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→21st century: Stephen Weigel's ''Ĥ̶̩̠̐Ä̶̝͙́̓Ȑ̸̢͒K̷̥̩͌͑!̵̙͆̄ THE BIBLICALLY ACCURATE ANGELS SING!'' (2025): Add live performance in Munich, Germany (2026)
 
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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 3 × 7
{{ED intro}}
| Step size = 28.57143¢
| Fifth = 25\42 (714¢)
| Major 2nd = 8\42 = (229¢) (→[[21edo|4\21]])
| Minor 2nd = 1\42 = (29¢)
| Augmented 1sn = 7\42 = (200¢) (→[[6edo|1\6]])
}}


The '''42edo''' is the equal division of the octave into 42 equal parts of 28.6 [[cent]]s each. It has a fifth (the step of which being coprime to its cardinality, this being a first for a composite equal division of cardinality 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 supporting [[augene]] temperament.
== 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.


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 on the [[optimal ET sequence]] of the [[Augmented family#eugene|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 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 {{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 * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D
=== 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.


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.
{{Harmonics in equal|42}}
 
=== Subsets and supersets ===
Since 42 factors into {{factorization|42}}, 42edo contains subset edos {{EDOs| 2, 3, 6, 7, 14, and 21 }}.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-4"
{| class="wikitable center-all right-2 left-4"
|-
|-
! #
! #
! Cents
! Cents
! colspan="3" |[[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| P1
| P1
| perfect unison
| perfect unison
Line 33: Line 31:
|-
|-
| 1
| 1
| 28.571
| 28.6
| ^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
Line 39: Line 37:
|-
|-
| 2
| 2
| 57.143
| 57.1
| ^^1, ^m2
| ^^1, ^m2
| double-up 1sn, upminor 2nd
| dup 1sn, upminor 2nd
| ^^D, ^Eb
| ^^D, ^Eb
|-
|-
| 3
| 3
| 85.714
| 85.7
| ^^m2
| ^^m2
| double-up minor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
|-
|-
| 4
| 4
| 114.286
| 114.3
| v~2
| ^<sup>3</sup>m
| downmid 2nd
| trupminor 2nd
| ^<sup>3</sup>Eb
| ^<sup>3</sup>Eb
|-
|-
| 5
| 5
| 143.857
| 143.9
| ^~2
| v<sup>3</sup>M
| upmid 2nd
| trudmajor 2nd
| vvD#
| v<sup>3</sup>E
|-
|-
| 6
| 6
| 171.429
| 171.4
| vvM2
| vvM2
| double-down major 2nd
| dudmajor 2nd
| vD#
| vvE
|-
|-
| 7
| 7
| 200.000
| 200.0
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
| D#
| vE
|-
|-
| 8
| 8
| 228.571
| 228.6
| M2
| M2
| major 2nd
| major 2nd
Line 81: Line 79:
|-
|-
| 9
| 9
| 257.143
| 257.1
| m3
| m3
| minor 3rd
| minor 3rd
Line 87: Line 85:
|-
|-
| 10
| 10
| 285.714
| 285.7
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| Gb
| ^F
|-
|-
| 11
| 11
| 314.286
| 314.3
| ^^m3
| ^^m3
| double-up minor 3rd
| dupminor 3rd
| ^Gb
| ^^F
|-
|-
| 12
| 12
| 342.857
| 342.9
| v~3
| ^<sup>3</sup>m3
| downmid 3rd
| trupminor 3rd
| ^^Gb
| ^<sup>3</sup>F
|-
|-
| 13
| 13
| 371.429
| 371.4
| ^~3
| v<sup>3</sup>M3
| upmid 3rd
| trudmajor 3rd
| v<sup>3</sup>F#
| v<sup>3</sup>F#
|-
|-
| 14
| 14
| 400.000
| 400.0
| vvM3
| vvM3
| double-down major 3rd
| dudmajor 3rd
| vvF#
| vvF#
|-
|-
| 15
| 15
| 428.571
| 428.6
