Xenharmonic Wiki

39edo: Difference between revisions

← Older edit
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,262 edits
No edit summary
added more patent val discussion
 
(33 intermediate revisions by 10 users not shown)
Line 3: Line 3:


== Theory ==
== Theory ==
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]], we get a system which [[tempering out|tempers out]] the [[diesis]] (128/125) and the [[amity comma]] (1600000/1594323). We have two choices for a [[map]] for [[7/1|7]], but the sharp one works better with the [[3/1|3]] and [[5/1|5]], which adds [[64/63]] and [[126/125]] to the list. [[Tempering out]] both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39et has a fine [[11/1|11]], and adding it to consideration we find that the equal temperament tempers out [[99/98]] and [[121/120]] also. This choice for 39et is the 39d [[val]] {{val| 39 62 91 '''110''' 135 }}.
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a tuning of [[7/1|7]]. The sharp one yields [[superpyth]] temperament, while the flat (patent) one yields [[semaphore]] (and also [[hemifamity]]) temperament.


A particular anecdote with this system was made in the ''Teliochordon'', in 1788 by {{w|Charles Clagget}} (Ireland, 1740?–1820), a little extract [http://ml.oxfordjournals.org/content/76/2/291.extract.jpg here].
As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.  
 
As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39et is [[quasisuper]]. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody because the [[diatonic semitone]] is [[quartertone]]-sized, which results in a very strange-sounding [[5L 2s|diatonic scale]]. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.  


Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.  
Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.  


39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).
39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).
39edo is a reasonable tuning of [[triforce]] beyond 15edo, and optimizes both its semaphore and augmented components by tuning the fifth sharp.


=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|39}}
{{Harmonics in equal|39|columns=11}}
{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}
 
