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Major third (diatonic interval category): Difference between revisions

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{{Infobox|Title=Diatonic major third|Data 5=400c|Data 8=1, 4, 5|Header 9=Interval regions|Header 8=Positions in major scale|Data 7=Mediant|Data 6=[[Diatonic minor third]]|Header 7=Function on root|Header 6=Chromatically adjacent interval|Data 4=343c - 480c|Header 1=MOS|Data 3=+4 generators|Data 2=Major 2-diastep|Header 5=Basic tuning|Header 4=Tuning range|Header 3=Generator span|Header 2=Other names|Data 1=[[5L 2s]]|Data 9=[[Neutral third]], [[Major third (interval region)|Major third]], [[Perfect fourth]]|Header 10=Associated just intervals|Data 10=[[5/4]], [[81/64]]}}''This article is about the diatonic interval category. For the interval region, see [[Major third (interval region)]].''
{{Infobox
| Title = Diatonic major third
| Header 1 = MOS | Data 1 = [[5L 2s]]
| Header 2 = Other names | Data 2 = Major 2-diastep
| Header 3 = Generator span | Data 3 = +4 generators
| Header 4 = Tuning range | Data 4 = 343c – 480c
| Header 5 = Basic tuning | Data 5 = 400c
| Header 6 = Chromatically adjacent interval | Data 6 = [[Minor third (diatonic interval category)|Diatonic minor third]]
| Header 7 = Function on root | Data 7 = Mediant
| Header 8 = Positions in major scale | Data 8 = 1, 4, 5
| Header 9 = Interval regions | Data 9 = [[Neutral third]], [[Major third (interval region)|Major third]], [[Perfect fourth]]
| Header 10 = Associated just intervals | Data 10 = [[5/4]], [[81/64]]
}}
A '''major third''' ('''M3'''), in the diatonic scale, is an interval that spans two scale steps in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 4 fifths [[octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 343 to 480 [[cent|¢]] ([[7edo|2\7]] to [[5edo|2\5]]). In [[just intonation]], an interval may be classified as a major third if it is reasonably mapped to 2\7 and [[24edo|8\24]] (precisely two steps of the diatonic scale and four steps of the chromatic scale). The use of 24edo's 8\24 as the mapping criteria here rather than [[12edo]]'s 4\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].


A '''major third (M3)''', in the diatonic scale, is an interval that spans two scale steps in the [[5L 2s|diatonic]] scale with the major (wider) quality. It is generated by stacking 4 fifths [[Octave reduction|octave reduced]], and depending on the specific tuning, it ranges from 343 to 480 [[Cent|¢]] ([[7edo|2\7]] to [[5edo|2\5]]). In [[just intonation]], an interval may be classified as a major third if it is reasonably mapped to 2\7 and [[24edo|8\24]] (precisely two steps of the
The major third can be stacked with a [[minor third (diatonic interval)|diatonic minor third]] to form a perfect fifth, and as such is often involved in chord structures in diatonic harmony.
 
diatonic scale and four steps of the chromatic scale). The use of 24edo's 8\24 as the mapping criteria here rather than [[12edo]]'s 4\12 better captures the characteristics of many intervals in the [[11-limit|11-]] and [[13-limit]].
 
The major third can be stacked with a [[Minor third (diatonic interval)|diatonic minor third]] to form a perfect fifth, and as such is often involved in chord structures in diatonic harmony.


In [[TAMNAMS]], this interval is called the '''major 2-diastep'''.
In [[TAMNAMS]], this interval is called the '''major 2-diastep'''.
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== Tunings ==
== Tunings ==
Being an abstract MOS degree, and not a specific interval, the diatonic major third doesn't have a fixed tuning, but instead has a range of ways it can be tuned, based on the tuning of the generator used in making the scale.
Being an abstract mos degree, and not a specific interval, the diatonic major third does not have a fixed tuning, but instead has a range of ways it can be tuned, based on the tuning of the generator used in making the scale.


The tuning range of the diatonic major third ranges from 342.8 cents to 480 cents. The generator for a given tuning in cents, n, for the diatonic major third can be found by (n+2400)/4. For example, the third 384c gives us (384+2400)/4 = 2784/4 = 696c, corresponding to 50edo.
The tuning range of the diatonic major third ranges from 342.8 cents to 480 cents. The generator for a given tuning in cents, n, for the diatonic major third can be found by (''n'' + 2400)/4. For example, the third 384c gives us (384 + 2400)/4 = 2784/4 = 696{{c}}, corresponding to 50edo.


