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| The ''42 equal division'' divides the octave into 42 equal parts of 28.571 [[cent]]s 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 [[12edo|12]], which means it [[tempering_out|tempers out]] 128/125. In the [[7-limit]], it tempers out 64/63 and [[126/125]], making it a tuning supporting [[Augmented_family|augene temperament]]. | | The '''42edo''' is the equal division of the octave into 42 equal parts of 28.571 [[cent]]s each. It has a third (the size of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 7n) and a fifth both over 12 cents sharp, using the same 400 cent interval to represent [[5/4]] as does [[12edo|12]], which means it [[tempering out|tempers out]] 128/125. In the [[7-limit]], it tempers out 64/63 and [[126/125]], making it a tuning supporting [[augene]] temperament. |
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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]]. On this subgroup 42 has the same [[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]]. On this subgroup 42 has the same [[comma]]s as 84. |
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| D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D | | D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D |
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| 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. | | 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<sup>3</sup>G# - v<sup>5</sup>B# - v<sup>3</sup>D# - vF# or as ^<sup>3</sup>Ab - ^C - ^<sup>3</sup>Eb - ^<sup>5</sup>Gb. This is a double-down double-up-seven chord, written either as v<sup>3</sup>G#vv,^^7 or as ^<sup>3</sup>Abvv,^^7. |
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| ==Intervals of 42edo== | | ==Intervals of 42edo== |
|
| |
|
| {| class="wikitable" | | {| class="wikitable center-all right-2 left-4" |
| |- | | |- |
| ! |Degree | | ! # |
| ! |Cents | | ! Cents |
| ! colspan="3" |[[Ups and Downs Notation|ups and downs notation]] | | ! colspan="3" |[[Ups and Downs Notation]] |
| |- | | |- |
| | style="text-align:center;" |0
| | | 0 |
| | style="text-align:center;" |0.000
| | | 0.000 |
| | style="text-align:center;" |P1
| | | P1 |
| | style="text-align:center;" |perfect unison
| | | perfect unison |
| | style="text-align:center;" |D
| | | D |
| |- | | |- |
| | style="text-align:center;" |1
| | | 1 |
| | style="text-align:center;" |29.571
| | | 29.571 |
| | style="text-align:center;" |^1, m2
| | | ^1, m2 |
| | style="text-align:center;" |up unison, minor 2nd
| | | up unison, minor 2nd |
| | style="text-align:center;" |^D, Eb
| | | ^D, Eb |
| |- | | |- |
| | style="text-align:center;" |2
| | | 2 |
| | style="text-align:center;" |57.143
| | | 57.143 |
| | style="text-align:center;" |^^1, ^m2
| | | ^^1, ^m2 |
| | style="text-align:center;" |double-up 1sn, upminor 2nd
| | | double-up 1sn, upminor 2nd |
| | style="text-align:center;" |^^D, ^Eb
| | | ^^D, ^Eb |
| |- | | |- |
| | style="text-align:center;" |3
| | | 3 |
| | style="text-align:center;" |85.714
| | | 85.714 |
| | style="text-align:center;" |^^m2
| | | ^^m2 |
| | style="text-align:center;" |double-up minor 2nd
| | | double-up minor 2nd |
| | style="text-align:center;" | ^^Eb
| | | ^^Eb |
| |- | | |- |
| | style="text-align:center;" |4
| | | 4 |
| | style="text-align:center;" |114.286
| | | 114.286 |
| | style="text-align:center;" |v~2
| | | v~2 |
| | style="text-align:center;" |downmid 2nd
| | | downmid 2nd |
| | style="text-align:center;" |^<span style="font-size: 90%; vertical-align: super;">3</span>Eb
| | | ^<sup>3</sup>Eb |
| |- | | |- |
| | style="text-align:center;" | 5
| | | 5 |
| | style="text-align:center;" |143.857
| | | 143.857 |
| | style="text-align:center;" |^~2
