View source for Perfect fifth

[[File:Perfect fifth on C.png|thumb|Perfect fifth<br />[[File:Perfect fifth on C.mid|thumb|center|Equal tempered]][[File:Just perfect fifth on C.mid|thumb|center|Just]]]]
{{Infobox interval|
       main_interval_name = perfect fifth|
       inverse = [[perfect fourth]]|
       complement = [[perfect fourth]]|
       other_names = diapente|
       abbreviation = P5 |
       semitones = 7 |
       interval_class = 5 |
       just_interval = 3:2|
       cents_equal_temperament = 700|
       cents_24T_equal_temperament = |
       cents_just_intonation = 701.955{{refn|<math>1200 \times \log_2(1.5)</math>}}
}}

[[File:The perfect fifth.gif|thumb|The perfect fifth with two strings]]

{{Image frame|content=<score>  { <<
 \new Staff \with{ \magnifyStaff #4/3 } \relative c' { 
  \key c \major \clef treble \override Score.TimeSignature #'stencil = ##f \time 3/4
   <g' d'> <b fis'> <d, a'>
}
 \new Staff \with{ \magnifyStaff #4/3 } \relative c' { 
  \key c \major \clef bass \override Score.TimeSignature #'stencil = ##f \time 3/4
   <c, g'> <a e'> <f' c'>
} >> } 
</score>|width=|align=|caption=Examples of perfect fifth intervals}}

In [[music theory]], a '''perfect fifth''' is the [[Interval (music)|musical interval]] corresponding to a pair of [[pitch (music)|pitches]] with a frequency ratio of 3:2, or very nearly so.

In [[classical music]] from [[Western culture]], a fifth is the interval from the first to the last of the first five consecutive [[Musical note|note]]s in a [[diatonic scale]].<ref>[[Don Michael Randel]] (2003), "Interval", ''Harvard Dictionary of Music'', fourth edition (Cambridge, Massachusetts: Harvard University Press): p. 413.</ref> The perfect fifth (often abbreviated '''P5''') spans seven [[semitone]]s, while the [[Tritone|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 (music)|harmonic series]] as the interval between the second and third harmonics. In a diatonic scale, the [[dominant (music)|dominant]] note is a perfect fifth above the [[tonic (music)|tonic]] note.

The perfect fifth is more [[consonance and dissonance|consonant]], or stable, than any other interval except the [[unison]] and the [[octave]]. It occurs above the [[root (chord)|root]] of all [[Major chord|major]] and [[Minor chord|minor]] chords (triads) and their [[extended chords|extensions]]. Until the late 19th century, it was often referred to by one of its Greek names, ''diapente''.<ref>{{cite book|title=A Dictionary of Christian Antiquities|author1=William Smith|author1-link=William Smith (lexicographer)|author2=Samuel Cheetham|author2-link=Samuel Cheetham (priest)|location=London|publisher=John Murray|year=1875|page=550|isbn=9780790582290|url=https://books.google.com/books?id=1LIPFk6oFVkC&q=diatessaron+diapason+diapente+fourth+fifth&pg=PA550}}</ref> Its [[Inversion (interval)|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 <span id="term_perfect_anchor" class="anchor"></span>==

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 and dissonance|consonance]].<ref>{{cite book |last1=Piston |first1=Walter |author1-link=Walter Piston |last2=de&#x202f;Voto |first2=Mark |year=1987 |title=Harmony |edition=5th |place=New York, NY |publisher=W.W. Norton |page=15 |isbn=0-393-95480-3 |quote=Octaves, perfect intervals, thirds, and sixths are classified as being 'consonant intervals', but thirds and sixths are qualified as 'imperfect consonances'.}}</ref> 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 interval|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{{Music|sharp}}–E{{Music|flat}}).

