Editing Quartal and quintal harmony

{{short description|Types of harmonic structures in music}}
{{redirects|Fourth chord||Eleventh chord#Fourth}}
In [[music]], '''quartal harmony''' is the building of [[Harmony|harmonic]] structures built from the [[Interval (music)|intervals]] of the [[perfect fourth]], the [[tritone|augmented fourth]] and the [[diminished fourth]]. For instance, a three-note quartal chord on C can be built by stacking perfect fourths, C–F–B{{music|flat}}. 

:<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' { 
  \clef treble \time 4/4 <c f bes>1
} }
</score>

'''Quintal harmony''' is harmonic structure preferring the [[perfect fifth]], the [[augmented fifth]] and the [[tritone|diminished fifth]]. For instance, a three-note quintal chord on C can be built by stacking perfect fifths, C–G–D. 

:<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' { 
  \clef treble \time 4/4 <c g' d'>1
} }
</score>

== Properties ==
{{Image frame|content=<score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' { 
  \clef treble \time 4/4 <a d g c f bes es>1 <bes d f a c es g>
} }
</score>|width=300|caption=Notes in a quartal chord on A can be arranged to form a [[thirteenth|thirteenth chord]] on B{{music|b}}.{{sfn|Benward and Saker|2009|loc=279}}}}
The terms ''quartal'' and ''quintal'' imply a contrast, either compositional or perceptual, with traditional harmonic constructions based on thirds: listeners familiar with music of the [[common practice period]] are guided by [[tonalities]] constructed with familiar elements: the chords that make up major and minor scales, all in turn built from [[major third|major]] and [[minor third]]s.

Regarding chords built from perfect fourths alone, composer [[Vincent Persichetti]] writes that:
{{quote|Chords by perfect fourth are ambiguous in that, like all chords built by equidistant intervals ([[Diminished seventh chord|diminished seventh chords]] or [[Augmented triad|augmented triads]]), any member can function as the [[Root (chord)|root]]. The indifference of this rootless harmony to tonality places the burden of [[Key (music)|key]] verification upon the [[Part (music)|voice]] with the most active melodic line.{{sfn|Persichetti|1961|loc=94}}|sign=|source=}}

''Quintal harmony'' (the harmonic layering of fifths specifically) is a lesser-used term, and since the fifth is the [[Inversion (interval)|inversion]] or [[Complement (music)|complement]] of the fourth, it is usually considered indistinct from ''quartal harmony''. Because of this relationship, any quartal chord can be rewritten as a quintal chord by changing the order of its pitches.

Like tertian chords, a given quartal or quintal chord can be written with different [[Voicing (music)|voicings]], some of which obscure its quartal structure. For instance, the quartal chord, C–F–B{{music|flat}}, can be written as
<blockquote><score sound="1">
{
\override Score.TimeSignature #'stencil = ##f
\relative c' { 
  \clef treble \time 4/4 <c f bes>1 <f bes c> <bes c f> <f bes c f>
} }
</score></blockquote>

==History==
In the [[Medieval music|Middle Ages]], simultaneous notes a fourth apart were heard as a consonance.  During the [[common practice period]] (between about 1600 and 1900), this interval came to be heard either as a [[Consonance and dissonance|dissonance]] (when appearing as a [[Suspension (music)|suspension]] requiring resolution in the [[voice leading]]) or as a [[Consonance and dissonance|consonance]] (when the root of the chord appears in parts higher than the fifth of the chord).  In the later 19th century, during the [[Emancipation of the dissonance|breakdown of tonality]] in [[classical music]], all intervallic relationships were once again reassessed.  Quartal harmony was developed in the early [[20th-century classical music|20th century]] as a result of this breakdown and reevaluation of tonality.

===Precursors===
The [[Tristan chord]] is made up of the notes F{{music|natural}}, B{{music|natural}}, D{{music|sharp}} and G{{music|sharp}} and is the first chord heard in [[Richard Wagner|Wagner]]'s [[opera]] ''[[Tristan und Isolde]]''.  

:<score override_ogg="Wagner Tristan opening (orchestral).ogg">
    {
      \new PianoStaff <<
        \new Staff <<
            \new Voice \relative c'' {
                \clef treble \key a \minor \time 6/8
                \voiceOne \partial8 r8 R2. \once \override NoteHead.color = #red gis4.->(~ gis4 a8 ais8-> b4~ b8) r r
                }
            \new Voice \relative c' {
                \override DynamicLineSpanner.staff-padding = #4.5
                \once \override DynamicText.X-offset = #-5
                \voiceTwo \partial8 a\pp( f'4.~\< f4 e8 \once \override NoteHead.color = #red dis2.)(\> d!4.)~\p d8 r r
                }
            >>
        \new Staff <<
            \relative c {
                \clef bass \key a \minor \time 6/8
                \partial8 r8 R2. \once \override NoteHead.color = #red <f b>2.( <e gis>4.)~ <e gis>8 r r
                }
            >>
    >> }
</score>

The bottom two notes make up an augmented fourth, while the upper two make up a perfect fourth. This layering of fourths in this context has been seen as highly significant.  The chord had been found in earlier works,{{sfn|Vogel|1962|loc=12}} notably [[Ludwig van Beethoven|Beethoven]]'s [[Piano Sonata No. 18 (Beethoven)|Piano Sonata No. 18]], but Wagner's use was significant, first because it is seen as moving away from traditional [[tonal harmony]] and even towards [[atonality]], and second because with this chord Wagner actually provoked the sound or structure of musical harmony to become more predominant than its [[Functional harmony|function]], a notion which was soon after to be explored by [[Claude Debussy|Debussy]] and others.{{sfn|Erickson|1975|p={{Page needed|date=June 2012}}}}  

Despite the layering of fourths, it is rare to find musicologists identifying this chord as "quartal harmony" or even as "proto-quartal harmony", since Wagner's musical language is still essentially built on thirds, and even an ordinary [[dominant seventh]] chord can be laid out as augmented fourth plus perfect fourth (F–B–D–G). Wagner's unusual chord is really a device to draw the listener into the musical-dramatic argument which the composer is presenting to us.

