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{{Harmonics in equal|42}}
{{Harmonics in equal|42}}
=== Octave stretch ===
42edo’s inaccurate 3rd and 5th harmonics can be greatly 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.
The following table compares three stretched tunings of 42edo:
{| class="wikitable mw-collapsible mw-collapsed"
|-
! Tuning
! [[42ed257/128]]
! [[1ed28.7c|APS720jot]]
! [[zpi|189zpi]]
! [[42edo]]
|-
! Steps / octave
| ~41.77
| ~41.81
| ~41.83
| 42.00
|-
! Approximation<br>of harmonics
| great: 5<br>good: 2, 3, 7 <br>okay: <br>bad: 11, 13
| great: 5<br>good: 2, 3 <br>okay: 7, 11, 13 <br>bad:
| great: 5<br>good: 2, 13 <br>okay: 3, 11 <br>bad: 7
| great: 2, 7<br>good: <br>okay: 11 <br>bad: 3, 5, 13
|-
| colspan="5" | <span style="font-size: 0.75em;"><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></span>
|}
The following table compares three compressed tunings of 42edo:
{| class="wikitable mw-collapsible mw-collapsed"
|-
! Tuning
! [[42edo]]
! [[1ed28.5c|APS715jot]]
! [[191zpi]]
! [[ed255/128#42ed255/128|42ed255/128]]
|-
! Steps / octave
| 42.00
| ~42.10
| ~42.19
| ~42.24
|-
! Approximation <br>of harmonics
| great: 2, 7 <br>good: <br>okay: 11 <br>bad: 3, 5, 13
| great: 2 <br>good: 5, 7, 13 <br>okay: 3, 11 <br>bad:
| great: 5, 11, 13 <br>good: 2, 3 <br>okay: <br>bad: 7
| great: 3, 5, 11 <br>good: 2 <br>okay: 13 <br>bad: 7
|-
| colspan="5" | <span style="font-size: 0.75em;"><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></span>
|}
For a more detailed comparison see [[Table of stretched 42edo tunings]].


=== Subsets and supersets ===
=== Subsets and supersets ===
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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:
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 * * * * * * * 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.


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 key or fret 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:
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
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</imagemap>
</imagemap>


== JI approximation ==
== Approximation to JI ==
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals}}
=== Zeta peak index ===
{{ZPI
| zpi = 190
| steps = 41.9988672418690
| step size = 28.5721991759747
| tempered height = 2.689436
| pure height = 2.688894
| integral = 0.303886
| gap = 9.541545
| octave = 1200.03236539094
| consistent = 8
| distinct = 8
}}


== Regular temperament properties ==
== Regular temperament properties ==
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| 11.42
| 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 ==
== Scales ==
; [[MOS scale]]s
{{main|List of MOS scales in 42edo}}
{{main|List of MOS scales in 42edo}}
; [[MOS scale]]s
* Eugene/Tritikleismic[9]: '''3 8 3 3 8 3 3 8 3'''
* 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'''
* Eugene/Tritikleismic[15]: '''3 3 2 3 3 3 3 2 3 3 3 3 2 3 3'''
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* Seville/Sevond[14] 2nd mode: '''5 1 5 1 5 1 5 1 5 1 5 1 5 1'''
* 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'''
* 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  
; Subsets of MOS scales  
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** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
** Undecimal lydian-aeolian pentatonic: '''8 14 3 11 6'''
** Yokai pentatonic: '''3 14 8 3 14'''
** 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 ==
; Fretted instruments
=== Lumatone ===
; [[Skip fretting system 42 3 11]]
; Lumatone
{{main|Lumatone mapping for 42edo}}
{{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 ==
== Music ==
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=== 21st century ===
=== 21st century ===
; [[Bryan Deister]]
; [[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=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]]
; [[James Kukula]]
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* [https://www.youtube.com/watch?v=ORy7nv6SnN8 ''Glory of Them''] (2024)
* [https://www.youtube.com/watch?v=ORy7nv6SnN8 ''Glory of Them''] (2024)


[[Category:Augene]]
; [[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}}
{{Todo|review|add rank 2 temperaments table}}
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