MUS 312 Form & Analysis

L. The Vertical Dimension:  Chords & Simultaneities

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The music of the tonal era is almost exclusively tertian--harmonies consisting of stacked thirds.  Much of the music of the 20th century is also tertian; but there is a good deal of music using chords built from 2nds, 4ths, and from combinations of various intervals.

A result of the unlimited array of 20th-century harmonic material is that the distinction between chord tones and nonchord tones is difficult, or sometimes impossible, to make.  Also, chords sometimes seem to result, more or less, from the coincidental combination of harmonically independent lines.  For these reasons, many theorists prefer at times to use terms such as verticality, simultaneity, or note complex, instead of chord.  Although chord may be used freely.

Nine Chord Types in Brief:

In the most simple terms, there are four possibilities for chord construction:

  • Secundal chords (also tone clusters)
  • Tertian chords (including 9ths, 11ths, 13ths, and alterations)
  • Quartal/Quintal chords
  • Mixed-interval chords

Tertian chords, the most traditional of the four types, have been subjected to some new variations:

  • Added notes
  • Split-chord members
  • Open 5ths (Chord of Omission)

One special case, especially important in the early part of the 20th century:

  • Whole-tone chords

Finally, the possibility of juxtaposing two or more aurally distinguishable sonorities:

  • Polychords

*It is frequently the case that a particular sonority may be open to more than one interpretation, particularly with mixed-interval chords, which can be arranged to resemble secundal, tertian, or quartal chords.  One must be sensitive to the musical context and the voicing in attempting to choose the best analytical approach.

Conventional Tertian Sonorities

Tertian triads and 7th chords are an important, though perhaps less preponderant, part of the harmonic vocabulary of 20th-century art music.  Some composers make greater use of them than others.  For example, works by composers such as Rachmaninoff, Menotti, Copland might be expected to contain a high proportion of triads and 7th chords, whereas other composers, such as Hindemith, tend to reserve the pure sound of a triad for important cadences or even for the end of a movement, etc.  Still, other composers rarely make use of these traditional sounds.

Tertian sonorities "taller" than the 7th chord--e.g., 9th, 11th & 13th chords--are not an important part of the harmonic vocabulary before the late 19th or early 20th centuries.

In theory, any diatonic triad can be extended to a 13th chord before its root is duplicated.  In practice, however, it is the dominant and secondary dominant chords, and to a lesser extent the supertonic and submediant chords, that tend to be singled out for this treatment.  Chromatic alterations, especially of chords with a dominant function, are often used.

Chords taller than a 7th are frequently incomplete, posing certain problems in analysis.  The answer to analyzing such sonorities depends on how one hears the sonority.  There is often more than one possible interpretation.

Tertian Chords with Added Notes

Though the possibility of a triad's having a note added a 6th above the root was recognized by theorists as early as the 18th century, chords with added notes did not become an accepted part of the harmonic vocabulary until the 20th century.  These chords are sometimes called chords of addition.

The basic chords are usually triads, and the added notes are usually 2nds, 6ths, sometimes 4ths (always figured above the root).  Any triad with an added 6th could also be analyzed as a 7th chord, but the context will usually settle the issue.  E.g., a C6 could be analyzed as Ami7.  Similarly, a triad with an added 2nd or 4th could be interpreted as an incomplete 9th or 11th chord, especially if voiced with the added note above the triad.  Since the root is the same in either case, the distinction is not a crucial one.  For all practical purposes, a chord with an added 2nd or 4th can be considered the same as with an added 9th or 11th.

Added notes are a feature of what is sometimes called "wrong-note style," in which the listener's conventional expectations are almost met, but not quite.  The result is often humorous.

Tertian Chords with Split Chord Members

A special kind of added-note chord features one or more chord members that are "split" by adding a note a minor 2nd away.  Common examples are triads and 7th chords with split 3rds; but split roots, 5ths and 7ths also occur.

There is no standard analytical symbol for split chord members.  Here, the exclamation point (!) will be used. E.g., C7(3!) = C E Eb G Bb.  In traditional popular/jazz notation, the preceding chord would be analyzed as C7(#9).  With split chord members, any correct enharmonic spelling is acceptable. E.g., E or Fb.

