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39edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{ED intro}}


== 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 cents 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 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 }}.


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].
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].
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As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. 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.  
As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}. 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 cents 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).
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! Approximate ratios*
! Approximate ratios*
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[Nearest just interval]]<br>(Ratio, cents, error)
! colspan="3" | [[Nearest just interval]] <br />(Ratio, cents, error)
|-
|-
| 0
| 0
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| 30.8
| 30.8
| ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]
| ''[[36/35]]'', [[50/49]], [[55/54]], [[56/55]], [[81/80]]
| ^1,<br>vm2
| ^1, <br />vm2
| up unison, <br>downminor 2nd
| up unison, <br />downminor 2nd
| ^D, <br>vEb
| ^D, <br />vEb
| 57/56
| 57/56
| 30.64
| 30.64
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| 246.2
| 246.2
| [[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
| 15/13
| 247.74
| 247.74
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| 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
| 7/5
| 582.51
| 582.51
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| 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
| 10/7
| 617.49
| 617.49
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| 953.8
| 953.8
| [[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
| 26/15
| 952.26
| 952.26
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| 1169.2
| 1169.2
| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]
| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]
| ^M7,<br>v8
| ^M7, <br />v8
| upmajor 7th,<br>down 8ve
| upmajor 7th, <br />down 8ve
| ^C#, <br>vD
| ^C#, <br />vD
| 112/57
| 112/57
| 1169.36
| 1169.36
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal <br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods <br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
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|-
|-
| 3
| 3
| 8\39<br>(5\39)
| 8\39 <br />(5\39)
| 246.2<br>(153.8)
| 246.2 <br />(153.8)
| [[Triforce]] (39)
| [[Triforce]] (39)
| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[9L&nbsp;6s]], [[15L&nbsp;9s]]
| [[3L&nbsp;3s]], [[6L&nbsp;3s]], [[9L&nbsp;6s]], [[15L&nbsp;9s]]
|-
|-
| 3
| 3
| 16\39<br>(3\39)
| 16\39 <br />(3\39)
| 492.3<br>(92.3)
| 492.3 <br />(92.3)
| [[Augene]] (39d)
| [[Augene]] (39d)
| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[12L&nbsp;3s]], [[12L&nbsp;15s]]
| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[3L&nbsp;9s]], [[12L&nbsp;3s]], [[12L&nbsp;15s]]
|-
|-
| 3
| 3
| 17\39<br>(4\39)
| 17\39 <br />(4\39)
| 523.1<br>(123.0)
| 523.1 <br />(123.0)
| [[Deflated]] (39bd)
| [[Deflated]] (39bd)
| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[9L&nbsp;3s]], [[9L&nbsp;12s]], [[9L&nbsp;21s]]
| [[3L&nbsp;3s]], [[3L&nbsp;6s]], [[9L&nbsp;3s]], [[9L&nbsp;12s]], [[9L&nbsp;21s]]
|-
|-
| 13
| 13
| 16\39<br>(1\39)
| 16\39 <br />(1\39)
| 492.3<br>(30.8)
| 492.3 <br />(30.8)
| [[Tridecatonic]]
| [[Tridecatonic]]
| [[13L&nbsp;13s]]
| [[13L&nbsp;13s]]
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