42edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
→21st century: Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'' (2025): Add live performance in Munich, Germany (2026)
 
(116 intermediate revisions by 18 users not shown)
Line 1: Line 1:
The ''42 equal division'' divides the octave into 42 equal parts of 28.571 [[cent|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}}
{{ED intro}}


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.
== Theory ==
42edo has a [[patent val]] [[3/2|fifth]] (the step of which is not from [[7edo]], this being a first for edos of the form 7''n'') and a third both over 12 cents sharp, using the same 400-cent interval to represent [[5/4]] as does [[12edo]], 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 [[support]]ing the [[augene]] temperament.


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 on the [[optimal ET sequence]] of the [[Augmented family#eugene|eugene]], [[joan]], [[lemba]], [[neutron]], [[qeema]], [[seville]], [[sevond]], [[skateboard]], [[tritikleismic]] and [[vines]] temperaments.


D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D
42edo is a diatonic edo because its 5th falls between {{nowrap|4\7 {{=}} 686{{c}}}} and {{nowrap|3\5 {{=}} 720{{c}}}}. 42edo is one of the most difficult diatonic edos to notate, because no other diatonic edo's fifth is as sharp (see [[47edo]] for the opposite extreme).


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.
=== Odd harmonics ===
While not an accurate tuning on the full [[7-limit]], 42edo 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.


==Intervals of 42edo==
{{Harmonics in equal|42}}


{| class="wikitable"
=== Subsets and supersets ===
Since 42 factors into {{factorization|42}}, 42edo contains subset edos {{EDOs| 2, 3, 6, 7, 14, and 21 }}.
 
