Quartal Harmony

A close-up photograph of the fretboard of an electric bass guitar being played. The fretboard of an electric bass guitar. The four strings of the bass guitar and upright double bass are E1, A1, D2 and G2, creating a quartal tetrad.
(Feliciano Guimarães | CC BY-2.0)

Most popular music today is based on chords which are built from major and minor thirds, but chords can be built from other intervals as well.

Types of Harmony

Harmonic systems can be identified by basic diatonic intervals. Tertial harmony, the system used in most commercially popular music, uses chords based on major thirds and minor thirds.

F triads: F diminished, consisting of F, A flat, C flat, two minor thirds. F minor, F, A flat, C, has a minor third on bottom and a major third on top. F major, F, A, C, has a major third on bottom and a minor third on top. F augmented, F, A, C sharp, has two major thirds.
Figure 1: The four possible tertial triads, constructed of major and minor thirds.

Tertial harmony does not include diminished thirds and augmented thirds, which are enharmonically equivalent to major seconds and perfect fourths.

Augmented sixth chords in C major: Italian, French and German.
Figure 2: The three common augmented sixth chords. These chords contain the interval of an augmented sixth, which inverts to a diminished third, giving them a slightly non-tertial quality.

In 12-TET, we can consider building systems of harmony based upon intervals beside thirds.

Secundal "Harmony"

In contast to tertial harmony, secundal harmony would include chords built from major seconds and minor seconds. These chords, however, are more commonly referred to as tone clusters.

Measures 106 through 114 of “The Battle of Manassas” by Thomas Wiggins. In measures 109, 110, 112 and 114, a symbol in the left hand indicates tone clusters.
Figure 3: Measures 106–114 of The Battle of Manassas by American composer Thomas Wiggins, portraying cannon fire through the use of low-pitched tone clusters. The symbols in the left hand part are explained with a note in the score that reads: "The Cannon is played by striking with both hands, (if both are at liberty; if not, with the left hand alone) and with the flat of the hand, as many notes as possible, and with as much force as possible, at the bass of the piano."

In keyboard music, because of the density of notes involved, these chords are sometimes written with an alternate notation.

Measure 22 of The Tides of Manaunaun by Henry Cowell. The left hand part consists entirely of two-octave tone clusters, notated in traditional notation with a vertical black bar connecting the top and bottom noteheads. In braille music notation, this is notated as a standard interval, with a three-cell tone cluster symbol placed between the root note symbol and the interval symbol.
Figure 4: Measure 22 of The Tides of Manaunaun by American composer Henry Cowell. Cowell's notation for tone clusters was explained in the notes to the score; clusters with a natural sign are meant to be played on the white notes of the piano, while notes with a flat sign are meant to be played using only the black notes. The size of the tone clusters in this measure require the use of the pianist's forearm.

However, if we consider harmony to be a balance of the individual voices with their combined sound, then the term "secundal harmony" can be considered a misnomer. Instead of being heard as a combination of individual notes, tone clusters are more timbral in nature, and tend to be heard as a single, approximately-pitched sound.

Higher-Interval Harmonies

Quartal Harmony is built upon perfect fourths, and discussed further below.

Harmony built upon intervals larger than a fourth are not possible, as these intervals are simply inversions of smaller intervals. For example, hexal harmony would involve chords built upon major sixths and minor sixths; these chords are simply open voicings of tertial chords.

Two chords: a “hexal” triad containing C4, A4 and F5, and a tertial triad containing F4, A4 and C5.
Figure 5: The "hexal" chord on the left is simply the tertial triad on the right, placed in an open voicing.

Thus, quintal chords — chords built upon perfect fifths — exist as distinct chords, but can be understood as voicings of quartal chords, are considered to be a part of quartal harmony.

Two chords: a quintal triad containing C4, G4 and D5, and a quartal triad containing D4, G4 and C5.
Figure 6: Three pitch classes, C, D and G, spelled as a quintal triad on the left and as a quartal triad on the right.

Quartal Harmony

Quartal and Quintal Chords

Quartal chords are chords built by stacking perfect fourths. As in tertial harmony, quartal chords can include triads, tetrads, and so on.

Two chords: a quartal triad containing F4, B flat 4, and E flat 5, and a quartal tetrad containing C sharp 4, F sharp 4, B4, and E5.
Figure 7: Two quartal chords: a triad and a tetrad.

Likewise, quintal chords are built by stacking perfect fifths.

