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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}}
{{EDO intro|42}}
{{ED intro}}


== Theory ==
== 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 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).  
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).  


=== Odd harmonics ===
=== 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.
{{Harmonics in equal|42}}
{{Harmonics in equal|42}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 42 factors into {{factorization|42}}, 42edo contains subset edos {{EDOs| 2, 3, 6, 7, 14, and 21 }}.  
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 30: Line 31:
|-
|-
| 1
| 1
| 28.571
| 28.6
| ^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
Line 36: Line 37:
|-
|-
| 2
| 2
| 57.143
| 57.1
| ^^1, ^m2
| ^^1, ^m2
| dup 1sn, upminor 2nd
| dup 1sn, upminor 2nd
Line 42: Line 43:
|-
|-
| 3
| 3
| 85.714
| 85.7
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
Line 48: Line 49:
|-
|-
| 4
| 4
| 114.286
| 114.3
| ^<sup>3</sup>m
| ^<sup>3</sup>m
| trupminor 2nd
| trupminor 2nd
Line 54: Line 55:
|-
|-
| 5
| 5
| 143.857
| 143.9
| v<sup>3</sup>M
| v<sup>3</sup>M
| trudmajor 2nd
| trudmajor 2nd
Line 60: Line 61:
|-
|-
| 6
| 6
| 171.429
| 171.4
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
Line 66: Line 67:
|-
|-
| 7
| 7
| 200.000
| 200.0
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 72: Line 73:
|-
|-
| 8
| 8
| 228.571
| 228.6
| M2
| M2
| major 2nd
| major 2nd
Line 78: Line 79:
|-
|-
| 9
| 9
| 257.143
| 257.1
| m3
| m3
| minor 3rd
| minor 3rd
Line 84: Line 85:
|-
|-
| 10
| 10
| 285.714
| 285.7
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 90: Line 91:
|-
|-
| 11
| 11
| 314.286
| 314.3
| ^^m3
| ^^m3
| dupminor 3rd
| dupminor 3rd
Line 96: Line 97:
|-
|-
| 12
| 12
| 342.857
| 342.9
| ^<sup>3</sup>m3
| ^<sup>3</sup>m3
| trupminor 3rd
| trupminor 3rd
Line 102: Line 103:
|-
|-
| 13
| 13
| 371.429
| 371.4
| v<sup>3</sup>M3
| v<sup>3</sup>M3
| trudmajor 3rd
| trudmajor 3rd
Line 108: Line 109:
|-
|-
| 14
| 14
| 400.000
| 400.0
| vvM3
| vvM3
| dudmajor 3rd
| dudmajor 3rd
Line 114: Line 115:
|-
|-
| 15
| 15
| 428.571
| 428.6
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 120: Line 121:
|-
|-
| 16
| 16
| 457.143
| 457.1
| M3, v4
| M3, v4
| major 3rd, down 4th
| major 3rd, down 4th
Line 126: Line 127:
|-
|-
| 17
| 17
| 485.714
| 485.7
| P4
| P4
| perfect 4th
| perfect 4th
Line 132: Line 133:
|-
|-
| 18
| 18
| 514.286
| 514.3
| ^4
| ^4
| up 4th
| up 4th
Line 138: Line 139:
|-
|-
| 19
| 19
| 543.857
| 543.9
| ^^4
| ^^4
| dup 4th
| dup 4th
Line 144: Line 145:
|-
|-
| 20
| 20
| 571.429
| 571.4
| ^<sup>3</sup>4, ^^d5
| ^<sup>3</sup>4, ^^d5
| trup 4th, dupdim 5th
| trup 4th, dupdim 5th
Line 150: Line 151:
|-
|-
| 21
| 21
| 600.000
| 600.0
| v<sup>3</sup>A4, ^<sup>3</sup>d5
| v<sup>3</sup>A4, ^<sup>3</sup>d5
| trudaug 4th, trupdim 5th
| trudaug 4th, trupdim 5th
Line 156: Line 157:
|-
|-
| 22
| 22
| 628.571
| 628.6
| vvA4, v<sup>3</sup>5
| vvA4, v<sup>3</sup>5
| dudaug 4th, trud 5th
| dudaug 4th, trud 5th
Line 162: Line 163:
|-
|-
| 23
| 23
| 657.143
| 657.1
| vv5
| vv5
| dud 5th
| dud 5th
Line 168: Line 169:
|-
|-
| 24
| 24