== Intervals ==
{| 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
| 0.0
| 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
| 28.6
| 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
| 57.1
| style="text-align:center;" | ^^1, ^m2
| ^^1, ^m2
| style="text-align:center;" | double-up 1sn, upminor 2nd
| dup 1sn, upminor 2nd
| style="text-align:center;" | ^^D, ^Eb
| ^^D, ^Eb
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" |85.714
| 85.7
| style="text-align:center;" | ^^m2
| ^^m2
| style="text-align:center;" | double-up minor 2nd
| dupminor 2nd
| style="text-align:center;" | ^^Eb
| ^^Eb
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" |114.286
| 114.3
| style="text-align:center;" | v~2
| ^<sup>3</sup>m
| style="text-align:center;" | downmid 2nd
| trupminor 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
| 143.9
| style="text-align:center;" | ^~2
| v<sup>3</sup>M
| style="text-align:center;" | upmid 2nd
| trudmajor 2nd
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>E
| v<sup>3</sup>E
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" |171.429
| 171.4
| style="text-align:center;" | vvM2
| vvM2
| style="text-align:center;" | double-down major 2nd
| dudmajor 2nd
| style="text-align:center;" | vvE
| vvE
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | 200
| 200.0
| style="text-align:center;" | vM2
| vM2
| style="text-align:center;" | downmajor 2nd
| downmajor 2nd
| style="text-align:center;" | vE
| vE
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" |228.571
| 228.6
| 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
| 257.1
| 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
| 285.7
| style="text-align:center;" | ^m3
| ^m3
| style="text-align:center;" | upminor 3rd
| upminor 3rd
| style="text-align:center;" | ^F
| ^F
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | 314.286
| 314.3
| style="text-align:center;" | ^^m3
| ^^m3
| style="text-align:center;" | double-up minor 3rd
| dupminor 3rd
| style="text-align:center;" | ^^F
| ^^F
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" |342.857
| 342.9
| style="text-align:center;" | v~3
| ^<sup>3</sup>m3
| style="text-align:center;" | downmid 3rd
| trupminor 3rd
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>F
| ^<sup>3</sup>F
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | 371.429
| 371.4
| style="text-align:center;" | ^~3
| v<sup>3</sup>M3
| style="text-align:center;" | upmid 3rd
| trudmajor 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
| 400.0
| style="text-align:center;" | vvM3
| vvM3
| style="text-align:center;" | double-down major 3rd
| dudmajor 3rd
| style="text-align:center;" | vvF#
| vvF#
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" |428.571
| 428.6
| 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
| 457.1
| 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
| 485.7
| 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>
| 514.3
| style="text-align:center;" | ^4
| ^4
| style="text-align:center;" | up 4th
| up 4th
| style="text-align:center;" | ^G
| ^G
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | 543.857
| 543.9
| style="text-align:center;" | ^^4
| ^^4
| style="text-align:center;" | double-up 4th
| dup 4th
| style="text-align:center;" | ^^G
| ^^G
|-
|-
| style="text-align:center;" | 20
| 20
| style="text-align:center;" |571.429
| 571.4
| style="text-align:center;" | v~4
| ^<sup>3</sup>4, ^^d5
| style="text-align:center;" | downmid 4th
| trup 4th, dupdim 5th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>G
| ^<sup>3</sup>G, ^^Ab
|-
|-
| style="text-align:center;" | 21
| 21
| style="text-align:center;" | 600
| 600.0
| style="text-align:center;" | ^~4, v~5
| v<sup>3</sup>A4, ^<sup>3</sup>d5
| style="text-align:center;" | upmid 4th, downmid 5th
| trudaug 4th, trupdim 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
| 628.6
| style="text-align:center;" | ^~5
| vvA4, v<sup>3</sup>5
| style="text-align:center;" | upmid 5th
| dudaug 4th, trud 5th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>A
| vvG#, v<sup>3</sup>A
|-
|-
| style="text-align:center;" | 23
| 23
| style="text-align:center;" | 657.143
| 657.1
| style="text-align:center;" | vv5
| vv5
| style="text-align:center;" | double-down 5th
| dud 5th
| style="text-align:center;" | vvA
| vvA
|-
|-
| style="text-align:center;" | 24
| 24
| style="text-align:center;" |685.714
| 685.7
| style="text-align:center;" | v5
| v5
| style="text-align:center;" | down 5th
| down 5th
| style="text-align:center;" | vA
| vA
|-
|-
| style="text-align:center;" | 25
| 25
| style="text-align:center;" | 714.286
| 714.3
| 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
| 742.9
| 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
| 771.4
| 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
| 800.0
| style="text-align:center;" | ^^m6
| ^^m6
| style="text-align:center;" | double-up minor 6th
| dupminor 6th
| style="text-align:center;" | ^^Bb
| ^^Bb
|-
|-
| style="text-align:center;" | 29
| 29
| style="text-align:center;" | 829.571
| 828.6
| style="text-align:center;" | v~6
| ^<sup>3</sup>m6
| style="text-align:center;" | downmid 6th
| trupminor 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
| 857.1
| style="text-align:center;" | ^~6
| v<sup>3</sup>M6
| style="text-align:center;" | upmid 6th
| trudmajor 6th
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>B
| v<sup>3</sup>B
|-
|-
| style="text-align:center;" | 31
| 31
| style="text-align:center;" | 885.714
| 885.7
| style="text-align:center;" | vvM6
| vvM6
| style="text-align:center;" | double-down major 6th
| dudmajor 6th
| style="text-align:center;" | vvB
| vvB
|-
|-
| style="text-align:center;" | 32
| 32
| style="text-align:center;" |914.286
| 914.3
| style="text-align:center;" | vM6
| vM6
| style="text-align:center;" | downmajor 6th
| downmajor 6th
| style="text-align:center;" | vB
| vB
|-
|-
| style="text-align:center;" | 33
| 33
| style="text-align:center;" |942.857
| 942.9
| 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
| 971.4
| 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
| 1000.0
| style="text-align:center;" | ^m7
| ^m7
| style="text-align:center;" | upminor 7th
| upminor 7th
| style="text-align:center;" | ^C
| ^C
|-
|-
| style="text-align:center;" | 36
| 36
| style="text-align:center;" |1028.571
| 1028.6
| style="text-align:center;" | ^^m7
| ^^m7
| style="text-align:center;" | double-up minor 7th
| dupminor 7th
| style="text-align:center;" | ^^C
| ^^C
|-
|-
| style="text-align:center;" | 37
| 37
| style="text-align:center;" | 1057.143
| 1057.1
| style="text-align:center;" | v~7
| ^<sup>3</sup>m7
| style="text-align:center;" | downmid 7th
| trupminor 7th
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>C
| ^<sup>3</sup>C
|-
|-
| style="text-align:center;" | 38
| 38
| style="text-align:center;" |1085.714
| 1085.7
| style="text-align:center;" | ^~7
| v<sup>3</sup>M7
| style="text-align:center;" | upmid 7th
| trudmajor 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
| 1114.3
| style="text-align:center;" | vvM7
| vvM7
| style="text-align:center;" | double-down major 7th
| dudmajor 7th
| style="text-align:center;" | vvC#
| vvC#
|-
|-
| style="text-align:center;" | 40
| 40
| style="text-align:center;" |1142.857
| 1142.9
| 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
| 1171.4
| 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
| 1200.0
| 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:Equal divisions of the octave]]
 
