1. Properties

2. History
    Precursors  20th- and 21st-century classical music   Schoenberg    Others   Jazz  Rock music 

3. Examples of quartal pieces
    Classical  Jazz  Folk  Rock 

4. See also

5. References

6. Further reading

7. External links

In music, quartal harmony is the building of harmonic structures built from the intervals of the perfect fourth, the 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}}.

\\override Score.TimeSignature #'stencil = ##f

\\relative c' {

} }

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

\\override Score.TimeSignature #'stencil = ##f

\\relative c' {

} }

Properties

    {      \ew PianoStaff <<        \ew Staff <<            \ew 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                }            \ew 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                }            >>        \ew Staff <<            \\relative c {                \\clef bass \\key a \\minor \\time 6/8                \\partial8 r8 R2. \\once \\override NoteHead.color = #red 2.( 4.)~ 8 r r                }            >>    >> }

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 ({{harvnb|Vogel|1962|loc=12}}), notably Beethoven's 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 function, a notion which was soon after to be explored by Debussy and others ({{harvnb|Erickson|1975}}, {{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. Scriabin used a self-developed system of transposition using fourth-chords, like his Mystic chord (shown below) in his Piano Sonata No. 6.

{

\\override Score.TimeSignature #'stencil = ##f

\\relative c' {

} }

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 show that he intended to develop the Mystic chord into a huge chord incorporating all twelve notes of the chromatic scale {{harv|Morrison|1998|loc=316}}.

In France, Erik Satie experimented with planing in the stacked fourths (not all perfect) of his 1891 score for Le Fils des étoiles {{harv|Solomon|2003}}. Paul Dukas's 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, and Anton Webern {{harv|Herder|1987|loc=78}}.

Schoenberg

Arnold Schoenberg's Chamber Symphony Op. 9 (1906) displays quartal harmony: the first measures construct a five-part fourth chord with the notes C–F–B{{music|flat}}–E{{music|flat}}–A{{music|flat}} distributed over several instruments.

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:

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!" ({{harvnb|Webern|1963|loc=48}}; "So was mußt du auch machen!"{{Citation needed|date=September 2014}})

Others

In his Theory of Harmony {{harv|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 Puccini, are not far off."

French composer Maurice Ravel used quartal chords in Sonatine (1906) and Ma mère l'oye (1910), while American Charles Ives used quartal chords in his song "The Cage" (1906).

Hindemith constructed large parts of his symphonic work 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," 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." 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 tones."

The works of the Filipino composer Eliseo M. Pajaro (1915–1984) are characterised by quartal and quintal harmonies, as well as by dissonant counterpoint and polychords {{harv|Kasilag|2001}}.

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

Jazz

The style of jazz, having an eclectic harmonic orbit, was in its early days overtaken (until perhaps the Swing of the 1930s) by the vocabulary of 19th-century European music.{{Clarify|date=January 2015}} Important influences come thereby from opera, operetta, military bands as well as from the piano music of Classical and Romantic composers, and even that of the Impressionists. Jazz musicians had a clear interest in harmonic richness of colour, for which quartal harmony provided possibilities, as used by pianists and arrangers like Jelly Roll Morton, Duke Ellington, Art Tatum, Bill Evans {{harv|Hester|2000|loc=199}} Milt Buckner {{harv|Hester|2000|loc=199}} Chick Corea ({{harvnb|Herder|1987|loc=78}}; {{harvnb|Scivales|2005|loc=203}}) Herbie Hancock ({{harvnb|Herder|1987|loc=78}}; {{harvnb|Scivales|2005|loc=203}}) and especially McCoy Tyner ({{harvnb|Herder|1987|loc=78}}; {{harvnb|Scivales|2005|loc=205}}).

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 brass 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.

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". 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 {{harv|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 voicings {{harv|Corozine|2002|loc=12}}. 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 {{harv|Floyd|2004|loc=4}}.

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 {{harv|Floyd|2004|loc=4}} (an octave and fourth equals an 11th). Jazz guitarists cited as using modern quartal harmony include Jim Hall (especially Sonny Rollins's The Bridge), George Benson ("Skydive"), Pat Martino, Jack Wilkins ("Windows"), Joe Diorio, Howard Roberts ("Impressions"), Kenny Burrell ("So What"), Wes Montgomery ("Little Sunflower"), Henry Johnson, Russell Malone, Jimmy Bruno, Howard Alden, Paul Bollenback, Mark Whitfield, and Rodney Jones {{harv|Floyd|2004|loc=4}}.