At the beginning of the 20th century, quartal harmony finally became an important element of harmony. [[Alexander Scriabin|Scriabin]] used a self-developed system of transposition using fourth-chords, like his [[Mystic chord]] (shown below) in his [[Piano Sonata No. 6 (Scriabin)|Piano Sonata No. 6]]. 

:<score sound="1"> {
\override Score.TimeSignature #'stencil = ##f
\relative c' { 
  \clef treble \time 4/4 <c fis bes e a d>1
} }
</score>

Scriabin wrote this chord in his sketches alongside other quartal passages and more traditional [[tertian]] passages, often passing between systems, for example widening the six-note quartal sonority (C–F{{music|sharp}}–B{{music|flat}}–E–A–D) into a seven-note chord (C–F{{music|sharp}}–B{{music|flat}}–E–A–D–G). Scriabin's sketches for his unfinished work ''[[Mysterium (Scriabin)|Mysterium]]'' show that he intended to develop the Mystic chord into a huge chord incorporating all twelve notes of the [[chromatic scale]].{{sfn|Morrison|1998|loc=316}}

In France, [[Erik Satie]] experimented with [[Parallel harmony|planing]] in the stacked fourths (not all perfect) of his 1891 score for ''[[Le Fils des étoiles]]''.{{sfn|Solomon|2003}} [[Paul Dukas]]'s ''[[The Sorcerer's Apprentice (Dukas)|The Sorcerer's Apprentice]]'' (1897) has a rising repetition in fourths, as the tireless work of out-of-control walking brooms causes the water level in the house to "rise and rise".

===20th- and 21st-century classical music===
Composers who use the techniques of quartal harmony include [[Claude Debussy]], [[Francis Poulenc]], [[Alexander Scriabin]], [[Alban Berg]], [[Leonard Bernstein]], [[Arnold Schoenberg]], [[Igor Stravinsky]], [[Maurice Ravel]], [[Joe Hisaishi]] and [[Anton Webern]].{{sfn|Herder|1987|loc=78}}

==== Schoenberg ====
[[Arnold Schoenberg]]'s [[Chamber Symphony No. 1|Chamber Symphony Op. 9]] (1906) displays quartal harmony: the first measure and a half construct a five-part fourth chord with the notes (highlighted in red in the illustration) A–D{{music|sharp}}–F–B{{music|flat}}–E{{music|flat}}–A{{music|flat}} distributed over the five stringed instruments (the viola must tune down the lowest string by a minor third, and read in the unfamiliar tenor clef).

[[File:SchoenbergOp9.png|thumb|center|upright=1.4|Vertical quartal-harmony in the string parts of the opening measures of Arnold Schoenberg's Chamber Symphony Op. 9[[File:SchoenbergOp9.mid]]]]

[[File:Schönberg Kammersymhonie 9 for wikipedia.png|thumb|upright=1.25|Six-note horizontal fourth chord in Arnold Schoenberg's Chamber Symphony Op. 9|alt=]]

The composer then picks out this vertical quartal harmony in a horizontal sequence of fourths from the horns, eventually leading to a passage of triadic quartal harmony (i.e., chords of three notes, each layer a fourth apart).{{citation needed|date=September 2012}}

Schoenberg was also one of the first to write on the theoretical consequences of this harmonic innovation. In his ''Theory of Harmony'' (''Harmonielehre'') of 1911, he wrote:
{{quote|The construction of chords by superimposing fourths can lead to a chord that contains all the twelve notes of the [[chromatic scale]]; hence, such construction does manifest a possibility for dealing systematically with those harmonic phenomena that already exist in the works of some of us: seven, eight, nine, ten, eleven, and twelve-part chords… But the quartal construction makes possible, as I said, accommodation of all phenomena of harmony.{{sfn|Schoenberg|1978|loc=406–407}}}}

For [[Anton Webern]], the importance of quartal harmony lay in the possibility of building new sounds. After hearing Schoenberg's Chamber Symphony, Webern wrote "You must write something like that, too!"{{sfn|Webern|1963|loc=48}}

==== Others ====
In his ''Theory of Harmony'':{{sfn|Schoenberg|1978|loc=407}} "Besides myself my students Dr. Anton Webern and [[Alban Berg]] have written these harmonies (fourth chords), but also the Hungarian [[Béla Bartók]] or the Viennese [[Franz Schreker]], who both go a similar way to Debussy, Dukas and perhaps also [[Giacomo Puccini|Puccini]], are not far off."<!--- ("Außer mir haben meine Schüler Dr. Anton Webern und Alban Berg solche Klänge [gemeint sind Quartenklänge] geschrieben. Aber auch der Ungar Béla Bartók oder der Wiener Franz Schreker, die beide einen ähnlichen Weg gehen wie Debussy, Dukas und vielleicht auch Puccini, sind wohl nicht weit davon entfernt.")--->

[[File:Bartok Mikrokosmos Quartes for wikipedia.png|thumb|center|upright=1.8|Fourths in Béla Bartók's ''[[Mikrokosmos (Béla Bartók)|Mikrokosmos V]]'', No. 131, ''Fourths'' (''Quartes'')[[File:Bartok Mikrokosmos Quartes for wikipedia.mid]]]]
French composer [[Maurice Ravel]] used quartal chords in [[Sonatine (Ravel)|Sonatine]] (1906) and ''[[Ma mère l'Oye]]'' (1910), while American [[Charles Ives]] used quartal chords in his song "The Cage" (1906).
{| align="center"
|[[File:Ravel Ma Mere l'Oye Laideronnette.PNG|thumb|center|upright=1.65|Quartal harmony in "Laideronnette" from Ravel's ''[[Ma mère l'Oye]]''. The top line uses the [[pentatonic scale]]{{sfn|Benward and Saker|2009|loc=37}}[[File:Ravel Ma Mere l'Oye Laideronnette.mid]]]]
|[[File:Ives - 114 Songs, The Cage introduction.png|thumb|center|upright=1.4|Introduction to Charles Ives's "The Cage" from ''114 Songs''{{sfn|Reisberg|1975|loc=345}}[[File:Ives - 114 Songs, The Cage introduction.mid]]]]
|}
Hindemith constructed large parts of his symphonic work ''[[Symphony: Mathis der Maler]]'' by means of fourth and fifth intervals. These steps are a restructuring of fourth chords (C–D–G becomes the fourth chord D–G–C), or other mixtures of fourths and fifths (D{{music|sharp}}–A{{music|sharp}}–D{{music|sharp}}–G{{music|sharp}}–C{{music|sharp}} in measure 3 of the example).

