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m Proper sharp/flat enharmonicity gets even more important up here, where it can cut the amount of ups and downs you need to notate things correctly in half.
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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|7-limit]], it tempers out 64/63 and [[126/125]], making it a tuning supporting [[Augmented_family|augene temperament]].
{{Infobox ET
| Prime factorization = 42 (prime)
| Step size = 29.571¢
| Fifth = 25\42 = 714.286¢
| Major 2nd = 8\42 = 228.571¢ (→[[21edo|4\21]])
| Minor 2nd = 1\42 = 29.571¢
| Augmented 1sn = 7\42 = 200¢
}}


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]]s as 84.
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]].
 
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.


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:
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:
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{| class="wikitable"
{| class="wikitable"
|-
|-
! | Degree
! |Degree
! | Cents
! |Cents
! colspan="3" |[[Ups and Downs Notation|ups and downs notation]]
! colspan="3" |[[Ups and Downs Notation|ups and downs notation]]
|-
|-
| style="text-align:center;" | 0
| style="text-align:center;" |0
| style="text-align:center;" |0.000
| style="text-align:center;" |0.000
| style="text-align:center;" | P1
| style="text-align:center;" |P1
| style="text-align:center;" | perfect unison
| style="text-align:center;" |perfect unison
| style="text-align:center;" | D
| style="text-align:center;" |D
|-
|-
| style="text-align:center;" | 1
| style="text-align:center;" |1
| style="text-align:center;" | 29.571
| style="text-align:center;" |29.571
| style="text-align:center;" | ^1, m2
| style="text-align:center;" |^1, m2
| style="text-align:center;" | up unison, minor 2nd
| style="text-align:center;" |up unison, minor 2nd
| style="text-align:center;" | ^D, Eb
| style="text-align:center;" |^D, Eb
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" |2
| style="text-align:center;" |57.143
| style="text-align:center;" |57.143
| style="text-align:center;" | ^^1, ^m2
| style="text-align:center;" |^^1, ^m2
| style="text-align:center;" | double-up 1sn, upminor 2nd
| style="text-align:center;" |double-up 1sn, upminor 2nd
| style="text-align:center;" | ^^D, ^Eb
| style="text-align:center;" |^^D, ^Eb
|-
|-
| style="text-align:center;" | 3
| style="text-align:center;" |3
| style="text-align:center;" |85.714
| style="text-align:center;" |85.714
| style="text-align:center;" | ^^m2
| style="text-align:center;" |^^m2
| style="text-align:center;" | double-up minor 2nd
| style="text-align:center;" |double-up minor 2nd
| style="text-align:center;" | ^^Eb
| style="text-align:center;" | ^^Eb
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" |4
| style="text-align:center;" |114.286
| style="text-align:center;" |114.286
| style="text-align:center;" | v~2
| style="text-align:center;" |v~2
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" |downmid 2nd
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Eb
| 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
| style="text-align:center;" | 143.857
| style="text-align:center;" |143.857
| style="text-align:center;" | ^~2
| style="text-align:center;" |^~2
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" |upmid 2nd
| style="text-align:center;" | vvD#
| style="text-align:center;" |vvD#
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" |6
| style="text-align:center;" |171.429
| style="text-align:center;" |171.429
| style="text-align:center;" | vvM2
| style="text-align:center;" |vvM2
| style="text-align:center;" | double-down major 2nd
| style="text-align:center;" |double-down major 2nd
| style="text-align:center;" | vD#
| style="text-align:center;" |vD#
|-
|-
| style="text-align:center;" | 7
| style="text-align:center;" |7
| style="text-align:center;" | 200
| style="text-align:center;" |200
| style="text-align:center;" | vM2
| style="text-align:center;" |vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" |downmajor 2nd
| style="text-align:center;" | D#
| style="text-align:center;" |D#
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" |8