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 123: Line 121:
|-
|-
| 16
| 16
| 457.143
| 457.1
| M3, v4
| M3, v4
| major 3rd, down 4th
| major 3rd, down 4th
Line 129: Line 127:
|-
|-
| 17
| 17
| 485.714
| 485.7
| P4
| P4
| perfect 4th
| perfect 4th
Line 135: Line 133:
|-
|-
| 18
| 18
| 514.286
| 514.3
| ^4
| ^4
| up 4th
| up 4th
| A'''b'''
| ^G
|-
|-
| 19
| 19
| 543.857
| 543.9
| ^^4
| ^^4
| double-up 4th
| dup 4th
| ^Ab
| ^^G
|-
|-
| 20
| 20
| 571.429
| 571.4
| v~4
| ^<sup>3</sup>4, ^^d5
| downmid 4th
| trup 4th, dupdim 5th
| ^^Ab
| ^<sup>3</sup>G, ^^Ab
|-
|-
| 21
| 21
| 600.000
| 600.0
| ^~4, v~5
| v<sup>3</sup>A4, ^<sup>3</sup>d5
| upmid 4th, downmid 5th
| trudaug 4th, trupdim 5th
| v<sup>3</sup>G#, ^<sup>3</sup>Ab
| v<sup>3</sup>G#, ^<sup>3</sup>Ab
|-
|-
| 22
| 22
| 628.571
| 628.6
| ^~5
| vvA4, v<sup>3</sup>5
| upmid 5th
| dudaug 4th, trud 5th
| vvG#
| vvG#, v<sup>3</sup>A
|-
|-
| 23
| 23
| 657.143
| 657.1
| vv5
| vv5
| double-down 5th
| dud 5th
| vG#
| vvA
|-
|-
| 24
| 24
| 685.714
| 685.7
| v5
| v5
| down 5th
| down 5th
| G#
| vA
|-
|-
| 25
| 25
| 714.286
| 714.3
| P5
| P5
| perfect 5th
| perfect 5th
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|-
|-
| 26
| 26
| 742.857
| 742.9
| ^5, m6
| ^5, m6
| up 5th, minor 6th
| up 5th, minor 6th
Line 189: Line 187:
|-
|-
| 27
| 27
| 771.429
| 771.4
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 195: Line 193:
|-
|-
| 28
| 28
| 800.000
| 800.0
| ^^m6
| ^^m6
| double-up minor 6th
| dupminor 6th
| ^^Bb
| ^^Bb
|-
|-
| 29
| 29
| 828.571
| 828.6
| v~6
| ^<sup>3</sup>m6
| downmid 6th
| trupminor 6th
| ^<sup>3</sup>Bb
| ^<sup>3</sup>Bb
|-
|-
| 30
| 30
| 857.143
| 857.1
| ^~6
| v<sup>3</sup>M6
| upmid 6th
| trudmajor 6th
| ^^A#
| v<sup>3</sup>B
|-
|-
| 31
| 31
| 885.714
| 885.7
| vvM6
| vvM6
| double-down major 6th
| dudmajor 6th
| ^A#
| vvB
|-
|-
| 32
| 32
| 914.286
| 914.3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| A#
| vB
|-
|-
| 33
| 33
| 942.857
| 942.9
| M6
| M6
| major 6th
| major 6th
Line 231: Line 229:
|-
|-
| 34
| 34
| 971.429
| 971.4
| m7
| m7
| minor 7th
| minor 7th
Line 237: Line 235:
|-
|-
| 35
| 35
| 1000.000
| 1000.0
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| D#
| ^C
|-
|-
| 36
| 36
| 1028.571
| 1028.6
| ^^m7
| ^^m7
| double-up minor 7th
| dupminor 7th
| ^D#
| ^^C
|-
|-
| 37
| 37
| 1057.143
| 1057.1
| v~7
| ^<sup>3</sup>m7
| downmid 7th
| trupminor 7th
| ^^D#
| ^<sup>3</sup>C
|-
|-
| 38
| 38
| 1085.714
| 1085.7
| ^~7
| v<sup>3</sup>M7
| upmid 7th
| trudmajor 7th
| v<sup>3</sup>C#
| v<sup>3</sup>C#
|-
|-
| 39
| 39
| 1114.286
| 1114.3
| vvM7
| vvM7
| double-down major 7th
| dudmajor 7th
| vvC#
| vvC#
|-
|-
| 40
| 40
| 1142.857
| 1142.9
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 273: Line 271:
|-
|-
| 41
| 41
| 1171.429
| 1171.4
| M7, v8
| M7, v8
| major 7th, down 8ve
| major 7th, down 8ve
Line 279: Line 277:
|-
|-
| 42
| 42
| 1200.000
| 1200.0
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 285: Line 283:
|}
|}


Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[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]].
 
== 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:Augene]]
[[Category:Augmented]]
[[Category:Equal divisions of the octave]]
{{Todo|review|add rank 2 temperaments table}}
[[Category:Subgroup]]
Retrieved from "https://en.xen.wiki/w/42edo"