42edo: Difference between revisions

Jkarimak (talk | contribs)
Intervals of 42edo: Removed 7mus column
TallKite (talk | contribs)
updated ups/down notation
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While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 [[k*N_subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo|84edo]]. On this subgroup 42 has the same [[Comma|comma]]s as 84.
While not an accurate tuning on the full 7-limit, it does an excellent job on the 2.9.15.7.33.39 [[k*N_subgroups|2*42 subgroup]], having the same tuning on it as does [[84edo|84edo]]. On this subgroup 42 has the same [[Comma|comma]]s as 84.


42edo is a [[Ups_and_Downs_Notation#Other EDOs|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:
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 * * * * * * * * 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#v<span style="font-size: 90%; vertical-align: super;">3</span> - B#v<span style="font-size: 90%; vertical-align: super;">5</span> - D#v<span style="font-size: 90%; vertical-align: super;">3</span> - F#v or as Ab^<span style="font-size: 90%; vertical-align: super;">3</span> - C^ - Eb^<span style="font-size: 90%; vertical-align: super;">3</span> - Gb^<span style="font-size: 90%; vertical-align: super;">5</span>. This is a dot-double-down double-up-seven chord, written either as G#v<span style="font-size: 90%; vertical-align: super;">3</span>.vv,^^7 or as Ab^<span style="font-size: 90%; vertical-align: super;">3</span>.vv,^^7.
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 v<span style="font-size: 90%; vertical-align: super;">3</span>G# - v<span style="font-size: 90%; vertical-align: super;">5</span>B# - v<span style="font-size: 90%; vertical-align: super;">3</span>D# - vF# or as ^<span style="font-size: 90%; vertical-align: super;">3</span>Ab - ^C - ^<span style="font-size: 90%; vertical-align: super;">3</span>Eb - ^<span style="font-size: 90%; vertical-align: super;">5</span>Gb. This is a double-down double-up-seven chord, written either as v<span style="font-size: 90%; vertical-align: super;">3</span>G#vv,^^7 or as ^<span style="font-size: 90%; vertical-align: super;">3</span>Abvv,^^7.


==Intervals of 42edo==
==Intervals of 42edo==
Line 28: Line 28:
| style="text-align:center;" | up 1sn, minor 2nd
| style="text-align:center;" | up 1sn, minor 2nd
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | D^, Eb
| style="text-align:center;" | ^D, Eb
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
Line 34: Line 34:
| style="text-align:center;" | double-up 1sn, upminor 2nd
| style="text-align:center;" | double-up 1sn, upminor 2nd
| style="text-align:center;" | ^^1, ^m2
| style="text-align:center;" | ^^1, ^m2
| style="text-align:center;" | D^^, Eb^
| style="text-align:center;" | ^^D, ^Eb
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 3
Line 40: Line 40:
| style="text-align:center;" | double-up minor 2nd
| style="text-align:center;" | double-up minor 2nd
| style="text-align:center;" | ^^m2
| style="text-align:center;" | ^^m2
| style="text-align:center;" | Eb^^
| style="text-align:center;" | ^^Eb
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
Line 46: Line 46:
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | v~2
| style="text-align:center;" | v~2
| style="text-align:center;" | Eb^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Eb
|-
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 5
Line 52: Line 52:
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | ^~2
| style="text-align:center;" | ^~2
| style="text-align:center;" | Ev<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
Line 58: Line 58:
| style="text-align:center;" | double-down major 2nd
| style="text-align:center;" | double-down major 2nd
| style="text-align:center;" | vvM2
| style="text-align:center;" | vvM2
| style="text-align:center;" | Evv
| style="text-align:center;" | vvE
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 7
Line 64: Line 64:
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | vM2
| style="text-align:center;" | vM2
| style="text-align:center;" | Ev
| style="text-align:center;" | vE
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
Line 82: Line 82:
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | ^m3
| style="text-align:center;" | ^m3
| style="text-align:center;" | F^
| style="text-align:center;" | ^F