[[Category:subgroup]]
== Notation ==
=== Ups and downs notation ===
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♯ is next to E. The notation requires ups and downs with three arrows, and if chords are to be spelled correctly four or more arrows may be required in certain cases. For example, a {{dash|1/1, 5/4, 3/2, 9/5|med}} chord with a root on the edostep midway between G and A would be written either as {{dash|v<sup>3</sup>G♯–v<sup>5</sup>B♯, v<sup>3</sup>D♯, vF♯|med}} or as {{dash|^<sup>3</sup>A♭, ^C, ^<sup>3</sup>E♭, ^<sup>5</sup>G♭}}. This is a dud dup-seven chord, written either as v<sup>3</sup>G♯vv,^^7 or as ^<sup>3</sup>A♭vv,^^7.
 
In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc.
{{Ups and downs sharpness}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
 
{{sharpness-sharp7}}
 
=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[35edo #Second-best fifth notation|35b]].
 
===== Evo flavor =====
<imagemap>
File:42-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:42-EDO_Evo_Sagittal.svg]]
</imagemap>
 
===== Revo flavor =====
<imagemap>
File:42-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 663 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:42-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Second-best fifth notation ====
This notation uses the same sagittal sequence as [[47edo#Sagittal notation|47edo]], and is a superset of the notations for edos [[21edo #Sagittal notation|21]], [[14edo #Sagittal notation|14]], and [[7edo #Sagittal notation|7]].
 
<imagemap>
File:42b_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 663 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 300 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:42b_Sagittal.svg]]
</imagemap>
 
== Approximation to JI ==
{{Q-odd-limit intervals}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 67 -42 }}
| {{Mapping| 42 67 }}
| −3.89
| 3.88
| 13.57
|-
| 2.3.5
| 128/125, 5000000/4782969
| {{Mapping| 42 67 98 }}
| −4.55
| 3.30
| 11.55
|-
| 2.3.5.7
| 64/63, 126/125, 6860/6561
| {{Mapping| 42 67 98 118 }}
| −3.65
| 3.26
| 11.42
|}
 
== Octave stretch or compression ==
42edo’s inaccurate 3rd and 5th harmonics can be improved through [[stretched and compressed tuning|stretching or compressing]] octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.
 
