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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|42}}
{{ED intro}}


== Theory ==
== Theory ==
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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 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.
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}}
=== Octave stretch ===
42edo’s inaccurate 3rd and 5th harmonics can be greatly improved through [[octave stretching]] or [[octave shrinking]]. 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"
|+
!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" | <small><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></small>
|}
The following table compares three compressed tunings of 42edo:
{| class="wikitable"
|+
!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" | <small><div style="text-align: center;">''“great” = 0-13% relative error • “good” = 13-27% • “okay” = 27-40% • “bad” = 40-50%''</div></small>
|}
For a more detailed comparison see [[Table of stretched 42edo tunings]].


=== 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
| D
| D
|-
|-
|1
| 1
|28.571
| 28.6
|^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
|^D, Eb
| ^D, Eb
|-
|-
|2
| 2
|57.143
| 57.1
|^^1, ^m2
| ^^1, ^m2
|dup 1sn, upminor 2nd
| dup 1sn, upminor 2nd
| ^^D, ^Eb
| ^^D, ^Eb
|-
|-
|3
| 3
|85.714
| 85.7
|^^m2
| ^^m2
|dupminor 2nd
| dupminor 2nd
|^^Eb
| ^^Eb
|-
|-
| 4
| 4
|114.286
| 114.3
|^<sup>3</sup>m
| ^<sup>3</sup>m
|trupminor 2nd
| trupminor 2nd
|^<sup>3</sup>Eb
| ^<sup>3</sup>Eb
|-
|-
|5
| 5
|143.857
| 143.9
|v<sup>3</sup>M
| v<sup>3</sup>M
|trudmajor 2nd
| trudmajor 2nd
|v<sup>3</sup>E
| v<sup>3</sup>E
|-
|-
|6
| 6
|171.429
| 171.4
|vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
|vvE
| vvE
|-
|-
|7
| 7
|200.000
| 200.0
|vM2
| vM2
|downmajor 2nd
| downmajor 2nd
|vE
| vE
|-
|-
| 8
| 8
|228.571
| 228.6
|M2
| M2
|major 2nd
| major 2nd
| E
| E
|-
|-
|9
| 9
|257.143
| 257.1
|m3
| m3
|minor 3rd
| minor 3rd
|F
| F
|-
|-
|10
| 10
| 285.714
| 285.7
|^m3
| ^m3
| upminor 3rd
| upminor 3rd
|^F
| ^F
|-
|-
|11
| 11
|314.286
| 314.3
|^^m3
| ^^m3
|dupminor 3rd
| dupminor 3rd
|^^F
| ^^F
|-
|-
|12
| 12
|342.857
| 342.9
|^<sup>3</sup>m3
| ^<sup>3</sup>m3
| trupminor 3rd
| trupminor 3rd
|^<sup>3</sup>F
| ^<sup>3</sup>F
|-
|-
|13
| 13
|371.429
| 371.4
|v<sup>3</sup>M3
| v<sup>3</sup>M3
|trudmajor 3rd
| trudmajor 3rd
|v<sup>3</sup>F#
| v<sup>3</sup>F#
|-
|-
|14
| 14
|400.000
| 400.0
|vvM3
| vvM3
|dudmajor 3rd
| dudmajor 3rd
|vvF#
| vvF#
|-
|-
|15
| 15
|428.571
| 428.6
|vM3
| vM3
|downmajor 3rd
| downmajor 3rd
| vF#
| vF#
|-
|-
|16
| 16
| 457.143
| 457.1
|M3, v4
| M3, v4
|major 3rd, down 4th
| major 3rd, down 4th
|F#, vG
| F#, vG
|-
|-
| 17
| 17
|485.714
| 485.7
|P4
| P4
|perfect 4th
| perfect 4th
|G
| G
|-
|-
|18
| 18
| 514.286
| 514.3
|^4
| ^4
|up 4th
| up 4th
|^G
| ^G
|-
|-
|19
| 19
|543.857
| 543.9
| ^^4
| ^^4
|dup 4th
| dup 4th
|^^G
| ^^G
|-
|-
|20
| 20
|571.429
| 571.4
|^<sup>3</sup>4, ^^d5
| ^<sup>3</sup>4, ^^d5
| trup 4th, dupdim 5th
| trup 4th, dupdim 5th
| ^<sup>3</sup>G, ^^Ab
| ^<sup>3</sup>G, ^^Ab
|-
|-
|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
|v<sup>3</sup>G#, ^<sup>3</sup>Ab
| v<sup>3</sup>G#, ^<sup>3</sup>Ab
|-
|-
|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
|vvG#, v<sup>3</sup>A
| vvG#, v<sup>3</sup>A
|-
|-
|23
| 23
|657.143
| 657.1
|vv5
| vv5
|dud 5th
| dud 5th
|vvA
| vvA
|-
|-
|24
| 24
| 685.714
| 685.7
|v5
| v5
|down 5th
| down 5th
| vA
| vA
|-
|-
|25
| 25
|714.286
| 714.3
|P5
| P5
|perfect 5th
| perfect 5th
|A
| A
|-
|-
|26
| 26
|742.857
| 742.9
|^5, m6
| ^5, m6
|up 5th, minor 6th
| up 5th, minor 6th
|^A, Bb
| ^A, Bb
|-
|-
|27
| 27
|771.429
| 771.4
|^m6
| ^m6
|upminor 6th
| upminor 6th
|^Bb
| ^Bb
|-
|-
|28
| 28
|800.000
| 800.0
|^^m6
| ^^m6
|dupminor 6th
| dupminor 6th
|^^Bb
| ^^Bb
|-
|-
|29
| 29
|828.571
| 828.6
|^<sup>3</sup>m6
| ^<sup>3</sup>m6
| trupminor 6th
| trupminor 6th
| ^<sup>3</sup>Bb
| ^<sup>3</sup>Bb
|-
|-
|30
| 30
|857.143
| 857.1
|v<sup>3</sup>M6
| v<sup>3</sup>M6
|trudmajor 6th
| trudmajor 6th
|v<sup>3</sup>B
| v<sup>3</sup>B
|-
|-
| 31
| 31
|885.714
| 885.7
|vvM6
| vvM6
|dudmajor 6th
| dudmajor 6th
|vvB
| vvB
|-
|-
|32
| 32
|914.286
| 914.3
|vM6
| vM6
|downmajor 6th
| downmajor 6th
|vB
| vB
|-
|-
|33
| 33
|942.857
| 942.9
|M6
| M6
|major 6th
| major 6th
|B
| B
|-
|-
| 34
| 34
|971.429
| 971.4
|m7
| m7
|minor 7th
| minor 7th
|C
| C
|-
|-
|35
| 35
|1000.000
| 1000.0
|^m7
| ^m7
|upminor 7th
| upminor 7th
|^C
| ^C
|-
|-
|36
| 36
|1028.571
| 1028.6
|^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
|^^C
| ^^C
|-
|-
|37
| 37
|1057.143
| 1057.1
|^<sup>3</sup>m7
| ^<sup>3</sup>m7
|trupminor 7th
| trupminor 7th
|^<sup>3</sup>C
| ^<sup>3</sup>C
|-
|-
|38
| 38
|1085.714
| 1085.7
|v<sup>3</sup>M7
| v<sup>3</sup>M7
|trudmajor 7th
| trudmajor 7th
| v<sup>3</sup>C#
| v<sup>3</sup>C#
|-
|-
|39
| 39
| 1114.286
| 1114.3
|vvM7
| vvM7
|dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
|-
|-
|40
| 40
|1142.857
| 1142.9
|vM7
| vM7
|downmajor 7th
| downmajor 7th
|vC#
| vC#
|-
|-
|41
| 41
| 1171.429
| 1171.4
|M7, v8
| M7, v8
|major 7th, down 8ve
| major 7th, down 8ve
|C#, vD
| C#, vD
|-
|-
|42
| 42
| 1200.000
| 1200.0
|P8
| P8
| perfect 8ve
| perfect 8ve
|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]].
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 ==
=== Ups and downs 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:
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 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.
In this table, dup is equivalent to quidsharp, trup is equivalent to quudsharp, trudsharp is equivalent to quup, dudsharp is equivalent to quip, etc.
{{sharpness-sharp7a}}