=== As a tuning of other temperaments ===
39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. Alternatively, the patent val tempers out 49/48 to yield semaphore. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 39 factors into {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.
Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3 right-9 right-10"
{| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"
|-
|-
! Steps
! rowspan="2" | Steps
! Cents
! rowspan="2" | Cents
! Approximate ratios*
! rowspan="2" | Ratios of the<br>[[2.3.5.11 subgroup]]
! colspan="3" | [[Ups and downs notation]]
! colspan="2" | Intervals of 7
! colspan="3" | [[Nearest just interval]] <br>(Ratio, cents, error)
! colspan="3" rowspan="2" | [[Ups and downs notation]]
|-
! Patent val
! 39d val
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| colspan=3 | [[1/1]]
| P1
| P1
| perfect unison
| perfect unison
| D
| D
| 1/1
| 0.00
| None
|-
|-
| 1
| 1
| 30.8
| 30.8
| ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]
| [[55/54]], [[81/80]]
| ''[[28/27]]'', [[64/63]]
| ''[[36/35]]'', [[50/49]], ''[[56/55]]''
| ^1, <br>vm2
| ^1, <br>vm2
| up unison, <br>downminor 2nd
| up unison, <br>downminor 2nd
| ^D, <br>vEb
| ^D, <br>vEb
| 57/56
| 30.64
| +0.1271
|-
|-
| 2
| 2
| 61.5
| 61.5
| [[28/27]], [[33/32]], ''[[49/48]]''
| [[33/32]]
| ''[[21/20]]'', [[36/35]]
| [[28/27]], ''[[49/48]]''
| m2
| m2
| minor 2nd
| minor 2nd
| Eb
| Eb
| 29/28
| 60.75
| +0.7872
|-
|-
| 3
| 3
| 92.3
| 92.3
| ''[[16/15]]'', [[21/20]], ''[[25/24]]''
| ''[[16/15]]'', ''[[25/24]]''
| ''[[50/49]]''
| [[21/20]]
| ^m2
| ^m2
| upminor 2nd
| upminor 2nd
| ^Eb
| ^Eb
| 39/37
| 91.14
| +1.1691
|-
|-
| 4
| 4
| 123.1
| 123.1
|
|
| [[15/14]]
| [[15/14]]
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
| ^^Eb
| ^^Eb
| 44/41
| 122.26
| +0.8214
|-
|-
| 5
| 5
| 153.8
| 153.8
| [[11/10]], [[12/11]]
| [[11/10]], [[12/11]]
| ''[[15/14]]''
|
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
| vvE
| vvE
| 35/32
| 155.14
| -1.2934
|-
|-
| 6
| 6
| 184.6
| 184.6
| [[10/9]]
| [[10/9]]
|
|
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
| vE
| vE
| 10/9
| 182.40
| +2.2117
|-
|-
| 7
| 7
| 215.4
| 215.4
| [[9/8]], ''[[8/7]]''
| [[9/8]]
|
| ''[[8/7]]''
| M2
| M2
| major 2nd
| major 2nd
| E
| E
| 17/15
| 216.69
| -1.3021
|-
|-
| 8
| 8
| 246.2
| 246.2
|
| [[8/7]], ''[[7/6]]''
| [[81/70]]
| [[81/70]]
| ^M2, <br>vm3
| ^M2, <br>vm3
| upmajor 2nd, <br>downminor 3rd
| upmajor 2nd, <br>downminor 3rd
| ^E, <br>vF
| ^E, <br>vF
| 15/13
| 247.74
| -1.5873
|-
|-
| 9
| 9
| 276.9
| 276.9
|
| ''[[81/70]]''
| [[7/6]]
| [[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
| F
| F
| 27/23
| 277.59
| -0.6676
|-
|-
| 10
| 10
| 307.7
| 307.7
| [[6/5]]
| [[6/5]]
|
|
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
| ^F
| ^F
| 43/36
| 307.61
| +0.0846
|-
|-
| 11
| 11
| 338.5
| 338.5
| [[11/9]]
| [[11/9]]
|
|
| ^^m3
| ^^m3
| dupminor 3rd
| dupminor 3rd
| ^^F
| ^^F
| 17/14
| 336.13
| +2.3320
|-
|-
| 12
| 12
| 369.2
| 369.2
| [[27/22]]
| [[27/22]]
|
|
| vvM3
| vvM3
| dudmajor 3rd
| dudmajor 3rd
| vvF#
| vvF#
| 26/21
| 369.75
| -0.5160
|-
|-
| 13
| 13
| 400.0
| 400.0
| [[5/4]]
| [[5/4]]
| ''[[14/11]]''
|
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
| vF#
| vF#
| 34/27
| 399.09
| +0.9096
|-
|-
| 14
| 14
| 430.8
| 430.8
|
| ''[[35/27]]''
| [[9/7]], [[14/11]]
| [[9/7]], [[14/11]]
| M3
| M3
| major 3rd
| major 3rd
| F#
| F#
| 41/32
| 429.06
| +1.7068
|-
|-
| 15
| 15
| 461.5
| 461.5
|
| ''[[9/7]]''
| [[35/27]]
| [[35/27]]
| v4
| v4
| down 4th
| down 4th
| vG
| vG
| 30/23
| 459.99
| +1.5441
|-
|-
| 16
| 16
| 492.3
| 492.3
| [[4/3]]
| [[4/3]]
|
|
| P4
| P4
| perfect 4th
| perfect 4th
| G
| G
| 85/64
| 491.27
| +1.0386
|-
|-
| 17
| 17
| 523.1
| 523.1
| [[27/20]]
| [[27/20]]
|
|
| ^4
| ^4
| up 4th
| up 4th
| ^G
| ^G