* Good stretched options: [[ed6|108ed6]], [[ed5|97ed5]], [[zpi|189zpi]], [[ed12|150ed12]]
* Good compressed options: [[ed7|118ed7]], [[ed12|151ed12]], [[ed6|109ed6]], [[zpi|191zpi]]
 
== Scales ==
; [[MOS scale]]s
{{main|List of MOS scales in 42edo}}
* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
* Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3'''
* Lemba[16]: '''3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3'''
* Qeema/Skateboard[15]: '''2 5 2 2 2 5 2 2 2 5 2 2 2 5 2'''
* Qeema/Skateboard[19]: '''2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2'''
* Seville/Sevond[14] 1st mode: '''1 5 1 5 1 5 1 5 1 5 1 5 1 5'''
* Seville/Sevond[14] 2nd mode: '''5 1 5 1 5 1 5 1 5 1 5 1 5 1'''
* Seville/Sevond[21]: '''1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4'''
 
 
; Subsets of MOS scales
''(Names used are [[Template:Idiosyncratic|idiosyncratic]].)''
* Eugene/Tritikleismic[9]
** Groovy aeolian pentatonic: '''11 6 8 3 14'''
** [[Otonal]] mixolydian pentatonic: '''14 3 8 11 6'''
** Pseudo-[[equipentatonic]]: '''11 6 8 6 11'''
** Septimal melodic minor pentatonic: '''8 3 14 14 3'''
** Septimal Picardy pentatonic: '''8 6 11 3 14'''
** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
** Yokai pentatonic: '''3 14 8 3 14'''
 
 
; Approximations of [[gamelan]] scales:
* 5-tone pelog: 4 5 15 3 15
* 7-tone pelog: 4 5 9 6 3 10 5
* 5-tone slendro: 8 9 8 9 8
 
 
; Other scales
* 12-tone 6&7edo scale: 6 1 5 2 4 3 3 4 2 5 1 6
 
== Instruments ==
=== Lumatone ===
{{main|Lumatone mapping for 42edo}}
 
=== Skip fretting ===
'''[[Skip fretting]] system 42 3 11''': One way to play [[42edo]] on a [[14edo]] guitar is to tune the strings 11\42, or approximately a [[just]] 6/5, apart. All examples on this page are for 7-string guitar.
 
; Prime intervals
1/1: string 2 open
 
2/1: string 5 fret 3
 
3/2: string 4 fret 1 and string 7 fret 4
 
5/4: string 3 fret 1
 
7/4: string 1 fret 1 and string 4 fret 4
 
11/8: string 7 fret 2
 
13/8: string 3 fret 6
 
17/16: string 1 fret 5
 
19/16: string 1 fret 7
 
23/16: string 4 open and string 7 fret 3
 
29/16: string 5 fret 1
 
31/16: string 1 fret 3 and string 4 fret 6
 
; Chords
Minor 7th: 100123X
 
== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=Wh4GmL5Xw8Q "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)
 
; {{W|Bing Crosby}}
* ''[https://soundcloud.com/puffinwrangler/sets/white-christmas White Christmas]'' - 42edo reimagining by [[Todd Harrop]] (2024)
 
=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=PJw8gZyNPjg ''improv 42edo''] (2023)
* [https://www.youtube.com/watch?v=ljaSpsQP2qc ''Improvisation in 42edo''] (2023), transcribed by [[Stephen Weigel]] (2024)
* [https://www.youtube.com/watch?v=cL6CY3U9mHM ''42edo groove''] (2025)
* ''A Hunger Awakes - 42edo'' (2026)
** [https://www.youtube.com/shorts/B90JT_SxSSE <nowiki>[short]</nowiki>] (Lumatone view)
** [https://www.youtube.com/watch?v=VwHqWffglj4 <nowiki>[full version]</nowiki>] (music video with stop-motion by [[Jelly Eyes]])
* ''Waltz in 42edo'' (2026)
** [https://www.youtube.com/shorts/D_YgzRJFg8I <nowiki>[short]</nowiki>] (Lumatone view)
** [https://www.youtube.com/watch?v=QyglWQ_0bIk <nowiki>[full version]</nowiki>]
 
; [[James Kukula]]
* ''[https://app.box.com/s/70bwqa09oq84rqehc90j5fttgqjmbxme Circulating and Traversing]'' (2024) - see the ''[https://interdependentscience.blogspot.com/2024/12/circulating-and-traversing.html composer’s notes]''
 
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=vdjhC9i5KF4 ''Four Short Experiments in Octave Stretched 42edo''] (2024)
 
; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/through-the-dark Through the Dark]'' (2024) - uses mostly [[Augene]][15] with some chromaticism
 
; [[Mundoworld]]
* [https://www.youtube.com/watch?v=ORy7nv6SnN8 ''Glory of Them''] (2024)
 
; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=tLmaQK10aYM ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!''] (2025; mostly in 42edo, but also some in 40edo)
** [https://www.youtube.com/watch?v=NE77rwCsGHw live performance of the above in Munich, Germany] (2026)
 
[[Category:Augmented]]
{{Todo|review|add rank 2 temperaments table}}

Latest revision as of 03:26, 31 May 2026

← 41edo 42edo 43edo →
Prime factorization 2 × 3 × 7
Step size 28.5714 ¢ 
Fifth 25\42 (714.286 ¢)
Semitones (A1:m2) 7:1 (200 ¢ : 28.57 ¢)
Dual sharp fifth 25\42 (714.286 ¢)
Dual flat fifth 24\42 (685.714 ¢) (→ 4\7)
Dual major 2nd 7\42 (200 ¢) (→ 1\6)
Consistency limit 7
Distinct consistency limit 7

42 equal divisions of the octave (abbreviated 42edo or 42ed2), also called 42-tone equal temperament (42tet) or 42 equal temperament (42et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 42 equal parts of about 28.6 ¢ each. Each step represents a frequency ratio of 21/42, or the 42nd root of 2.

Theory

42edo has a patent val fifth (the step of which is not from 7edo, this being a first for edos of the form 7n) and a third both over 12 cents sharp, using the same 400-cent interval to represent 5/4 as does 12edo, which means it tempers out 128/125. In the 7-limit, it tempers out 64/63 and 126/125, making it a tuning supporting the augene temperament.

42edo is on the optimal ET sequence of the eugene, joan, lemba, neutron, qeema, seville, sevond, skateboard, tritikleismic and vines temperaments.

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 fifth is as sharp (see 47edo for the opposite extreme).

Odd harmonics

While not an accurate tuning on the full 7-limit, 42edo 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.


Approximation of odd harmonics in 42edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +12.3 +13.7 +2.6 -3.9 -8.5 -12.0 -2.6 +9.3 -11.8 -13.6 +0.3
Relative (%) +43.2 +47.9 +9.1 -13.7 -29.6 -41.8 -8.9 +32.7 -41.3 -47.7 +1.0
Steps
(reduced) 		67
(25) 		98
(14) 		118
(34) 		133
(7) 		145
(19) 		155
(29) 		164
(38) 		172
(4) 		178
(10) 		184
(16) 		190
(22)

Subsets and supersets

Since 42 factors into 2 × 3 × 7, 42edo contains subset edos 2, 3, 6, 7, 14, and 21.

