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| Subregions = [[Submajor third]] <br> [[Supermajor third]] <br> [[Ultramajor third]]
| Subregions = [[Submajor third]] <br> [[Supermajor third]] <br> [[Ultramajor third]]
}}{{Wikipedia}}
}}{{Wikipedia}}
A '''major third''' ('''M3'''), as an [[interval region]], is typically near 400{{c}} in size, distinct from the [[minor third]] of roughly 300{{c}} and the [[neutral third]] of roughly 350{{c}}. A rough tuning range for the major third is about 370 to 440{{c}} according to [[Margo Schulter]]'s theory of interval regions. ''Major third'' in this sense refers both to the ~350–450{{c}} range as a whole, and to a specific subdivision within it (~370–415{{c}}) as opposed to supermajor thirds; major thirds sharp of this are often called "supermajor thirds".  
A major third (M3) is an interval close to 400 cents in size, distinct from the [[minor third]] of roughly 300{{c}} and the [[neutral third]] of roughly 350{{c}}. In the diatonic scale, it is the larger of the two "thirds" - intervals spanning 3 degrees or 2 scale steps.


This article covers intervals between 360 and 460{{c}}. The outer range of this might be too extreme to call "major thirds", but this is done so that one can find what they're looking for easily.   
== As an interval region ==
As an [[interval region]], a major third is typically near 400{{c}} in size. A rough tuning range for the major third is about 370 to 440{{c}} according to [[Margo Schulter]]'s theory of interval regions. ''Major third'' in this sense refers both to the ~350–450{{c}} range as a whole, and to a specific subdivision within it (~370–415{{c}}) as opposed to supermajor thirds; major thirds sharp of this are often called "supermajor thirds".
 
This section covers intervals between 360 and 460{{c}}. The outer range of this might be too extreme to call "major thirds", but this is done so that one can find what they're looking for easily.   
 
=== In MOS scales ===
Intervals between 360 and 480 cents generate the following [[mos]] scales:
 
These tables start from the last monolarge mos generated by the interval range.
 
Scales with more than 12 notes are not included.
 
{| class="wikitable"
|-
! Range
! colspan="5" | Mos
|-
| 360–400{{c}}
| [[1L 2s]]
| [[3L 1s]]
| [[3L 4s]]
| colspan="2" | [[3L 7s]]
|-
| 400–436{{c}}
| rowspan="3" | [[1L 1s]]
| rowspan="3" | [[2L 1s]]
| rowspan="3" | [[3L 2s]]
| rowspan="2" | [[3L 5s]]
| [[3L 8s]]
|-
| 436–450{{c}}
| [[8L 3s]]
|-
| 450–480{{c}}
| colspan="2" | [[5L 3s]]
|}
 
== As a diatonic interval category ==
As a diatonic interval category, a major third 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]]).{{Infobox|Title=Diatonic major third|Header 1=MOS|Data 1=[[5L&nbsp;2s]]|Header 2=Other names|Data 2=Major 2-diastep|Header 3=Generator span|Data 3=+4 generators|Header 4=Tuning range|Data 4=343–480{{c}}|Header 5=Basic tuning|Data 5=400{{c}}|Header 6=Function on root|Data 6=Mediant|Header 7=Interval regions|Data 7=[[Neutral third (interval region)|Neutral third]], [[major third (interval region)|major third]], ([[naiadic]])|Header 8=Associated just intervals|Data 8=[[5/4]], [[81/64]]|Header 9=Octave complement|Data 9=[[Minor sixth (diatonic interval category)|Minor sixth]]}}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]].
 
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.
 
In [[TAMNAMS]], this interval is called the '''major 2-diastep'''.
 
The diminished fourth is enharmonic with the major third, ranging from 240 to 514{{c}} (2\5 to 3\7). It is generated by stacking 8 fourths octave reduced, and is as such not found in the diatonic scale. Regardless, in TAMNAMS, it may be called the '''diminished 3-diastep'''.
 
In [[just intonation]], an interval may be classified as a diminished fourth if it is reasonably mapped to '''three''' steps of the diatonic scale and four steps of the chromatic scale, or formally 3\7 and [[24edo|8\24]].
 
=== 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 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. This is similar for the diminished fourth.
 
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.
 
