Perfect fifth

In Western tonal music, a perfect fifth is the musical interval between two notes separated by seven semitones, usually three whole tones and a semitone; it is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Depending on the tuning system in use, it is likely to be tuned to a frequency ratio of exactly or approximately 3:2. The perfect fifth (often abbreviated P5) spans seven semitones, (e.g., in C major, C to G); other fifths in diatonic music are the diminished fifth, of six semitones, (e.g., in C major, B to F); and the augmented fifth, of eight semitones, , (e.g., in C melodic minor ascending, E♭ to B).

The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth. A perfect fifth is at the start of "Twinkle, Twinkle, Little Star"; the pitch of the first "twinkle" is the root note and the pitch of the second "twinkle" is a perfect fifth above it.

Alternative definitions The term perfect identifies the perfect fifth as belonging to the group of perfect intervals (including the unison, perfect fourth, and octave), so called because of their simple pitch relationships and their high degree of consonance. When an instrument with only twelve notes to an octave (such as the piano) is tuned using Pythagorean tuning, one of the twelve fifths (the wolf fifth) sounds severely discordant and can hardly be qualified as "perfect", if this term is interpreted as "highly consonant". However, when using correct enharmonic spelling, the wolf fifth in Pythagorean tuning or meantone temperament is actually not a perfect fifth but a diminished sixth (for instance G♯–E♭). Perfect intervals are also defined as those natural intervals whose inversions are also natural, where natural, as opposed to altered, designates those intervals between a base note and another note in the major diatonic scale starting at that base note (for example, the intervals from C to C, D, E, F, G, A, B, C, with no sharps or flats); this definition leads to the perfect intervals being only the unison, fourth, fifth, and octave, without appealing to degrees of consonance. The term perfect has also been used as a synonym of just, to distinguish intervals tuned to ratios of small integers from those that are "tempered" or "imperfect" in various other tuning systems, such as equal temperament. The perfect unison has a pitch ratio 1:1, the perfect octave 2:1, the perfect fourth 4:3, and the perfect fifth 3:2. Within this definition, other intervals may also be called perfect, for example a perfect third (5:4) or a perfect major sixth (5:3).

Other qualities In addition to perfect, there are two other kinds, or qualities, of fifths: the diminished fifth, which is one chromatic semitone smaller, and the augmented fifth, which is one chromatic semitone larger. In terms of semitones, these are equivalent to the tritone (or augmented fourth), and the minor sixth, respectively.

Pitch ratio

The justly tuned pitch ratio of a perfect fifth is 3:2 (also known, in early music theory, as a hemiola), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune. Keyboard instruments such as the piano normally use an equal-tempered version of the perfect fifth, enabling the instrument to play in all keys. In 12-tone equal temperament, the frequencies of the tempered perfect fifth are in the ratio ( 2 12 ) 7 {\displaystyle ({\sqrt[{12}]{2}})^{7}} or approximately 1.498307. An equally tempered perfect fifth, defined as 700 cents, is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents. Kepler explored musical tuning in terms of integer ratios, and defined a "lower imperfect fifth" as a 40:27 pitch ratio, and a "greater imperfect fifth" as a 243:160 pitch ratio. His lower perfect fifth ratio of 1.48148 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.4983 (relative to the ideal 1.50). Hermann von Helmholtz uses the ratio 301:200 (708 cents) as an example of an imperfect fifth; he contrasts the ratio of a fifth in equal temperament (700 cents) with a "perfect fifth" (3:2), and discusses the audibility of the beats that result from such an "imperfect" tuning.

Use in harmony

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