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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 one possible 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]]. 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 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.  
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=== 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 24: Line 24:
{| class="wikitable center-all right-2 left-3 right-9 right-10"
{| class="wikitable center-all right-2 left-3 right-9 right-10"
|-
|-
! Steps
! rowspan="2" | Steps
! Cents
! rowspan="2" | Cents
! Approximate ratios*
! rowspan="2" | Approximate ratios (2.3.5.11)
! 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]]
! colspan="3" rowspan="2" | [[Nearest just interval]] <br>(Ratio, cents, error)
|-
!Patent val
!Superpyth (39d) val
|-
|-
| 0
| 0
| 0.0
| 0.0
| [[1/1]]
| [[1/1]]
|
|
| P1
| P1
| perfect unison
| perfect unison
Line 42: Line 48:
| 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
Line 52: Line 60:
| 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
Line 62: Line 72:
| 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
Line 72: Line 84:
| 4
| 4
| 123.1
| 123.1
| [[15/14]]
|
|
|[[15/14]]
| ^^m2
| ^^m2
| dupminor 2nd
| dupminor 2nd
Line 83: Line 97:
| 153.8
| 153.8
| [[11/10]], [[12/11]]
| [[11/10]], [[12/11]]
|''15/14''
|
| vvM2
| vvM2
| dudmajor 2nd
| dudmajor 2nd
Line 93: Line 109:
| 184.6
| 184.6
| [[10/9]]
| [[10/9]]
|
|
| vM2
| vM2
| downmajor 2nd
| downmajor 2nd
Line 102: Line 120:
| 7
| 7
| 215.4
| 215.4
| [[9/8]], ''[[8/7]]''
| [[9/8]]
|
|''[[8/7]]''
| M2
| M2
| major 2nd
| major 2nd
Line 112: Line 132:
| 8
| 8
| 246.2
| 246.2
| [[81/70]]
|
|8/7, ''7/6''
|[[81/70]]
| ^M2, <br>vm3
| ^M2, <br>vm3
| upmajor 2nd, <br>downminor 3rd
| upmajor 2nd, <br>downminor 3rd
Line 122: Line 144:
| 9
| 9
| 276.9
| 276.9
| [[7/6]]
|
|''81/70''
|[[7/6]]
| m3
| m3
| minor 3rd
| minor 3rd
Line 133: Line 157:
| 307.7
| 307.7
| [[6/5]]
| [[6/5]]
|
|
| ^m3
| ^m3
| upminor 3rd
| upminor 3rd
Line 143: Line 169:
| 338.5
| 338.5
| [[11/9]]
| [[11/9]]
|
|
| ^^m3
| ^^m3
| dupminor 3rd
| dupminor 3rd
Line 153: Line 181:
| 369.2
| 369.2
| [[27/22]]
| [[27/22]]
|
|
| vvM3
| vvM3
| dudmajor 3rd
| dudmajor 3rd
Line 163: Line 193:
| 400.0
| 400.0
| [[5/4]]
| [[5/4]]
|''14/11''
|
| vM3
| vM3
| downmajor 3rd
| downmajor 3rd
Line 172: Line 204:
| 14
| 14
| 430.8
| 430.8
| [[9/7]], [[14/11]]
|
|''35/27''
|[[9/7]], [[14/11]]
| M3
| M3
| major 3rd
| major 3rd
Line 182: Line 216:
| 15
| 15
| 461.5
| 461.5
| [[35/27]]
|
|''9/7''
|[[35/27]]
| v4
| v4
| down 4th
| down 4th
Line 193: Line 229:
| 492.3
| 492.3