Hindemith was, however, not a proponent of an explicit quartal harmony. In his 1937 writing ''Unterweisung im Tonsatz'' (''The Craft of Musical Composition'',{{harvnb|Hindemith|1937}}) he wrote that "notes have a family of relationships, that are the bindings of tonality, in which the ranking of intervals is unambiguous,"<!--- ("dass die Töne eine Familienzugehörigkeit besitzen, die sich in der Bindung an tonale Haupttöne äußert, die eine unzweideutige Rangliste der Tonverwandschaften aufstellt.")---> so much so, indeed, that in the art of triadic composition "...the musician is bound by this, as the painter to his primary colours, the architect to the three dimensions."<!--- ("... der Musiker ist an ihn gebunden, wie der Maler an die primären Farben, der Architekt an die drei Dimensionen.")---> He lined up the harmonic and melodic aspects of music in a row in which the octave ranks first, then the fifth and the third, and then the fourth. "The strongest and most unique harmonic interval after the octave is the fifth, the prettiest nevertheless is the third by right of the chordal effects of its [[Combination tone]]s."<!--- ("Das stärkste und eindeutige harmonische Intervall ist nächst der alleinstehenden  Oktave die Quinte, das schönste jedoch die Terz wegen ihrer in den Kombinationstönen begründeten Akkordwirkung.")--->

[[File:Hindemith, Flute Sonata, II quartal harmony.png|thumb|center|upright=1.8|Quartal harmony in Hindemith's Flute Sonata, II with tonal center on B established by descent in left hand in [[Dorian mode|Dorian]] and repeated B's and F{{music|sharp}}'s{{sfn|Kostka|Payne|Almén|2013|loc=Chapter 26: Materials and techniques, Chord structures, Quartal and secundal harmony, 469–470<!--  Meaningful subheadings simplify reference in translations and other editions (p. 498 in 3rd ed.) -->}}[[File:Hindemith, Flute Sonata, II quartal harmony.mid]]]]
The works of the Filipino composer {{ill|Eliseo M. Pajaro|it|Eliseo Pajaro|nl|Eliseo Pajaro|tl|Eliseo M. Pajaro}} (1915–1984) are characterised by quartal and quintal harmonies, as well as by dissonant counterpoint and polychords.{{sfn|Kasilag|2001}}

As a transition to the history of jazz, [[George Gershwin]] may be mentioned. In the first movement of his [[Concerto in F (Gershwin)|Concerto in F]] altered fourth chords descend chromatically in the right hand with a chromatic scale leading upward in the left hand.

===Jazz===
{{listen|filename=Adam Cuerden - Edinburgh in August.mp3|title="Edinburgh in August"|description=In [[Adam Cuerden]]'s "Edinburgh in August", the opening section is made up entirely of quartal chords, which also appear frequently in the piece.}}
Jazz is often understood as a synthesis of the European common practice harmonic vocabulary with textural paradigms from West African folk music—but it would be an oversimplification to describe jazz as sharing the same fundamental theory of harmony as European music. Important influences come from [[opera]] as well as from the instrumental work of Classical- and Romantic-era composers, and even that of the Impressionists. From the beginning, jazz musicians expressed a particular interest in rich harmonic colours, for which non-tertiary harmony was a means of exploration, as used by pianists and [[arranger]]s like [[Jelly Roll Morton]], [[Duke Ellington]], [[Art Tatum]], [[Bill Evans]],{{sfn|Hester|2000|loc=199}} [[Milt Buckner]],{{sfn|Hester|2000|loc=199}} [[Chick Corea]],{{sfn|Herder|1987|loc=78}}{{sfn|Scivales|2005|loc=203}} [[Herbie Hancock]],{{sfn|Herder|1987|loc=78}}{{sfn|Scivales|2005|loc=203}} and especially [[McCoy Tyner]].{{sfn|Herder|1987|loc=78}}{{sfn|Scivales|2005|loc=205}}

[[File:II V I for wikipedia.png|thumb|right|upright=1.4|[[ii–V–I progression|ii–V–I cadence]][[File:II V9 I.MID]]{{nobreak|fourth-suspension or [[sus chord]]}}[[File:II V9sus I.MID]]]]
[[File:SenorBlues.png|thumb|upright=1.8|Typical [[hard bop]] brass part, from [[Horace Silver]]'s "Señor Blues"]]

The [[hard bop]] of the 1950s made new applications of quartal harmony accessible to jazz.{{Citation needed|date=December 2009}} [[Quintet]] writing in which two melodic instruments (commonly trumpet and saxophone) may proceed in fourths, while the piano (as a uniquely harmonic instrument) lays down chords, but sparsely, only hinting at the intended harmony. This style of writing, in contrast with that of the previous decade, preferred a moderate tempo. Thin-sounding unison bebop horn sections occur frequently, but these are balanced by bouts of very refined [[polyphony]] such as is found in [[cool jazz]].

[[File:So What chord.png|thumb|upright|The [[So What chord|"So What" chord]] uses three intervals of a fourth.]]

On his watershed record ''[[Kind of Blue]]'', [[Miles Davis]] with pianist [[Bill Evans]] used a chord consisting of three perfect fourth intervals and a major third on the composition "[[So What (composition)|So What]]". This particular voicing is sometimes referred to as a [[So What chord]], and can be analyzed (without regard for added sixths, ninths, etc.) as a minor seventh with the root on the bottom, or as a major seventh with the third on the bottom.{{sfn|Levine|1989|loc=97}}

From the outset of the 1960s, the employment of quartal possibilities had become so familiar that the musician now felt the fourth chord existed as a separate entity, self standing and free of any need to resolve. The pioneering of quartal writing in later jazz and rock, like the pianist [[McCoy Tyner]]'s work with saxophonist [[John Coltrane]]'s "classic quartet", was influential throughout this epoch. [[Oliver Nelson]] was also known for his use of fourth chord [[voicing (music)|voicings]].{{sfn|Corozine|2002|loc=12}} Tom Floyd claims that the "foundation of 'modern quartal harmony'" began in the era when the [[Charlie Parker]]–influenced John Coltrane added classically trained pianists Bill Evans and McCoy Tyner to his ensemble.{{sfn|Floyd|2004|loc=4}}{{Clarification needed|reason=Is this referring to Bill Evans playing with the Miles Davis Quintet?|date=July 2022}}

