Perfect fifth

Perfect fifth  Play (help·info) equal tempered and  Play (help·info) 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 (music)#Number for more details), and the perfect fifth 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).

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. Up until the late 19th century it was often referred to by its Greek name, diapente,^[1] and abbreviated P5. 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 is tuned using Pythagorean tuning or meantone temperament, 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  Play (help·info). 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  Play (help·info). The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second (D♮, 9/8).^[8]

The idealized pitch ratio of a perfect fifth is 3:2, 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. Something close to the idealized 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 is felt to be "in tune". Idealized perfect fifths are employed in just intonation.

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.^[9] 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.^[10]

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

It was the first accepted harmony (besides the octave) in Gregorian chant, a very early formal style of musical composition.^{[citation needed]} 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".^[11] Hermann von Helmholtz argues that some intervals, namely the perfect fourth, fifth, and octave, "are found in all the musical scales known." though his editor notes the fourth and fifth may be interchangeable or indeterminate.^[12]

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.^[13]

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 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:  Play K'antu (help·info),  Play Pacha Siku (help·info).

Use in tuning and tonal systems

A perfect fifth in just intonation, a just fifth, corresponds to a frequency ratio of 3:2 (702 cents); while in 12-tone equal temperament, where each semitone spans 100 cents, a perfect fifth is equal to 700 cents, about two cents smaller than the just 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 not as adjacency but as the number of perfect fifths required to get from one note to another.

References

1. ^ William Smith and Samuel Cheetham (1875). A Dictionary of Christian Antiquities. London: John Murray. p. 550. http://books.google.com/books?id=1LIPFk6oFVkC&pg=PA550&dq=diatessaron+diapason+diapente+fourth+fifth. 
2. ^ Walter Piston and Mark DeVoto (1987), Harmony, 5th ed. (New York: W. W. Norton), p. 15. ISBN 0393954803. 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. http://books.google.com/books?id=QsAPAAAAYAAJ&pg=PA16&dq=intitle:Harmony+perfect-interval#PPA17,M1. 
4. ^ Charles Knight (1843). Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge. Society for the Diffusion of Useful Knowledge. p. 356. http://books.google.com/books?id=muBPAAAAMAAJ&pg=PA356&dq=%22perfect+fifth%22+%22imperfect+fifth%22+tempered. 
5. ^ John Stillwell (2006). Yearning for the Impossible. A K Peters, Ltd.. p. 21. ISBN 156881254X. http://books.google.com/books?id=YonsAAWIA7sC&pg=PA21&dq=%22perfect+fifth%22+%22imperfect+fifth%22+tempered. 
6. ^ Llewelyn Southworth Lloyd (1970). Music and Sound. Ayer Publishing. p. 27. ISBN 0836951883. http://books.google.com/books?id=LxTwmfDvTr4C&pg=PA27&dq=%22perfect+third%22++%22perfect+major%22. 
7. ^ John Broadhouse (1892). Musical Acoustics. W. Reeves. p. 277. http://books.google.com/books?id=l9c5AAAAIAAJ&pg=PA277&dq=%22perfect+major+sixth%22+ratio. 
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. ^ Johannes Kepler (2004). Stephen W. Hawking. ed. Harmonies of the World. Running Press. p. 22. ISBN 0762420189. http://books.google.com/books?id=br1sKvZPIKQC&pg=PA22&dq=%22perfect+fifth%22+%22imperfect+fifth%22. 
10. ^ Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green. pp. 199, 313. http://books.google.com/books?id=po6fAAAAMAAJ&dq=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor%3AHelmholtz%20tempered&pg=PA199#v=onepage&q=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor:Helmholtz%20tempered&f=false. 
11. ^ Hauptmann, Moritz (1888). The nature of harmony and metre, Volume 1888, Part 1, p.xx.
12. ^ Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. pp. 253. http://books.google.com/books?id=po6fAAAAMAAJ&dq=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor%3AHelmholtz%20tempered&pg=PA199#v=onepage&q=%22perfect%20fifth%22%20%22imperfect%20fifth%22%20inauthor:Helmholtz%20tempered&f=false. 
13. ^ Piston and DeVoto (1987), p. 503–505.

See also

• All fifths
• Circle of fifths