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39edo: Difference between revisions

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'''39-EDO''', '''39-ED2''' or '''39-tET''' divides the [[octave]] in 39 equal parts of 30.76923 [[Cent|cents]] each one.
{{Infobox ET
| Prime factorization = 3 * 13
| Step size = 30.769
| Fifth type = 23\39 = 707.692¢
| Major 2nd = 7\39 = 215¢
| Minor 2nd = 2\39 = 62¢
| Augmented 1sn = 5\39 = 154¢
}}


== Theory ==
== Theory ==
 
{| class="wikitable center-all"
If we take 22\39 as a fifth, 39edo can be used in [[Mavila|mavila temperament]], 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 basical scale for notation and theory, suited in [[16edo|16-ED2]], and allied systems: [[25edo|25-ED2]] [1/3-tone 3;2]; [[41edo|41-ED2]] [1/5-tone 5;3]; and [[57edo|57-ED2]] [1/7-tone 7;4]. [[Hornbostel temperaments]] is included too with: [[23edo|23-ED2]] [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] & [[62edo|62-ED2]] [1/8-tone 8;3].  
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
|  +5.74
|  +13.7
|  -15.0
|  +2.5
| -9.8
| -12.6
|10.2
|-
![[Relative error|relative]] (%)
| 0.0
|  +19
|  +44
|  -49
|  +8
| -32
| -41
|33
|-
! colspan="2" |[[nearest edomapping]]
|39
|23
|13
|31
|18
|27
|3
|10
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| -13
| -19
| -6
| -9
| +12
| +14
|}
'''39-EDO''', '''39-ED2''' or '''39-tET''' divides the [[octave]] in 39 equal parts. If we take 22\39 as a fifth, 39edo can be used in [[Mavila|mavila temperament]], 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 basical scale for notation and theory, suited in [[16edo|16-ED2]], and allied systems: [[25edo|25-ED2]] [1/3-tone 3;2]; [[41edo|41-ED2]] [1/5-tone 5;3]; and [[57edo|57-ED2]] [1/7-tone 7;4]. [[Hornbostel temperaments]] is included too with: [[23edo|23-ED2]] [1/3-tone 3;1]; 39-ED2 [1/5-tone 5;2] & [[62edo|62-ED2]] [1/8-tone 8;3].


However, its 23\39 fifth, 5.737 cents sharp, is in much better tune than the mavila fifth which like all mavila fifths is very, very flat, in this case, 25 cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, [[128/125]], and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds [[64/63]] and [[126/125]] to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting [[Augene|augene temperament]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out [[99/98]] and [[121/120]] also. This better choice for 39et is {{val|39 62 91 110 135}}.
However, its 23\39 fifth, 5.737 cents sharp, is in much better tune than the mavila fifth which like all mavila fifths is very, very flat, in this case, 25 cents flat. Together with its best third which is the familiar 400 cents of 12 equal, we get a system which tempers out the diesis, [[128/125]], and the amity comma, 1600000/1594323. We have two choices for a map for 7, but the sharp one works better with the 3 and 5, which adds [[64/63]] and [[126/125]] to the list. Tempering out both 128/125 and 64/63 makes 39EDO, in some few ways, allied to 12-ET in supporting [[Augene|augene temperament]], and is in fact, an excellent choice for an augene tuning, but one difference is that 39 has a fine 11, and adding it to consideration we find that 39-EDO tempers out [[99/98]] and [[121/120]] also. This better choice for 39et is {{val|39 62 91 110 135}}.
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== Just approximation ==
== Just approximation ==
=== Selected just intervals ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +5.7
| +13.7
| -15.0
| +2.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +18.6
| +44.5
| -48.7
| +8.2
|}
=== Temperament measures ===
=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 39et.  
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 39et.  
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