Fifth

The diatonic scales generated by different fifths (in 130edo tuning).

For intervals with a denominator of 5, see 6/5, 7/5, and 8/5.

A fifth is an interval that spans, or could reasonably span, four steps of the 7-form. The fifth is the generator of the diatonic scale, and one of the most important intervals in not only Western harmony but many musical systems worldwide as well. This is because of the concordance of 3/2, the "perfect fifth" of about 702 cents. The fifth, together with the third and the root note, comprise a tertian triad. Fifths are generally considered to be approximations to, chromatic alterations of, or detunings of 3/2, but other "imperfect" fifths exist.

Name

The term fifth comes from conventional music theory. In the diatonic interval naming convention, intervals are 1-indexed, so the heptatonic 4-step is referred to as a fifth due to spanning five notes including both the notes in the interval.

Fifths in diatonic music theory and interval regions are not considered major or minor; instead 3/2 is referred to as the perfect fifth. This is because not only is the perfect fifth considered a "perfect consonance" (as it is one of the simplest interval ratios), it is also the generator of the MOS diatonic scale and therefore is almost always the same quality.

In heptatonic scales

The concept of a fifth may be defined in terms of a heptatonic moment-of-symmetry scale, which always has two varieties of fifth:

• Diatonic (5L 2s) is generated by the fifth, and therefore has six perfect fifths ranging from 686 to 720 cents, with the basic tuning at 700 cents, and one diminished fifth ranging from 480 to 686 cents, with the basic tuning at 600 cents. Due to the perfect fifth being the generator, diatonic scales that attempt to approximate 3/2 have a wide range of tunings; this is the reason why many temperaments are defined by detuning 3/2 to reach some other target, as a slightly detuned 3/2 can massively alter the diatonic scale it generates.
• Antidiatonic (2L 5s) is also generated by the fifth, and has six perfect fifths, but in antidiatonic they range from 600 to 686 cents, with the basic tuning at 667 cents, and are distinguished from augmented fifths found at 686-1200 cents (with the basic tuning at 800 cents). Usually, antidiatonic is used without a good approximation of 3/2, but both choices for approximations (the perfect fifth, as in mavila, and the augmented fifth, as in gravity) lead to softer antidiatonic tunings.
• Onyx (1L 6s) has four major fifths, ranging from 686 to 1200 cents (the basic tuning being 750 cents), and three minor fifths, ranging from 0 to 686 cents (with the basic tuning being 600 cents). Usually, it is the major fifth that is intended to approximate 3/2, and this is the reason why softer onyx scales are common, like that of porcupine.
• Archaeotonic (6L 1s) has four minor fifths, ranging from 600 to 686 cents (the basic tuning being 646 cents), and three major fifths, ranging from 686 to 800 cents (the basic tuning being 738 cents). It is again the major fifth that is generally intended to approximate 3/2, and thus why softer archaeotonic scales are common, like that of tetracot.
• Mosh (3L 4s) has five major fifths, ranging from 686 to 800 cents, with the basic tuning at 720 cents. In contrast to this are two minor fifths, ranging from 400c to 686c, with the basic tuning at 600 cents. The major fifth approximates 3/2 when tuned (as is common) slightly soft, like in rastmic.
• Smitonic (4L 3s) has five minor fifths, ranging from 600 to 686c (the basic tuning being 655 cents) and two major fifths, ranging from 686 to 900 cents (the basic tuning being 764 cents). Surprisingly, one of the most common temperaments to utilize this scale structure, kleismic, does not assign 3/2 to the fifth at all, but rather to the minor sixth. One example of a temperament that sets 3/2 to the major fifth is sixix.

As an interval region

As an interval region, fifths tend to exclude their overlap with fourths, which belongs to the separate tritone region. Because of this, fifths can be seen as ranging from about 660 to 740 cents; the interordinal intervals between 740 and 760 cents can also function as fifths.