Jazz guitarists cited as using chord voicings using quartal harmony include [[Johnny Smith]], [[Tal Farlow]], [[Chuck Wayne]], [[Barney Kessel]], [[Joe Pass]], [[Jimmy Raney]], [[Wes Montgomery]]—however, all in a traditional manner, as major 9th, 13th and minor 11th chords{{sfn|Floyd|2004|loc=4}} (an octave and fourth equals an 11th). Jazz guitarists cited as using modern quartal harmony include [[Jim Hall (musician)|Jim Hall]] (especially [[Sonny Rollins]]'s ''[[The Bridge (Sonny Rollins album)|The Bridge]]''), [[George Benson]] ("Sky Dive"), [[Pat Martino]], [[Jack Wilkins]] ("Windows"), [[Joe Diorio]], [[Howard Roberts]], [[Kenny Burrell]], [[Wes Montgomery]], [[Henry Johnson (guitarist)|Henry Johnson]], [[Russell Malone]], [[Jimmy Bruno]], [[Howard Alden]], [[Bill Frisell]], [[Paul Bollenback]], [[Mark Whitfield]], and [[Rodney Jones (guitarist)|Rodney Jones]].{{sfn|Floyd|2004|loc=4}}

Quartal harmony was also explored as a possibility under new experimental [[Scale (music)|scale]] models as they were "discovered" by jazz.{{Citation needed|date=December 2009}} Musicians began to work extensively with the so-called [[Mode (music)|church modes]] of old European music, and they became firmly situated in their compositional process. Jazz was well-suited to incorporate the medieval use of fourths to thicken lines into its improvisation. The pianists [[Herbie Hancock]], and [[Chick Corea]] are two musicians well known for their modal experimentation. Around this time, a style known as [[free jazz]] also came into being, in which quartal harmony had extensive use, owing to the wandering nature of its harmony.

[[File:Maiden Voyage2.png|thumb|center|upright=1.8|Fourths in Herbie Hancock's "[[Maiden Voyage (composition)|Maiden Voyage]]"{{citation needed|date=January 2011}}]]

In jazz, the way chords were built from a scale came to be called [[voicing (music)|voicing]], and specifically quartal harmony was referred to as fourth voicing.

[[File:Quartal voicings.png|thumb|center|upright=1.8|[[ii–V–I turnaround]] with fourth voicings: all chords are in fourth voicings.[[File:Quartal voicings.mid]] They are often ambiguous as, for example, the Dm11 and G9sus chords are here voiced identically and will thus be distinguished for the listener by the [[root (chord)|root]] movement of the [[bassist]].{{sfn|Boyd|1997|loc=94}}]]

Thus when the m11 and the dominant 7th sus (9sus above) chords in quartal voicings are used together they tend to "blend into one overall sound" sometimes referred to as modal voicings, and both may be applied where the m11 chord is called for during extended periods such as the entire chorus.{{sfn|Boyd|1997|loc=95}}

===Rock music===
[[File:Robert Fripp.jpg|right|thumb|Disliking the sound of thirds (in equal-temperament tuning), Robert Fripp builds chords with perfect intervals in his new standard tuning.]]
{{See also|Progressive rock|Symphonic rock}}

Quartal and quintal harmony have been used by [[Robert Fripp]], [[guitarist]] of [[King Crimson]]. Fripp dislikes minor thirds and especially major thirds in [[equal temperament]] tuning, which is used by non-experimental guitars. The perfect fourths and fifths of [[just intonation]] are well approximated in equal temperament tuning, and perfect fifths and octaves are highly consonant intervals. Fripp builds chords using perfect fifths, fourths, and octaves in his [[new&nbsp;standard tuning]]&nbsp;(NST), a [[regular tuning]] having perfect fifths between its successive [[open string (music)|open string]]s.{{sfn|Mulhern|1986|p={{Page needed|date=January 2013}}}}<!--The specific page citation is needed here; the inclusive page numbers of the interview are found in the list of References, 88–103, so the publication is not unpaginated, even if the version on the weblink is.-->

The 1971 album ''[[Tarkus]]'' by [[Emerson, Lake & Palmer]] depends on quartal harmony throughout,{{sfn|Macan|1997|loc=55}} including a recurrent elaboration on the classical [[Alberti bass]] pattern, in this case consisting of three broken quartal three-note chords, the first two of which are also a perfect fourth apart, and the third a semitone higher than the first. Keith Emerson uses programmatic quintal harmony in several places for extended rapid obbligato passages where human fingering would be impracticable, the first on Hammond organ and the second on Modular Moog, in a similar manner to the mutation stops on pipe organs, such as the "Twelfth" at 2 2/3' pitch played against a 4' "Principal" (which plays the eighth note). In the second instance, the triad is both quartal and quintal, being 1+4+5.

[[Ray Manzarek]] of [[The Doors]] was another keyboard player and composer who put classical and jazz elements, including quartal harmonies, into the service of rock music. The keyboard solo of "[[Riders on the Storm]]", for instance, has several passages where the melody line is doubled at an interval of a perfect fourth, and extensive use of (E dorian) minor chord voicings featuring the seven and three, spaced by that same interval, as the prominent notes.{{citation needed|date=December 2021}}