Perfect intervals are also defined as those natural intervals whose [[Inversion (interval)|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]], [[perfect fourth|fourth]], fifth, and [[octave]], without appealing to degrees of consonance.<ref>{{cite book | title = Harmony and Analysis | author =Kenneth McPherson Bradley | publisher = C. F. Summy| year = 1908 | page = 17 | url = https://books.google.com/books?id=QsAPAAAAYAAJ&q=intitle:Harmony+perfect-interval&pg=PA16 }}</ref>

The term ''perfect'' has also been used as a synonym of ''[[just interval|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]].<ref>{{cite book|title=Penny Cyclopaedia|author=Charles Knight|author-link=Charles Knight (publisher)|publisher=Society for the Diffusion of Useful Knowledge|year=1843|page=356|url=https://books.google.com/books?id=muBPAAAAMAAJ&q=%22perfect+fifth%22+%22imperfect+fifth%22+tempered&pg=PA356}}</ref><ref>{{cite book | title = Yearning for the Impossible |author=[[John Stillwell]]| publisher = A. K. Peters| year = 2006 | isbn = 1-56881-254-X | page = [https://archive.org/details/yearningforimpos0000stil/page/21 21] | url = https://archive.org/details/yearningforimpos0000stil | url-access = registration | quote = perfect fifth imperfect fifth tempered. }}</ref> The perfect unison has a [[interval ratio|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)<ref>{{cite book | title = Music and Sound | author = Llewelyn Southworth Lloyd | publisher = Ayer Publishing | year = 1970 | isbn = 0-8369-5188-3 | page = 27 | url = https://books.google.com/books?id=LxTwmfDvTr4C&q=%22perfect+third%22++%22perfect+major%22&pg=PA27 }}</ref> or a perfect [[major sixth]] (5:3).<ref>{{cite book | title = Musical Acoustics | author = John Broadhouse | publisher = W. Reeves | year = 1892 | page = [https://archive.org/details/musicalacoustic00broagoog/page/n292 277] | url = https://archive.org/details/musicalacoustic00broagoog | quote = perfect major sixth ratio. }}</ref>

==Other qualities==

In addition to perfect, there are two other kinds, or qualities, of fifths: the [[diminished fifth]], which is one [[semitone|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==
[[File:Just perfect fifth on D.png|thumb|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]] above middle C: (A{{music|natural}}, 5/3).<ref name="Fonville">{{cite journal|author-link=John Fonville|author=John Fonville|title=[[Ben Johnston (composer)|Ben Johnston]]'s Extended Just Intonation: A Guide for Interpreters|pages=109 (106–137)|journal=[[Perspectives of New Music]]|volume=29|issue=2|date=Summer 1991|doi=10.2307/833435 |jstor=833435}}</ref>[[File:Just perfect fifth on D.mid]]]]
[[File:Just perfect fifth below A.png|thumb|Just perfect fifth below A. The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second above middle C: (D{{music|natural}}, 9/8).<ref name="Fonville"/>[[File:Just perfect fifth below A.mid]]]]

The [[Just intonation|justly tuned]] [[interval ratio|pitch ratio]] of a perfect fifth is 3:2 (also known, in early music theory, as a ''[[hemiola]]''),<ref>
{{cite dictionary
 | title = Harvard Dictionary of Music
 | edition = 2nd
 | author = [[Willi Apel]]
 | location=Cambridge, Massachusetts
 | publisher = Harvard University Press
 | year = 1972
 | isbn = 0-674-37501-7 <!--did not originally have isbn-13-->
 | page = 382
 | entry = Hemiola, hemiolia
 | entry-url = https://archive.org/details/harvarddictionar0000apel/page/382
 | entry-url-access = registration
 }}</ref><ref>{{cite dictionary |editor-first=Don Michael |editor-last=Randel|editor-link=Don Michael Randel|date=2003 |entry=Hemiola, hemiola |dictionary=Harvard Dictionary of Music |edition=4th |location=Cambridge, Massachusetts |publisher=Harvard University Press |page=389 |entry-url=https://books.google.com/books?id=02rFSecPhEsC&pg=PA389 |isbn=0-674-01163-5|title=The Harvard Dictionary of Music: Fourth Edition }}</ref> 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 temperament|equal-tempered]] version of the perfect fifth, enabling the instrument to play in all [[Key (music)|keys]]. In 12-tone equal temperament, the frequencies of the tempered perfect fifth are in the ratio <math>(\sqrt [12]{2})^7</math> or approximately 1.498307. An equally tempered perfect fifth, defined as 700 [[Cent (music)|cents]], is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents.