Quartal harmony was also explored as a possibility under new experimental scale models as they were "discovered" by jazz.{{Citation needed|date=December 2009}} Musicians began to work extensively with the so-called 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 due to the wandering nature of its harmony.

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

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 {{harv|Boyd|1997|loc=95}}.

Rock music

Quartal and quintal harmony have been used by Robert Fripp, who has described himself as the rhythm guitarist of King Crimson. Fripp dislikes minor thirds and especially major thirds in equal temperament tuning, which is used by non-experimental guitars. Of course, just intonation's perfect octaves, perfect fifths, and perfect fourths 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 standard tuning (NST), a regular tuning having perfect fifths between its successive open-strings ({{harvnb|Mulhern|1986}}, {{Page needed|date=January 2013}}).

Tarkus by Emerson, Lake & Palmer uses quartal harmony {{harv|Macon|1997|loc=55}}.

Examples of quartal pieces

Classical

• William Albright

Sonata for Alto Saxophone and Piano {{harv|Lewis|1985|loc=443}}

• Alban Berg

Sonata for Piano, op. 1 {{harv|Lambert|1996|loc=118}}

Wozzeck ({{harvnb|Lambert|1996|loc=118}}; {{harvnb|Reisberg|1975|loc=344–46}})

• Carlos Chávez

Sinfonía de Antígona (Symphony No. 1), uses quartal harmony throughout {{harv|Orbón|1987|loc=83}}

Sinfonía india (Symphony No. 2), the A-minor Sonora melody beginning in b. 183 is accompanied by quartal harmonies {{harv|Leyva|2010|loc=56}}

• Aaron Copland

Of Mice and Men {{harv|Bick|2005|loc=446, 448, 451}}

• Claude Debussy

"La cathédrale engloutie", beginning and ending {{harv|Reisberg|1975|loc=343–44}}

• Norman Dello Joio

Suite for Piano{{citation needed|date=April 2012}}

• Caspar Diethelm

Piano Sonata No. 7 {{harv|Kroeger|1969}}

• Alberto Ginastera

12 American Preludes, Prelude #7{{citation needed|date=April 2012}}

• Carlos Guastavino

"Donde habite el olvido" {{harv|Kulp|2006|loc=207}}

• Howard Hanson

Symphony No. 2 ("Romantic") {{harv|Perone|1993|loc=8}}

• Walter Hartley

Bacchanalia for Band {{harv|Spieth|1978}}

• Charles Ives

"The Cage" (1906) ({{harvnb|Carr|1989|loc=135}}; {{harvnb|Lambert|1990|loc=44}}; {{harvnb|Lambert|1996|loc=118}}; {{harvnb|Murphy|2008|loc=179, 181, 183, 185–86, 190–91}}; {{harvnb|Reisberg|1975|loc=344–45}}; {{harvnb|Scott|1994|loc=458}})

Central Park in the Dark {{harv|Scott|1994|loc=458}}

"Harpalus" {{harv|Scott|1994|loc=458}}

Psalm 24, verse 5 ({{harvnb|Lambert|1990|loc=67}}; {{harvnb|Scott|1994|loc=458}})

Psalm 90 {{harv|Scott|1994|loc=458}}

"Walking" {{harv|Scott|1994|loc=458}}

• Aram Khachaturian

Toccata{{citation needed|date=April 2012}}

• Benjamin Lees

String Quartet No. 2, Adagio {{harv|Cowell|1956|loc=243}}

• Darius Milhaud

Sonatina for flute & piano, Op. 76 {{harv|Cardew-Fanning|n.d.}}

• Walter Piston

Clarinet Concerto {{harv|Archibald|1969|loc=825}}

Ricercare for Orchestra {{harv|Archibald|1969|loc=825}}

• Maurice Ravel

Ma mère l'oye : "Mouvt de Marche" of "Laideronnette" ({{harvnb|Murphy, Melcher, and Warch|1973}}, {{Page needed|date=April 2012}})

• Ned Rorem

King Midas, cantata {{harv|Sjoerdsma|1972}}

• Erik Satie

Le Fils des étoiles ({{harvnb|Carpenter|n.d.}}; {{harvnb|Reisberg|1975|loc=347}})