| | | ^~2 |
| | style="text-align:center;" |upmid 2nd
| | | upmid 2nd |
| | style="text-align:center;" |vvD#
| | | vvD# |
| |- | | |- |
| | style="text-align:center;" |6
| | | 6 |
| | style="text-align:center;" |171.429
| | | 171.429 |
| | style="text-align:center;" |vvM2
| | | vvM2 |
| | style="text-align:center;" |double-down major 2nd
| | | double-down major 2nd |
| | style="text-align:center;" |vD#
| | | vD# |
| |- | | |- |
| | style="text-align:center;" |7
| | | 7 |
| | style="text-align:center;" |200
| | | 200.000 |
| | style="text-align:center;" |vM2
| | | vM2 |
| | style="text-align:center;" |downmajor 2nd
| | | downmajor 2nd |
| | style="text-align:center;" |D#
| | | D# |
| |- | | |- |
| | style="text-align:center;" |8
| | | 8 |
| | style="text-align:center;" |228.571
| | | 228.571 |
| | style="text-align:center;" |M2
| | | M2 |
| | style="text-align:center;" |major 2nd
| | | major 2nd |
| | style="text-align:center;" |E
| | | E |
| |- | | |- |
| | style="text-align:center;" |9
| | | 9 |
| | style="text-align:center;" |257.143
| | | 257.143 |
| | style="text-align:center;" |m3
| | | m3 |
| | style="text-align:center;" |minor 3rd
| | | minor 3rd |
| | style="text-align:center;" |F
| | | F |
| |- | | |- |
| | style="text-align:center;" |10
| | | 10 |
| | style="text-align:center;" |285.714
| | | 285.714 |
| | style="text-align:center;" |^m3
| | | ^m3 |
| | style="text-align:center;" |upminor 3rd
| | | upminor 3rd |
| | style="text-align:center;" | Gb
| | | Gb |
| |- | | |- |
| | style="text-align:center;" |11
| | | 11 |
| | style="text-align:center;" |314.286
| | | 314.286 |
| | style="text-align:center;" |^^m3
| | | ^^m3 |
| | style="text-align:center;" |double-up minor 3rd
| | | double-up minor 3rd |
| | style="text-align:center;" | ^Gb
| | | ^Gb |
| |- | | |- |
| | style="text-align:center;" |12
| | | 12 |
| | style="text-align:center;" |342.857
| | | 342.857 |
| | style="text-align:center;" |v~3
| | | v~3 |
| | style="text-align:center;" |downmid 3rd
| | | downmid 3rd |
| | style="text-align:center;" |^^Gb
| | | ^^Gb |
| |- | | |- |
| | style="text-align:center;" |13
| | | 13 |
| | style="text-align:center;" |371.429
| | | 371.429 |
| | style="text-align:center;" |^~3
| | | ^~3 |
| | style="text-align:center;" |upmid 3rd
| | | upmid 3rd |
| | style="text-align:center;" |v<span style="font-size: 90%; vertical-align: super;">3</span>F#
| | | v<sup>3</sup>F# |
| |- | | |- |
| | style="text-align:center;" |14
| | | 14 |
| | style="text-align:center;" |400
| | | 400.000 |
| | style="text-align:center;" |vvM3
| | | vvM3 |
| | style="text-align:center;" |double-down major 3rd
| | | double-down major 3rd |
| | style="text-align:center;" |vvF#
| | | vvF# |
| |- | | |- |
| | style="text-align:center;" |15
| | | 15 |
| | style="text-align:center;" |428.571
| | | 428.571 |
| | style="text-align:center;" |vM3
| | | vM3 |
| | style="text-align:center;" |downmajor 3rd
| | | downmajor 3rd |
| | style="text-align:center;" | vF#
| | | vF# |
| |- | | |- |
| | style="text-align:center;" |16
| | | 16 |
| | style="text-align:center;" |457.143
| | | 457.143 |
| | style="text-align:center;" |M3, v4
| | | M3, v4 |
| | style="text-align:center;" |major 3rd, down 4th
| | | major 3rd, down 4th |
| | style="text-align:center;" |F#, vG
| | | F#, vG |
| |- | | |- |
| | style="text-align:center;" |17
| | | 17 |
| | style="text-align:center;" |485.714
| | | 485.714 |
| | style="text-align:center;" |P4
| | | P4 |
| | style="text-align:center;" |perfect 4th
| | | perfect 4th |
| | style="text-align:center;" |G
| | | G |
| |- | | |- |
| | style="text-align:center;" |18
| | | 18 |
| | style="text-align:center;" |51<span style="line-height: 1.5;">4.286</span> | | | 514.286 |
