View source for Major third (diatonic interval category)

{{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)]].''

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]].

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'''.

== Scale info ==
The diatonic scale contains three major thirds. In the Ionian mode, major thirds are found on the first, fourth, and fifth degrees of the scale; the other four degrees have minor thirds. This roughly equal distribution leads to diatonic tonality being largely based on the distinction between major and minor thirds and triads.

== 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.

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.

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

== In regular temperaments ==

=== 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:
{| class="wikitable"
|+
!Just interval
!Cents
!Temperament
!Tempered comma
!Generator (eigenmonzo tuning)
|-
|[[27/22]]
|355c
|[[Io]]
|[[33/32]]
|Perfect fifth ≈ 689c
|-
|[[16/13]]
|359c
|[[Superflat]]
|[[1053/1024]]
|Perfect fifth ≈ 690c
|-
|[[21/17]]
|366c
|Temperament of 459/448
|459/448
|Perfect fifth ≈ 692c
|-
|[[5/4]]
|386c
|[[Meantone]]
|[[81/80]]
|Perfect fifth ≈ 697c
|-
|[[81/64]]
|408c
|[[Pythagorean tuning|Pythagorean]]
|[[1/1]]
|Perfect fifth ≈ 702c
|-
|[[14/11]]
|418c
|[[Parapyth]]/[[Pentacircle]]
|[[896/891]]
|Perfect fifth ≈ 705c
|-
|[[9/7]]
|435c
|[[Superpyth|Archy/Superpyth]]
|[[64/63]]
|Perfect fifth ≈ 709c
|-
|[[13/10]]
|454c
|[[Oceanfront]]/Temperament of 416/405
|[[416/405]]
|Perfect fifth ≈ 714c
|}