.429
|181.714
|219.429 (DB.6DB<sub>16</sub>)
|219.429 (DB.6DB<sub>16</sub>)
| style="text-align:center;" | double-down major 2nd
| style="text-align:center;" | double-down major 2nd
Line 75: Line 70:
| style="text-align:center;" | 7
| style="text-align:center;" | 7
| style="text-align:center;" | 200
| style="text-align:center;" | 200
|212
|256 (100<sub>16</sub>)
|256 (100<sub>16</sub>)
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | downmajor 2nd
Line 83: Line 77:
| style="text-align:center;" | 8
| style="text-align:center;" | 8
| style="text-align:center;" |228.571
| style="text-align:center;" |228.571
|242.286
|292.571 (124.925<sub>16</sub>)
|292.571 (124.925<sub>16</sub>)
| style="text-align:center;" | major 2nd
| style="text-align:center;" | major 2nd
Line 91: Line 84:
| style="text-align:center;" | 9
| style="text-align:center;" | 9
| style="text-align:center;" |257.143
| style="text-align:center;" |257.143
|282.571
|329.143 (149.249<sub>16</sub>)
|329.143 (149.249<sub>16</sub>)
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | minor 3rd
Line 99: Line 91:
| style="text-align:center;" | 10
| style="text-align:center;" | 10
| style="text-align:center;" |285.714
| style="text-align:center;" |285.714
|302.857
|365.714 (16D.B6E<sub>16</sub>)
|365.714 (16D.B6E<sub>16</sub>)
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | upminor 3rd
Line 107: Line 98:
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:center;" | 314.286
| style="text-align:center;" | 314.286
|333.143
|402.286 (192.492<sub>16</sub>)
|402.286 (192.492<sub>16</sub>)
| style="text-align:center;" | double-up minor 3rd
| style="text-align:center;" | double-up minor 3rd
Line 115: Line 105:
| style="text-align:center;" | 12
| style="text-align:center;" | 12
| style="text-align:center;" |342.857
| style="text-align:center;" |342.857
|363.429
|438.857 (1B6.DB7<sub>16</sub>)
|438.857 (1B6.DB7<sub>16</sub>)
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | downmid 3rd
Line 123: Line 112:
| style="text-align:center;" | 13
| style="text-align:center;" | 13
| style="text-align:center;" | 371.429
| style="text-align:center;" | 371.429
|393.714
|475.429 (1DB.6DB<sub>16</sub>)
|475.429 (1DB.6DB<sub>16</sub>)
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | upmid 3rd
Line 131: Line 119:
| style="text-align:center;" | 14
| style="text-align:center;" | 14
| style="text-align:center;" | 400
| style="text-align:center;" | 400
|424
|512 (200<sub>16</sub>)
|512 (200<sub>16</sub>)
| style="text-align:center;" | double-down major 3rd
| style="text-align:center;" | double-down major 3rd
Line 139: Line 126:
| style="text-align:center;" | 15
| style="text-align:center;" | 15
| style="text-align:center;" |428.571
| style="text-align:center;" |428.571
|454.286
|548.571 (224.925<sub>16</sub>)
|548.571 (224.925<sub>16</sub>)
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | downmajor 3rd
Line 147: Line 133:
| style="text-align:center;" | 16
| style="text-align:center;" | 16
| style="text-align:center;" |457.143
| style="text-align:center;" |457.143
|484.571
|585.143 (249.249<sub>16</sub>)
|585.143 (249.249<sub>16</sub>)
| style="text-align:center;" | major 3rd, down 4th
| style="text-align:center;" | major 3rd, down 4th
Line 155: Line 140:
| style="text-align:center;" | 17
| style="text-align:center;" | 17
| style="text-align:center;" | 485.714
| style="text-align:center;" | 485.714
|514.857
|621.714 (26D.B6E<sub>16</sub>)
|621.714 (26D.B6E<sub>16</sub>)
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | perfect 4th
Line 163: Line 147:
| style="text-align:center;" | 18
| style="text-align:center;" | 18
| style="text-align:center;" |51<span style="line-height: 1.5;">4.286</span>
| style="text-align:center;" |51<span style="line-height: 1.5;">4.286</span>
|545.143
|658.286 (292.492<sub>16</sub>)
|658.286 (292.492<sub>16</sub>)
| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
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| style="text-align:center;" | 19