Several example tunings are provided below:
Several example tunings are provided below:
{| class="wikitable"
{| class="wikitable"
|+Tunings of the major third
|+Tunings of the major third
!Tuning
! Tuning
!Step ratio
! Step ratio
!Edo
! Edo
!Cents
! Cents
|-
|-
|Equalized
| Equalized
|1:1
| 1:1
|7
| 7
|343c
| 343c
|-
|-
|Supersoft
| Supersoft
|4:3
| 4:3
|26
| 26
|369c
| 369c
|-
|-
|Soft
| Soft
|3:2
| 3:2
|19
| 19
|379c
| 379c
|-
|-
|Semisoft
| Semisoft
|5:3
| 5:3
|31
| 31
|387c
| 387c
|-
|-
|Basic
| Basic
|2:1
| 2:1
|12
| 12
|400c
| 400c
|-
|-
|Semihard
| Semihard
|5:2
| 5:2
|29
| 29
|414c
| 414c
|-
|-
|Hard
| Hard
|3:1
| 3:1
|17
| 17
|424c
| 424c
|-
|-
|Superhard
| Superhard
|4:1
| 4:1
|22
| 22
|436c
| 436c
|-
|-
|Collapsed
| Collapsed
|1:0
| 1:0
|5
| 5
|480c
| 480c
|}
|}


== In regular temperaments ==
== In regular temperaments ==
=== P5 = 3/2 ===
=== P5 = 3/2 ===
If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major third leads to the following temperaments:
If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major third leads to the following temperaments:
{| class="wikitable"
{| class="wikitable"
|+
|+
!Just interval
! Just interval
!Cents
! Cents
!Temperament
! Temperament
!Tempered comma
! Tempered comma
!Generator (eigenmonzo tuning)
! Generator (eigenmonzo tuning)
|-
|-
|[[27/22]]
| [[27/22]]
|355c
| 355{{c}}
|[[Io]]
| [[Io]]
|[[33/32]]
| [[33/32]]
|Perfect fifth ≈ 689c
| Perfect fifth ≈ 689c
|-
|-
|[[16/13]]
| [[16/13]]
|359c
| 359{{c}}
|[[Superflat]]
| [[Superflat]]
|[[1053/1024]]
| [[1053/1024]]
|Perfect fifth ≈ 690c
| Perfect fifth ≈ 690c
|-
|-
|[[21/17]]
| [[21/17]]
|366c
| 366{{c}}
|Temperament of 459/448
| Temperament of 459/448
|459/448
| 459/448
|Perfect fifth ≈ 692c
| Perfect fifth ≈ 692c
|-
|-
|[[5/4]]
| [[5/4]]
|386c
| 386{{c}}
|[[Meantone]]
| [[Meantone]]
|[[81/80]]
| [[81/80]]
|Perfect fifth ≈ 697c
| Perfect fifth ≈ 697c
|-
|-
|[[81/64]]
| [[81/64]]
|408c
| 408{{c}}
|[[Pythagorean tuning|Pythagorean]]
| [[Pythagorean tuning|Pythagorean]]
|[[1/1]]
| [[1/1]]
|Perfect fifth ≈ 702c
| Perfect fifth ≈ 702c
|-
|-
|[[14/11]]
| [[14/11]]
|418c
| 418{{c}}
|[[Parapyth]]/[[Pentacircle]]
| [[Parapyth]]/[[pentacircle]]
|[[896/891]]
| [[896/891]]
|Perfect fifth ≈ 705c
| Perfect fifth ≈ 705c
|-
|-
|[[9/7]]
| [[9/7]]
|435c
| 435{{c}}
|[[Superpyth|Archy/Superpyth]]
| [[Superpyth|Archy/superpyth]]
|[[64/63]]
| [[64/63]]
|Perfect fifth ≈ 709c
| Perfect fifth ≈ 709c
|-
|-
|[[13/10]]
| [[13/10]]
|454c
| 454{{c}}
|[[Oceanfront]]/Temperament of 416/405
| [[Oceanfront]]/Temperament of 416/405
|[[416/405]]
| [[416/405]]
|Perfect fifth ≈ 714c
| Perfect fifth ≈ 714c
|}
|}


== In just notation systems ==
== In just notation systems ==
Due to the way the primes 7 and 11 are notated, in many systems of notation for just intonation, the interval [[14/11]] is not considered to be a major third, but instead belongs to the [[enharmonic]] category of diminished fourth.
Due to the way the primes 7 and 11 are notated, in many systems of notation for just intonation, the interval [[14/11]] is not considered to be a major third, but instead belongs to the [[enharmonic]] category of diminished fourth.
== See also ==
* [[Major third]] (disambiguation page)
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