| 23/17
| 523.32
| -0.2420
|-
|-
| 18
| 18
| 553.8
| 553.8
| [[11/8]]
| [[11/8]]
| ''[[7/5]]''
|
| ^^4
| ^^4
| dup 4th
| dup 4th
| ^^G
| ^^G
| 11/8
| 551.32
| +2.5283
|-
|-
| 19
| 19
| 584.6
| 584.6
|
|
| [[7/5]]
| [[7/5]]
| vvA4, <br>^d5
| vvA4, <br>^d5
| dudaug 4th, <br>updim 5th
| dudaug 4th, <br>updim 5th
| vvG#, <br>^Ab
| vvG#, <br>^Ab
| 7/5
| 582.51
| +2.1032
|-
|-
| 20
| 20
| 615.4
| 615.4
|
|
| [[10/7]]
| [[10/7]]
| vA4, <br>^^d5
| vA4, <br>^^d5
| downaug 4th, <br>dupdim 5th
| downaug 4th, <br>dupdim 5th
| vG#, <br>^^Ab
| vG#, <br>^^Ab
| 10/7
| 617.49
| -2.1032
|-
|-
| 21
| 21
| 646.2
| 646.2
| [[16/11]]
| [[16/11]]
| ''[[10/7]]''
|
| vv5
| vv5
| dud 5th
| dud 5th
| vvA
| vvA
| 16/11
| 648.68
| -2.5283
|-
|-
| 22
| 22
| 676.9
| 676.9
| [[40/27]]
| [[40/27]]
|
|
| v5
| v5
| down 5th
| down 5th
| vA
| vA
| 34/23
| 676.68
| +0.2420
|-
|-
| 23
| 23
| 707.7
| 707.7
| [[3/2]]
| [[3/2]]
|
|
| P5
| P5
| perfect 5th
| perfect 5th
| A
| A
| 128/85
| 708.73
| -1.0386
|-
|-
| 24
| 24
| 738.5
| 738.5
|
| ''[[14/9]]''
| [[54/35]]
| [[54/35]]
| ^5
| ^5
| up 5th
| up 5th
| A^
| A^
| 23/15
| 740.01
| -1.5441
|-
|-
| 25
| 25
| 769.2
| 769.2
|
| ''[[54/35]]''
| [[11/7]], [[14/9]]
| [[11/7]], [[14/9]]
| m6
| m6
| minor 6th
| minor 6th
| Bb
| Bb
| 64/41
| 770.94
| -1.7068
|-
|-
| 26
| 26
| 800.0
| 800.0
| [[8/5]]
| [[8/5]]
| ''[[11/7]]''
|
| ^m6
| ^m6
| upminor 6th
| upminor 6th
| ^Bb
| ^Bb
| 27/17
| 800.91
| -0.9096
|-
|-
| 27
| 27
| 830.8
| 830.8
| [[44/27]]
| [[44/27]]
|
|
| ^^m6
| ^^m6
| dupminor 6th
| dupminor 6th
| ^^Bb
| ^^Bb
| 21/13
| 830.25
| +0.5160
|-
|-
| 28
| 28
| 861.5
| 861.5
| [[18/11]]
| [[18/11]]
|
|
| vvM6
| vvM6
| dudmajor 6th
| dudmajor 6th
| vvB
| vvB
| 28/17
| 863.87
| -2.3320
|-
|-
| 29
| 29
| 892.3
| 892.3
| [[5/3]]
| [[5/3]]
|
|
| vM6
| vM6
| downmajor 6th
| downmajor 6th
| vB
| vB
| 72/43
| 892.39
| -0.0846
|-
|-
| 30
| 30
| 923.1
| 923.1
|
| ''[[140/81]]''
| [[12/7]]
| [[12/7]]
| M6
| M6
| major 6th
| major 6th
| B
| B
| 46/27
| 922.41
| +0.6676
|-
|-
| 31
| 31
| 953.8
| 953.8
|
| [[7/4]], ''[[12/7]]''
| [[140/81]]
| [[140/81]]
| ^M6, <br>vm7
| ^M6, <br>vm7
| upmajor 6th, <br>downminor 7th
| upmajor 6th, <br>downminor 7th
| ^B, <br>vC
| ^B, <br>vC
| 26/15
| 952.26
| +1.5873
|-
|-
| 32
| 32
| 984.6
| 984.6
| ''[[7/4]]'', [[16/9]]
| [[16/9]]
|
| ''[[7/4]]''
| m7
| m7
| minor 7th
| minor 7th
| C
| C
| 30/17
| 983.31
| +1.3021
|-
|-
| 33
| 33
| 1015.4
| 1015.4
| [[9/5]]
| [[9/5]]
|
|
| ^m7
| ^m7
| upminor 7th
| upminor 7th
| ^C
| ^C
| 9/5
| 1017.60
| -2.2117
|-
|-
| 34
| 34
| 1046.2
| 1046.2
| [[11/6]], [[20/11]]
| [[11/6]], [[20/11]]
| ''[[28/15]]''
|
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
| ^^C
| ^^C
| 64/35
| 1044.86
| +1.2934
|-
|-
| 35
| 35
| 1076.9
| 1076.9
|
|
| [[28/15]]
| [[28/15]]
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
| vvC#
| vvC#
| 41/22
| 1077.74
| -0.8214
|-
|-
| 36
| 36
| 1107.7
| 1107.7
| ''[[15/8]]'', [[40/21]], ''[[48/25]]''
| ''[[15/8]]'', ''[[48/25]]''
| ''[[49/25]]''
| [[40/21]]
| vM7
| vM7
| downmajor 7th
| downmajor 7th
| vC#
| vC#
| 74/39
| 1108.86
| -1.1691
|-
|-
| 37
| 37
| 1138.5
| 1138.5
| [[27/14]], ''[[96/49]]'', [[64/33]]
| [[64/33]]
| [[35/18]], ''[[40/21]]''
| [[27/14]], ''[[96/49]]''
| M7
| M7
| major 7th
| major 7th
| C#
| C#
| 56/29
| 1139.25
| -0.7872
|-
|-
| 38
| 38
| 1169.2
| 1169.2
| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]
| [[108/55]], [[160/81]]
| [[63/32]], ''[[27/14]]''
| ''[[35/18]]'', [[49/25]]
| ^M7, <br>v8
| ^M7, <br>v8
| upmajor 7th, <br>down 8ve
| upmajor 7th, <br>down 8ve
| ^C#, <br>vD
| ^C#, <br>vD
| 112/57
| 1169.36
| -0.1271
|-
|-
| 39
| 39
| 1200.0
| 1200.0
| [[2/1]]
| colspan=3 | [[2/1]]
| P8
| P8
| perfect 8ve
| perfect 8ve
| D
| D
| 2/1
| 1200.00
| None
|}
|}
<nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic''