| style="text-align:center;" | 19
| style="text-align:center;" | 543.857
| style="text-align:center;" | 543.857
|575.429
|694.857 (2B6.DB7<sub>16</sub>)
|694.857 (2B6.DB7<sub>16</sub>)
| style="text-align:center;" | double-up 4th
| style="text-align:center;" | double-up 4th
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| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:center;" |571.429
| style="text-align:center;" |571.429
|605.714
|731.429 (2DB.6DB<sub>16</sub>)
|731.429 (2DB.6DB<sub>16</sub>)
| style="text-align:center;" | triple-up 4th
| style="text-align:center;" | triple-up 4th
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| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:center;" | 600
| style="text-align:center;" | 600
|6
|768 (300<sub>16</sub>)
|768 (300<sub>16</sub>)
| style="text-align:center;" | triple-down aug 4th, triple-up dim 5th
| style="text-align:center;" | triple-down aug 4th, triple-up dim 5th
Line 195: Line 175:
| style="text-align:center;" | 22
| style="text-align:center;" | 22
| style="text-align:center;" |628.571
| style="text-align:center;" |628.571
|666.286
|804.571 (324.925<sub>16</sub>)
|804.571 (324.925<sub>16</sub>)
| style="text-align:center;" | triple-down 5th
| style="text-align:center;" | triple-down 5th
Line 203: Line 182:
| style="text-align:center;" | 23
| style="text-align:center;" | 23
| style="text-align:center;" | 657.143
| style="text-align:center;" | 657.143
|696.571
|841.143 (349.249<sub>16</sub>)
|841.143 (349.249<sub>16</sub>)
| style="text-align:center;" | double-down 5th
| style="text-align:center;" | double-down 5th
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| style="text-align:center;" | 24
| style="text-align:center;" | 24
| style="text-align:center;" |685.714
| style="text-align:center;" |685.714
|726.857
|877.714 (36D.B6E<sub>16</sub>)
|877.714 (36D.B6E<sub>16</sub>)
| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
Line 219: Line 196:
| style="text-align:center;" | 25
| style="text-align:center;" | 25
| style="text-align:center;" | 714.286
| style="text-align:center;" | 714.286
|757.143
|950.286 (392.492<sub>16</sub>)
|950.286 (392.492<sub>16</sub>)
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | perfect 5th
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| style="text-align:center;" | 26
| style="text-align:center;" | 26
| style="text-align:center;" |742.857
| style="text-align:center;" |742.857
|787.429
|950.857 (3B6.DB7<sub>16</sub>)
|950.857 (3B6.DB7<sub>16</sub>)
| style="text-align:center;" | up 5th, minor 6th
| style="text-align:center;" | up 5th, minor 6th
Line 235: Line 210:
| style="text-align:center;" | 27
| style="text-align:center;" | 27
| style="text-align:center;" |771.429
| style="text-align:center;" |771.429
|817.714
|987.429 (3DB.6DB<sub>16</sub>)
|987.429 (3DB.6DB<sub>16</sub>)
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | upminor 6th
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| style="text-align:center;" | 28
| style="text-align:center;" | 28
| style="text-align:center;" | 800
| style="text-align:center;" | 800
|848
|1024 (400<sub>16</sub>)
|1024 (400<sub>16</sub>)
| style="text-align:center;" | double-up minor 6th
| style="text-align:center;" | double-up minor 6th
Line 251: Line 224:
| style="text-align:center;" | 29
| style="text-align:center;" | 29
| style="text-align:center;" | 829.571
| style="text-align:center;" | 829.571
|878.286
|1060.571 (424.925<sub>16</sub>)
|1060.571 (424.925<sub>16</sub>)
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | downmid 6th
Line 259: Line 231:
| style="text-align:center;" | 30
| style="text-align:center;" | 30
| style="text-align:center;" |857.143
| style="text-align:center;" |857.143
|908.571
|1097.143 (449.249<sub>16</sub>)
|1097.143 (449.249<sub>16</sub>)
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | upmid 6th
Line 267: Line 238:
| style="text-align:center;" | 31
| style="text-align:center;" | 31
| style="text-align:center;" | 885.714
