Perfect fourth

This page is about the interval region. For the just perfect fourth, see 4/3.

← Major third (interval region)

Interval region

Semi-augmented fourth Tritone →

Name	Perfect fourth
Lower bound	450¢ – 480¢
Upper bound	514¢ – 550¢
Generated MOSes	5L 2s, 2L 5s, 5L 3s, 7L 2s, 2L 7s
Example JI intervals
Intervals	4/3 (498¢)
Related regions
Complement	Perfect fifth

A perfect fourth (P4) is the smaller and most common of the two "fourths" - intervals spanning 4 degrees or 3 scale steps in the diatonic scale. It is found between the 1st and 4th degrees of all diatonic modes except Lydian. Another diatonic interval around the same size is the augmented third (A3). More generally, an interval close to 500 cents can be called a perfect fourth.

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Perfect fourth

As an interval region

As an interval region, the perfect fourth is typically near 500 ¢ in size, distinct from the semiaugmented fourth of roughly 550 ¢. A rough tuning range for the perfect fourth is about 470 to 530 ¢ according to Margo Schulter's theory of interval regions. Another common range is the stricter range from 480 to 514 ¢, which generates a diatonic scale.

This article covers intervals from 450 to 540 ¢.

In MOS scales

Intervals between 450 and 545 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
450–480 ¢	1L 1s	2L 1s	3L 2s	5L 3s
480–500 ¢			2L 3s	5L 2s	5L 7s
500–514 ¢					7L 5s
514–533 ¢				2L 5s	7L 2s
533–545 ¢					2L 7s	9L 2s

As a diatonic interval category

A perfect fourth is an interval that spans three steps of the diatonic scale with a perfect quality, i.e. the quality that exists in all but one modes. Depending on the specific tuning, it ranges from 480 to 514 ¢ (2\5 to 3\7).

In just intonation, the just perfect fourth is 4/3. Other intervals are also classified as perfect fourths, sometimes called wolf fourths or imperfect fourths, if they are reasonably mapped to three steps of the diatonic scale and five steps of the chromatic scale. The perfect fourth is the complement of the perfect fifth, and defines the subdominant function against the tonic.

In TAMNAMS, this interval is called the perfect 3-diastep.

The augmented third is enharmonic with the perfect fourth, ranging from 343 to 720 cents (2/7 to 3/5). It is generated by stacking 11 fifths octave-reduced, and is thus not found in the diatonic scale. Regardless, in TAMNAMS, it may be called the augmented 2-diastep.

In just intonation, an interval may be classified as an augmented third if it is reasonably mapped to two steps of the diatonic scale and five steps of the chromatic scale, or formally 2\7 and 10\24.

Scale info

The diatonic scale contains six perfect fourths. In the Ionian mode, perfect fourths are found on all but the fourth degree of the scale, which has an augmented fourth. The strong presence of perfect fourths as opposed to their augmented counterparts leads to fourths (along with fifths) being the most fundamental intervals for diatonic harmony.

In just intonation

By prime limit

The only "perfect" fourth in JI is the Pythagorean perfect fourth of 4/3, about 498 ¢ in size, which corresponds to the mos-based interval category of the diatonic perfect fourth and is the octave complement of the perfect fifth of 3/2. However, various "out of tune" fourths exist, such as the Pythagorean wolf fourth 177147/131072, which is sharp of 4/3 by one Pythagorean comma, and is about 522 ¢ in size.

Other "out of tune" fourths in higher limits include:

• The 5-limit acute fourth is a ratio of 27/20, and is about 520 ¢
• The 7-limit subfourth is a ratio of 21/16, and is about 471 ¢.
• The 11-limit augmented fourth is a ratio of 15/11, and is about 537 ¢, and may better be analyzed as an ultrafourth.
  • There is also an 11-limit grave fourth, which is a ratio of 33/25, and is about 480 ¢.
• The 13-limit infrafourth is a ratio of 13/10, and is about 454 ¢, but it might be better analyzed as an ultramajor third. Despite that, it is also here for completeness.

By delta

See Delta-N ratio.

Delta 1		Delta 3		Delta 4		Delta 5		Delta 6
4/3	498 ¢	13/10	454 ¢	15/11	537 ¢	19/14	529 ¢	23/17	523 ¢
				17/13	464 ¢	21/16	471 ¢	25/19	475 ¢

In edos

The following table lists the best tuning of 4/3, as well as other fourths if present, in various significant edos.

Edo	4/3	Other fourths
5	480 ¢
7	514 ¢
12	500 ¢
15	480 ¢
16	525 ¢	450 ¢ ≈ 13/10
17	494 ¢
19	506 ¢
22	491 ¢
24	500 ¢	450 ¢ ≈ 13/10, 550 ¢ ≈ 15/11
25	480 ¢	528 ¢ ≈ 27/20
26	508 ¢	462 ¢ ≈ 21/16, 13/10
27	489 ¢	533 ¢ ≈ 15/11
29	496 ¢	455 ¢ ≈ 13/10, 537 ¢ ≈ 15/11
31	503 ¢	464 ¢ ≈ 21/16
34	494 ¢	458 ¢ ≈ 13/10, 529 ¢ ≈ 27/20, 15/11
41	498 ¢	468 ¢ ≈ 21/16, 526 ¢ ≈ 27/20
53	498 ¢	452 ¢ ≈ 13/10, 476 ¢ ≈ 21/16, 521 ¢ ≈ 27/20

In temperaments

The simplest perfect 4th ratio is 4/3. The following notable temperaments are generated by it:

Temperaments that use 4/3 as a generator

• Archy, the temperament flattening 4/3 such that three 4/3's stack to 7/6 (and furthermore, in superpyth, where eight stack to 6/5) octave-reduced
• Schismic, the temperament tuning 4/3 such that eight 4/3's stack to 5/4 octave-reduced
• Meantone, the temperament sharpening 4/3 such that three 4/3's stack to 6/5 octave-reduced
• Mavila, the temperament sharpening 4/3 such that three 4/3's stack to 5/4 octave-reduced
• Various historical well temperaments generated by tempered 4/3's or 3/2's, equivalent to 12edo as compton and meantone

Temperaments that use wolf fourths as generators

• Buzzard, the temperament generated by sharpening 21/16 so that four of it stack to 3/1.
• Gravity, the temperament generated by flattening 27/20 so that two of it stack to 20/11, three of which in turn reach 3/2.