Perfect fourth

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

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Perfect fourth

A perfect fourth (P4) 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 3\7 and 10\24 (precisely three steps of the diatonic scale and five steps of the chromatic scale). The use of 24edo's 10\24 as the mapping criteria here rather than 12edo's 5\12 better captures the characteristics of many intervals in the 11- and 13-limit.

As a concrete interval region, it 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 550 ¢, in order to cover the range of intervals without needing extra articles for subfourths or superfourths.

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 ¢.
  • 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 ¢	545 ¢ ≈ 15/11
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, 541 ¢ ≈ 15/11
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, 543 ¢ ≈ 15/11

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

• Compton, the temperament of the Pythagorean comma, equivalent to 12edo
  • The 3-limit circular temperaments in general
• Archy, the temperament flattening 4/3 such that three 4/3s stack to 7/6
• Meantone, the temperament sharpening 4/3 such that three 4/3s stack to 6/5
• Mavila, the temperament sharpening 4/3 such that three 4/3s stack to 5/4
• Various historical Well temperaments generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone

In moment-of-symmetry 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.

MOSes with more than 12 notes are not included.

Range	MOS
450-480c	1L 1s	2L 1s	3L 2s	5L 3s
480-500c			2L 3s	5L 2s	5L 7s
500-514c					7L 5s
514-533c				2L 5s	7L 2s
533-545c					2L 7s	9L 2s