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
Line 88: Line 88:
| style="text-align:center;" | double-up minor 3rd
| style="text-align:center;" | double-up minor 3rd
| style="text-align:center;" | ^^m3
| style="text-align:center;" | ^^m3
| style="text-align:center;" | F^^
| style="text-align:center;" | ^^F
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
Line 94: Line 94:
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | v~3
| style="text-align:center;" | v~3
| style="text-align:center;" | F^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 13
Line 100: Line 100:
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | ^~3
| style="text-align:center;" | ^~3
| style="text-align:center;" | F#v<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>F#
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
Line 106: Line 106:
| style="text-align:center;" | double-down major 3rd
| style="text-align:center;" | double-down major 3rd
| style="text-align:center;" | vvM3
| style="text-align:center;" | vvM3
| style="text-align:center;" | F#vv
| style="text-align:center;" | vvF#
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 15
Line 112: Line 112:
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | vM3
| style="text-align:center;" | vM3
| style="text-align:center;" | F#v
| style="text-align:center;" | vF#
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
Line 118: Line 118:
| style="text-align:center;" | major 3rd, down 4th
| style="text-align:center;" | major 3rd, down 4th
| style="text-align:center;" | M3, v4
| style="text-align:center;" | M3, v4
| style="text-align:center;" | F#, Gv
| style="text-align:center;" | F#, vG
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 17
Line 130: Line 130:
| style="text-align:center;" | up 4th
| style="text-align:center;" | up 4th
| style="text-align:center;" | ^4
| style="text-align:center;" | ^4
| style="text-align:center;" | G^
| style="text-align:center;" | ^G
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" | 19
Line 136: Line 136:
| style="text-align:center;" | double-up 4th
| style="text-align:center;" | double-up 4th
| style="text-align:center;" | ^^4
| style="text-align:center;" | ^^4
| style="text-align:center;" | G^^
| style="text-align:center;" | ^^G
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
| style="text-align:center;" |571.429
| style="text-align:center;" |571.429
| style="text-align:center;" | triple-up 4th
| style="text-align:center;" | downmid 4th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4
| style="text-align:center;" | v~4
| style="text-align:center;" | G^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" | 21
| style="text-align:center;" | 600
| style="text-align:center;" | 600
| style="text-align:center;" | triple-down aug 4th, triple-up dim 5th
| style="text-align:center;" | upmid 4th, downmid 5th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A4, ^<span style="font-size: 90%; vertical-align: super;">3</span>d5
| style="text-align:center;" | ^~4, v~5
| style="text-align:center;" | G#v<span style="font-size: 90%; vertical-align: super;">3</span>, Ab^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>G#, ^<span style="font-size: 90%; vertical-align: super;">3</span>Ab
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
| style="text-align:center;" |628.571
| style="text-align:center;" |628.571
| style="text-align:center;" | triple-down 5th
| style="text-align:center;" | upmid 5th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5
| style="text-align:center;" | ^~5
| style="text-align:center;" | Av<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" | 23
Line 160: Line 160:
| style="text-align:center;" | double-down 5th
| style="text-align:center;" | double-down 5th
| style="text-align:center;" | vv5
| style="text-align:center;" | vv5
| style="text-align:center;" | Avv
| style="text-align:center;" | vvA
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
Line 166: Line 166:
| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
| style="text-align:center;" | v5
| style="text-align:center;" | v5
| style="text-align:center;" | Av
| style="text-align:center;" | vA
|-
|-
| style="text-align:center;" | 25
| style="text-align:center;" | 25
Line 178: Line 178:
| style="text-align:center;" | up 5th, minor 6th
| style="text-align:center;" | up 5th, minor 6th