===Sagittal notation===
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
====Best fifth notation====
This notation uses the same sagittal sequence as [[35edo#Second-best fifth notation|35b]].
=====Evo flavor=====


{{sharpness-sharp7}}
=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[35edo #Second-best fifth notation|35b]].
===== Evo flavor =====
<imagemap>
<imagemap>
File:42-EDO_Evo_Sagittal.svg
File:42-EDO_Evo_Sagittal.svg
Line 362: Line 314:
</imagemap>
</imagemap>


=====Revo flavor=====
===== Revo flavor =====
 
<imagemap>
<imagemap>
File:42-EDO_Revo_Sagittal.svg
File:42-EDO_Revo_Sagittal.svg
Line 373: Line 324:
</imagemap>
</imagemap>


====Second-best fifth notation====
==== Second-best fifth notation ====
This notation uses the same sagittal sequence as [[47edo#Sagittal notation|47-EDO]], and is a superset of the notations for EDOs [[21edo#Sagittal notation|21]], [[14edo#Sagittal notation|14]], and [[7edo#Sagittal notation|7]].
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>
<imagemap>
Line 385: Line 336:
</imagemap>
</imagemap>


==Scales==
== 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
|}


;[[MOS scale]]s
== Octave stretch or compression ==
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.
 
What follows is a comparison of stretched- and compressed-octave 42edo tunings.
 