Two chords: a quintal triad containing E flat 4, B flat 4, and F5, and a quintal tetrad containing B3, F sharp 4, C sharp 5, and G sharp 5.
Figure 8: Two quintal chords: a triad and a tetrad.

Quartal Textures

Quartal harmony can be used in a tonal soundscape, but lacks the familiar harmonic functional roles of diatonic tertial harmony.

An excerpt from “On a Mountain Road at Summer's End” by Tui St. George Tucker. The passage, which does not use measures or barlines, features quintal triads in the left hand combined with tertial dyads in the right hand.
Figure 9: The end of line 3 on page 2 of "On a Mountain Road at Summer's End" from Notes From the Blue Mountains by American composer Tui St. George Tucker. The quintal chords in the left hand are combined with tertial harmonies in the right hand.

Quartal and quintal chords are often juxtaposed, and vary slightly in their level of dissonance.

An orchestral reduction of measures 48 through 51 from “One Summer's Day” by Joe Hisiashi. Measures 48 and 49 contain an A quintal and C quintal chord respectively, while the harmony changes in measure 50 to a C quartal chord.
Figure 10: An orchestral reduction of measures 48–51 of One Summer's Day by Japanese composer Joe Hisaishi, from the 2001 film Spirited Away. In this passage, a C quintal chord moves to a quartal chord on the same root, creating a different, slightly more dissonant texture that leads to a harmonic resolution.

Quartal harmony can be effectively combined with tertial harmony, in traditionally functional or pantriadic textures.

Measures 19 through 23 from the second movement of Paul Hindemith's Trombone Sonata, featuring an F sharp quartal chord followed by an A minor seventh chord in the piano part.
Figure 11: Measures 19–23 from the second movement of German composer Paul Hindemith's 1941 Trombone Sonata. The chromatically ascending bass note in the piano creates a quartal chord in measure 21, which moves to a minor seventh chord in the following measure.

Other Sonorities

Power Chords

Removing the third from a major or minor tertial triad and doubling the octave creates a chord comprised of only a perfect fourth and a perfect fifth. This chord, often called a power chord, is common in certain genres of rock music, where it helps create modal ambiguity.

Measures 1 and 2 of `I Love Rock and Roll` by Joan Jett and the Blackhearts, a primarily eighth-note guitar riff comprised completely of power chords and open octaves.
Figure 12: Measures 1 and 2 showing the primary riff of I Love Rock and Roll by the American band Joan Jett and the Blackhearts. This chord configuration is sometimes referred to as an open fifth chord in notated music, but is generally called a power chord in popular music styles.

Mixed Harmonies

Thirds and fourths can be combined in the same chord to create different sonorities. These complex harmonies are often found in jazz music beginning in the 1960s.

An F six-nine chord, with notes F4, A4, C5, D5 and G5, and an F dominant seven sharp nine, with notes F4, A4, C5, E flat 5 and G sharp 5. Both chords have a sounding perfect fourth between the top two pitches.
Figure 13: Two common jazz chords, the F6/9 and Fdom7#9. Both of these chords contain a combination of thirds and fourths.

Another chord commonly heard in jazz is a four-note quartal chord with fifth note placed a major third above the top note. Because of its use by American composer and jazz pianist Bill Evans in the 1959 recording of American composer and trumpeter Miles Davis' chart So What, this chord is often referred to by jazz musicians as the So What chord.

The “So What” chord, consisting of the notes C4, F4, B flat 4, E flat 5, and G5.
Figure 14: A So What chord, built on C.

Quartal Harmony: Summary

  • The system of harmony which uses chords built from major and minor thirds is tertial harmony.
  • Chords which are built from seconds are called tone clusters.
    • Because tone clusters are timbral in nature and do not balance individual voices with combined sounds, no real "secundal harmony" exists in the 12-TET system.
  • Harmonic systems built on intervals higher than a fourth — quintal, hexal, and septal harmony, for example, are not distinct from harmonies based on those intervals' inversions.
    • For example, since fifths invert to fourths, quintal harmony is identical to quartal harmomy.
  • Quartal chords are chords built by stacking perfect fourths.
    • Quintal chords are chords built by stacking perfect fifths.
    • Quartal and quintal chords can be used in a tonal piece, but do not follow traditional patterns of chord function.
    • Quartal, quintal, and tertial chords can be combined and juxtaposed to exploit their varying levels of consonance and dissonance.
    • Jazz often uses chords which combine aspects of tertial and quartal harmony.

Exercises

Exercise 1: Analyzing Erik Satie's Le Fils Des Étoiles

Exercise 2: Harmonizing a Melody