| [[4/3]]
| [[4/3]]
|
|
| P4
| P4
| perfect 4th
| perfect 4th
Line 203: Line 241:
| 523.1
| 523.1
| [[27/20]]
| [[27/20]]
|
|
| ^4
| ^4
| up 4th
| up 4th
Line 213: Line 253:
| 553.8
| 553.8
| [[11/8]]
| [[11/8]]
|''7/5''
|
| ^^4
| ^^4
| dup 4th
| dup 4th
Line 222: Line 264:
| 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
Line 232: Line 276:
| 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
Line 243: Line 289:
| 646.2
| 646.2
| [[16/11]]
| [[16/11]]
|''10/7''
|
| vv5
| vv5
| dud 5th
| dud 5th
Line 253: Line 301:
| 676.9
| 676.9
| [[40/27]]
| [[40/27]]
|
|
| v5
| v5
| down 5th
| down 5th
Line 263: Line 313:
| 707.7
| 707.7
| [[3/2]]
| [[3/2]]
|
|
| P5
| P5
| perfect 5th
| perfect 5th
Line 272: Line 324:
| 24
| 24
| 738.5
| 738.5
| [[54/35]]
|
|''14/9''
|[[54/35]]
| ^5
| ^5
| up 5th
| up 5th
Line 282: Line 336:
| 25
| 25
| 769.2
| 769.2
| [[11/7]], [[14/9]]
|
|''54/35''
|[[11/7]], [[14/9]]
| m6
| m6
| minor 6th
| minor 6th
Line 293: Line 349:
| 800.0
| 800.0
| [[8/5]]
| [[8/5]]
|''11/7''
|
| ^m6
| ^m6
| upminor 6th
| upminor 6th
Line 303: Line 361:
| 830.8
| 830.8
| [[44/27]]
| [[44/27]]
|
|
| ^^m6
| ^^m6
| dupminor 6th
| dupminor 6th
Line 313: Line 373:
| 861.5
| 861.5
| [[18/11]]
| [[18/11]]
|
|
| vvM6
| vvM6
| dudmajor 6th
| dudmajor 6th
Line 323: Line 385:
| 892.3
| 892.3
| [[5/3]]
| [[5/3]]
|
|
| vM6
| vM6
| downmajor 6th
| downmajor 6th
Line 332: Line 396:
| 30
| 30
| 923.1
| 923.1
| [[12/7]]
|
|''140/81''
|[[12/7]]
| M6
| M6
| major 6th
| major 6th
Line 342: Line 408:
| 31
| 31
| 953.8
| 953.8
| [[140/81]]
|
|7/4, ''12/7''
|[[140/81]]
| ^M6, <br>vm7
| ^M6, <br>vm7
| upmajor 6th, <br>downminor 7th
| upmajor 6th, <br>downminor 7th
Line 352: Line 420:
| 32
| 32
| 984.6
| 984.6
| ''[[7/4]]'', [[16/9]]
| [[16/9]]
|
|''[[7/4]]''
| m7
| m7
| minor 7th
| minor 7th
Line 363: Line 433:
| 1015.4
| 1015.4
| [[9/5]]
| [[9/5]]
|
|
| ^m7
| ^m7
| upminor 7th
| upminor 7th
Line 373: Line 445:
| 1046.2
| 1046.2
| [[11/6]], [[20/11]]
| [[11/6]], [[20/11]]
|''28/15''
|
| ^^m7
| ^^m7
| dupminor 7th
| dupminor 7th
Line 382: Line 456:
| 35
| 35
| 1076.9
| 1076.9
| [[28/15]]
|
|
|[[28/15]]
| vvM7
| vvM7
| dudmajor 7th
| dudmajor 7th
Line 392: Line 468:
| 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
Line 402: Line 480:
| 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
Line 412: Line 492:
| 38
| 38
| 1169.2
| 1169.2
| ''[[35/18]]'', [[49/25]], [[55/28]], [[108/55]], [[160/81]]
| [[55/28]], [[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
Line 423: Line 505:
| 1200.0
| 1200.0
| [[2/1]]
| [[2/1]]
|
|
| P8
| P8
| perfect 8ve
| perfect 8ve
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