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| {{Infobox
| | #REDIRECT [[Major third (interval region)]] |
| | Title = Diatonic major third
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| | Header 1 = MOS | Data 1 = [[5L 2s]]
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| | Header 2 = Other names | Data 2 = Major 2-diastep
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| | Header 3 = Generator span | Data 3 = +4 generators
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| | Header 4 = Tuning range | Data 4 = 343–480{{c}}
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| | Header 5 = Basic tuning | Data 5 = 400{{c}}
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| | Header 6 = Function on root | Data 6 = Mediant
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| | Header 7 = Interval regions | Data 7 = [[Neutral third]], [[major third (interval region)|major third]], [[naiadic]]
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| | Header 8 = Associated just intervals | Data 8 = [[5/4]], [[81/64]]
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| | Header 9 = Octave complement | Data 9 = [[Minor sixth (diatonic interval category)|Minor sixth]]
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| }}
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| A '''major third''' ('''M3''') 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 two steps of the diatonic scale and four steps of the chromatic scale, or formally 2\7 and [[24edo|8\24]]. 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]].
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| The major third can be stacked with a [[minor third (diatonic interval category)|minor third]] to form a perfect fifth, and as such is often involved in chord structures in diatonic harmony.
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| In [[TAMNAMS]], this interval is called the '''major 2-diastep'''.
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| == Scale info ==
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| 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.
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| == Tunings ==
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| 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.
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| The tuning range of the diatonic major third ranges from 342.8 to 480{{c}}. The generator for a given tuning in cents, ''n'', for the diatonic major third can be found by {{nowrap| (''n'' + 2400)/4 }}. For example, the third 384{{c}} gives us {{nowrap| (384 + 2400)/4 {{=}} 2784/4 {{=}} 696{{c}} }}, corresponding to 50edo.
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| Several example tunings are provided below:
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| {| class="wikitable center-all left-1"
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| |+ style="font-size: 105%;" | Tunings of the major third
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| |-
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| ! Tuning
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| ! Step ratio
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| ! Edo
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| ! Cents
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| |-
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| | Equalized
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| | 1:1
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| | 7
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| | 343{{c}}
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| |-
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| | Supersoft
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| | 4:3
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| | 26
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| | 369{{c}}
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| |-
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| | Soft
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| | 3:2
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| | 19
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| | 379{{c}}
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| |-
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| | Semisoft
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| | 5:3
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| | 31
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| | 387{{c}}
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| |-
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| | Basic
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| | 2:1
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| | 12
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| | 400{{c}}
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| |-
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| | Semihard
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| | 5:2
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| | 29
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| | 414{{c}}
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| |-
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| | Hard
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| | 3:1
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| | 17
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| | 424{{c}}
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| |-
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| | Superhard
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| | 4:1
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| | 22
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| | 436{{c}}
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| |-
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| | Collapsed
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| | 1:0
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| | 5
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| | 480{{c}}
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| |}
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| == In regular temperaments ==
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| === P5 {{=}} 3/2 ===
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| If the diatonic perfect fifth is treated as [[3/2]], approximating various intervals with the diatonic major third leads to the following temperaments:
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| {| class="wikitable center-2 center-5"
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| |-
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| ! Just<br>interval
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| ! Cents
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| ! Temperament
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| ! Vanishing<br>comma
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| ! Generator<br>(eigenmonzo tuning)
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| |-
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| | [[27/22]]
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| | 355{{c}}
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| | [[Io]]
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| | [[33/32]]
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| | {{nowrap| P5 ≈ 689{{c}} }}
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| |-
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| | [[16/13]]
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| | 359{{c}}
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| | [[Superflat]]
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| | [[1053/1024]]
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| | {{nowrap| P5 ≈ 690{{c}} }}
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| |-
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| | [[21/17]]
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| | 366{{c}}
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| | Temperament of 459/448
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| | 459/448
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| | {{nowrap| P5 ≈ 692{{c}} }}
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| |-
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| | [[5/4]]
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| | 386{{c}}
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| | [[Meantone]]
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| | [[81/80]]
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| | {{nowrap| P5 ≈ 697{{c}} }}
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| |-
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| | [[81/64]]
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| | 408{{c}}
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| | [[Pythagorean tuning]]
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| | [[1/1]]
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| | {{nowrap| P5 ≈ 702{{c}} }}
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| |-
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| | [[14/11]]
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| | 418{{c}}
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| | [[Pepperoni]]
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| | [[896/891]]
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| | {{nowrap| P5 ≈ 705{{c}} }}
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| |-
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| | [[9/7]]
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| | 435{{c}}
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| | [[Superpyth|Archy/superpyth]]
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| | [[64/63]]
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| | {{nowrap| P5 ≈ 709{{c}} }}
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| |-
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| | [[13/10]]
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| | 454{{c}}
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| | [[Oceanfront]] / temperament of 416/405
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| | [[416/405]]
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| | {{nowrap| P5 ≈ 714{{c}} }}
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| |}
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| == In just notation systems ==
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| 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.
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| == See also ==
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| * [[Major third]] (disambiguation page)
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| [[Category:Diatonic interval categories]] | | [[Category:Diatonic interval categories]] |