| | style="text-align:center;" |^4
| | | ^4 |
| | style="text-align:center;" |up 4th
| | | up 4th |
| | style="text-align:center;" |A'''b'''
| | | A'''b''' |
| |- | | |- |
| | style="text-align:center;" |19
| | | 19 |
| | style="text-align:center;" |543.857
| | | 543.857 |
| | style="text-align:center;" |^^4
| | | ^^4 |
| | style="text-align:center;" |double-up 4th
| | | double-up 4th |
| | style="text-align:center;" |^Ab
| | | ^Ab |
| |- | | |- |
| | style="text-align:center;" |20
| | | 20 |
| | style="text-align:center;" |571.429
| | | 571.429 |
| | style="text-align:center;" |v~4
| | | v~4 |
| | style="text-align:center;" |downmid 4th
| | | downmid 4th |
| | style="text-align:center;" |^^Ab
| | | ^^Ab |
| |- | | |- |
| | style="text-align:center;" |21
| | | 21 |
| | style="text-align:center;" |600
| | | 600.000 |
| | style="text-align:center;" |^~4, v~5
| | | ^~4, v~5 |
| | style="text-align:center;" |upmid 4th, downmid 5th
| | | upmid 4th, downmid 5th |
| | 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
| | | v<sup>3</sup>G#, ^<sup>3</sup>Ab |
| |- | | |- |
| | style="text-align:center;" |22
| | | 22 |
| | style="text-align:center;" |628.571
| | | 628.571 |
| | style="text-align:center;" |^~5
| | | ^~5 |
| | style="text-align:center;" |upmid 5th
| | | upmid 5th |
| | style="text-align:center;" |vvG#
| | | vvG# |
| |- | | |- |
| | style="text-align:center;" |23
| | | 23 |
| | style="text-align:center;" |657.143
| | | 657.143 |
| | style="text-align:center;" |vv5
| | | vv5 |
| | style="text-align:center;" |double-down 5th
| | | double-down 5th |
| | style="text-align:center;" |vG#
| | | vG# |
| |- | | |- |
| | style="text-align:center;" |24
| | | 24 |
| | style="text-align:center;" | 685.714
| | | 685.714 |
| | style="text-align:center;" |v5
| | | v5 |
| | style="text-align:center;" | down 5th
| | | down 5th |
| | style="text-align:center;" | G#
| | | G# |
| |- | | |- |
| | style="text-align:center;" |25
| | | 25 |
| | style="text-align:center;" |714.286
| | | 714.286 |
| | style="text-align:center;" |P5
| | | P5 |
| | style="text-align:center;" |perfect 5th
| | | perfect 5th |
| | style="text-align:center;" | A
| | | A |
| |- | | |- |
| | style="text-align:center;" |26
| | | 26 |
| | style="text-align:center;" |742.857
| | | 742.857 |
| | style="text-align:center;" |^5, m6
| | | ^5, m6 |
| | style="text-align:center;" |up 5th, minor 6th
| | | up 5th, minor 6th |
| | style="text-align:center;" |^A, Bb
| | | ^A, Bb |
| |- | | |- |
| | style="text-align:center;" |27
| | | 27 |
| | style="text-align:center;" |771.429
| | | 771.429 |
| | style="text-align:center;" |^m6
| | | ^m6 |
| | style="text-align:center;" |upminor 6th
| | | upminor 6th |
| | style="text-align:center;" |^Bb
| | | ^Bb |
| |- | | |- |
| | style="text-align:center;" |28
| | | 28 |
| | style="text-align:center;" |800
| | | 800.000 |
| | style="text-align:center;" |^^m6
| | | ^^m6 |
| | style="text-align:center;" |double-up minor 6th
| | | double-up minor 6th |
| | style="text-align:center;" |^^Bb
| | | ^^Bb |
| |- | | |- |
| | style="text-align:center;" |29
| | | 29 |
| | style="text-align:center;" |829.571
| | | 829.571 |
| | style="text-align:center;" |v~6
| | | v~6 |
| | style="text-align:center;" |downmid 6th
| | | downmid 6th |
| | style="text-align:center;" |^<span style="font-size: 90%; vertical-align: super;">3</span>Bb
| | | ^<sup>3</sup>Bb |
| |- | | |- |
| | style="text-align:center;" |30
| | | 30 |
| | style="text-align:center;" |857.143
| | | 857.143 |
| | style="text-align:center;" |^~6
| | | ^~6 |
| | style="text-align:center;" |upmid 6th
| | | upmid 6th |
| | style="text-align:center;" |^^A#
| | | ^^A# |
| |- | | |- |
| | style="text-align:center;" |31
| | | 31 |