== Examples of quartal pieces ==

===Classical===
*[[William Albright (musician)|William Albright]]
*:Sonata for Alto Saxophone and Piano{{sfn|Lewis|1985|loc=443}}
*[[Alban Berg]]
*:Sonata for Piano, Op. 1{{sfn|Lambert|1996|loc=118}}
*:''[[Wozzeck]]''{{sfn|Lambert|1996|loc=118}}{{sfn|Reisberg|1975|loc=344–46}}
*[[Carlos Chávez]]
*:''[[Sinfonía de Antígona]]'' (Symphony No. 1), uses quartal harmony throughout{{sfn|Orbón|1987|loc=83}}
*:''[[Sinfonía india]]'' (Symphony No. 2), the A-minor Sonora melody beginning in b. 183 is accompanied by quartal harmonies{{sfn|Leyva|2010|loc=56}}
*[[Aaron Copland]]
*:''Of Mice and Men''{{sfn|Bick|2005|loc=446, 448, 451}}
[[File:Parallel fourths evoking organum in Debussy's "The Sunken Cathedral" opening 01.wav|thumb|upright=2|Parallel fourths evoking organum in Debussy's ''La cathédrale engloutie'' opening]]
*[[Claude Debussy]]
*:"[[La cathédrale engloutie]]", beginning and ending{{sfn|Reisberg|1975|loc=343–344}}
*[[Norman Dello Joio]]
*:Suite for Piano{{citation needed|date=April 2012}}
*[[Caspar Diethelm]]
*:Piano Sonata No. 7{{sfn|Kroeger|1969}}
*[[Alberto Ginastera]]
*:''12 American Preludes'', Prelude #7{{citation needed|date=April 2012}}
*[[Carlos Guastavino]]
*:"Donde habite el olvido"{{sfn|Kulp|2006|loc=207}}
*[[Howard Hanson]]
*:[[Symphony No. 2 (Hanson)|Symphony No. 2 ("Romantic")]]{{sfn|Perone|1993|loc=8}}
*[[Walter Hartley]]
*:''Bacchanalia for Band''{{sfn|Spieth|1978}}
*[[Charles Ives]]
*:"The Cage" (1906){{sfn|Carr|1989|loc=135}}{{sfn|Lambert|1990|loc=44}}{{sfn|Lambert|1996|loc=118}}{{sfn|Murphy|2008|loc=179, 181, 183, 185–186, 190–191}}{{sfn|Reisberg|1975|loc=344–345}}{{sfn|Scott|1994|loc=458}}
*:''Central Park in the Dark''{{sfn|Scott|1994|loc=458}}
*:"Harpalus"{{sfn|Scott|1994|loc=458}}
*:Psalm 24, verse 5{{sfn|Lambert|1990|loc=67}}{{sfn|Scott|1994|loc=458}}
*:Psalm 90{{sfn|Scott|1994|loc=458}}
*:"Walking"{{sfn|Scott|1994|loc=458}}
*[[Aram Khachaturian]]
*:[[Toccata (Khachaturian)|Toccata]]{{citation needed|date=April 2012}}
*[[Benjamin Lees]]
*:String Quartet No. 2, Adagio{{sfn|Cowell|1956|loc=243}}
*[[Darius Milhaud]]
*:Sonatina for flute & piano, Op. 76{{sfn|Cardew-Fanning|n.d.}}
*[[Walter Piston]]
*:Clarinet Concerto{{sfn|Archibald|1969|loc=825}}
*:''Ricercare for Orchestra''{{sfn|Archibald|1969|loc=825}}
*[[Einojuhani Rautavaara]]
*:"Kvartit" (Fourths), Op. 42, [[Études (Rautavaara)]]{{Citation needed|date=November 2019}}
*[[Maurice Ravel]]
*:''[[Ma mère l'oye]]'' : "Mouvt de Marche" of  "Laideronnette"{{sfn|Murphy, Melcher, and Warch|1973|p={{Page needed|date=April 2012}}}}
*[[Ned Rorem]]
*:''King Midas'', cantata{{sfn|Sjoerdsma|1972}}
*[[Erik Satie]]
*:''[[Le Fils des étoiles]]''{{sfn|Carpenter|n.d.}}{{sfn|Reisberg|1975|loc=347}}
*[[Arnold Schoenberg]]
*:''[[The Book of the Hanging Gardens]]''{{sfn|Domek|1979|loc=112–113, 117}}
*:''[[Chamber Symphony No. 1|Chamber Symphony]]'', Op. 9{{sfn|Reisberg|1975|loc=344–345}}{{sfn|Sanderson|n.d.}}) slow section,{{sfn|Rubin|2005}} b. 1–3{{sfn|Lambert|1990|loc=68}}
*:[[Wind Quintet (Schoenberg)|Wind Quintet]], Op. 26{{sfn|Corson and Christensen|1984}}
*[[Cyril Scott]]
*:''Diatonic Study'' (1914){{sfn|Stein|1979|loc=18}}
*[[Nikos Skalkottas]]
*:Suite No. 3 for Piano{{sfn|Dickinson|1963}}
*[[Stephen Sondheim]]
*:Piano Sonata{{sfn|Swayne|2002|loc=285–287, 290}}
*[[Karlheinz Stockhausen]]
*:''[[Klavierstücke (Stockhausen)#Klavierstück IX|Klavierstück IX]]''{{sfn|Reisberg|1975|loc=349–350}}<!--[0 5 6 11]-->
*[[Howard Swanson]]
*:"Saw a Grave"{{sfn|Moe|1981–82|loc=70}}
*[[Heitor Villa-Lobos]]
*:[[Nonet (Villa-Lobos)|Nonet]] (1923){{sfn|Béhague|1994|loc=70, 72}}
*[[Anton Webern]]
*:[[Variations for piano (Webern)|Variations for Piano]], Op. 27{{sfn|Reisberg|1975|loc=348}}<!--Viennese trichord-->
*[[John Williams]] 
*:[[Star Wars (Main Title)|Star Wars - Main Title]] (1977){{sfn|Richards|2013a}}
*:[[Superman music|Superman - Main Title]] (1978){{sfn|Richards|2013b}}

===Jazz===
*[[Miles Davis]]
*:''[[Kind of Blue]]''{{citation needed|date=May 2020}}
*[[Herbie Hancock]]
*:"[[Maiden Voyage (composition)|Maiden Voyage]]"{{citation needed|date=May 2020}}
*[[Eddie Harris]]
*:"Freedom Jazz Dance"{{citation needed|date=May 2020}}
*[[McCoy Tyner]]
*:"Contemplation"{{citation needed|date=May 2020}}
*:"[[Passion Dance]]"{{citation needed|date=May 2020}}

===Folk===
On her 1968 debut album ''[[Song to a Seagull]]'', [[Joni Mitchell]] used quartal and quintal harmony in "Dawntreader", and she used quintal harmony in the title track ''Song to a Seagull''.{{sfn|Whitesell|2008|loc=131, 202–203}}

