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=== Octave stretch ===
=== Octave stretch ===
42edo’s poor 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.
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.  


Stretched versions of 42edo include:
The following is a table comparing stretched and shrunk tunings of 42edo:
* [[Ed257/128#42ed257/128|42ed257/128]] (~41.77edo, improves harmonics 3 & 5 at the expense of harmonic 11)
* [[zpi|189zpi]] (~41.83edo) (~41.77edo, improves harmonics 3 & 5 at the expense of harmonics 7 & 11)


Compressed versions of 42edo include:
'''Key'''
* [[zpi|191zpi]] (~42.19edo, improves harmonics 3, 5 & 13 at the expense of harmonic 7)
*“Great” = <20% relative error
* [[Ed255/128#42ed255/128|42ed255/128]] (~42.24edo, improves harmonics 3, 5 & 13 at the expense of harmonic 7)
*“Good” = 20 to 30% relative error
 
*“Poor” = >30% relative error
{| class="wikitable sortable mw-collapsible"
{| class="wikitable mw-collapsible mw-collapsed"
|+
|+
!Relative error of
!Tuning
!42ed257/128
![[Ed257/128#42ed257/128|42ed257/128]] <br>(~41.77edo)
!42edo
![[zpi|189zpi]] <br>(~41.83edo)
!42ed255/128
![[42edo]]
![[zpi|191zpi]] <br>(~42.19edo)
![[Ed255/128#42ed255/128|42ed255/128]] <br>(~42.24edo)
|-
|-
!2nd harmonic
!Summary
!great: 3, 5 <br>good: 2, 7 <br>poor: 11, 13
!great: 2, 5 <br>good: 3, 11, 13 <br>poor: 7
!great: 2, 7 <br>good: 11 <br>poor: 3, 5, 13
!great: 2, 3, 5, 11, 13 <br>good: - <br>poor: 7
!great: 3, 5, 11 <br>good: 2 <br>poor: 7, 13
|-
|2/1 relative error
|23.5%
|23.5%
|17.2%
|0.0%
|0.0%
|18.8%
|23.9%
|23.9%
|-
|-
!3rd harmonic
|3/1 rel. err.
|19.6%
|19.6%
|29.6%
|43.2%
|43.2%
|13.4%
|5.4%
|5.4%
|-
|-
!5th harmonic
|5/1 rel. err.
|2.4%
| 2.4%
|12.2%
|47.9%
|47.9%
| 4.3%
|7.5%
|7.5%
|-
|-
!7th harmonic
| 7/1 rel. err.
|24.9%
|24.9%
|42.6%
|9.1%
|9.1%
| 43.6%
|42.2%
|42.2%
|-
|-
!11th harmonic
|11/1 rel. err.
|48.3%
| 48.3%
|29.6%
|29.8%
| 29.6%
|5.5%
|12.1%
|12.1%
|-
|-
!13th harmonic
| 13/1 rel. err.
|45.1%
|45.1%
|21.7%
|41.8%
|41.8%
|11.3%
|30.1%
|30.1%
|}
|}


