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 is near 700 cents in size, distinct from diminished fifths (a type of tritone, about 600 cents). A rough tuning range for the perfect fifth is about 650 to 750 cents, though this is extremely wide; some might prefer to restrict it to around 670-730 cents. Another common range is the even stricter diatonic range, from ~686 to 720 cents, which corresponds to diatonic perfect fifths that can be used to generate 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

• Compton, the temperament of the Pythagorean comma, equivalent to 12edo
  • The 3-limit circular temperaments in general
• Archy, the temperament sharpening 3/2 such that four 3/2s stack to 9/7
• Meantone, the temperament flattening 3/2 such that four 3/2s stack to 5/4
• 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