Perfect fifth

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

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Perfect fifth

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 ¢ (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 ¢ in size, distinct from semidiminished fifths of rougly 650 ¢. 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 ¢, which generates a diatonic scale.

This article covers intervals from 650 to 750 ¢, which is very wide, but is done to avoid having to make articles for the tiny interval regions that would otherwise be required to cover the entire span of intervals.

In just intonation

The only "perfect" fifth in JI is the Pythagorean perfect fifth of 3/2, about 702 ¢ 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 ¢ 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 ¢
• The 7-limit superfifth is a ratio of 32/21, and is about 729 ¢.
• The 11-limit diminished fifth is a ratio of 22/15, and is about 663 ¢.
  • There is also an 11-limit acute fifth, which is a ratio of 50/33, and is about 720 ¢.
• The 13-limit ultrafifth is a ratio of 20/13, and is about 746 ¢, 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	720 ¢
7	686 ¢
12	700 ¢
15	720 ¢
16	675 ¢	750 ¢ ≈ 20/13
17	706 ¢
19	694 ¢
22	709 ¢	654 ¢ ≈ 22/15
24	700 ¢	750 ¢ ≈ 20/13, 650 ¢ ≈ 22/15
25	720 ¢	672 ¢ ≈ 40/27
26	692 ¢	738 ¢ ≈ 32/21, 20/13
27	711 ¢	666 ¢ ≈ 22/15
29	704 ¢	745 ¢ ≈ 20/13, 663 ¢ ≈ 22/15
31	697 ¢	736 ¢ ≈ 32/21, 659 ¢ ≈ 22/15
34	706 ¢	742 ¢ ≈ 20/13, 671 ¢ ≈ 40/27, 22/15
41	702 ¢	732 ¢ ≈ 32/21, 674 ¢ ≈ 40/27
53	702 ¢	748 ¢ ≈ 20/13, 724 ¢ ≈ 32/21, 679 ¢ ≈ 40/27, 657 ¢ ≈ 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

In moment-of-symmetry scales

Intervals between 654 and 750 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
720-750c	1L 1s	2L 1s	3L 2s	5L 3s
700-720c			2L 3s	5L 2s	5L 7s
686-700c					7L 5s
667-686c				2L 5s	7L 2s
654-667c					2L 7s	9L 2s