Major third

A major third (M3) is an interval that spans two steps of 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 cents (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.

As a concrete interval region, it is typically near 400 cents in size, distinct from the minor third of roughly 300 cents and the neutral third of roughly 350 cents. A rough tuning range for the major third is about 370 to 440 cents according to Margo Schulter's theory of interval regions. Major third in this sense refers both to the ~350-450 cent range as a whole, and to a specific subdivision within it (~370–415 cents) as opposed to supermajor thirds; major thirds sharp of this are often called "supermajor thirds".

In just intonation

By prime limit

3-limit intervals in the range of major thirds include the Pythagorean major third of 81/64, about 408 cents in size, which corresponds to the mos-based interval category of the diatonic major third and is generated by stacking four just perfect fifths of 3/2, and the Pythagorean diminished fourth of 8192/6561, which is flat of 81/64 by one Pythagorean comma, and is about 384 cents in size.

Much simpler major thirds exist in higher limits, however, for example:

• The 5-limit classical major third is a ratio of 5/4, and is about 386 cents.
• The 7-limit supermajor third is a ratio of 9/7, and is about 435 cents.
• The 11-limit neogothic major third is a ratio of 14/11, and is about 418 cents.
• The 13-limit ultramajor third is a ratio of 13/10, and is about 454 cents.
  • There is also a 13-limit submajor third, which is a ratio of 26/21, and is about 370 cents.
• The 17-limit submajor third is a ratio of 21/17, and is about 366 cents.

By delta

Delta 1	Cents	Delta 2	Cents	Delta 3	Cents	Delta 4	Cents	Delta 5	Cents
5/4	386c	9/7	435c	13/10	454c	19/15	409c	22/17	446c
				14/11	418c	21/17	366c	23/18	424c
								24/19	404c
								26/21	370c

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 edos.

EDO	5/4	9/7	Other major thirds
12	400c
15	400c	**
16	375c	450c
17	***	424c
19	379c	442c
22	382c	436c
24	400c	450c
25	384c	432c
26	369c	415c
27	400c	444c
29	372c	455c	414c ≈ 81/64, 14/11
31	388c	426c
34	388c	424c	459c ≈ 13/10
41	381c	439c	410c ≈ 81/64
53	385c	430c	362c ≈ 21/17, 408c ≈ 81/64, 452c ≈ 13/10

** These edos have an approximation to 9/7, but it's sharper than 460 cents, not really a major third.

*** These edos have an approximation to 5/4, but it's flatter than 360 cents, not really a major third.

In regular temperaments

The two simplest major 3rd ratios are 5/4 and 9/7. The following notable temperaments are generated by them:

Temperaments that use 5/4 as a generator

• Magic, which generates 3/2 by stacking five 5/4s (octave-reduced).
• Augmented, which splits the octave into three equal parts, each representing 5/4.
• Father, a very inaccurate temperament which equates 4/3 and 5/4 as a single "fourth-third" interval
• Dicot, a somewhat inaccurate temperament which equates 5/4 and 6/5, tempering out the chromatic semitone that usually separates them

Temperaments that use 9/7 as a generator

• Sensi, which equates 5/3 with a stack of two sharp major thirds representing 9/7.