| style="text-align:center;" |228.571
| style="text-align:center;" |228.571
| style="text-align:center;" | M2
| style="text-align:center;" |M2
| style="text-align:center;" | major 2nd
| style="text-align:center;" |major 2nd
| style="text-align:center;" | E
| style="text-align:center;" |E
|-
|-
| style="text-align:center;" | 9
| style="text-align:center;" |9
| style="text-align:center;" |257.143
| style="text-align:center;" |257.143
| style="text-align:center;" | m3
| style="text-align:center;" |m3
| style="text-align:center;" | minor 3rd
| style="text-align:center;" |minor 3rd
| style="text-align:center;" | F
| style="text-align:center;" |F
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" |10
| style="text-align:center;" |285.714
| style="text-align:center;" |285.714
| style="text-align:center;" | ^m3
| style="text-align:center;" |^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" |upminor 3rd
| style="text-align:center;" | Gb
| style="text-align:center;" | Gb
|-
|-
| style="text-align:center;" | 11
| style="text-align:center;" |11
| style="text-align:center;" | 314.286
| style="text-align:center;" |314.286
| style="text-align:center;" | ^^m3
| style="text-align:center;" |^^m3
| style="text-align:center;" | double-up minor 3rd
| style="text-align:center;" |double-up minor 3rd
| style="text-align:center;" | ^Gb
| style="text-align:center;" | ^Gb
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" |12
| style="text-align:center;" |342.857
| style="text-align:center;" |342.857
| style="text-align:center;" | v~3
| style="text-align:center;" |v~3
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" |downmid 3rd
| style="text-align:center;" | ^^Gb
| style="text-align:center;" |^^Gb
|-
|-
| style="text-align:center;" | 13
| style="text-align:center;" |13
| style="text-align:center;" | 371.429
| style="text-align:center;" |371.429
| style="text-align:center;" | ^~3
| style="text-align:center;" |^~3
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" |upmid 3rd
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>F#
| 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
| style="text-align:center;" | 400
| style="text-align:center;" |400
| style="text-align:center;" | vvM3
| style="text-align:center;" |vvM3
| style="text-align:center;" | double-down major 3rd
| style="text-align:center;" |double-down major 3rd
| style="text-align:center;" | vvF#
| style="text-align:center;" |vvF#
|-
|-
| style="text-align:center;" | 15
| style="text-align:center;" |15
| style="text-align:center;" |428.571
| style="text-align:center;" |428.571
| style="text-align:center;" | vM3
| style="text-align:center;" |vM3
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" |downmajor 3rd
| style="text-align:center;" | vF#
| style="text-align:center;" | vF#
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" |16
| style="text-align:center;" |457.143
| style="text-align:center;" |457.143
| style="text-align:center;" | M3, v4
| style="text-align:center;" |M3, v4
| style="text-align:center;" | major 3rd, down 4th
| style="text-align:center;" |major 3rd, down 4th
| style="text-align:center;" | F#, vG
| style="text-align:center;" |F#, vG
|-
|-
| style="text-align:center;" | 17
| style="text-align:center;" |17
| style="text-align:center;" | 485.714
| style="text-align:center;" |485.714
| style="text-align:center;" | P4
| style="text-align:center;" |P4
| style="text-align:center;" | perfect 4th
| style="text-align:center;" |perfect 4th
| style="text-align:center;" | G
| style="text-align:center;" |G
|-
|-
| 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>
| style="text-align:center;" | ^4
| style="text-align:center;" |^4
| style="text-align:center;" | up 4th
| style="text-align:center;" |up 4th
| style="text-align:center;" | A'''b'''
| style="text-align:center;" |A'''b'''
|-
|-
| style="text-align:center;" | 19
| style="text-align:center;" |19
| style="text-align:center;" | 543.857
| style="text-align:center;" |543.857
| style="text-align:center;" | ^^4
| style="text-align:center;" |^^4
| style="text-align:center;" | double-up 4th
| style="text-align:center;" |double-up 4th
| style="text-align:center;" | ^Ab