| 685.714
| 685.7
| v5
| v5
| down 5th
| down 5th
Line 174: Line 175:
|-
|-
| 25
| 25
| 714.286
| 714.3
| P5
| P5
| perfect 5th
| perfect 5th
Line 180: Line 181:
|-
|-
| 26
| 26
| 742.857
| 742.9
| ^5, m6
| ^5, m6
| up 5th, minor 6th
| up 5th, minor 6th
Line 186: Line 187:
|-
|-
| 27
| 27
| 771.429
| 771.4
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 192: Line 193:
|-
|-
| 28
| 28
| 800.000
| 800.0
| ^^m6
| ^^m6
| dupminor 6th
| dupminor 6th
Line 198: Line 199:
|-
|-
| 29
| 29
| 828.571
| 828.6
| ^<sup>3</sup>m6
| ^<sup>3</sup>m6
| trupminor 6th
| trupminor 6th
Line 204: Line 205:
|-
|-
| 30
| 30
| 857.143
| 857.1
| v<sup>3</sup>M6
| v<sup>3</sup>M6
| trudmajor 6th
| trudmajor 6th
Line 210: Line 211:
|-
|-
| 31
| 31
| 885.714
| 885.7
| vvM6
| vvM6
| dudmajor 6th
| dudmajor 6th
Line 216: Line 217:
|-
|-
| 32
| 32
| 914.286
| 914.3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 222: Line 223:
|-
|-
| 33
| 33
| 942.857
| 942.9
| M6
| M6
| major 6th
| major 6th
Line 228: Line 229:
|-
|-
| 34
| 34
| 971.429
| 971.4
| m7
| m7
| minor 7th
| minor 7th
Line 234: Line 235:
|-
|-
| 35
| 35
| 1000.000
| 1000.0
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 240: Line 241:
|-
|-
| 36
| 36
| 1028.571
| 1028.6
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
Line 246: Line 247:
|-
|-
| 37
| 37
| 1057.143
| 1057.1
| ^<sup>3</sup>m7
| ^<sup>3</sup>m7
| trupminor 7th
| trupminor 7th
Line 252: Line 253:
|-
|-
| 38
| 38
| 1085.714
| 1085.7
| v<sup>3</sup>M7
| v<sup>3</sup>M7
| trudmajor 7th
| trudmajor 7th
Line 258: Line 259:
|-
|-
| 39
| 39
| 1114.286
| 1114.3
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
Line 264: Line 265:
|-
|-
| 40
| 40
| 1142.857
| 1142.9
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 270: Line 271:
|-
|-
| 41
| 41
| 1171.429
| 1171.4
| M7, v8
| M7, v8
| major 7th, down 8ve
| major 7th, down 8ve
Line 276: Line 277:
|-
|-
| 42
| 42
| 1200.000
| 1200.0
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 282: 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 ==
== 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:
=== 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]]


D * * * * * * * * E F * * * * * * * * G * * * * * * * * A * * * * * * * * B C * * * * * * * * D
== 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'''


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 dud dup-seven chord, written either as v<sup>3</sup>G#vv,^^7 or as ^<sup>3</sup>Abvv,^^7.
 
; 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 ==
== Instruments ==
; Lumatone
=== 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


See [[Lumatone mapping for 42edo]]
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 ==
== Music ==
* ''[https://m.youtube.com/watch?v=ljaSpsQP2qc Improvisation in 42edo]'' - composed and played by [[Bryan Deister]] (May 2023), transcribed by [[Stephen Weigel]] (Sept 2024)
=== Modern renderings ===
* ''[https://m.youtube.com/watch?v=ORy7nv6SnN8 Glory of Them]'' - [[Mundoworld]] (July 2024)
; {{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:Todo:add rank 2 temperaments table]]
{{Todo|review|add rank 2 temperaments table}}
Retrieved from "https://en.xen.wiki/w/42edo"