Editing Quartal and quintal harmony (section)

===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.