===Rock===
* [[Emerson, Lake & Palmer]]
*: ''[[Tarkus]]''{{sfn|Macan|1997|loc=55}}
*[[Frank Zappa]]
*:"[[Zoot Allures]]"{{sfn|Mermikides|2014|loc=31}}
*[[XTC]]
*:"Rook" (composed by [[Andy Partridge]], from the album ''[[Nonsuch (album)|Nonsuch]]''){{sfn|Anon.|n.d.}}

==See also==
* [[Secundal]]
* [[Polychord]]
* [[Viennese trichord]]
* [[Traditional sub-Saharan African harmony]]

==References==
{{Reflist|15em}}

'''Sources'''
{{div col|colwidth=45em}}
* {{wikicite|ref={{harvid|Anon.|n.d.}}|reference=Anon. (n.d.). "[http://chalkhills.org/reelbyreal/s_Rook.html Rook]", ''ChalkHills.org''. Analysis and guitar tab.}}
* {{wikicite|ref={{harvid|Archibald|1969}}|reference=Archibald, Bruce (1969). "Variations for Cello and Orchestra (1966) by Walter Piston; Concerto for Clarinet and Orchestra (1967) by Walter Piston; ''Ricercare for Orchestra'' (1967) by Walter Piston". ''[[Notes (journal)|Notes]]'', second series 25, no. 4 (June): 824–26.}}
*{{wikicite|ref={{harvid|Béhague|1994}}|reference=[[Béhague, Gerard]]. 1994. ''Villa-Lobos: The Search for Brazil's Musical Soul''. Austin: Institute of Latin American Studies, University of Texas at Austin, 1994. {{ISBN|0-292-70823-8}}.}}
* {{wikicite|ref={{harvid|Benward and Saker|2009}}|reference=Benward, Bruce,  and Nadine Saker (2009). ''Music in Theory and Practice'', Vol. II. {{ISBN|978-0-07-310188-0}}.}}{{Full citation needed|date=June 2012}}
* {{wikicite|ref={{harvid|Bick|2005}}|reference=Bick, Sally (2005). "''Of Mice and Men'': Copland, Hollywood, and American Musical Modernism". ''[[American Music (journal)|American Music]]'' 23, no. 4 (Winter): 426–72.}}
* {{wikicite|ref={{harvid|Boyd|1997}}|reference=Boyd, Bill (1997). ''Jazz Chord Progressions''.{{Full citation needed|date=June 2012}}<!--Place and publisher needed.--> {{ISBN|0-7935-7038-7}}.}}
* {{wikicite|ref={{harvid|Cardew-Fanning|n.d.}}|reference=Cardew-Fanning, Neil (n.d.). "[http://www.allmusic.com/work/sonatina-for-flute--piano-op-76-c50158/description Sonatina for flute & piano, Op. 76]", [[AllMusic]].}}
* {{wikicite|ref={{harvid|Carpenter|n.d.}}|reference=Carpenter, Alexander (n.d.). "[http://www.allmusic.com/work/le-fils-des-toiles-chaldean-pastoral-3-preludes-for-piano-c18057/description Le fils des étoiles, Chaldean pastoral, 3 preludes for piano]", [[AllMusic]].}}
* {{wikicite|ref={{harvid|Carr|1989}}|reference=Carr, Cassandra I. (1989). "Charles Ives's Humor as Reflected in His Songs". ''[[American Music (journal)|American Music]]'' 7, no. 2 (Summer): 123–39.}}
* {{cite book |last=Corozine |first=Vince |title=Arranging Music for the Real World: Classical and Commercial Aspects |year=2002 |publisher=Mel Bay |location=Pacific, Missouri|isbn=978-0-7866-4961-7 |oclc=50470629}}
* {{wikicite|ref={{harvid|Corson and Christensen|1984}}|reference=Corson, Langdon, and Roy Christensen (1984). ''Arnold Schoenberg's Woodwind Quintet, Op. 26: Background and Analysis''. Nashville: Gasparo Co.}}
* {{wikicite|ref={{harvid|Cowell|1956}}|reference=[[Henry Cowell|Cowell, Henry]] (1956). "Current Chronicle: United States: New York" ''[[The Musical Quarterly]]'' 42, no. 2 (April): 240–44.}}
* {{wikicite|ref={{harvid|Dickinson|1963}}|reference=Dickinson, Peter (1963). "Suite for Piano No 3 by Nikos Skalkottas". ''[[The Musical Times]]'' 104, no. 1443 (May): 357.}}
* {{wikicite|ref={{harvid|Domek|1979}}|reference=Domek, Richard C. (1979). "Some Aspects of Organization in Schoenberg's Book of the Hanging Gardens, opus 15". ''College Music Symposium'' 19, no. 2 (Fall): 111–28.}}
* {{wikicite|ref={{harvid|Erickson|1975}}|reference=[[Robert Erickson|Erickson, Robert]] (1975). ''Sound Structure in Music''. Oakland, California: University of California Press. {{ISBN|0-520-02376-5}}.}}
* {{wikicite|ref={{harvid|Floyd|2004}}|reference=Floyd, Tom (2004). ''Quartal Harmony & Voicings for Guitar''. Pacific, Missouri: Mel Bay. {{ISBN|0-7866-6811-3}}.}}
* {{wikicite|ref={{harvid|Herder|1987}}|reference=Herder, Ronald (1987). ''1000 Keyboard Ideas''.{{Full citation needed|date=June 2012}}<!--Place and publisher needed.--> {{ISBN|0-943748-48-8}}.}}
* {{cite book|last=Hindemith|first=Paul|author-link=Paul Hindemith|title=Unterweisung im Tonsatz|year=1937|volume=1}}{{Full citation needed|date=January 2013|reason=Place and publisher needed.}}
* {{wikicite|ref={{harvid|Hester|2000}}|reference=Hester, Karlton E. (2000). ''From Africa to Afrocentric Innovations Some Call "Jazz": The Creation of Free, Fusion and Reconstructive Modern Styles (1950–2000)''. Santa Cruz, California: Histeria Records & Pub. Co. {{ISBN|1-58684-054-1}}.}}
* {{wikicite|ref={{harvid|Kasilag|2001}}|reference=Kasilag, Lucrecia R. (2001). "Pajaro, Eliseo (Morales)". ''The New Grove Dictionary of Music and Musicians'', (2nd ed.) edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]]. London: Macmillan Publishers.}}
* {{cite book|last1=Kostka|first1=Stefan|author1-link=Stefan Kostka|last2=Payne|first2=Dorothy|last3=Almén|first3=Byron|title=Tonal Harmony with an Introduction to Twentieth-Century Music|edition=7th|year=2013|location=New York|publisher=McGraw-Hill|isbn=978-0-07-131828-0}}
* {{wikicite|ref={{harvid|Kroeger|1969}}|reference=Kroeger, Karl (1969). "Caspar Diethelm: Klaviersonate VII". ''Notes'', second series 26, no. 2 (December): 363.}}
* {{wikicite|ref={{harvid|Kulp|2006}}|reference=Kulp, Jonathan (2006). "Carlos Guastavino: A Re-Evaluation of His Harmonic Language". ''Latin American Music Review'' / ''Revista de Música Latinoamericana'' 27, no. 2 (Autumn-Winter): 196–219.}}
*{{wikicite|ref={{harvid|Lambert|1990}}|reference= Lambert, J. Philip (1990). "Interval Cycles as Compositional Resources in the Music of Charles Ives". ''Music Theory Spectrum'' 12, no. 1 (Spring): 43–82.}}
* {{wikicite|ref={{harvid|Lambert|1996}}|reference=Lambert, J. Phillip (1996). "Ives and Berg: 'Normative' Procedures and Post-Tonal Alternatives". In ''Charles Ives and the Classical Tradition'', edited by Geoffrey Block and J. Peter Burkholder, 105–30. New Haven: Yale University Press.}}
* {{cite book|last=Levine|first=Mark|year=1989|title=The Jazz Piano Book|place=Petaluma, California|publisher=Sher Music|isbn=978-0-9614701-5-9}}
* {{wikicite|ref={{harvid|Lewis|1985}}|reference=Lewis, Robert Hall (1985). "New Music Festival 1985: Bowling Green State University". ''[[Perspectives of New Music]]'' 24, no. 1 (Fall–Winter): 440–43.}}
* {{wikicite|ref={{harvid|Leyva|2010}}|reference=Leyva, Jesse (2010). "Carlos Chávez: An Examination of His Compositional Style with a Conductor's Analysis of ''Sinfonía India'' as Arranged for Concert Band by Frank Erickson". DMA diss. Tempe: Arizona State University.}}
