Xenharmonic Wiki

42edo: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
18,543 edits
Stretch compression
 
(22 intermediate revisions by 7 users not shown)
Line 7: Line 7:
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" | <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"
|-
! 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 ===
Line 76: Line 22:
! #
! #
! 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 85: Line 31:
|-
|-
| 1
| 1
| 28.571
| 28.6
| ^1, m2
| ^1, m2
| up unison, minor 2nd
| up unison, minor 2nd
Line 91: Line 37:
|-
|-
| 2
| 2
| 57.143
| 57.1
| ^^1, ^m2
| ^^1, ^m2
| dup 1sn, upminor 2nd
| dup 1sn, upminor 2nd
Line 97: Line 43:
|-
|-
| 3
| 3
| 85.714
| 85.7
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
Line 103: Line 49:
|-
|-
| 4
| 4
| 114.286
| 114.3
| ^<sup>3</sup>m
| ^<sup>3</sup>m
| trupminor 2nd
| trupminor 2nd
Line 109: Line 55:
|-
|-
| 5
| 5
| 143.857
| 143.9
| v<sup>3</sup>M
| v<sup>3</sup>M
| trudmajor 2nd
| trudmajor 2nd
Line 115: Line 61:
|-
|-
| 6
| 6
| 171.429
| 171.4
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
Line 121: Line 67:
|-
|-
| 7
| 7
| 200.000
| 200.0
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 127: Line 73:
|-
|-
| 8
| 8
| 228.571
| 228.6
| M2
| M2
| major 2nd
| major 2nd
Line 133: Line 79:
|-
|-
| 9
| 9
| 257.143
| 257.1
| m3
| m3
| minor 3rd
| minor 3rd
Line 139: Line 85:
|-
|-
| 10
| 10
| 285.714
| 285.7
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 145: Line 91:
|-
|-
| 11
| 11
| 314.286
| 314.3
| ^^m3
| ^^m3
| dupminor 3rd
| dupminor 3rd
Line 151: Line 97:
|-
|-
| 12
| 12
| 342.857
| 342.9
| ^<sup>3</sup>m3
| ^<sup>3</sup>m3
| trupminor 3rd
| trupminor 3rd
Line 157: Line 103:
|-
|-
| 13
| 13
| 371.429
| 371.4
| v<sup>3</sup>M3
| v<sup>3</sup>M3
| trudmajor 3rd
| trudmajor 3rd
Line 163: Line 109:
|-
|-
| 14
| 14
| 400.000
| 400.0
| vvM3
| vvM3
| dudmajor 3rd
| dudmajor 3rd
Line 169: Line 115:
|-
|-
| 15
| 15
| 428.571
| 428.6
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 175: Line 121:
|-
|-
| 16
| 16
| 457.143
| 457.1
| M3, v4
| M3, v4
| major 3rd, down 4th
| major 3rd, down 4th
Line 181: Line 127:
|-
|-
| 17
| 17
| 485.714
| 485.7
| P4
| P4
| perfect 4th
| perfect 4th
Line 187: Line 133:
|-
|-
| 18
| 18
| 514.286
| 514.3
| ^4
| ^4
| up 4th
| up 4th
Line 193: Line 139:
|-
|-
| 19
| 19
| 543.857
| 543.9
| ^^4
| ^^4
| dup 4th
| dup 4th
Line 199: 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 205: 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 211: 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 217: Line 163:
|-
|-
| 23
| 23
| 657.143
| 657.1
| vv5
| vv5
| dud 5th
| dud 5th
Line 223: Line 169:
|-
|-
| 24
| 24
| 685.714
| 685.7
| v5
| v5
| down 5th
| down 5th
Line 229: Line 175:
|-
|-
| 25
| 25
| 714.286
| 714.3
| P5
| P5
| perfect 5th
| perfect 5th
Line 235: Line 181:
|-
|-
| 26
| 26
| 742.857
| 742.9
| ^5, m6
| ^5, m6
| up 5th, minor 6th
| up 5th, minor 6th
Line 241: Line 187:
|-
|-
| 27
| 27
| 771.429
| 771.4
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 247: Line 193:
|-
|-
| 28
| 28
| 800.000
| 800.0
| ^^m6
| ^^m6
| dupminor 6th
| dupminor 6th
Line 253: Line 199:
|-
|-
| 29
| 29
| 828.571
| 828.6
| ^<sup>3</sup>m6
| ^<sup>3</sup>m6
| trupminor 6th
| trupminor 6th
Line 259: Line 205:
|-
|-
| 30
| 30
| 857.143
| 857.1
| v<sup>3</sup>M6
| v<sup>3</sup>M6
| trudmajor 6th
| trudmajor 6th
Line 265: Line 211:
|-
|-
| 31
| 31
| 885.714
| 885.7
| vvM6
| vvM6
| dudmajor 6th
| dudmajor 6th
Line 271: Line 217:
|-
|-
| 32
| 32
| 914.286
| 914.3
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 277: Line 223:
|-
|-
| 33
| 33
| 942.857
| 942.9
| M6
| M6
| major 6th
| major 6th
Line 283: Line 229:
|-
|-
| 34
| 34
| 971.429
| 971.4
| m7
| m7
| minor 7th
| minor 7th
Line 289: Line 235:
|-
|-
| 35
| 35
| 1000.000
| 1000.0
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 295: Line 241:
|-
|-
| 36
| 36
| 1028.571
| 1028.6
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
Line 301: Line 247:
|-
|-
| 37
| 37
| 1057.143
| 1057.1
| ^<sup>3</sup>m7
| ^<sup>3</sup>m7
| trupminor 7th
| trupminor 7th
Line 307: Line 253:
|-
|-
| 38
| 38
| 1085.714
| 1085.7
| v<sup>3</sup>M7
| v<sup>3</sup>M7
| trudmajor 7th
| trudmajor 7th
Line 313: Line 259:
|-
|-
| 39
| 39
| 1114.286
| 1114.3
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
Line 319: Line 265:
|-
|-
| 40
| 40
| 1142.857
| 1142.9
| vM7
| vM7
| downmajor 7th
| downmajor 7th
Line 325: Line 271:
|-
|-
| 41
| 41
| 1171.429
| 1171.4
| M7, v8
| M7, v8
| major 7th, down 8ve
| major 7th, down 8ve
Line 331: Line 277:
|-
|-
| 42
| 42
| 1200.000
| 1200.0
| P8
| P8
| perfect 8ve
| perfect 8ve
Line 337: 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 ==
Line 343: Line 289:
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 {{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.
{{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:
Line 353: Line 302:
=== 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 =====
Line 376: Line 325:


==== 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 386: Line 335:
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
|}
== 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 ==
== Scales ==
Line 398: Line 457:
* 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  
Line 409: Line 469:
** 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 ==
; Lumatone
=== 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 ==
Line 424: Line 521:
=== 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 439: Line 537:


[[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"