The tuning range of the diatonic diminished fourth ranges from 240 to 514{{c}}. The generator for a given tuning in cents, n, for the diminished fourth can be found by (n + 3600)/8. For example, the diminished fourth 384{{c}} gives us (384 + 3600)/8 = 3984/8 = 498{{c}}, corresponding to 200edo.
 
Several example tunings are provided below:
{| class="wikitable center-all left-1"
|+Tunings of the major third and diminished fourth
!Tuning
!Step ratio
!Edo
!Major third
!Diminished fourth
|-
|Equalized
|1:1
|7
|343{{c}}
|514{{c}}
|-
|Supersoft
|4:3
|26
|369{{c}}
|462{{c}}
|-
|Soft
|3:2
|19
|379{{c}}
|442{{c}}
|-
|Semisoft
|5:3
|31
|387{{c}}
|426{{c}}
|-
|Basic
|2:1
|12
|400{{c}}
|400{{c}}
|-
|Semihard
|5:2
|29
|414{{c}}
|372{{c}}
|-
|Hard
|3:1
|17
|424{{c}}
|353{{c}}
|-
|Superhard
|4:1
|22
|436{{c}}
|327{{c}}
|-
|Collapsed
|1:0
|5
|480{{c}}
|240{{c}}
|}


== In just intonation ==
== In just intonation ==
=== By prime limit ===
=== By prime limit ===
The simplest 3-limit interval in the range of major thirds is the Pythagorean major third of [[81/64]], 408{{c}} in size, which is generated by [[stacking]] four just perfect fifths of [[3/2]].
The simplest 3-limit major third is the Pythagorean major third of [[81/64]], 408{{c}} in size, which is generated by [[stacking]] four just perfect fifths of [[3/2]]. There is also a Pythagorean diminished fourth of about 384{{c}}.


Much [[odd limit|simpler]] major thirds exist in higher [[prime limit|limits]], however, for example:
Much [[odd limit|simpler]] major thirds and diminished fourths exist in higher [[prime limit|limits]], however, for example:


* The 5-limit '''classical major third''' is a ratio of [[5/4]], and is about 386{{c}}.
* The 5-limit '''classical major third''' is a ratio of [[5/4]], and is about 386{{c}}.
* The 7-limit '''(septimal) supermajor third''' is a ratio of [[9/7]], and is almost exactly 435{{c}}.
* The 7-limit '''(septimal) supermajor third''' is a ratio of [[9/7]], and is almost exactly 435{{c}}.
* The 11-limit '''neogothic major third''' is a ratio of [[14/11]], and is about 418{{c}}.
* The 11-limit '''neogothic major third''' is a ratio of [[14/11]], and is about 418{{c}}. (Note that this is often considered an imperfect or diminished fourth.)
* The 13-limit '''(tridecimal) ultramajor third''' is a ratio of [[13/10]], and is about 454{{c}}.
* The 13-limit '''(tridecimal) ultramajor third''' is a ratio of [[13/10]], and is about 454{{c}}.
** There is also a 13-limit '''(tridecimal) submajor third''', which is a ratio of [[26/21]], and is about 370{{c}}.
** There is also a 13-limit '''(tridecimal) submajor third''', which is a ratio of [[26/21]], and is about 370{{c}}.
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== In edos ==
== In edos ==
The following table lists the best tuning of 5/4 and 9/7, as well as other major thirds if present, in various significant [[edo]]s.
The following table lists the best tuning of 5/4 and 9/7, alongside the diatonic major third in various significant [[edo]]s.