• Arnold Schoenberg

The Book of the Hanging Gardens {{harv|Domek|1979|loc=112–13, 117}}

Chamber Symphony, Op. 9 ({{harvnb|Reisberg|1975|loc=344–45}}; {{harvnb|Sanderson|n.d.}}), slow section {{harv|Rubin|2005}}, b. 1–3 {{harv|Lambert|1990|loc=68}}

Wind Quintet, op. 26 {{harv|Corson and Christensen|1984}}

• Cyril Scott

Diatonic Study (1914) {{harv|Stein|1979|loc=18}}

• Nikos Skalkottas

Suite No. 3 for Piano {{harv|Dickinson|1963}}

• Stephen Sondheim

Piano Sonata {{harv|Swayne|2002|loc=285–87, 290}}

• Karlheinz Stockhausen

Klavierstück IX {{harv|Reisberg|1975|loc=349–50}}

• Howard Swanson

"Saw a Grave" {{harv|Moe|1981–82|loc=70}}

• Heitor Villa-Lobos

Nonet (1923) {{harv|Béhague|1994|loc=70, 72}}

• Anton Webern

Variations for Piano, Op. 27 {{harv|Reisberg|1975|loc=348}}

Jazz

• Miles Davis

Kind of Blue

• Herbie Hancock

"Maiden Voyage"

• Eddie Harris

"Freedom Jazz Dance"

• McCoy Tyner

"Contemplation"

"Passion Dance"

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 {{harv|Whitesell|2008|loc=131 and 202–203}}.

Rock

• Emerson, Lake & Palmer

Tarkus {{harv|Macon|1997|p=55}}

• Frank Zappa

"Zoot Allures" {{harv|Mermikides|2014|loc=31}}

• XTC

"Rook" (composed by Andy Partridge, from the album Nonsuch) {{harv|Anon.|n.d.}}

See also

• Secundal
• Polychord
• Viennese trichord
• Traditional sub-Saharan African harmony

References

• {{wikicite|ref={{harvid|Anon.|n.d.}}|reference=Anon. (n.d.). "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, 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 23, no. 4 (Winter): 426–72.}}
• {{wikicite|ref={{harvid|Boyd|1997}}|reference=Boyd, Bill (1997). Jazz Chord Progressions.{{Full citation needed|date=June 2012}} {{ISBN|0-7935-7038-7}}.}}
• {{wikicite|ref={{harvid|Cardew-Fanning|n.d.}}|reference=Cardew-Fanning, Neil (n.d.). "Sonatina for flute & piano, Op. 76", AllMusic.com.}}
• {{wikicite|ref={{harvid|Carpenter|n.d.}}|reference=Carpenter, Alexander (n.d.). "Le fils des étoiles, Chaldean pastoral, 3 preludes for piano", AllMusic.com.}}
• {{wikicite|ref={{harvid|Carr|1989}}|reference=Carr, Cassandra I. (1989). "Charles Ives’s Humor as Reflected in His Songs". 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, MO |isbn=978-0-7866-4961-7 |oclc=50470629 |ref=harv}}
• {{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=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.}}
• 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=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.{{Full citation needed|date=June 2012}} {{ISBN|0-7866-6811-3}}.}}
• {{wikicite|ref={{harvid|Herder|1987}}|reference=Herder, Ronald (1987). 1000 Keyboard Ideas.{{Full citation needed|date=June 2012}} {{ISBN|0-943748-48-8}}.}}
• {{cite book|last=Hindemith|first=Paul|title=Unterweisung im Tonsatz|year=1937|volume=1|ref=harv}}{{Full citation needed|date=January 2013}}
• {{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){{Full citation needed|date=June 2012}}. {{ISBN|1-58684-054-1}}.}}
• {{wikicite|ref={{harvid|Josh|2010}}|reference=Josh (2010) "4th Chords and Quartal Harmony". FretterVerse.com (30 June, accessed 6 September 2014).}}
• {{wikicite|ref={{harvid|Kasilag|2001}}|reference=Kasilag, Lucrecia R. (2001). "Pajaro, Eliseo (Morales)". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.}}
• {{cite book