Intervals

# Cents Ups and downs notation
0 0.0 P1 perfect unison D
1 28.6 ^1, m2 up unison, minor 2nd ^D, Eb
2 57.1 ^^1, ^m2 dup 1sn, upminor 2nd ^^D, ^Eb
3 85.7 ^^m2 dupminor 2nd ^^Eb
4 114.3 ^3m trupminor 2nd ^3Eb
5 143.9 v3M trudmajor 2nd v3E
6 171.4 vvM2 dudmajor 2nd vvE
7 200.0 vM2 downmajor 2nd vE
8 228.6 M2 major 2nd E
9 257.1 m3 minor 3rd F
10 285.7 ^m3 upminor 3rd ^F
11 314.3 ^^m3 dupminor 3rd ^^F
12 342.9 ^3m3 trupminor 3rd ^3F
13 371.4 v3M3 trudmajor 3rd v3F#
14 400.0 vvM3 dudmajor 3rd vvF#
15 428.6 vM3 downmajor 3rd vF#
16 457.1 M3, v4 major 3rd, down 4th F#, vG
17 485.7 P4 perfect 4th G
18 514.3 ^4 up 4th ^G
19 543.9 ^^4 dup 4th ^^G
20 571.4 ^34, ^^d5 trup 4th, dupdim 5th ^3G, ^^Ab
21 600.0 v3A4, ^3d5 trudaug 4th, trupdim 5th v3G#, ^3Ab
22 628.6 vvA4, v35 dudaug 4th, trud 5th vvG#, v3A
23 657.1 vv5 dud 5th vvA
24 685.7 v5 down 5th vA
25 714.3 P5 perfect 5th A
26 742.9 ^5, m6 up 5th, minor 6th ^A, Bb
27 771.4 ^m6 upminor 6th ^Bb
28 800.0 ^^m6 dupminor 6th ^^Bb
29 828.6 ^3m6 trupminor 6th ^3Bb
30 857.1 v3M6 trudmajor 6th v3B
31 885.7 vvM6 dudmajor 6th vvB
32 914.3 vM6 downmajor 6th vB
33 942.9 M6 major 6th B
34 971.4 m7 minor 7th C
35 1000.0 ^m7 upminor 7th ^C
36 1028.6 ^^m7 dupminor 7th ^^C
37 1057.1 ^3m7 trupminor 7th ^3C
38 1085.7 v3M7 trudmajor 7th v3C#
39 1114.3 vvM7 dudmajor 7th vvC#
40 1142.9 vM7 downmajor 7th vC#
41 1171.4 M7, v8 major 7th, down 8ve C#, vD
42 1200.0 P8 perfect 8ve 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.

Notation

Ups and downs notation

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♯ is next to E. The notation requires ups and downs with three arrows, and if chords are to be spelled correctly four or more arrows may be required in certain cases. For example, a 1/1 – 5/4 – 3/2 – 9/5 chord with a root on the edostep midway between G and A would be written either as v3G♯–v5B♯ – v3D♯ – vF♯ or as ^3A♭ – ^C – ^3E♭ – ^5G♭. This is a dud dup-seven chord, written either as v3G♯vv,^^7 or as ^3A♭vv,^^7.

In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sharp symbol   
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   
Flat symbol
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 35b.

Evo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation
Revo flavor
Sagittal notationPeriodic table of EDOs with sagittal notationapotome-fraction notation

Second-best fifth notation

This notation uses the same sagittal sequence as 47edo, and is a superset of the notations for edos 21, 14, and 7.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

Approximation to JI

The following tables show how 15-odd-limit intervals are represented in 42edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 42edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 1.356 4.7
15/8, 16/15 2.554 8.9
7/4, 8/7 2.603 9.1
13/10, 20/13 2.929 10.3
13/11, 22/13 3.495 12.2
9/8, 16/9 3.910 13.7
13/12, 24/13 4.284 15.0
11/9, 18/11 4.551 15.9
15/14, 28/15 5.157 18.0
15/11, 22/15 5.906 20.7
11/10, 20/11 6.424 22.5
9/7, 14/9 6.513 22.8
11/6, 12/11 7.780 27.2
13/9, 18/13 8.046 28.2
11/8, 16/11 8.461 29.6
15/13, 26/15 9.402 32.9
7/6, 12/7 9.728 34.0
9/5, 10/9 10.975 38.4
11/7, 14/11 11.063 38.7
7/5, 10/7 11.084 38.8
13/8, 16/13 11.956 41.8
3/2, 4/3 12.331 43.2
5/4, 8/5 13.686 47.9
13/7, 14/13 14.013 49.0
15-odd-limit intervals in 42edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 1.356 4.7
7/4, 8/7 2.603 9.1
13/11, 22/13 3.495 12.2
11/8, 16/11 8.461 29.6
7/6, 12/7 9.728 34.0
9/5, 10/9 10.975 38.4
11/7, 14/11 11.063 38.7
7/5, 10/7 11.084 38.8
13/8, 16/13 11.956 41.8
3/2, 4/3 12.331 43.2
5/4, 8/5 13.686 47.9
13/7, 14/13 14.559 51.0
11/6, 12/11 20.792 72.8
9/7, 14/9 22.059 77.2
11/10, 20/11 22.147 77.5
15/14, 28/15 23.414 82.0
13/12, 24/13 24.287 85.0
9/8, 16/9 24.661 86.3
13/10, 20/13 25.643 89.7
15/8, 16/15 26.017 91.1
11/9, 18/11 33.122 115.9
15/11, 22/15 34.478 120.7
13/9, 18/13 36.618 128.2
15/13, 26/15 37.973 132.9

