Major third

A major third (M3) is an interval that is near 400 cents in size, distinct from the minor third of roughly 300 cents. A rough tuning range for the major third is about 360 to 460 cents, though this is extremely wide; some might prefer to restrict it to around 370-440 cents as in Schulter's theory of interval regions. Flat of major thirds (but sharp of minor thirds) are neutral thirds.

"Major third" refers both to the ~370-450 cent range as a whole, and to a specific subdivision within it (about ~370-415 cents); major thirds sharp of this are often called "supermajor thirds".

"Major third" may also refer to the diatonic major third, which is an interval generated by stacking 4 fifths and is not the subject of this article.

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).
• 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

• TBD