| | style="text-align:center;" | 885.714
| | | 885.714 |
| | style="text-align:center;" |vvM6
| | | vvM6 |
| | style="text-align:center;" |double-down major 6th
| | | double-down major 6th |
| | style="text-align:center;" |^A#
| | | ^A# |
| |- | | |- |
| | style="text-align:center;" |32
| | | 32 |
| | style="text-align:center;" |914.286
| | | 914.286 |
| | style="text-align:center;" |vM6
| | | vM6 |
| | style="text-align:center;" |downmajor 6th
| | | downmajor 6th |
| | style="text-align:center;" | A#
| | | A# |
| |- | | |- |
| | style="text-align:center;" |33
| | | 33 |
| | style="text-align:center;" | 942.857
| | | 942.857 |
| | style="text-align:center;" |M6
| | | M6 |
| | style="text-align:center;" | major 6th
| | | major 6th |
| | style="text-align:center;" |B
| | | B |
| |- | | |- |
| | style="text-align:center;" |34
| | | 34 |
| | style="text-align:center;" |971.429
| | | 971.429 |
| | style="text-align:center;" | m7
| | | m7 |
| | style="text-align:center;" |minor 7th
| | | minor 7th |
| | style="text-align:center;" |C
| | | C |
| |- | | |- |
| | style="text-align:center;" |35
| | | 35 |
| | style="text-align:center;" | 1000
| | | 1000.000 |
| | style="text-align:center;" |^m7
| | | ^m7 |
| | style="text-align:center;" |upminor 7th
| | | upminor 7th |
| | style="text-align:center;" |D#
| | | D# |
| |- | | |- |
| | style="text-align:center;" |36
| | | 36 |
| | style="text-align:center;" |1028.571
| | | 1028.571 |
| | style="text-align:center;" |^^m7
| | | ^^m7 |
| | style="text-align:center;" |double-up minor 7th
| | | double-up minor 7th |
| | style="text-align:center;" |^D#
| | | ^D# |
| |- | | |- |
| | style="text-align:center;" |37
| | | 37 |
| | style="text-align:center;" |1057.143
| | | 1057.143 |
| | style="text-align:center;" |v~7
| | | v~7 |
| | style="text-align:center;" |downmid 7th
| | | downmid 7th |
| | style="text-align:center;" |^^D#
| | | ^^D# |
| |- | | |- |
| | style="text-align:center;" |38
| | | 38 |
| | style="text-align:center;" |1085.714
| | | 1085.714 |
| | style="text-align:center;" |^~7
| | | ^~7 |
| | style="text-align:center;" |upmid 7th
| | | upmid 7th |
| | style="text-align:center;" |v<span style="font-size: 90%; vertical-align: super;">3</span>C#
| | | v<sup>3</sup>C# |
| |- | | |- |
| | style="text-align:center;" |39
| | | 39 |
| | style="text-align:center;" |1114.286
| | | 1114.286 |
| | style="text-align:center;" |vvM7
| | | vvM7 |
| | style="text-align:center;" |double-down major 7th
| | | double-down major 7th |
| | style="text-align:center;" |vvC#
| | | vvC# |
| |- | | |- |
| | style="text-align:center;" |40
| | | 40 |
| | style="text-align:center;" |1142.857
| | | 1142.857 |
| | style="text-align:center;" |vM7
| | | vM7 |
| | style="text-align:center;" |downmajor 7th
| | | downmajor 7th |
| | style="text-align:center;" |vC#
| | | vC# |
| |- | | |- |
| | style="text-align:center;" |41
| | | 41 |
| | style="text-align:center;" |1171.429
| | | 1171.429 |
| | style="text-align:center;" |M7, v8
| | | M7, v8 |
| | style="text-align:center;" | major 7th, down 8ve
| | | major 7th, down 8ve |
| | style="text-align:center;" |C#, vD
| | | C#, vD |
| |- | | |- |
| | style="text-align:center;" |42
| | | 42 |
| | style="text-align:center;" |1200
| | | 1200.000 |
| | style="text-align:center;" |P8
| | | P8 |
| | style="text-align:center;" |perfect 8ve
| | | perfect 8ve |
| | style="text-align:center;" |D
| | | 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]] | | Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and Downs Notation #Chords and Chord Progressions]]. |
| | |
| | [[Category:Augene]] |
| [[Category:Equal divisions of the octave]] | | [[Category:Equal divisions of the octave]] |
| [[Category:subgroup]] | | [[Category:Subgroup]] |