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [67 -42⟩ [⟨42 67]] −3.89 3.88 13.57
2.3.5 128/125, 5000000/4782969 [⟨42 67 98]] −4.55 3.30 11.55
2.3.5.7 64/63, 126/125, 6860/6561 [⟨42 67 98 118]] −3.65 3.26 11.42

Octave stretch or compression

42edo’s inaccurate 3rd and 5th harmonics can be improved through stretching or compressing octaves. Both approaches work about equally well but in opposite directions, giving two quite different flavors of tuning to play with.

Scales

MOS scales
Main article: List of MOS scales in 42edo
  • Eugene/Tritikleismic[9]: 3 8 3 3 8 3 3 8 3
  • Eugene/Tritikleismic[15]: 3 3 2 3 3 3 3 2 3 3 3 3 2 3 3
  • Lemba[16]: 3 2 3 2 3 3 2 3 3 2 3 2 3 3 2 3
  • Qeema/Skateboard[15]: 2 5 2 2 2 5 2 2 2 5 2 2 2 5 2
  • Qeema/Skateboard[19]: 2 2 3 2 2 2 2 3 2 2 2 3 2 2 2 2 3 2 2
  • Seville/Sevond[14] 1st mode: 1 5 1 5 1 5 1 5 1 5 1 5 1 5
  • Seville/Sevond[14] 2nd mode: 5 1 5 1 5 1 5 1 5 1 5 1 5 1
  • Seville/Sevond[21]: 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4 1 1 4


Subsets of MOS scales

(Names used are idiosyncratic.)

  • Eugene/Tritikleismic[9]
    • Groovy aeolian pentatonic: 11 6 8 3 14
    • Otonal mixolydian pentatonic: 14 3 8 11 6
    • Pseudo-equipentatonic: 11 6 8 6 11
    • Septimal melodic minor pentatonic: 8 3 14 14 3
    • Septimal Picardy pentatonic: 8 6 11 3 14
    • Undecimal lydian-aeolian pentatonic: 8 14 3 11 6
    • Yokai pentatonic: 3 14 8 3 14


Approximations of gamelan scales
  • 5-tone pelog: 4 5 15 3 15
  • 7-tone pelog: 4 5 9 6 3 10 5
  • 5-tone slendro: 8 9 8 9 8


Other scales
  • 12-tone 6&7edo scale: 6 1 5 2 4 3 3 4 2 5 1 6

Instruments

Lumatone

Main article: Lumatone mapping for 42edo

Skip fretting

Skip fretting system 42 3 11: One way to play 42edo on a 14edo guitar is to tune the strings 11\42, or approximately a just 6/5, apart. All examples on this page are for 7-string guitar.

Prime intervals

1/1: string 2 open

2/1: string 5 fret 3

3/2: string 4 fret 1 and string 7 fret 4

5/4: string 3 fret 1

7/4: string 1 fret 1 and string 4 fret 4

11/8: string 7 fret 2

13/8: string 3 fret 6

17/16: string 1 fret 5

19/16: string 1 fret 7

23/16: string 4 open and string 7 fret 3

29/16: string 5 fret 1

31/16: string 1 fret 3 and string 4 fret 6

Chords

Minor 7th: 100123X

Music

Modern renderings

Johann Sebastian Bach
Bing Crosby

21st century

Bryan Deister
James Kukula
Budjarn Lambeth
Herman Miller
Mundoworld
Stephen Weigel
Retrieved from "https://en.xen.wiki/index.php?title=42edo&oldid=231369"