Perfect fourth

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

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

In just intonation

By prime limit

The only "perfect" fourth in JI is the Pythagorean perfect fourth of 4/3, about 498 cents 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 cents 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 cents
• The 7-limit subfourth is a ratio of 21/16, and is about 471 cents.
• The 11-limit augmented fourth is a ratio of 15/11, and is about 537 cents.
  • There is also an 11-limit grave fourth, which is a ratio of 33/25, and is about 480 cents.
• The 13-limit infrafourth is a ratio of 13/10, and is about 454 cents, but it might be better analyzed as an ultramajor third. Despite that, it is also here for completeness.

By delta

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

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	480c
7	514c
12	500c
15	480c
16	525c	450c ≈ 13/10
17	494c
19	506c
22	491c	545c ≈ 15/11
24	500c	450c ≈ 13/10, 550c ≈ 15/11
25	480c	528c ≈ 27/20
26	508c	462c ≈ 21/16, 13/10
27	489c	533c ≈ 15/11
29	496c	455c ≈ 13/10, 537c ≈ 15/11
31	503c	464c ≈ 21/16, 541c ≈ 15/11
34	494c	458c ≈ 13/10, 529c ≈ 27/20, 15/11
41	498c	468c ≈ 21/16, 526c ≈ 27/20
53	498c	452c ≈ 13/10, 476c ≈ 21/16, 521c ≈ 27/20, 543c ≈ 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