| style="text-align:center;" |^Ab
|-
|-
| 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;" | v~4
| style="text-align:center;" |v~4
| style="text-align:center;" | downmid 4th
| style="text-align:center;" |downmid 4th
| style="text-align:center;" | ^^Ab
| style="text-align:center;" |^^Ab
|-
|-
| style="text-align:center;" | 21
| style="text-align:center;" |21
| style="text-align:center;" | 600
| style="text-align:center;" |600
| style="text-align:center;" | ^~4, v~5
| style="text-align:center;" |^~4, v~5
| style="text-align:center;" | upmid 4th, downmid 5th
| style="text-align:center;" |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
| 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;" | ^~5
| style="text-align:center;" |^~5
| style="text-align:center;" | upmid 5th
| style="text-align:center;" |upmid 5th
| style="text-align:center;" | vvG#
| style="text-align:center;" |vvG#
|-
|-
| style="text-align:center;" | 23
| style="text-align:center;" |23
| style="text-align:center;" | 657.143
| style="text-align:center;" |657.143
| style="text-align:center;" | vv5
| style="text-align:center;" |vv5
| style="text-align:center;" | double-down 5th
| style="text-align:center;" |double-down 5th
| style="text-align:center;" | vG#
| style="text-align:center;" |vG#
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" |24
| style="text-align:center;" |685.714
| style="text-align:center;" | 685.714
| style="text-align:center;" | v5
| style="text-align:center;" |v5
| style="text-align:center;" | down 5th
| style="text-align:center;" | down 5th
| style="text-align:center;" | G#
| style="text-align:center;" | G#
|-
|-
| style="text-align:center;" | 25
| style="text-align:center;" |25
| style="text-align:center;" | 714.286
| style="text-align:center;" |714.286
| style="text-align:center;" | P5
| style="text-align:center;" |P5
| style="text-align:center;" | perfect 5th
| style="text-align:center;" |perfect 5th
| style="text-align:center;" | A
| style="text-align:center;" | A
|-
|-
| style="text-align:center;" | 26
| style="text-align:center;" |26
| style="text-align:center;" |742.857
| style="text-align:center;" |742.857
| style="text-align:center;" | ^5, m6
| style="text-align:center;" |^5, m6
| style="text-align:center;" | up 5th, minor 6th
| style="text-align:center;" |up 5th, minor 6th
| style="text-align:center;" | ^A, Bb
| style="text-align:center;" |^A, Bb
|-
|-
| style="text-align:center;" | 27
| style="text-align:center;" |27
| style="text-align:center;" |771.429
| style="text-align:center;" |771.429
| style="text-align:center;" | ^m6
| style="text-align:center;" |^m6
| style="text-align:center;" | upminor 6th
| style="text-align:center;" |upminor 6th
| style="text-align:center;" | ^Bb
| style="text-align:center;" |^Bb
|-
|-
| style="text-align:center;" | 28
| style="text-align:center;" |28
| style="text-align:center;" | 800
| style="text-align:center;" |800
| style="text-align:center;" | ^^m6
| style="text-align:center;" |^^m6
| style="text-align:center;" | double-up minor 6th
| style="text-align:center;" |double-up minor 6th
| style="text-align:center;" | ^^Bb
| style="text-align:center;" |^^Bb
|-
|-
| style="text-align:center;" | 29
| style="text-align:center;" |29
| style="text-align:center;" | 829.571
| style="text-align:center;" |829.571
| style="text-align:center;" | v~6
| style="text-align:center;" |v~6
| style="text-align:center;" | downmid 6th
| style="text-align:center;" |downmid 6th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>Bb
| 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
| style="text-align:center;" |857.143
| style="text-align:center;" |857.143
| style="text-align:center;" | ^~6
| style="text-align:center;" |^~6
| style="text-align:center;" | upmid 6th
| style="text-align:center;" |upmid 6th
| style="text-align:center;" | ^^A#
| style="text-align:center;" |^^A#
|-
|-
| style="text-align:center;" | 31
| style="text-align:center;" |31
| style="text-align:center;" | 885.714
| style="text-align:center;" | 885.714
| style="text-align:center;" | vvM6
| style="text-align:center;" |vvM6
| style="text-align:center;" | double-down major 6th
| style="text-align:center;" |double-down major 6th
