Major third

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Major third

A major third (M3) in the diatonic scale is an interval that spans two scale steps 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.

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

This article covers intervals between 360 and 460 ¢. 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 just intonation

By prime limit

3-limit intervals in the range of major thirds include the Pythagorean major third of 81/64, 407.8 ¢ 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 ¢ 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 ¢.
• The 7-limit (septimal) supermajor third is a ratio of 9/7, and is almost exactly 435 ¢.
• The 11-limit neogothic major third is a ratio of 14/11, and is almost exactly 417.5 ¢.
• The 13-limit (tridecimal) ultramajor third is a ratio of 13/10, and is about 454 ¢.
  • There is also a 13-limit (tridecimal) submajor third, which is a ratio of 26/21, and is about 370 ¢.
• The 17-limit (septendecimal) submajor third is a ratio of 21/17, and is about 366 ¢.

By delta

See Delta-N ratio.

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

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

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

** These edos have an approximation to 5/4, but it's flatter than 360 ¢, 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

• Würschmidt, which generates 6/1 by stacking eight 5/4's, so that 128/125 flat of 5/4 represents a neutral third.
• Magic, which generates 3/1 by stacking five 5/4's.
• Dicot, an exotemperament which generates 3/2 by stacking two 5/4's so that the mapping dictates that 5/4 and 6/5 are equated.
• Father, an exotemperament which equates 4/3 and 5/4 as a single "fourth-third" interval, from which it derives its name.
• Augmented, which splits the octave into three equal parts, each representing 5/4.

Temperaments that use 9/7 as a generator

• 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.