Unlike other interval regions, the fifth region can be more precisely defined in terms of the MOSes that fifths generate, between 667 and 750 cents such that all fifths generate an armotonic, diatonic, or oneirotonic scale; more strictly fifths can be limited entirely to diatonic fifths.

Fifths as scale generators

Fifths are some of the most common scale generators for MOSes, not only due to the prevalence of 3/2 but also the large number of usable MOS generators around the fifth region (see Golden sequences and tuning).

• Fifths flat of 686 cents generate the aforementioned antidiatonic scale, and proceed to also generate armotonic. If these fifths are not interpreted as 3/2, a temperament option is trismegistus temperament, or gravity for a near-equiheptatonic scale. If they are, an option is mavila.
• A fifth of exactly 686 cents generates 7edo, and is associated with whitewood temperament and found in many multiples of 7edo.
• Fifths between 686 and 700 cents generate soft MOS diatonic scales and m-chromatic scales, and are associated with meantone and deeptone.
• Fifths between 700 and 720 cents generate hard MOS diatonic scales and p-chromatic scales, and are associated with the just Pythagorean tuning of 701.96c (and thus schismic) and superpyth. Also of note here is the argent fifth of 702.94 cents, tuned such that it and its complement are in the logarithmic ratio of the square root of 2, and thus functions similarly to the golden oneirotonic fifth but is reasonably interpretable as 3/2.
• A fifth of exactly 720 cents generates 5edo, and is associated with blackwood temperament and found in many multiples of 5edo.
• Fifths sharper than 720 cents generate oneirotonic scales. Of particular note is the generator of about 741.6 cents, which generates golden oneirotonic and thus creates the "anti-cluster temperament" where all notes at all MOS sizes are as evenly spaced as possible and the step counts of the MOSes create the Fibonacci sequence. Regular temperament interpretations include trienstonic, A-team, and the exotemperament father.

In just intonation

Diminished fifths are covered at Tritone.

3-limit

The definitive 3-limit fifth is 3/2 at 702c, which is covered in its own article. There is also a 3-limit "wolf fifth" of about 678c (actually a diminished sixth) that is flat of 3/2 by a Pythagorean comma and is found in the dodecatonic Pythagorean scale. The 2:3 dyad is the fundamental consonance of the 3-limit.

5-limit

The 5-limit wolf fifth is 40/27 (about 680c), and is flat of 3/2 by a syntonic comma. It appears in the justly tuned zarlino scale.

7-limit

A 7-limit imperfect fifth appears as 32/21 and is about 730 cents. It is, unlike the previous intervals that appear primarily as wolf intervals, rather functionally significant as the septimal superfifth, although its complement the septimal subfourth is more common in fifth-based harmony. It is equated to two 7/4s stacked in slendric temperament.

11-limit

The 11-limit intervals 16/11 (the reduced 11th subharmonic, ~648c) and 22/15 (~663c) serve as very flat fifths or sharp tritones. Intervals in their range do not even generate armotonic; they generate balzano and may be called "zavala" fifths in that context.

Additionally, there is the interval 11/7 of about 782 cents, which is notated as a wolf fifth (specifically, an 11-sharp 7-sharp fifth) in JI notation systems that do not have a Pythagorean comma accidental, despite being closer to a sixth in size.

Higher limits

Just intervals near to the perfect fifth are usually of interest due to serving as more accurate interpretations of a severely detuned 3/2 in certain temperaments. Due to this, the choice thereof is often more independent of prime limit than in the case of other intervals. Of particular note here are 28/19 (which is nearly an optimally tuned trismegistus generator of about 671 cents and allows trismegistus to extend to prime 19), 25/17 (about 668 cents, represented extremely well in 9edo), and 34/23 (about 677 cents) on the flat end, and 20/13 (also serving as an inframinor sixth, about 746 cents), 23/15 (about 740 cents), and 26/17 (about 736 cents).