[[Johannes Kepler|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.<ref>{{cite book|title=Harmonies of the World|author=[[Johannes Kepler]]|editor=[[Stephen Hawking]]|publisher=Running Press|year=2004|isbn=0-7624-2018-9|page=22}}</ref> 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 [[beat (acoustics)|beats]] that result from such an "imperfect" tuning.<ref>{{cite book | title = On the Sensations of Tone as a Physiological Basis for the Theory of Music |author=[[Hermann von Helmholtz]]| publisher = Longmans, Green | year = 1912 | url = https://archive.org/details/onsensationston01helmgoog | quote = perfect fifth imperfect fifth Helmholtz tempered| pages = [https://archive.org/details/onsensationston01helmgoog/page/n220 199], 313|isbn=9781419178931 }}</ref>

==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".<ref>W. E. Heathcote (1888), "Introductory Essay", in [[Moritz Hauptmann]], [https://archive.org/details/bub_gb_a8o5AAAAIAAJ <!-- quote="first octave" "then fifth". --> ''The Nature of Harmony and Metre''], translated and edited by W. E. Heathcote (London: Swan Sonnenschein), p. xx.</ref> 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.<ref>{{cite book | title = On the Sensations of Tone as a Physiological Basis for the Theory of Music | publisher = Longmans, Green |author=[[Hermann von Helmholtz]]| year = 1912 | url = https://archive.org/details/onsensationston01helmgoog | quote = perfect fifth imperfect fifth Helmholtz tempered| page = [https://archive.org/details/onsensationston01helmgoog/page/n274 253]| isbn = 9781419178931 }}</ref>

The perfect fifth is a basic element in the construction of major and minor [[triad (music)|triad]]s, and their [[extended chords|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 [[Consonance and dissonance|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 [[trumpet]]s, a [[piccolo trumpet]], and one [[French horn|horn]] play a five-tone B-flat quintal chord.{{sfn|Piston|DeVoto|1987|pp=503–505}}

==Bare fifth, open fifth, or empty fifth==
{{Image frame|content=<score sound="1">
{
  \set Staff.midiInstrument = "electric guitar (clean)"
  \omit Score.MetronomeMark \tempo 4=160
  \repeat unfold 16 { <e b e'>8-. } \bar "|."
}
</score>|caption=E5 power chord in eighth notes}}
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#Pérotin|Viderunt omnes]]'' and ''Sederunt Principes'', [[Guillaume de Machaut]]'s ''[[Messe de Nostre Dame]]'', the [[Kyrie]] in [[Wolfgang Amadeus Mozart|Mozart]]'s ''[[Requiem (Mozart)|Requiem]]'', and the first movement of [[Anton Bruckner|Bruckner]]'s ''[[Symphony No. 9 (Bruckner)|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]], [[heavy metal music|metal]], and [[punk music]], [[distortion (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 chord]]s''. Power chords often include octave doubling (i.e., their bass note is doubled one octave higher, e.g. F3–C4–F4).
[[File:Antara.mid|thumb|100px|''pacha siku'']]
{{stack|[[File:Kantu.mid|thumb|100px|''k'antu'']]}}

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.

Western composers may use the interval to give a passage an exotic flavor.<ref>Scott Miller, "[http://www.newlinetheatre.com/kingandichapter.html Inside ''The King and I'']", ''[[New Line Theatre]]'', accessed December 28, 2012</ref> Empty fifths are also sometimes used to give a [[cadence (music)|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 temperament|meantone]] tuning.{{Citation needed|date=February 2018|reason=The more normal explanation is that just major thirds are the aim, and the tuning divides this interval into equal halves. Who says the fifths are the basis, and not an accidental byproduct?}}

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.

==See also==
* [[All fifths tuning]]

==References==
{{Reflist}}

{{Intervals}}

{{DEFAULTSORT:Perfect Fifth}}
[[Category:Fifths (music)]]
[[Category:Perfect intervals]]
[[Category:3-limit tuning and intervals]]