Major third

A major third 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 tempered fifths and is not the subject of this article.

In just intonation

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.

In tempered scales

The following table lists the best tuning of 5/4 and 9/7*, as well as other major thirds if present, in several significant EDOs.

*Note that 9/7 may not always be the actual best note in the EDO, this is because of the way EDO tunings try to match how intervals in just intonation are stacked to form new intervals. See Val for more information. Essentially, this is what 9/7 should be based on the best tuning of 7 itself.

EDO	5/4	9/7	Other major thirds
12	400c
15	400c	**
16	375c	450c
17	***	424c
19	379c	442c
20	360c	**	420c ≈ 14/11
22	382c	436c
24	400c	450c
25	384c	432c
26	369c	415c
27	400c	444c
28	386c	428c
29	372c	455c	414c ≈ 81/64, 14/11
31	388c	426c
32	375c	**	413c ≈ 14/11, 450c ≈ 13/10
34	388c	459c	424c ≈ 14/11
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 is tuned sharper than 460 cents, and is thus definitively not a major third.

*** These edos have an approximation to 5/4, but it is tuned flatter than 360 cents, and is thus definitively not a major third.

In regular temperaments

The following lists go over the uses of 5/4 and 9/7 in temperaments.

This list is incomplete.

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