; [[ed6|108ed6]]
* Octave size: 1206.3{{c}}
Stretching the octave of 42edo by around 6{{c}} results in improved primes 3, 5 and 13, but worse primes 2, 7 and 11. This approximates all harmonics up to 16 within 13.3{{c}}. The tuning 108ed6 does this. So does the tuning [[ed5|97ed5]] whose octave differs by only 0.1{{c}}.
{{Harmonics in equal|108|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 108ed6}}
{{Harmonics in equal|108|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 108ed6 (continued)}}
 
; [[zpi|189zpi]]
* Step size: 28.689{{c}}, octave size: 1204.9{{c}}
Stretching the octave of 42edo by around 5{{c}} results in improved primes 3, 5 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. The tuning 189zpi does this.
{{Harmonics in cet|28.689|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 189zpi}}
{{Harmonics in cet|28.689|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 189zpi (continued)}}
 
; [[ed12|150ed12]]
* Octave size: 1204.5{{c}}
Stretcing the octave of 42edo by around 4.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.6{{c}}. The tuning 150ed12 does this.
{{Harmonics in equal|150|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 150ed12}}
{{Harmonics in equal|150|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 150ed12 (continued)}}
 
; [[equal tuning|145ed11]]
* Octave size: 1202.5{{c}}
Stretching the octave of 42edo by around 2.5{{c}} results in improved primes 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 11.9{{c}}. The tuning 145ed11 does this.
{{Harmonics in equal|145|11|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 145ed11}}
{{Harmonics in equal|145|11|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 145ed11 (continued)}}
 
; 42edo
* Step size: 28.571{{c}}, octave size: 1200.0{{c}}
Pure-octaves 42edo approximates all harmonics up to 16 within 13.7{{c}}. The tuning [[zpi|190zpi]] is almost exactly the same as pure-octaves 42edo, its octave differing by less than 0.05{{c}}.
{{Harmonics in equal|42|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42edo}}
{{Harmonics in equal|42|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42edo (continued)}}
 
; [[ed7|118ed7]]
* Step size: Octave size: 1199.1{{c}}
Compressing the octave of 42edo by around 1{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 13.2{{c}}. The tuning 118ed7 does this.
{{Harmonics in equal|118|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 118ed7}}
{{Harmonics in equal|118|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 118ed7 (continued)}}
 
; [[WE|42et, 13-limit WE tuning]]
* Step size: 28.534{{c}}, octave size: 1198.4{{c}}
Compressing the octave of 42edo by around 1.5{{c}} results in improved primes 3, 5, 7 and 13, but worse primes 2 and 11. This approximates all harmonics up to 16 within 13.9{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
 
Of the tunings discussed in this section, 13-limit WE and TE are the only ones to approximate all harmonics up to 10 within 10 cents, making them a good all-round choice.
{{Harmonics in cet|28.534|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning}}
{{Harmonics in cet|28.534|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42et, 13-limit WE tuning (continued)}}
 
; [[ed12|151ed12]]
* Step size: Octave size: 1196.6{{c}}
Compressing the octave of 42edo by around 3.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 13.7{{c}}. The tuning 151ed12 does this. So do the 7-limit [[WE]] and [[TE]] tunings of 42et, whose octaves are within 0.3{{c}} of 151ed12.
{{Harmonics in equal|151|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 151ed12}}
{{Harmonics in equal|151|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 151ed12 (continued)}}
 
; [[ed6|109ed6]]
* Octave size: 1195.2{{c}}
Compressing the octave of 42edo by around 5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 14.2{{c}}. The tuning 109ed6 does this.
{{Harmonics in equal|109|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 109ed6}}
{{Harmonics in equal|109|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 109ed6 (continued)}}
 
; [[zpi|191zpi]]
* Step size: 28.444{{c}}, octave size: 1194.6{{c}}
Compressing the octave of 42edo by around 5.5{{c}} results in improved primes 3, 5, 11 and 13, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.4{{c}}. The tuning 191zpi does this.
{{Harmonics in cet|28.444|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 191zpi}}
{{Harmonics in cet|28.444|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 191zpi (continued)}}
 
; [[67edt]]
* Step size: 28.387{{c}}, octave size: 1192.3{{c}}
Compressing the octave of 42edo by around 7.5{{c}} results in improved primes 3, 5 and 11, but worse primes 2 and 7. This approximates all harmonics up to 16 within 12.9{{c}}. The tuning 67edt does this.
{{Harmonics in equal|67|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 67edt}}
{{Harmonics in equal|67|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 67edt (continued)}}
 
== Scales ==
{{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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** Yokai pentatonic: '''3 14 8 3 14'''
** Yokai pentatonic: '''3 14 8 3 14'''


==Instruments==
;Lumatone


See [[Lumatone mapping for 42edo]]
; 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
 
== 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 ==
== 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)


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