• {{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, CA|publisher=Sher Music Co.|isbn=978-0-9614701-5-9|ref=harv}}
• {{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=Macon |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|ref=harv}}.
• {{wikicite|ref={{harvid|Mermikides|2014}}|reference=Mermikides, Milton (2014). "Extreme Guitar". Guitar Techniques magazine, Issue 230.{{Page needed|date=May 2014}}}}
• {{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}}}}
• {{wikicite|ref={{harvid|Mulhern|1986}}|reference=Mulhern, Tom (1986). "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, second edition. Englewood Cliffs, NJ: Prentice-Hall. {{ISBN|0-13-607515-0}}.
• Nattiez, Jean-Jacques (1990). Music and Discourse: Toward a Semiology of Music, translated by Carolyn Abbate. Princeton, N.J.: Princeton University Press. {{ISBN|0-691-02714-5}}.}}
• {{wikicite|ref={{harvid|Orbón|1987}}|reference=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.}}

• {{wikicite|ref={{harvid|Persichetti|1961}}|reference={{Cite book|title=Twentieth-Century Harmony: Creative Aspects and Practice|last=Persichetti|first=Vincent|publisher=W.W. Norton & Company, Inc.|year=1961|isbn=|location=New York|author-link=Vincent Persichetti}}}}
• {{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, 322–87. Englewood Cliffs, NJ: Prentice-Hall. {{ISBN|0-13-049346-5}}.}}
• {{wikicite|ref={{harvid|Rubin|2005}}|reference=Rubin, Justin Henry (2005). "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.). "Arnold Schoenberg: Verklärte Nacht; Chamber Symphony; Variations; 5 Pieces; 6 Songs; Erwartung", AllMusic.com.}}
• {{cite book|last=Schoenberg|first=Arnold|title=Harmonielehre|year=1922|edition=third|place=Vienna|publisher=universal edition|authorlink=Arnold Schoenberg|ref=harv}}
• {{cite book|last=Schoenberg|first=Arnold|year=1978|title=Theory of Harmony|others=translated by Roy E. Carter, based on the third edition (1922)|place=Berkeley and Los Angeles|publisher=University of California Press|ref=harv}} {{ISBN|0-520-04945-4}} (cloth); {{ISBN|0-520-04944-6}} (pbk).
• {{cite book|last=Scivales|first=Ricardo|year=2005|title=Jazz Piano—The Left Hand|place=|publisher=|isbn=978-1-929009-54-1|ref=harv}}.{{Full citation needed|date=June 2012}}
• {{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, second series 28, no. 4 (June): 782.}}
• {{wikicite|ref={{harvid|Solomon|2003}}|reference=Solomon, Larry J. 2003. "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, second series 34, no. 4 (June): 974.}}
• {{wikicite|ref={{harvid|Stein|1979}}|reference=Stein, Leon (1979). Structure & Style: The Study and Analysis of Musical Forms, second expanded edition. Princeton, N.J.: 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}}}}
• {{cite book|ref=harv|last=Webern|first=Anton|title=The Path to the New Music|others=edited by Willi Reich, translated by 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&lpg=PP1&dq=Joni%20Mitchell&pg=PP1#v=onepage&q&f=true|title=The Music of Joni Mitchell|location=Oxford and New York|publisher=Oxford University Press|isbn=978-0-19-530757-3|ref=harv}}

Further reading

• {{cite book |last=Baker|first=David N.|year=1983|title=Jazz Improvisation|place=Bloomington|publisher=Frangipani|isbn=978-0-89917-397-9|ref=harv}}
• [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=Persichetti |first=Vincent |authorlink=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|ref=harv}}
• {{cite book |last=Rosenthal|first=David H.|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=harv}}

External links

• 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]
• The Use of Quartal Harmony in Jazz Guitar

• Gaithersburg, md
• Gait lab
• Gaito Gazdanov
• Gaitsgory
• Gaitskell Commission
• Gaitskill Commission
• Gait training
• Gaius Aculeo
• Gaius Appuleius Diocles
• Gaius Apuleius Diocles
• Gaius Aquillius Florus
• Gaius Asinius Nicomachus Julianus
• Gaius Asinius Pollio (consul 23)
• Gaius Asinius Pollio (consul in 23 AD)
• Gaius Asinius Quadratus Protimus
• Gaius Ateius Capito (tribune)
• Gaius Bruttius Praesens
• Gaius Bruttius Praesens (consul AD 139)
• Gaius Caecilius Metellus
• Gaius Caecilius Metellus Caprarius
• Gaius Caeionius Rufius Volusianus
• Gaius Caesar (20 BC)
• Gaius Caligula
• Gaius Calpurnius Piso (consul 67 BC)
• Gaius Caninius Rebilus