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 ===
=== Stein–Zimmermann–Gould notation ===
39edo can be notated with [[ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5a}}
{{Sharpness-sharp5-szg}}
Another notation uses [[Alternative symbols for ups and downs notation #Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
 
{{Sharpness-sharp5}}
=== Kite's ups and downs notation ===
39edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Ups and downs sharpness}}
 
=== Sagittal notation ===
=== Sagittal notation ===
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].
Line 717: Line 682:
| 2/1
| 2/1
|}
|}
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 883: Line 853:
| [[13L&nbsp;13s]]
| [[13L&nbsp;13s]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Octave stretch or compression ==
== Octave stretch or compression ==
39edo's approximations of harmonics 3, 5, 7 and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[62edt]].
39edo is a [[zeta valley edo]] and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|101ed6]] or [[173zpi]].
 
What follows is a comparison of stretched- and compressed-octave 39edo tunings.
 
; [[zpi|171zpi]]
* Step size: 30.973{{c}}, octave size: 107.9{{c}}
Stretching the octave of 39edo by around 8{{c}} results in improved primes 5, 7, 11, 17, 19 and 23, but much worse primes 2 and 3. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 171zpi does this. Because it shares error evenly between 39edo's fifths, it is suited for use as a [[dual-fifth tuning]] of 39edo.
{{Harmonics in cet|30.973|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 171zpi}}
{{Harmonics in cet|30.973|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 171zpi (continued)}}
 
; 39edo
* Step size: 30.769{{c}}, octave size: 1200.00{{c}}
Pure-octaves 39edo approximates all harmonics up to 16 within 15.0{{c}}.
{{Harmonics in equal|39|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39edo}}
{{Harmonics in equal|39|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39edo (continued)}}
 
; [[WE|39et, 13-limit WE tuning]]  
* Step size: 30.757{{c}}, octave size: 1199.5{{c}}
Compressing the octave of 39edo by about half a cent results in improved primes 3, 5, 7 and 11, but a worse prime 13. This approximates all harmonics up to 16 within 14.4{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
{{Harmonics in cet|30.757|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning}}
{{Harmonics in cet|30.757|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 13-limit WE tuning (continued)}}