== 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
| 0.000
| P1
|P1
| perfect unison
|perfect unison
| D
| D
|-
|-
| 1
|1
| 28.571
|28.571
| ^1, m2
|^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
| ^D, Eb
|^D, Eb
|-
|-
| 2
|2
| 57.143
|57.143
| ^^1, ^m2
|^^1, ^m2
| dup 1sn, upminor 2nd
|dup 1sn, upminor 2nd
| ^^D, ^Eb
| ^^D, ^Eb
|-
|-
| 3
|3
| 85.714
|85.714
| ^^m2
|^^m2
| dupminor 2nd
|dupminor 2nd
| ^^Eb
|^^Eb
|-
|-
| 4
| 4
| 114.286
|114.286
| ^<sup>3</sup>m
|^<sup>3</sup>m
| trupminor 2nd
|trupminor 2nd
| ^<sup>3</sup>Eb
|^<sup>3</sup>Eb
|-
|-
| 5
|5
| 143.857
|143.857
| 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
|171.429
| vvM2
|vvM2
| dudmajor 2nd
| dudmajor 2nd
| vvE
|vvE
|-
|-
| 7
|7
| 200.000
|200.000
| vM2
|vM2
| downmajor 2nd
|downmajor 2nd
| vE
|vE
|-
|-
| 8
| 8
| 228.571
|228.571
| M2
|M2
| major 2nd
|major 2nd
| E
| E
|-
|-
| 9
|9
| 257.143
|257.143
| m3
|m3
| minor 3rd
|minor 3rd
| F
|F
|-
|-
| 10
|10
| 285.714
| 285.714
| ^m3
|^m3
| upminor 3rd
| upminor 3rd
| ^F
|^F
|-
|-
| 11
|11
| 314.286
|314.286
| ^^m3
|^^m3
| dupminor 3rd
|dupminor 3rd
| ^^F
|^^F
|-
|-
| 12
|12
| 342.857
|342.857
| ^<sup>3</sup>m3
|^<sup>3</sup>m3
| trupminor 3rd
| trupminor 3rd
| ^<sup>3</sup>F
|^<sup>3</sup>F
|-
|-
| 13
|13
| 371.429
|371.429
| 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
|400.000
| vvM3
|vvM3
| dudmajor 3rd
|dudmajor 3rd
| vvF#
|vvF#
|-
|-
| 15
|15
| 428.571
|428.571
| vM3
|vM3
| downmajor 3rd
|downmajor 3rd
| vF#
| vF#
|-
|-
| 16
|16
| 457.143
| 457.143
| M3, v4
|M3, v4
| major 3rd, down 4th
|major 3rd, down 4th
| F#, vG
|F#, vG
|-
|-
| 17
| 17
| 485.714
|485.714
| P4
|P4
| perfect 4th
|perfect 4th
| G
|G
|-
|-
| 18
|18
| 514.286
| 514.286
| ^4
|^4
| up 4th
|up 4th
| ^G
|^G
|-
|-
| 19
|19
| 543.857
|543.857
| ^^4
| ^^4
| dup 4th
|dup 4th
| ^^G
|^^G
|-
|-
| 20
|20
| 571.429
|571.429
| ^<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
|600.000
| 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
|628.571
| 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
|657.143
| vv5
|vv5
| dud 5th
|dud 5th
| vvA
|vvA
|-
|-
| 24
|24
| 685.714
| 685.714
| v5
|v5
| down 5th
|down 5th
| vA
| vA
|-
|-
| 25
|25
| 714.286
|714.286
| P5
|P5
| perfect 5th
|perfect 5th
| A
|A
|-
|-
| 26
|26
| 742.857
|742.857
| ^5, m6
|^5, m6
| up 5th, minor 6th
|up 5th, minor 6th
| ^A, Bb
|^A, Bb
|-
|-
| 27
|27
| 771.429
|771.429
| ^m6
|^m6
| upminor 6th
|upminor 6th
| ^Bb
|^Bb
|-
|-
| 28
|28
| 800.000
|800.000
| ^^m6
|^^m6
| dupminor 6th
|dupminor 6th
| ^^Bb
|^^Bb
|-
|-
| 29
|29
| 828.571
|828.571
| ^<sup>3</sup>m6
|^<sup>3</sup>m6
| trupminor 6th
| trupminor 6th
| ^<sup>3</sup>Bb
| ^<sup>3</sup>Bb
|-
|-
| 30
|30
| 857.143
|857.143
| 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
|885.714
| vvM6
|vvM6
| dudmajor 6th
|dudmajor 6th
| vvB
|vvB
|-
|-
| 32
|32
| 914.286
|914.286
| vM6
|vM6
| downmajor 6th
|downmajor 6th
| vB
|vB
|-
|-
| 33
|33
| 942.857
|942.857
| M6
|M6
| major 6th
|major 6th
| B
|B
|-
|-
| 34
| 34
| 971.429
|971.429
| m7
|m7
| minor 7th
|minor 7th
| C
|C
|-
|-
| 35
|35
| 1000.000
|1000.000
| ^m7
|^m7
| upminor 7th
|upminor 7th
| ^C
|^C
|-
|-
| 36
|36
| 1028.571
|1028.571
| ^^m7
|^^m7
| dupminor 7th
| dupminor 7th
| ^^C
|^^C
|-
|-
| 37
|37
| 1057.143
|1057.143
| ^<sup>3</sup>m7
|^<sup>3</sup>m7
| trupminor 7th
|trupminor 7th
| ^<sup>3</sup>C
|^<sup>3</sup>C
|-
|-
| 38
|38
| 1085.714
|1085.714
| 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
| 1114.286
| vvM7
|vvM7
| dudmajor 7th
|dudmajor 7th
| vvC#
| vvC#
|-
|-
| 40
|40
| 1142.857
|1142.857
| vM7
|vM7
| downmajor 7th
|downmajor 7th
| vC#
|vC#
|-
|-
| 41
|41
| 1171.429
| 1171.429
| M7, v8
|M7, v8
| major 7th, down 8ve
|major 7th, down 8ve
| C#, vD
|C#, vD
|-
|-
| 42
|42
| 1200.000
| 1200.000
| 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==
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:


Line 342: Line 362:
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.
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.


== Instruments ==
==Instruments==
; Lumatone
;Lumatone


See [[Lumatone mapping for 42edo]]
See [[Lumatone mapping for 42edo]]


== 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)
*''[https://m.youtube.com/watch?v=ljaSpsQP2qc Improvisation in 42edo]'' - composed and played by [[Bryan Deister]] (May 2023), transcribed by [[Stephen Weigel]] (Sept 2024)
* ''[https://m.youtube.com/watch?v=ORy7nv6SnN8 Glory of Them]'' - [[Mundoworld]] (July 2024)
*''[https://m.youtube.com/watch?v=ORy7nv6SnN8 Glory of Them]'' - [[Mundoworld]] (July 2024)


[[Category:Augene]]
[[Category:Augene]]
[[Category:Todo:add rank 2 temperaments table]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/42edo"