# Perfect fifth

Perfect fifth   equal tempered and   just.
perfect fifth
Inverse perfect fourth
Name
Other names diapente
Abbreviation P5
Size
Semitones 7
Interval class 5
Just interval 3:2
Cents
Equal temperament 700
24 equal temperament 700
Just intonation 702
Examples of perfect fifth intervals

In classical music from Western culture, a fifth is a musical interval encompassing five staff positions (see Interval number for more details), and the perfect fifth (often abbreviated P5) is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C, and there are five staff positions from C to G. Diminished and augmented fifths span the same number of staff positions, but consist of a different number of semitones (six and eight, respectively).

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.[1] Its inversion is the perfect fourth.

## 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.[2] Note that this interpretation of the term is not in all contexts compatible with the definition of perfect fifth given in the introduction. In fact, 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 dissonant 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 perfect, where natural, as opposed to altered, designates those intervals between a base note and the major diatonic scale starting at that 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.[3]

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.[4][5] 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)[6] or a perfect major sixth (5:3).[7]

## Other qualities of fifth

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.

## The pitch ratio of a fifth

Just perfect fifth on D  . The perfect fifth above D (A+, 27/16) is a syntonic comma (81/80 or 21.5 cents) higher than the just major sixth (A, 5/3).[8]
Just perfect fifth below A  . The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second (D, 9/8).[8]

The justly intoned pitch ratio of a perfect fifth is 3:2 (also known, in early music theory, as a hemiola[9][10]), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the cent system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. 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. Just perfect fifths are employed in just intonation. The 3:2 just perfect fifth arises in the C major scale between C and G.[11]

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.[12] His lower perfect fifth ratio of 1.4815 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.498 (relative to the ideal 1.50). 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.[13]

In keyboard instruments such as the piano, a slightly different version of the perfect fifth is normally used: in accordance with the principle of equal temperament, the perfect fifth is slightly narrowed to exactly 700 cents (seven semitones). (The narrowing is necessary to enable the instrument to play in all keys.) Many people can hear the slight deviation from the idealized perfect fifth when they play the interval on a piano.

## Use in harmony

Moritz Hauptmann describes the octave as a higher unity appearing as such within the triad, produced from the prime unity of first the octave, then fifth, then third, "which is the union of the former."[14] Hermann von Helmholtz argues that some intervals, namely the perfect fourth, fifth, and octave, "are found in all the musical scales known", though the editor of the English translation of his book notes the fourth and fifth may be interchangeable or indeterminate.[15]

The perfect fifth is a basic element in the construction of major and minor triads, and their extensions. Because these chords occur frequently in much music, the perfect fifth occurs just as often. However, since many instruments contain a perfect fifth as an overtone, it is not unusual to omit the fifth of a chord (especially in root position).

The perfect fifth is also present in seventh chords as well as "tall tertian" harmonies (harmonies consisting of more than four tones stacked in thirds above the root). The presence of a perfect fifth can in fact soften the dissonant intervals of these chords, as in the major seventh chord in which the dissonance of a major seventh is softened by the presence of two perfect fifths.

One can also build chords by stacking fifths, yielding quintal harmonies. Such harmonies are present in more modern music, such as the music of Paul Hindemith. This harmony also appears in Stravinsky's The Rite of Spring in the Dance of the Adolescents where four C Trumpets, a Piccolo Trumpet, and one Horn play a five-tone B-flat quintal chord.[16]

## Bare fifth, open fifth, or empty fifth

A bare fifth, open fifth or empty fifth is a chord containing only a perfect fifth with no third. The closing chord of the Kyrie in Mozart's Requiem and of the first movement of Bruckner's Ninth Symphony are both examples of pieces ending on an empty fifth. These "chords" are common in Sacred Harp singing and throughout rock music. In hard rock, metal, and punk music, overdriven or distorted guitar can make thirds sound muddy while the bare fifth remains crisp. In addition, fast chord-based passages are made easier to play by combining the four most common guitar hand shapes into one. Rock musicians refer to them as power chords and often include octave doubling (i.e., their bass note is doubled one octave higher, e.g. F3-C4-F4).

An empty fifth is sometimes used in traditional music, e.g., in Asian music and in some Andean music genres of pre-Columbian origin, such as k'antu and sikuri. The same melody is being led by parallel fifths and octaves during all the piece. Hear examples: , .

Western composers may use the interval to give a passage an exotic flavor.[17]

## Use in tuning and tonal systems

A perfect fifth in just intonation, a just fifth, corresponds to a frequency ratio of 3:2, while in 12-tone equally-tempered fifth, the frequencies are in the ratio $(\sqrt [12]{2})^7$, approximately 1.498307. Measured against an equally-tempered scale, where each semitone spans 100 cents, a just fifth is approximately 701.955001 cents, about two cents larger than the equally tempered fifth.

The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. A flattened perfect fifth is likewise the basis for meantone tuning.

The circle of fifths is a model of pitch space for the chromatic scale (chromatic circle), which considers nearness as the number of perfect fifths required to get from one note to another, rather than chromatic adjacency.

## References

1. ^ William Smith and Samuel Cheetham (1875). A Dictionary of Christian Antiquities. London: John Murray. p. 550.
2. ^ Walter Piston and Mark DeVoto (1987), Harmony, 5th ed. (New York: W. W. Norton), p. 15. ISBN 0-393-95480-3. Octaves, perfect intervals, thirds, and sixths are classified as being "consonant intervals", but thirds and sixths are qualified as "imperfect consonances".
3. ^ Kenneth McPherson Bradley (1908). Harmony and Analysis. C. F. Summy Co. p. 17.
4. ^ Charles Knight (1843). Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge. Society for the Diffusion of Useful Knowledge. p. 356.
5. ^ John Stillwell (2006). Yearning for the Impossible. A K Peters, Ltd. p. 21. ISBN 1-56881-254-X.
6. ^ Llewelyn Southworth Lloyd (1970). Music and Sound. Ayer Publishing. p. 27. ISBN 0-8369-5188-3.
7. ^ John Broadhouse (1892). Musical Acoustics. W. Reeves. p. 277.
8. ^ a b Fonville, John. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.109, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.
9. ^ Willi Apel (1969). Harvard dictionary of music (2nd ed.). Harvard University Press. p. 382. ISBN 978-0-674-37501-7.
10. ^ Don Michael Randel (ed.), New Harvard Dictionary of Music (Cambridge, MA: Belknap Press of Harvard University Press, 1986), p. 376.
11. ^ Paul, Oscar (1885). A manual of harmony for use in music-schools and seminaries and for self-instruction, p.165. Theodore Baker, trans. G. Schirmer.
12. ^ Johannes Kepler (2004). Stephen W. Hawking, ed. Harmonies of the World. Running Press. p. 22. ISBN 0-7624-2018-9.
13. ^ Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green. pp. 199, 313.
14. ^ Hauptmann, Moritz (1888). The nature of harmony and metre, Volume 1888, Part 1, p.xx.
15. ^ Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. p. 253.
16. ^ Piston and DeVoto (1987), p. 503–505.
17. ^ Scott Miller, "Inside The King and I", New Line Theatre, accessed December 28, 2012