Major third

A major third (M3) is the larger of the two thirds – intervals spanning 3 degrees or 2 scale steps in the diatonic scale. It is found between the 1st and 3rd notes of the major scale, hence its name. Another diatonic interval around the same size is the diminished fourth (d4). More generally, an interval close to 400 cents in size can be called a major third.

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Major third

Many JI intervals are called major thirds, but the term usually refers to one of these three intervals:

• 5/4, the classical major third of about 386 cents.
• 9/7, the septimal major third of about 435 cents.
• 81/64, the Pythagorean major third of about 408 cents.

As an interval region

← Neutral third

Major third

Perfect fourth →

Name	Major third
Lower bound	343¢ – 372¢
Upper bound	440¢ – 480¢
Generated MOSes	3L 4s, 7L 3s, 3L 7s, 3L 5s, 5L 3s
Example JI intervals
Intervals	5/4 (386.3¢) 9/7 (435.1¢)
Related regions
Complement	Minor sixth
Subregions	Submajor third Supermajor third Ultramajor third

As an interval region, a major third is typically near 400 ¢ in size. 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 section 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 mos scales

Intervals between 360 and 480 cents generate the following mos scales. These tables start from the last monolarge mos generated by the interval range. Scales with more than 12 notes are not included.

Range	Mos
360–400 ¢	1L 2s	3L 1s	3L 4s	3L 7s
400–436 ¢	1L 1s	2L 1s	3L 2s	3L 5s	3L 8s
436–450 ¢					8L 3s
450–480 ¢				5L 3s

As a diatonic interval category

Diatonic major third

MOS	5L 2s
Other names	Major 2-diastep
Generator span	+4 generators
Tuning range	343–480 ¢
Basic tuning	400 ¢
Function on root	Mediant
Interval regions	Neutral third, major third, (naiadic)
Associated just intervals	5/4, 81/64
Octave complement	Minor sixth

As a diatonic interval category, a major third 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 two steps of the diatonic scale and four steps of the chromatic scale, or formally 2\7 and 8\24. 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 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.

The diminished fourth is enharmonic with the major third, ranging from 240 to 514 ¢ (2\5 to 3\7). It is generated by stacking 8 fourths octave reduced, and is as such not found in the diatonic scale. Regardless, in TAMNAMS, it may be called the diminished 3-diastep.

In just intonation, an interval may be classified as a diminished fourth if it is reasonably mapped to three steps of the diatonic scale and four steps of the chromatic scale, or formally 3\7 and 8\24.

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. This is similar for the diminished fourth.

The tuning range of the diatonic major third ranges from 342.8 to 480 ¢. 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 384 ¢ gives us (384 + 2400)/4 = 2784/4 = 696 ¢, corresponding to 50edo.

The tuning range of the diatonic diminished fourth ranges from 240 to 514 ¢. The generator for a given tuning in cents, n, for the diminished fourth can be found by (n + 3600)/8. For example, the diminished fourth 384 ¢ gives us (384 + 3600)/8 = 3984/8 = 498 ¢, corresponding to 200edo.

Several example tunings are provided below:

Tunings of the major third and diminished fourth

Tuning	Step ratio	Edo	Major third	Diminished fourth
Equalized	1:1	7	343 ¢	514 ¢
Supersoft	4:3	26	369 ¢	462 ¢
Soft	3:2	19	379 ¢	442 ¢
Semisoft	5:3	31	387 ¢	426 ¢
Basic	2:1	12	400 ¢	400 ¢
Semihard	5:2	29	414 ¢	372 ¢
Hard	3:1	17	424 ¢	353 ¢
Superhard	4:1	22	436 ¢	327 ¢
Collapsed	1:0	5	480 ¢	240 ¢

In just intonation

By prime limit

The simplest 3-limit major third is the Pythagorean major third of 81/64, 408 ¢ in size, which is generated by stacking four just perfect fifths of 3/2. There is also a Pythagorean diminished fourth of about 384 ¢.

Much simpler major thirds and diminished fourths 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 about 418 ¢. (Note that this is often considered an imperfect or diminished fourth.)
• 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 ¢.
• The 23-limit vicesimoterial supermajor third is a ratio of 23/18, and is about 424 ¢.

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, alongside the diatonic major third in various significant edos.

Edo	5/4	9/7	Diatonic major third
12	400 ¢
15	400 ¢	*
16	375 ¢	450 ¢	*
17	*	424 ¢
19	379 ¢	442 ¢	379 ¢
22	382 ¢	436 ¢	436 ¢
24	400 ¢	450 ¢	400 ¢
25	384 ¢	432 ¢	*
26	369 ¢	415 ¢	369 ¢
27	400 ¢	444 ¢	444 ¢
29	372 ¢	455 ¢	414 ¢ ≈ 81/64, 14/11
31	387 ¢	426 ¢	387 ¢
34	388 ¢	424 ¢
41	381 ¢	439 ¢	410 ¢ ≈ 81/64
53	385 ¢	430 ¢	408 ¢ ≈ 81/64

* There is a valid interval in this edo, but it is well outside the range of a major third.

In regular temperaments

The two simplest major third 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.

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, and a stack of seven gives 6/1.
• Squares, generated by flat supermajor thirds representing 9/7 and 14/11, such that a stack of four gives 8/3.