{| class="wikitable"
{| class="wikitable"
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! 5/4
! 5/4
! 9/7
! 9/7
! Other major thirds
!Diatonic major third
|-
|-
| 12
| 12
| colspan="2" | 400{{c}}
| colspan="3" | 400{{c}}
|
|-
|-
| 15
| 15
| 400{{c}}
| 400{{c}}
| *
| colspan="2" | *
|
|-
|-
| 16
| 16
| 375{{c}}
| 375{{c}}
| 450{{c}}
| 450{{c}}
|  
|*
|-
|-
| 17
| 17
| **
| *
| 424{{c}}
| colspan="2" | 424{{c}}
|
|-
|-
| 19
| 19
| 379{{c}}
| 379{{c}}
| 442{{c}}
| 442{{c}}
|  
|379{{c}}
|-
|-
| 22
| 22
| 382{{c}}
| 382{{c}}
| 436{{c}}
| 436{{c}}
|  
|436{{c}}
|-
|-
| 24
| 24
| 400{{c}}
| 400{{c}}
| 450{{c}}
| 450{{c}}
|  
|400{{c}}
|-
|-
| 25
| 25
| 384{{c}}
| 384{{c}}
| 432{{c}}
| 432{{c}}
|  
|*
|-
|-
| 26
| 26
| 369{{c}}
| 369{{c}}
| 415{{c}}
| 415{{c}}
|  
|369{{c}}
|-
|-
| 27
| 27
| 400{{c}}
| 400{{c}}
| 444{{c}}
| 444{{c}}
|  
|444{{c}}
|-
|-
| 29
| 29
| 372{{c}}
| 372{{c}}
| 455{{c}}
| 455{{c}}
| {{nowrap|414{{c}} ≈ 81/64, 14/11}}
|{{nowrap|414{{c}} ≈ 81/64, 14/11}}
|-
|-
| 31
| 31
| 388{{c}}
| 387{{c}}
| 426{{c}}
| 426{{c}}
|  
|387{{c}}
|-
|-
| 34
| 34
| 388{{c}}
| 388{{c}}
| 424{{c}}
| colspan="2" | 424{{c}}
| {{nowrap|459{{c}} ≈ 13/10}}
|-
|-
| 41
| 41
| 381{{c}}
| 381{{c}}
| 439{{c}}
| 439{{c}}
| {{nowrap|410{{c}} ≈ 81/64}}
|{{nowrap|410{{c}} ≈ 81/64}}
|-
|-
| 53
| 53
| 385{{c}}
| 385{{c}}
| 430{{c}}
| 430{{c}}
| {{nowrap|362{{c}} ≈ 21/17|408{{c}} ≈ 81/64|452{{c}} ≈ 13/10}}
|408{{c}} ≈ 81/64
|}
|}
<nowiki/>* These edos have an approximation to 9/7, but it is sharper than 460{{c}}, not really a major third.
<nowiki>*</nowiki> There is a valid interval in this edo, but it is well outside the range of a major third.
 
<nowiki/>** These edos have an approximation to 5/4, but it is flatter than 360{{c}}, not really a major third.


<nowiki/>
== In regular temperaments ==
== In regular temperaments ==
The two simplest major third ratios are 5/4 and 9/7. The following notable temperaments are generated by them:
The two simplest major third ratios are 5/4 and 9/7. The following notable temperaments are generated by them:
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* [[Sensi]], generated by sharp supermajor thirds representing [[9/7]] and [[13/10]], such that a stack of two gives a major sixth approximating [[5/3]].
* [[Sensi]], generated by sharp supermajor thirds representing [[9/7]] and [[13/10]], such that a stack of two gives a major sixth approximating [[5/3]].
* [[Squares]], generated by flat supermajor thirds representing [[9/7]] and [[14/11]], such that a stack of four gives [[8/3]].
* [[Squares]], generated by flat supermajor thirds representing [[9/7]] and [[14/11]], such that a stack of four gives [[8/3]].
== In mos scales ==
Intervals between 360 and 480 cents generate the following [[mos]] scales:
These tables start from the last monolarge mos generated by the interval range.
Scales with more than 12 notes are not included.
{| class="wikitable"
|-
! Range
! colspan="5" | Mos
|-
| 360–400{{c}}
| [[1L&nbsp;2s]]
| [[3L&nbsp;1s]]
| [[3L&nbsp;4s]]
| colspan="2" | [[3L&nbsp;7s]]
|-
| 400–436{{c}}
| rowspan="3" | [[1L&nbsp;1s]]
| rowspan="3" | [[2L&nbsp;1s]]
| rowspan="3" | [[3L&nbsp;2s]]
| rowspan="2" | [[3L&nbsp;5s]]
| [[3L&nbsp;8s]]
|-
| 436–450{{c}}
| [[8L&nbsp;3s]]
|-
| 450–480{{c}}
| colspan="2" | [[5L&nbsp;3s]]
|}


== See also ==
== See also ==
Retrieved from "https://en.xen.wiki/w/Major_third"