* {{cite book |last=Macan |first=Edward L |year=1997 |title=Rocking the Classics: English Progressive Rock and the Counterculture |location=Oxford and New York |publisher=Oxford University Press |isbn=978-0-19-509887-7|url-access=registration |url=https://archive.org/details/rockingclassicse0000maca }}.
* {{wikicite|ref={{harvid|Mermikides|2014}}|reference=Mermikides, Milton (2014). "Extreme Guitar". ''Guitar Techniques magazine'', issue 230.{{Page needed|date=May 2014}}<!--Inclusive page numbers of the article are needed.-->}}
* {{wikicite|ref={{harvid|Moe|1981–82}}|reference=Moe, Orin (1981–82). "The Songs of Howard Swanson". ''[[Black Music Research Journal]]'' 2:57–71.}}
* {{wikicite|ref={{harvid|Morrison|1998}}|reference=Morrison, S. (1998). "Skryabin and the Impossible". ''[[Journal of the American Musicological Society]]'' 51, no. 2.{{Page needed|date=June 2012}}<!--Author's given name and inclusive page numbers needed.-->}}
* {{wikicite|ref={{harvid|Mulhern|1986}}|reference=Mulhern, Tom (1986). "[http://www.mulhern.com/articles/Fripp.html On the Discipline of Craft and Art: An Interview with Robert Fripp]". ''Guitar Player'' 20 (January): 88–103 (accessed 8 January 2013).}}
* {{wikicite|ref={{harvid|Murphy|2008}}|reference=Murphy, Scott (2008). "A Composite Approach to Ives's 'Cage'". ''Twentieth-Century Music'' 5:179–93.}}
* {{wikicite|ref={{harvid|Murphy, Melcher, and Warch|1973}}|reference=Murphy, Howard Ansley, Robert A. Melcher, and Willard F. Warch, eds. (1973). ''Music for Study: A Source Book of Excerpts'', (2nd ed.) Englewood Cliffs, New Jersey: Prentice-Hall. {{ISBN|0-13-607515-0}}.}}
* [[Jean-Jacques Nattiez|Nattiez, Jean-Jacques]] (1990). ''Music and Discourse: Toward a Semiology of Music'', translated by Carolyn Abbate. Princeton, New Jersey: Princeton University Press. {{ISBN|0-691-02714-5}}.
* {{wikicite|ref={{harvid|Orbón|1987}}|reference=[[Julián Orbón|Orbón, Julián]] (1987). "Las sinfonías de Carlos Chávez." (part 2). ''Pauta: Cuadernos de teoría y crítica musical'' 6, no. 22 (April–June): 81–91.}}
* {{wikicite|ref={{harvid|Perone|1993}}|reference=Perone, James E. (1993). ''Howard Hanson: A Bio-Bibliography''. Westport and London: Greenwood Press.}}
* {{cite book |last=Persichetti |first=Vincent |author-link=Vincent Persichetti |title=Twentieth-century Harmony: Creative Aspects and Practice |year=1961 |publisher=W. W. Norton |location=New York |isbn=978-0-393-09539-5 |oclc=398434|url-access=registration |url=https://archive.org/details/isbn_9780393095395}}
* {{wikicite|ref={{harvid|Reisberg|1975}}|reference=Reisberg, Horace (1975). "The Vertical Dimension in Twentieth-Century Music". In ''Aspects of 20th Century Music'', Gary E. Wittlich, coordinating editor, 322–387. Englewood Cliffs, New Jersey: Prentice-Hall. {{ISBN|0-13-049346-5}}.}}
* {{Cite web|last=Richards|first=Mark|date=9 March 2013a|title=John Williams Themes, Part 2 of 6: Star Wars, Main Title|url=https://www.filmmusicnotes.com/john-williams-themes-part-2-star-wars-main-title/|access-date=2020-10-28|website=Film Music Notes}}
* {{Cite web|last=Richards|first=Mark|date=15 July 2013b|title=John Williams' Superman Theme (Superman March)|url=https://www.filmmusicnotes.com/john-williams-superman-theme-superman-march/|access-date=2020-10-28|website=Film Music Notes}}
* {{wikicite|ref={{harvid|Rubin|2005}}|reference=Rubin, Justin Henry (2005). "[http://www.d.umn.edu/~jrubin1/JHR%20Quartal%201.htm Quartal Harmony]". University of Minnesota Duluth website ''d.umn.edu'' (accessed 26 April 2012).}}
* {{wikicite|ref={{harvid|Sanderson|n.d.}}|reference=Sanderson, Blair (n.d.). "[http://www.allmusic.com/album/arnold-schoenberg-verklrte-nacht-chamber-symphony-variations-5-pieces-6-songs-erwartung-w68725/review Arnold Schoenberg: Verklärte Nacht; Chamber Symphony; Variations; 5 Pieces; 6 Songs; Erwartung]", [[AllMusic]].}}
* {{cite book|last=Schoenberg|first=Arnold|title=Harmonielehre|year=1922|edition=3rd|place=Vienna|publisher=Universal Edition|author-link=Arnold Schoenberg}}
* {{cite book|last=Schoenberg|first=Arnold|year=1978|title=Theory of Harmony|translator=Roy E. Carter|publisher=University of California Press}} {{ISBN|0-520-04945-4}} (cloth); {{ISBN|0-520-04944-6}} (pbk). Based on {{harvnb|Schoenberg|1922}}.
* {{cite book|last=Scivales|first=Ricardo|year=2005|title=Jazz Piano—The Left Hand|publisher=Ekay Music |isbn=978-1-929009-54-1}}.{{Full citation needed|date=June 2012}}<!--Place and publisher needed.-->
* {{wikicite|ref={{harvid|Scott|1994}}|reference=Scott, Ann Besser (1994). "Medieval and Renaissance Techniques in the Music of Charles Ives: Horatio at the Bridge?" ''[[The Musical Quarterly]]'' 78, no. 3 (Autumn): 448–78.}}
* {{wikicite|ref={{harvid|Sjoerdsma|1972}}|reference=Sjoerdsma, Richard Dale (1972). "''King Midas; A Cantata for Voices and Piano on 10 Poems of Howard Moss'' by Ned Rorem". ''[[Notes (journal)|Notes]]'', second series 28, no. 4 (June): 782.}}
* {{wikicite|ref={{harvid|Solomon|2003}}|reference=Solomon, Larry J. 2003. "[https://web.archive.org/web/20071115223321/http://solomonsmusic.net/Satie.htm Satie, the First Modern]". Solomonsmusic.net (accessed 18 March 2016).}}{{Unreliable source?|certain=y|reason=Self-published website|date=April 2016}}
* {{wikicite|ref={{harvid|Spieth|1978}}|reference=Spieth, Donald (1978). "''Bacchanalia for Band'' by Walter S. Hartley". ''[[Notes (journal)|Notes]]'', second series 34, no. 4 (June): 974.}}
* {{wikicite|ref={{harvid|Stein|1979}}|reference=[[Leon Stein|Stein, Leon]] (1979). ''Structure & Style: The Study and Analysis of Musical Forms'', second expanded edition. Princeton, New Jersey: Summy-Birchard Music. {{ISBN|978-0-87487-164-7}}.}}
* {{wikicite|ref={{harvid|Swayne|2002}}|reference=Swayne, Steve (2002). "Sondheim's Piano Sonata". ''[[Journal of the Royal Musical Association]]'' 127, no. 2:258–304.}}
* {{wikicite|ref={{harvid|Vogel|1962}}|reference=Vogel, Martin (1962). ''Der Tristan-Akkord und die Krise der modernen Harmonielehre''{{Full citation needed|date=June 2012}}<!--Place, publisher needed.-->}}
* {{cite book|last=Webern|first=Anton|author-link=Anton Webern|title=The Path to the New Music|editor=Willi Reich|translator=Leo Black|place=Bryn Mawr|publisher=Theodore Presser, in association with Universal Edition|year=1963}}
* {{cite book|last=Whitesell|first=Lloyd|year=2008|url=https://books.google.com/books?id=xSmJbcWcYA0C&q=Joni%20Mitchell&pg=PP1|title=The Music of Joni Mitchell|location=Oxford and New York|publisher=Oxford University Press|isbn=978-0-19-530757-3}}
{{div col end}}

