Perfect fifth

A perfect fifth (P5), as a concrete interval region, 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.

← Semidiminished fifth Tritone

Perfect fifth

Minor sixth →

Name	Perfect fifth
Lower bound	650¢ – 686¢
Upper bound	720¢ – 750¢
Generated MOSes	5L 2s, 2L 5s, 5L 3s, 7L 2s, 2L 7s
Example JI intervals
Intervals	3/2 (702¢)
Related regions
Complement	Perfect fourth

English Wikipedia has an article on:

Perfect fifth

This article covers intervals from 660 to 750 ¢, but intervals between 650 and 660 cents have been "grandfathered in" due to the fact that superfourths and subfifths were not originally given their own articles.

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 mos scales

Intervals between 654 and 750 ¢ generate the following mos scales:

These tables start from the last monolarge mos generated by the interval range.

Scales with more than 12 notes are not included.

Range	Mos
720–750 ¢	1L 1s	2L 1s	3L 2s	5L 3s
700–720 ¢			2L 3s	5L 2s	5L 7s
686–700 ¢					7L 5s
667–686 ¢				2L 5s	7L 2s
654–667 ¢					2L 7s	9L 2s