| style="text-align:center;" | 885.714
|938.857
|1133.714 (46D.B6E<sub>16</sub>)
|1133.714 (46D.B6E<sub>16</sub>)
| style="text-align:center;" | double-down major 6th
| style="text-align:center;" | double-down major 6th
Line 275: Line 245:
| style="text-align:center;" | 32
| style="text-align:center;" | 32
| style="text-align:center;" |914.286
| style="text-align:center;" |914.286
|969.143
|1170.286 (492.492<sub>16</sub>)
|1170.286 (492.492<sub>16</sub>)
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | downmajor 6th
Line 283: Line 252:
| style="text-align:center;" | 33
| style="text-align:center;" | 33
| style="text-align:center;" |942.857
| style="text-align:center;" |942.857
|999.429
|1206.857 (4B6.DB7<sub>16</sub>)
|1206.857 (4B6.DB7<sub>16</sub>)
| style="text-align:center;" | major 6th
| style="text-align:center;" | major 6th
Line 291: Line 259:
| style="text-align:center;" | 34
| style="text-align:center;" | 34
| style="text-align:center;" |971.429
| style="text-align:center;" |971.429
|1029.714
|1243.429 (4DB.6DB<sub>16</sub>)
|1243.429 (4DB.6DB<sub>16</sub>)
| style="text-align:center;" | minor 7th
| style="text-align:center;" | minor 7th
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| style="text-align:center;" | 35
| style="text-align:center;" | 35
| style="text-align:center;" | 1000
| style="text-align:center;" | 1000
|1060
|1280 (500<sub>16</sub>)
|1280 (500<sub>16</sub>)
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | upminor 7th
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| style="text-align:center;" | 36
| style="text-align:center;" | 36
| style="text-align:center;" |1028.571
| style="text-align:center;" |1028.571
|1090.286
|1316.571 (524.925<sub>16</sub>)
|1316.571 (524.925<sub>16</sub>)
| style="text-align:center;" | double-up minor 7th
| style="text-align:center;" | double-up minor 7th
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| style="text-align:center;" | 37
| style="text-align:center;" | 37
| style="text-align:center;" | 1057.143
| style="text-align:center;" | 1057.143
|1120.571
|1353.143 (549.249<sub>16</sub>)
|1353.143 (549.249<sub>16</sub>)
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | downmid 7th
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| style="text-align:center;" | 38
| style="text-align:center;" | 38
| style="text-align:center;" |1085.714
| style="text-align:center;" |1085.714
|1150.857
|1389.714 (56D.B6E<sub>16</sub>)
|1389.714 (56D.B6E<sub>16</sub>)
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | upmid 7th
Line 331: Line 294:
| style="text-align:center;" | 39
| style="text-align:center;" | 39
| style="text-align:center;" |1114.286
| style="text-align:center;" |1114.286
|1181.143
|1426.286 (592.492<sub>16</sub>)
|1426.286 (592.492<sub>16</sub>)
| style="text-align:center;" | double-down major 7th
| style="text-align:center;" | double-down major 7th
Line 339: Line 301:
| style="text-align:center;" | 40
| style="text-align:center;" | 40
| style="text-align:center;" |1142.857
| style="text-align:center;" |1142.857
|1211.429
|1462.857 (5B6.DB7<sub>16</sub>)
|1462.857 (5B6.DB7<sub>16</sub>)
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | downmajor 7th
Line 347: Line 308:
| style="text-align:center;" | 41
| style="text-align:center;" | 41
| style="text-align:center;" | 1171.429
| style="text-align:center;" | 1171.429
|1241.714
|1499.429 (5DB.6DB<sub>16</sub>)
|1499.429 (5DB.6DB<sub>16</sub>)
| style="text-align:center;" | major 7th, down 8ve
| style="text-align:center;" | major 7th, down 8ve
Line 355: Line 315:
| style="text-align:center;" | 42
| style="text-align:center;" | 42
| style="text-align:center;" | 1200
| style="text-align:center;" | 1200
|1272
|1536 (600<sub>16</sub>)
|1536 (600<sub>16</sub>)
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" | perfect 8ve

Revision as of 13:34, 12 December 2019

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
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