Major third (diatonic interval category): Difference between revisions

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{{Infobox
{{Infobox
| Title = Diatonic major third
| Title = Diatonic major third
| Header 1 = MOS | Data 1 = [[5L 2s]]
| Header 1 = MOS | Data 1 = [[5L 2s]]
| Header 2 = Other names | Data 2 = Major 2-diastep
| Header 2 = Other names | Data 2 = Major 2-diastep
| Header 3 = Generator span | Data 3 = +4 generators
| Header 3 = Generator span | Data 3 = +4 generators
| Header 4 = Tuning range | Data 4 = 343c – 480c
| Header 4 = Tuning range | Data 4 = 343–480{{c}}
| Header 5 = Basic tuning | Data 5 = 400c
| Header 5 = Basic tuning | Data 5 = 400{{c}}
| Header 6 = Chromatically adjacent interval | Data 6 = [[Minor third (diatonic interval category)|Diatonic minor third]]
| Header 6 = Chromatically adjacent interval | Data 6 = [[Minor third (diatonic interval category)|Diatonic minor third]]
| Header 7 = Function on root | Data 7 = Mediant
| Header 7 = Function on root | Data 7 = Mediant
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| Header 10 = Associated just intervals | Data 10 = [[5/4]], [[81/64]]
| 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 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.
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.
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Several example tunings are provided below:
Several example tunings are provided below:
{| class="wikitable"
{| class="wikitable"
|+Tunings of the major third
|+ style="font-size: 105%;" | Tunings of the major third
|-
! Tuning
! Tuning
! Step ratio
! Step ratio
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| 1:1
| 1:1
| 7
| 7
| 343c
| 343{{c}}
|-
|-
| Supersoft
| Supersoft
| 4:3
| 4:3
| 26
| 26
| 369c
| 369{{c}}
|-
|-
| Soft
| Soft
| 3:2
| 3:2
| 19
| 19
| 379c
| 379{{c}}
|-
|-
| Semisoft
| Semisoft
| 5:3
| 5:3
| 31
| 31
| 387c
| 387{{c}}
|-
|-
| Basic
| Basic
| 2:1
| 2:1
| 12
| 12
| 400c
| 400{{c}}
|-
|-
| Semihard
| Semihard
| 5:2
| 5:2
| 29
| 29
| 414c
| 414{{c}}
|-
|-
| Hard
| Hard
| 3:1
| 3:1
| 17
| 17
| 424c
| 424{{c}}
|-
|-
| Superhard
| Superhard
| 4:1
| 4:1
| 22
| 22
| 436c
| 436{{c}}
|-
|-
| Collapsed
| Collapsed
| 1:0
| 1:0
| 5
| 5
| 480c
| 480{{c}}
|}
|}


== 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
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| [[Io]]
| [[Io]]
| [[33/32]]
| [[33/32]]
| Perfect fifth ≈ 689c
| {{nowrap|Perfect fifth ≈ 689{{c}}}}
|-
|-
| [[16/13]]
| [[16/13]]
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| [[Superflat]]
| [[Superflat]]
| [[1053/1024]]
| [[1053/1024]]
| Perfect fifth ≈ 690c
| {{nowrap|Perfect fifth ≈ 690{{c}}}}
|-
|-
| [[21/17]]
| [[21/17]]
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| Temperament of 459/448
| Temperament of 459/448
| 459/448
| 459/448
| Perfect fifth ≈ 692c
| {{nowrap|Perfect fifth ≈ 692{{c}}}}
|-
|-
| [[5/4]]
| [[5/4]]
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| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| Perfect fifth ≈ 697c
| {{nowrap|Perfect fifth ≈ 697{{c}}}}
|-
|-
| [[81/64]]
| [[81/64]]
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| [[Pythagorean tuning|Pythagorean]]
| [[Pythagorean tuning|Pythagorean]]
| [[1/1]]
| [[1/1]]
| Perfect fifth ≈ 702c
| {{nowrap|Perfect fifth ≈ 702{{c}}}}
|-
|-
| [[14/11]]
| [[14/11]]
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| [[Parapyth]]/[[pentacircle]]
| [[Parapyth]]/[[pentacircle]]
| [[896/891]]
| [[896/891]]
| Perfect fifth ≈ 705c
| {{nowrap|Perfect fifth ≈ 705{{c}}}}
|-
|-
| [[9/7]]
| [[9/7]]
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| [[Superpyth|Archy/superpyth]]
| [[Superpyth|Archy/superpyth]]
| [[64/63]]
| [[64/63]]
| Perfect fifth ≈ 709c
| {{nowrap|Perfect fifth ≈ 709{{c}}}}
|-
|-
| [[13/10]]
| [[13/10]]
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| [[Oceanfront]]/Temperament of 416/405
| [[Oceanfront]]/Temperament of 416/405
| [[416/405]]
| [[416/405]]
| Perfect fifth ≈ 714c
| {{nowrap|Perfect fifth ≈ 714{{c}}}}
|}
|}


Revision as of 13:55, 15 March 2025

Diatonic major third
MOS 5L 2s
Other names Major 2-diastep
Generator span +4 generators
Tuning range 343–480 ¢
Basic tuning 400 ¢
Chromatically adjacent interval Diatonic minor third
Function on root Mediant
Positions in major scale 1, 4, 5
Interval regions Neutral third, Major third, Perfect fourth
Associated just intervals 5/4, 81/64

A major third (M3), in the diatonic scale, is an interval that spans two scale steps in the diatonic scale with the major (wider) quality. It is generated by stacking 4 fifths octave reduced, and depending on the specific tuning, it ranges from 343 to 480 ¢ (2\7 to 2\5). In just intonation, an interval may be classified as a major third if it is reasonably mapped to 2\7 and 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- and 13-limit.

The major third can be stacked with a 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 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 = 696 ¢, corresponding to 50edo.

Several example tunings are provided below:

Tunings of the major third
Tuning Step ratio Edo Cents
Equalized 1:1 7 343 ¢
Supersoft 4:3 26 369 ¢
Soft 3:2 19 379 ¢
Semisoft 5:3 31 387 ¢
Basic 2:1 12 400 ¢
Semihard 5:2 29 414 ¢
Hard 3:1 17 424 ¢
Superhard 4:1 22 436 ¢
Collapsed 1:0 5 480 ¢

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:

Just interval Cents Temperament Tempered comma Generator (eigenmonzo tuning)
27/22 355 ¢ Io 33/32 Perfect fifth ≈ 689 ¢
16/13 359 ¢ Superflat 1053/1024 Perfect fifth ≈ 690 ¢
21/17 366 ¢ Temperament of 459/448 459/448 Perfect fifth ≈ 692 ¢
5/4 386 ¢ Meantone 81/80 Perfect fifth ≈ 697 ¢
81/64 408 ¢ Pythagorean 1/1 Perfect fifth ≈ 702 ¢
14/11 418 ¢ Parapyth/pentacircle 896/891 Perfect fifth ≈ 705 ¢
9/7 435 ¢ Archy/superpyth 64/63 Perfect fifth ≈ 709 ¢
13/10 454 ¢ Oceanfront/Temperament of 416/405 416/405 Perfect fifth ≈ 714 ¢

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.

See also

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