Perfect fifth

This page is about the interval region. For the just perfect fifth, see 3/2.

A perfect fifth (P5) is an interval that spans four 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 686 to 720 cents (4\7 to 3\5).

In just intonation, the just perfect fifth is 3/2. Other intervals are also classified as perfect fifths, sometimes called wolf fifths or imperfect fifths, if they are reasonably mapped to 4\7 and 14\24 (precisely four steps of the diatonic scale and seven steps of the chromatic scale). The use of 24edo's 14\24 as the mapping criteria here rather than 12edo's 7\12 better captures the characteristics of many intervals in the 11- and 13-limit.

As a concrete interval region, it is typically near 700 cents in size, distinct from semidiminished fifths of rougly 650 cents. A rough tuning range for the perfect fifth is about 670 to 730 cents according to Margo Schulter's theory of interval regions. Another common range is the stricter range from 686 to 720 cents, which generates a diatonic scale.

In just intonation

The only "perfect" fourth in JI is the Pythagorean perfect fifth of 3/2, about 702 cents in size, which corresponds to the mos-based interval category of the diatonic perfect fifth, and is the generator for Pythagorean tuning and the diatonic scale. However, various "out of tune" fifths exist, such as the Pythagorean wolf fifth 262144/177147, which is flat of 3/2 by one Pythagorean comma, and is about 678 cents in size.

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

• The 5-limit grave fifth is a ratio of 40/27, and is about 680 cents
• The 7-limit superfifth is a ratio of 32/21, and is about 729 cents.
• The 11-limit diminished fifth is a ratio of 22/15, and is about 663 cents.
  • There is also an 11-limit acute fifth, which is a ratio of 50/33, and is about 720 cents.
• The 13-limit ultrafifth is a ratio of 20/13, and is about 746 cents, but it might be better analyzed as an inframinor sixth. Despite that, it is also here for completeness.

In edos

The following table lists the best tuning of 3/2, as well as other fifths if present, in various significant edos.

Edo	3/2	Other fifths
5	720c
7	686c
12	700c
15	720c
16	675c	750c ≈ 20/13
17	706c
19	694c
22	709c	654c ≈ 22/15
24	700c	750c ≈ 20/13, 650c ≈ 22/15
25	720c	672c ≈ 40/27
26	692c	738c ≈ 32/21, 20/13
27	711c	666c ≈ 22/15
29	704c	745c ≈ 20/13, 663c ≈ 22/15
31	697c	736c ≈ 32/21, 659c ≈ 22/15
34	706c	742c ≈ 20/13, 671c ≈ 40/27, 22/15
41	702c	732c ≈ 32/21, 674c ≈ 40/27
53	702c	748c ≈ 20/13, 724c ≈ 32/21, 679c ≈ 40/27, 657c ≈ 22/15

In temperaments

The simplest perfect 5th ratio is 3/2. The following notable temperaments are generated by it:

Temperaments that use 3/2 as a generator

• Meantone, the temperament flattening 3/2 such that four 3/2s stack to 5/4
• Schismatic, the temperament slightly sharpening 3/2 such that nine 3/2s stack to 6/5
• Superpyth, the temperament sharpening 3/2 such that four 3/2s stack to 9/7
• Compton, the temperament of the Pythagorean comma, equivalent to 12edo
• Mavila, the temperament flattening 3/2 such that four 3/2s stack to 6/5
• Various historical well temperaments generated by tempered 4/3s or 3/2s, equivalent to 12edo as compton and meantone