==Further reading==
{{div col|colwidth=45em}}
* {{cite book |last=Baker|first=David N.|year=1983|title=Jazz Improvisation|place=Bloomington|publisher=Frangipani|isbn=978-0-89917-397-9|ref=none}}
* [https://independent.academia.edu/BernardFLOIRAT Floirat, Bernard] (2015). ''[https://www.academia.edu/12232101/Introduction_aux_accords_de_quartes_chez_Arnold_Sch%C5%93nberg Introduction aux accords de quartes chez Arnold Schoenberg]'', Paris, www.academia.edu.{{Unreliable source?|reason=self-published|date=March 2016}}
* {{cite book |last=Rosenthal|first=David H.|author-link=David H. Rosenthal|year=1993|title=Hard Bop, Jazz and Black Music 1955–1965|place=New York|publisher=Oxford University Press|isbn=978-0-19-508556-3|ref=none}}
{{div col end}}

==External links==
*[http://www.d.umn.edu/~jrubin1/JHR%20Quartal%201.htm Quartal harmony with notes and listening examples]
*[https://web.archive.org/web/20060503155941/http://www.schoenberg.at/6_archiv/music/works/op/compositions_op9_notes_e.htm Program notes for Arnold Schoenberg's Chamber Symphony for 15 Solo Instruments, Op. 9]
*[http://www.jazzguitar.be/blog/quartal-chords-harmony-voicings-for-guitar/ The Use of Quartal Harmony in Jazz Guitar]

{{Atonality}}
{{Chords}}
{{Modernism (music)}}

{{DEFAULTSORT:Quartal And Quintal Harmony}}
[[Category:Quartal harmony| ]]