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 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. Another common range is the even stricter diatonic range, from 480 to ~514 cents, which corresponds to diatonic perfect fourths that can be used to generate 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