Perfect fifth

Last updated
 Inverse perfect fourth diapente P5 7 5 3:2 700 701.955

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

Contents

In classical music from Western culture, a fifth is the interval from the first to the last of five consecutive notes in a diatonic scale. [1] The perfect fifth (often abbreviated P5) spans seven semitones, while the diminished fifth spans six and the augmented fifth spans eight semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C.

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. [2] 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 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. [3] 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 perfect, 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. [4]

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. [5] [6] 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) [7] or a perfect major sixth (5:3). [8]

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 ), [10] [11] 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 ${\displaystyle ({\sqrt[{12}]{2}})^{7}}$ 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. [12] 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). 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]

Use in harmony

W. E. Heathcote describes the octave as representing the prime unity within the triad, a higher unity produced from the successive process: "first Octave, then Fifth, then Third, which is the union of the two 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.

Chords can also be built 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 chords of Pérotin's Viderunt Omnes and Sederunt Principes, Guillaume de Machaut's Messe de Nostre Dame , the Kyrie in Mozart's Requiem , and the first movement of Bruckner's Ninth Symphony are all examples of pieces ending on an open fifth. These chords are common in Medieval music, sacred harp singing, and throughout rock music. In hard rock, metal, and punk music, overdriven or distorted electric guitar can make thirds sound muddy while the bare fifths remain 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 . Power chords 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. Examples: , .

Western composers may use the interval to give a passage an exotic flavor. [17] Empty fifths are also sometimes used to give a cadence an ambiguous quality, as the bare fifth does not indicate a major or minor tonality.

Use in tuning and tonal systems

The just perfect fifth, together with the octave, forms the basis of Pythagorean tuning. A slightly narrowed perfect fifth is likewise the basis for meantone tuning.[ citation needed ]

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.

Related Research Articles

An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals consist of members of a single harmonic series of a (lower) implied fundamental. For example, in the diagram, the notes G and middle C are both members of the harmonic series of the lowest C and their frequencies will be 3 and 4 times, respectively, the fundamental frequency; thus, their interval ratio will be 4:3. If the frequency of the fundamental is 50 Hertz, the frequencies of the two notes in question would be 150 and 200.

In music, there are two common meanings for tuning:

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

In music theory, an interval is the difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. According to this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. A tritone is also commonly defined as an interval spanning six semitones. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B and B–F. In twelve-equal temperament, the tritone divides the octave exactly in half as 6 of 12 semitones or 600 of 1200 cents.

A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth is the fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, because the note F is the fifth semitone above C, and there are four staff positions between C and F. Diminished and augmented fourths span the same number of staff positions, but consist of a different number of semitones.

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments.

The Bohlen–Pierce scale is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal tempered diatonic scale.

A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C; the interval between them is a semitone.

In classical music, a third is a musical interval encompassing three staff positions, and the major third is a third spanning four semitones. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as major because it is the larger of the two: the major third spans four semitones, the minor third three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees of a major scale are called major.

In music theory a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called minor because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

The twelfth root of two or is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, consonance is typically associated with sweetness, pleasantness, and acceptability; dissonance is associated with harshness, unpleasantness, or unacceptability, although this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. As Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.

Quarter-comma meantone, or 14-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80), with respect to its just intonation used in Pythagorean tuning ; the result is , or a fifth of cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

Twelve-tone equal temperament is the musical system that divides the octave into 12 parts, all of which are equally-tempered (equally-spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, ​112 the width of an octave, is called a semitone or half step.

Music theory has no axiomatic foundation in modern mathematics, although some interesting work has recently been done in this direction, yet the basis of musical sound can be described mathematically and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.

Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.

References

1. Don Michael Randel (2003), "Interval", Harvard Dictionary of Music, fourth edition (Cambridge, MA: Harvard University Press): p. 413.
2. William Smith and Samuel Cheetham (1875). A Dictionary of Christian Antiquities. London: John Murray. p. 550. ISBN   9780790582290.
3. 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".
4. Kenneth McPherson Bradley (1908). Harmony and Analysis. C. F. Summy Co. p. 17.
5. Charles Knight (1843). Penny Cyclopaedia of the Society for the Diffusion of Useful Knowledge. Society for the Diffusion of Useful Knowledge. p. 356.
6. John Stillwell (2006). . A K Peters, Ltd. p.  21. ISBN   1-56881-254-X. perfect fifth imperfect fifth tempered.
7. Llewelyn Southworth Lloyd (1970). Music and Sound. Ayer Publishing. p. 27. ISBN   0-8369-5188-3.
8. John Broadhouse (1892). Musical Acoustics. W. Reeves. p.  277. perfect major sixth ratio.
9. John Fonville (Summer 1991). "Ben Johnston's Extended Just Intonation: A Guide for Interpreters". Perspectives of New Music. 29 (2): 109. JSTOR   833435.
10. Willi Apel (1972). . Harvard Dictionary of Music (2nd ed.). Cambridge, MA: Harvard University Press. p. 382. ISBN   0-674-37501-7.
11. Randel, Don Michael, ed. (2003). "Hemiola, hemiola". Harvard Dictionary of Music (4th ed.). Cambridge, MA: Harvard University Press. p. 389. ISBN   0-674-01163-5.
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. perfect fifth imperfect fifth Helmholtz tempered.
14. W. E. Heathcote (1888), "Introductory Essay"", in Moritz Hauptmann, The Nature of Harmony and Metre , translated and edited by W. E. Heathcote (London: Swan Sonnenschein & Co.), p.xx.
15. Hermann von Helmholtz (1912). On the Sensations of Tone as a Physiological Basis for the Theory of Music. Longmans, Green. p.  253. perfect fifth imperfect fifth Helmholtz tempered.
16. Piston and DeVoto (1987), pp. 503–505.
17. Scott Miller, "Inside The King and I", New Line Theatre, accessed December 28, 2012