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | A^, Bb
| style="text-align:center;" | ^A, Bb
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" | 27
Line 184: Line 184:
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | ^m6
| style="text-align:center;" | ^m6
| style="text-align:center;" | Bb^
| style="text-align:center;" | ^Bb
|-
|-
| style="text-align:center;" | 28
| style="text-align:center;" | 28
Line 190: Line 190:
| style="text-align:center;" | double-up minor 6th
| style="text-align:center;" | double-up minor 6th
| style="text-align:center;" | ^^m6
| style="text-align:center;" | ^^m6
| style="text-align:center;" | Bb^^
| style="text-align:center;" | ^^Bb
|-
|-
| style="text-align:center;" | 29
| style="text-align:center;" | 29
Line 196: Line 196:
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | v~6
| style="text-align:center;" | v~6
| style="text-align:center;" | Bb^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Bb
|-
|-
| style="text-align:center;" | 30
| style="text-align:center;" | 30
Line 202: Line 202:
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | ^~6
| style="text-align:center;" | ^~6
| style="text-align:center;" | Bv<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B
|-
|-
| style="text-align:center;" | 31
| style="text-align:center;" | 31
Line 208: Line 208:
| style="text-align:center;" | double-down major 6th
| style="text-align:center;" | double-down major 6th
| style="text-align:center;" | vvM6
| style="text-align:center;" | vvM6
| style="text-align:center;" | Bvv
| style="text-align:center;" | vvB
|-
|-
| style="text-align:center;" | 32
| style="text-align:center;" | 32
Line 214: Line 214:
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | vM6
| style="text-align:center;" | vM6
| style="text-align:center;" | Bv
| style="text-align:center;" | vB
|-
|-
| style="text-align:center;" | 33
| style="text-align:center;" | 33
Line 232: Line 232:
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | ^m7
| style="text-align:center;" | ^m7
| style="text-align:center;" | C^
| style="text-align:center;" | ^C
|-
|-
| style="text-align:center;" | 36
| style="text-align:center;" | 36
Line 238: Line 238:
| style="text-align:center;" | double-up minor 7th
| style="text-align:center;" | double-up minor 7th
| style="text-align:center;" | ^^m7
| style="text-align:center;" | ^^m7
| style="text-align:center;" | C^^
| style="text-align:center;" | ^^C
|-
|-
| style="text-align:center;" | 37
| style="text-align:center;" | 37
Line 244: Line 244:
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | v~7
| style="text-align:center;" | v~7
| style="text-align:center;" | C^<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C
|-
|-
| style="text-align:center;" | 38
| style="text-align:center;" | 38
Line 250: Line 250:
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | ^~7
| style="text-align:center;" | ^~7
| style="text-align:center;" | C#v<span style="font-size: 90%; vertical-align: super;">3</span>
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>C#
|-
|-
| style="text-align:center;" | 39
| style="text-align:center;" | 39
Line 256: Line 256:
| style="text-align:center;" | double-down major 7th
| style="text-align:center;" | double-down major 7th
| style="text-align:center;" | vvM7
| style="text-align:center;" | vvM7
| style="text-align:center;" | C#vv
| style="text-align:center;" | vvC#
|-
|-
| style="text-align:center;" | 40
| style="text-align:center;" | 40
Line 262: Line 262:
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | vM7
| style="text-align:center;" | vM7
| style="text-align:center;" | C#v
| style="text-align:center;" | vC#
|-
|-
| style="text-align:center;" | 41
| style="text-align:center;" | 41
Line 268: Line 268:
| style="text-align:center;" | major 7th, down 8ve
| style="text-align:center;" | major 7th, down 8ve
| style="text-align:center;" | M7, v8
| style="text-align:center;" | M7, v8
| style="text-align:center;" | C#, Dv
| style="text-align:center;" | C#, vD
|-
|-
| style="text-align:center;" | 42
| style="text-align:center;" | 42
Line 276: Line 276:
| style="text-align:center;" | D
| style="text-align:center;" | 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|Ups and Downs Notation - Chords and Chord Progressions]].
[[Category:augene]]
[[Category:augene]]
[[Category:edo]]
[[Category:edo]]
[[Category:subgroup]]
[[Category:subgroup]]