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== 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 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]]. It also has a fine [[11/1|11]], and adding it to consideration the choice for 39et is the sharp-tending 39df [[val]] {{val| 39 62 91 '''110''' 135 '''145'''}}.
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.


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.  
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.  


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 ===
Line 16: Line 18:


=== As a tuning of other temperaments ===
=== 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. 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]].  
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 ===
Line 22: Line 24:


== 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-sharp5-szg}}
 
=== 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}}
{{Ups and downs sharpness}}
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}}


=== Sagittal notation ===
=== Sagittal notation ===
Line 721: Line 682:
| 2/1
| 2/1
|}
|}
== Approximation to JI ==
== Approximation to JI ==
=== Interval mappings ===
=== Interval mappings ===
Retrieved from "https://en.xen.wiki/w/39edo"