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=== 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}}
== JI approximation ==
{{Q-odd-limit intervals}}


=== Octave stretch ===
=== 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.  
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:
The following table compares three stretched tunings of 42edo:


{| class="wikitable"
{| class="wikitable mw-collapsible mw-collapsed"
|-
|-
! Tuning
! Tuning
! [[42ed257/128]]
! [[42ed257/128]]
! [[1ed28.7c|APS720jot]]
! [[11ed6/5]]
! [[zpi|189zpi]]
! [[zpi|189zpi]]
! [[42edo]]
! [[42edo]]
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The following table compares three compressed tunings of 42edo:
The following table compares three compressed tunings of 42edo:


{| class="wikitable"
{| class="wikitable mw-collapsible mw-collapsed"
|-
|-
! Tuning
! Tuning
! [[42edo]]
! [[42edo]]
! [[1ed28.5c|APS715jot]]
! [[Equal-step tuning|34ed7/4]]
! [[191zpi]]
! [[191zpi]]
! [[ed255/128#42ed255/128|42ed255/128]]
! [[AS|AS123/121]]
|-
|-
! Steps / octave
! Steps / octave
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| 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>
| 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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! #
! #
! 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
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|-
|-
| 1
| 1
| 28.571
| 28.6
| ^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
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|-
|-
| 2
| 2
| 57.143
| 57.1
| ^^1, ^m2
| ^^1, ^m2
| dup 1sn, upminor 2nd
| dup 1sn, upminor 2nd
Line 100: Line 96:
|-
|-
| 3
| 3
| 85.714
| 85.7
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
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|-
|-
| 4
| 4
| 114.286
| 114.3
| ^<sup>3</sup>m
| ^<sup>3</sup>m
| trupminor 2nd
| trupminor 2nd
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|-
|-
| 5
| 5
| 143.857
| 143.9
| v<sup>3</sup>M
| v<sup>3</sup>M
| trudmajor 2nd
| trudmajor 2nd
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|-
|-
| 6
| 6
| 171.429
| 171.4
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
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|-
|-
| 7
| 7
| 200.000
| 200.0
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
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|-
|-
| 8
| 8
| 228.571
| 228.6
| M2
| M2
| major 2nd
| major 2nd
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|-
|-
| 9
| 9
| 257.143
| 257.1
| m3
| m3
| minor 3rd
| minor 3rd
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|-
|-
| 10
| 10
| 285.714
| 285.7
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
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|-
|-
| 11
| 11
| 314.286
| 314.3
| ^^m3
| ^^m3
| dupminor 3rd
| dupminor 3rd
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|-
|-
| 12
| 12
| 342.857
| 342.9
| ^<sup>3</sup>m3
| ^<sup>3</sup>m3
| trupminor 3rd
| trupminor 3rd
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|-
|-
| 13
| 13
| 371.429
| 371.4
| v<sup>3</sup>M3
| v<sup>3</sup>M3
| trudmajor 3rd
| trudmajor 3rd
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|-
|-
| 14
| 14
| 400.000
| 400.0
| vvM3
| vvM3
| dudmajor 3rd
| dudmajor 3rd
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|-
|-
| 15
| 15
| 428.571
| 428.6
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
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|-
|-
| 16
| 16
| 457.143
| 457.1
| M3, v4
| M3, v4
| major 3rd, down 4th
| major 3rd, down 4th
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|-
|-
| 17
| 17
| 485.714
| 485.7
| P4
| P4
| perfect 4th
| perfect 4th
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|-
|-
| 18
| 18
| 514.286
| 514.3
| ^4
| ^4
| up 4th
| up 4th
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|-
|-
| 19
| 19
| 543.857
| 543.9
| ^^4
| ^^4
| dup 4th
| dup 4th
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|-
|-
| 20
| 20
| 571.429
| 571.4
| ^<sup>3</sup>4, ^^d5
| ^<sup>3</sup>4, ^^d5
| trup 4th, dupdim 5th
| trup 4th, dupdim 5th
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|-
|-
| 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
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|-
|-
| 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
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|-
|-
| 23
| 23
| 657.143
| 657.1
| vv5
| vv5
| dud 5th
| dud 5th
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|-
|-
| 24
| 24
| 685.714
| 685.7
| v5
| v5
| down 5th
| down 5th
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|-
|-
| 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
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|-
|-
| 27
| 27
| 771.429
| 771.4
| ^m6
| ^m6
| upminor 6th
| upminor 6th
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|-
|-
| 28
| 28
| 800.000
| 800.0
| ^^m6
| ^^m6
| dupminor 6th
| dupminor 6th
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|-
|-
| 29
| 29
| 828.571
| 828.6
| ^<sup>3</sup>m6
| ^<sup>3</sup>m6
| trupminor 6th
| trupminor 6th
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|-
|-
| 30
| 30
| 857.143
| 857.1
| v<sup>3</sup>M6
| v<sup>3</sup>M6
| trudmajor 6th
| trudmajor 6th
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|-
|-
| 31
| 31
| 885.714
| 885.7
| vvM6
| vvM6
| dudmajor 6th
| dudmajor 6th
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|-
|-
| 32
| 32
| 914.286
| 914.3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
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|-
|-
| 33
| 33
| 942.857
| 942.9
| M6
| M6
| major 6th
| major 6th
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|-
|-
| 34
| 34
| 971.429
| 971.4
| m7
| m7
| minor 7th
| minor 7th
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|-
|-
| 35
| 35
| 1000.000
| 1000.0
| ^m7
| ^m7
| upminor 7th
| upminor 7th
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|-
|-
| 36
| 36
| 1028.571
| 1028.6
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
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|-
|-
| 37
| 37
| 1057.143
| 1057.1
| ^<sup>3</sup>m7
| ^<sup>3</sup>m7
| trupminor 7th
| trupminor 7th
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|-
|-
| 38
| 38
| 1085.714
| 1085.7
| v<sup>3</sup>M7
| v<sup>3</sup>M7
| trudmajor 7th
| trudmajor 7th
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|-
|-
| 39
| 39
| 1114.286
| 1114.3
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
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|-
|-
| 40
| 40
| 1142.857
| 1142.9
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 328: Line 324:
|-
|-
| 41
| 41
| 1171.429
| 1171.4
| M7, v8
| M7, v8
| major 7th, down 8ve
| major 7th, down 8ve
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|-
|-
| 42
| 42
| 1200.000
| 1200.0
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 340: Line 336:
|}
|}


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]].
 
== 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| 22 35 51 62 }}
| −3.65
| 3.26
| 11.42
|}


== Notation ==
== Notation ==
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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 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.
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.
{{sharpness-sharp7a}}


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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=== Sagittal notation ===
=== Sagittal notation ===
==== Best fifth notation ====
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[35edo#Second-best fifth notation|35b]].
This notation uses the same sagittal sequence as [[35edo #Second-best fifth notation|35b]].


===== Evo flavor =====
===== Evo flavor =====
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==== 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>
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default [[File:42b_Sagittal.svg]]
default [[File:42b_Sagittal.svg]]
</imagemap>
</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
|}


== Scales ==
== Scales ==
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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


== 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)


Line 478: Line 517:


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