Perfect fifth
The perfect fifth (P5) is an interval that generally represents the frequency ratio 3/2. As 3/2 is the simplest ratio within the span of the octave, it is considered the most consonant octave-reduced interval after the octave itself, and some form of a perfect fifth is used in nearly all world musical tuning systems. It is the simplest octave-reduced prime harmonic, and is also a superparticular interval. The note a perfect fifth above the root serves as an important structural anchor for scales, similarly to the perfect fourth, and the perfect fifth serves as the bounding interval of the most common triads in conventional systems of harmony.
Diatonic scales
Most commonly, the term "perfect fifth" refers to the diatonic perfect fifth, that is, a generator of the MOS diatonic scale (5L 2s). As an interval in the abstract diatonic scale, it may range from 685.7 to 720 cents, depending on the tuning; it contrasts with the diatonic diminished fifth, a dissonant "tritone" interval, and with a broader melodic category of "fifths". Specifically, the diatonic scale contains six perfect fifths; in the Ionian mode, perfect fifths are found on all but the 7th degree of the scale, which instead has a diminished fifth. Note that not all diatonic perfect fifths necessarily represent 3/2, especially in larger EDOs where multiple diatonic fifths can be found.
Chains of fifths
The purely tuned interval 3/2 is approximately 702 cents in size, and the structure generated by it against the octave is known as Pythagorean tuning, or otherwise as the 3-limit. Due to the perfect fifth being foundational to so many musical structures, and especially due to the prevalence of diatonic, the tuning of 3/2 is generally one of the most important structural aspects of a tuning system. A common conceptualization of the set of intervals is in terms of a chain of fifths, and many notation systems designed for just intonation rely on defining "formal commas" to offset intervals of higher primes from a Pythagorean spine.
In temperaments
Further, the ploidacot system is a method of classifying temperaments and generated structures based on the position of the perfect fifth in that structure. Monocot temperaments are those which use the perfect fifth as a generator, of which there are many important ones, such as Meantone, Superpyth, and Schismic.
EDOs that approximate 3/2 well, in order of increasing accuracy, include 5, 7, 12, 29, 41, and 53edo.
Equal division
As 3/2 is such a profoundly concordant and simple interval, the idea of treating 3/2 as an equave has occasionally been advanced, leading to the notion of equal divisions of the fifth (EDFs) analogous to EDOs, and even to octaveless rank-2 structures that repeat at the perfect fifth. The most well-known EDFs are 9, 11, and 20 divisions of the fifth, which have the property of accurately representing 5/4 and 6/5; these are commonly referred to respectively as Carlos Alpha, Beta, and Gamma, though the original specifications for the Carlos scales included octaves for the purpose of transposition and slightly detuned the fifth and thus they strictly speaking are closer to rank-2 temperaments that split the fifth.
Regardless of the theoretical notion of fifth-equivalence, structures that divide the perfect fifth into a number of equal parts are extremely common throughout xenharmonic tuning systems - such as Rastmic (2 parts), Slendric (3 parts), Tetracot (4 parts), Miracle (6 parts), and Valentine (9 parts, related to the aforementioned Carlos Alpha) - and so the divisibility of the tuning of the fifth in an EDO hence can determine a lot about the harmonies that can be found in that EDO.
Tuning range
| Range (in cents) | Description | Notable edos |
|---|---|---|
| 685.7-720 | Range of fifths capable of generating a diatonic scale. | 4\7 = 685.7c, 7\12 = 700c, 3\5 = 720c |
| 666.7-685.7 | Range of intervals that generate a soft antidiatonic scale and may be considered fifths. | 9\16 = 675c |
| ~678-693 | Range of intervals that generate an equiheptatonic diatonic or antidiatonic scale. | 13\23 = 678.3c, 4\7 = 685.7c, 15\26 = 692.3c |
| 691.5-694.7 | Range of viable generators of Flattone temperament. | 26\45 = 693.3c, 15\26 = 692.3c |
| 694.7-700 | Range of viable generators of Septimal Meantone temperament. | 11\19 = 694.7c, 18\31 = 696.8c, 29\50 = 696c, 7\12 = 700c |
| 698.5-705.5 | Pythagorean tuning; fifths within a just noticeable difference of just intonation. Contains schismic tunings. | 7\12 = 700c; 24\41 = 702.4c; 31\53 = 701.9c |
| 703.4-705.9 | Range of "gentle" fifths, which generate neogothic intervals. | 17\29 = 703.4c; 10\17 = 705.9c |
| ~709-711 | Range of viable generators of Superpyth temperament. | 13\22 = 709.1c; 16\27 = 711.1c |
| ~711-715 | Range of viable generators of other Archy temperaments. | 22\37 = 713.5c |
| ~709-730 | Range of intervals that generate an equipentatonic pentic or antipentic scale. | 22\37 = 713.5c, 3\5 = 720c, 23\38 = 726.3c |