; [[ed6|101ed6]]
39edo can be usefully mapped onto the val 39dfgijk. The [[Tenney-Euclidean]] tuning of this regular temperament is 30.67475 cents per step, which is closely approximated by [[62edt]] and 173zpi.
* Octave size: 1197.8{{c}}
Compressing the octave of 101ed6 by around 2{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 12.8{{c}}. The tuning 101ed6 does this. So does [[zpi|172zpi]] whose octave differs by only 0.4{{c}}.
{{Harmonics in equal|101|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 101ed6}}
{{Harmonics in equal|101|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 101ed6 (continued)}}
 
; [[WE|39et, 2.3.5.11 WE tuning]]
* Step size: 30.703{{c}}, octave size: 1197.4{{c}}
Compressing the octave of 39edo by around 2.5{{c}} results in improved primes 3, 5 and 7, but worse primes 2, 11 and 13. This approximates all harmonics up to 16 within 11.4{{c}}. Its 2.3.5.11 WE tuning and 2.3.5.11 [[TE]] tuning both do this.
{{Harmonics in cet|30.703|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning}}
{{Harmonics in cet|30.703|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 39et, 2.3.5.11 WE tuning (continued)}}
 
; [[zpi|173zpi]]
* Step size: 30.672{{c}}, octave size: 1196.2{{c}}
Compressing the octave of 39edo by around 4{{c}} results in improved primes 3, 5, 7, 13, 17, 19 and 23, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.2{{c}}. The tuning 173zpi does this. So does [[62edt]] whose octave differs by only 0.2{{c}}.
{{Harmonics in cet|30.672|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 173zpi}}
{{Harmonics in cet|30.672|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 173zpi (continued)}}
 
; [[ed7|110ed7]]
* Octave size: 1194.4{{c}}
Compressing the octave of 39edo by around 5.5{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 14.4{{c}}. The tuning 110ed7 does this. So does [[equal tuning|145ed13]] whose octave differs by only 0.1{{c}}.
{{Harmonics in equal|110|7|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 110ed7}}
{{Harmonics in equal|110|7|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 110ed7 (continued)}}
 
; [[ed5|91ed5]]
* Octave size: 1194.1{{c}}
Compressing the octave of 39edo by around 6{{c}} results in improved primes 3, 5, 7, 13 and 17, but worse primes 2 and 11. This approximates all harmonics up to 16 within 15.3{{c}}. The tuning 91ed5 does this.
{{Harmonics in equal|91|5|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 91ed5}}
{{Harmonics in equal|91|5|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 91ed5 (continued)}}


== 39edo and world music ==
== 39edo and world music ==
Line 978: Line 900:


Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.
== Scales ==
* [[Quasisuper]][7] [[MOS scale]]: 7 7 2 7 7 7 2
* Quasisuper[7] [[5-limit|pental]] [[modmos]]: 7 6 3 7 6 7 3
* [[3L 6s]] modmos: 7 3 3 3 7 3 3 7 3
* Extended quasisuper: 4 3 6 3 4 3 6 4 3 3
* Quasisuper[22] MOS scale (resembles [[Indian]] [[sruti]]): 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 2
* Slendro approximations: 9 7 7 9 7 or 8 8 8 8 7 or 8 8 7 8 8
* An expressive [[oneirotonic]] subset: 9 6 9 9 6
* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''


== Instruments ==
== Instruments ==
=== Lumatone mapping ===
=== Lumatone mapping ===
See [[Lumatone mapping for 39edo]]
See [[Lumatone mapping for 39edo]]
Line 1,010: Line 941:


== Music ==
== Music ==
=== Modern renderings ===
; {{W|HOYO-MiX}}
* [https://www.youtube.com/shorts/4y11CWLIHNA "Sinner's Finale" from ''Genshin Impact OST''] (2023) – covered by [[Bryan Deister]] (2025)
=== 21st century ===
; [[Bryan Deister]]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
* [https://www.youtube.com/shorts/4y11CWLIHNA ''Sinner's Finale - Genshin Impact (microtonal cover in 39edo)''] (2025)
* [https://www.youtube.com/shorts/1T_xrZpUslQ ''39edo improv''] (2025)
* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)
* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)
 
; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "Resolute Prelude"
** "Residual Soliloquy"


; [[Randy Wells]]
; [[Randy Wells]]
Line 1,019: Line 962:


[[Category:Listen]]
[[Category:Listen]]
{{Todo|add scales list}}
Retrieved from "https://en.xen.wiki/w/39edo"