Perfect fourth

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

A perfect fourth is an interval that is near 500 cents in size, distinct from augmented fourths (a type of tritone, about 600 cents). A rough tuning range for the perfect fourth is about 450 to 550 cents, though this is extremely wide; some might prefer to restrict it to around 470-530 cents.

"Perfect fourth" may also refer to the diatonic perfect fourth, which is a tempered fourth used to generate the diatonic scale, and is not the subject of this article.

In just intonation

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.

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