| style="text-align:center;" | ^A#
| style="text-align:center;" |^A#
|-
|-
| style="text-align:center;" | 32
| style="text-align:center;" |32
| style="text-align:center;" |914.286
| style="text-align:center;" |914.286
| style="text-align:center;" | vM6
| style="text-align:center;" |vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" |downmajor 6th
| style="text-align:center;" | A#
| style="text-align:center;" | A#
|-
|-
| style="text-align:center;" | 33
| style="text-align:center;" |33
| style="text-align:center;" |942.857
| style="text-align:center;" | 942.857
| style="text-align:center;" | M6
| style="text-align:center;" |M6
| style="text-align:center;" | major 6th
| style="text-align:center;" | major 6th
| style="text-align:center;" | B
| style="text-align:center;" |B
|-
|-
| style="text-align:center;" | 34
| style="text-align:center;" |34
| style="text-align:center;" |971.429
| style="text-align:center;" |971.429
| style="text-align:center;" | m7
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" |minor 7th
| style="text-align:center;" | C
| style="text-align:center;" |C
|-
|-
| style="text-align:center;" | 35
| style="text-align:center;" |35
| style="text-align:center;" | 1000
| style="text-align:center;" | 1000
| style="text-align:center;" | ^m7
| style="text-align:center;" |^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" |upminor 7th
| style="text-align:center;" | D#
| style="text-align:center;" |D#
|-
|-
| style="text-align:center;" | 36
| style="text-align:center;" |36
| style="text-align:center;" |1028.571
| style="text-align:center;" |1028.571
| style="text-align:center;" | ^^m7
| style="text-align:center;" |^^m7
| style="text-align:center;" | double-up minor 7th
| style="text-align:center;" |double-up minor 7th
| style="text-align:center;" | ^D#
| style="text-align:center;" |^D#
|-
|-
| style="text-align:center;" | 37
| style="text-align:center;" |37
| style="text-align:center;" | 1057.143
| style="text-align:center;" |1057.143
| style="text-align:center;" | v~7
| style="text-align:center;" |v~7
| style="text-align:center;" | downmid 7th
| style="text-align:center;" |downmid 7th
| style="text-align:center;" | ^^D#
| style="text-align:center;" |^^D#
|-
|-
| style="text-align:center;" | 38
| style="text-align:center;" |38
| style="text-align:center;" |1085.714
| style="text-align:center;" |1085.714
| style="text-align:center;" | ^~7
| style="text-align:center;" |^~7
| style="text-align:center;" | upmid 7th
| style="text-align:center;" |upmid 7th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>C#
| 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
| style="text-align:center;" |1114.286
| style="text-align:center;" |1114.286
| style="text-align:center;" | vvM7
| style="text-align:center;" |vvM7
| style="text-align:center;" | double-down major 7th
| style="text-align:center;" |double-down major 7th
| style="text-align:center;" | vvC#
| style="text-align:center;" |vvC#
|-
|-
| style="text-align:center;" | 40
| style="text-align:center;" |40
| style="text-align:center;" |1142.857
| style="text-align:center;" |1142.857
| style="text-align:center;" | vM7
| style="text-align:center;" |vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" |downmajor 7th
| style="text-align:center;" | vC#
| style="text-align:center;" |vC#
|-
|-
| style="text-align:center;" | 41
| style="text-align:center;" |41
| style="text-align:center;" | 1171.429
| style="text-align:center;" |1171.429
| style="text-align:center;" | M7, v8
| style="text-align:center;" |M7, v8
| style="text-align:center;" | major 7th, down 8ve
| style="text-align:center;" | major 7th, down 8ve
| style="text-align:center;" | C#, vD
| style="text-align:center;" |C#, vD
|-
|-
| style="text-align:center;" | 42
| style="text-align:center;" |42
| style="text-align:center;" | 1200
| style="text-align:center;" |1200
| style="text-align:center;" | P8
| style="text-align:center;" |P8
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" |perfect 8ve
| 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]].
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]].
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