Open-5th Chords

While a large number of added-note chords are possible, there is one important "chord of omission," the triad without a 3rd.  Omitting the root or fifth from a triad, or omitting anything from a 7th chord, only results in another traditional sonority.  But the sound of an open 5th had been out of style for centuries, except for its occasional use in two-part counterpoint.

The sound of open 5ths rapidly becomes tiresome, so extended passages based on this chord are rare.  Typically they are used to create an impression of the Orient or of the distant past.

Quartal and Quintal Chords

Quartal chords are those built from 4ths, and quintal chords are those built from 5ths.  A quartal/quintal chord can have as few as three pitch classes, or it can have several.  It is sometimes possible to omit a member of a quartal or quintal chord without losing its character.  Various voicings and octave duplications are also used, but some voicings may destroy the character of the sonority.  Quintal chords have a more open and stable sound.

A convenient way to describe quartal and quintal chords is to use, for example, "3x4 on B" to mean a three-pitch-class quartal chord with B as the bottom pitch class (B,E,A).  5x4 on C = C, F, Bb, Eb, Ab. 7x5 on G = G, D, A, E, B, F#, C#, etc.  (The first number of the equation refers to the quantity of pitches, and the second number refers to the interval.)

Quartal and quintal chords are most often made up of perfect intervals, but augmented and diminished 4ths/5ths may be included.  The numerical analysis formula remains the same. E.g., 3x4 on C = C, F#, B or C, F, Bb.

A quartal/quintal chord is considered consonant if it contains 3-5 perfect 4ths or 5ths.  It is considered dissonant if it contains more than five factors, or one or more tritones.

Secundal Chords

Another possibility for chord construction is the secundal chord, a sonority built from major or minor 2nds, or from a combination of the two.  Such chords may be voiced as 7ths rather than 2nds, but this is the exception.  More often the notes of a secundal chord are placed adjacent to each other, an arrangement sometimes referred to by the terms "cluster" and "tone cluster."

In some keyboard works, special notation is used to indicate whether or not the black keys are to be included in the cluster.  Others require that the cluster be performed with the forearm or with a board.

A way of notating clusters that cannot easily be notated in terms of traditional analysis is to use the same system that is used with quartal/quintal analysis.  E.g., 5x2 on C = C, D, E, F, G.

Mixed-Interval Chords

A mixed-interval chord is on that did not originate as a series of 2nds, 3rds or 4ths, but instead combines two or more of those interval types (with inversions & compounds) to form a more complex sonority.  The possibilities are numerous.

Most mixed-interval chords are subject to other interpretations--that is, they could be arranged to look like secundal/quartal chords, etc.  In many cases, the context will suggest the correct analytical approach.  For example, if occasional secundal chords seem to be produced more or less coincidentally in an otherwise mixed-interval environment, then it might be better to analyze the entire passage in terms of mixed-interval chords.

This brings up the question as to how one goes about analyzing and labeling these sonorities, a complicated problem which has been tackled by various composers and theorists, such as Hindemith, Howard Hanson, Allen Forte. Because so many combinations of intervals are possible, a completely new system of chord classification had to be devised.

An analytical approach called pitch-set analysis is helpful along these lines.  Just as traditional triads are identifiable as units, and capable of being transposed, so are mixed-interval chords.

Simply stated, recurring pitch sets are identified, just as triads are identified in traditional analysis.  But in that traditional nomenclature typically does not work with the mixed-interval chord, each note of a determined pitch set is assigned a number chromatically from 0-11, with the fundamental being zero.  The first pitch-set would be Set 1 (consisting of any number of notes), followed by Set 2, and so on.

For example, if one recurring harmonic pattern is the notes F#, C, B, the F# (fundamental) would be assigned the number 0, B would be assigned 5 (5 half-steps above F#), C would be assigned 6 (6 half-steps above F#).  Pitch Set 1, therefore, would be [0,5,6]. Pitch Set 2 would consist of some other grouping of notes [0,?,?,?], and so on.  The assignment of Pitch Sets to harmonic groupings replaces the traditional chordal nomenclature.

The pitches are arranged in what is referred to as normal order, whereby the pitches are arranged as an ascending "scale."

Whole-Tone Chords

Any chord whose members could be obtained from a single whole-tone scale is a whole-tone chord.  A number of such chords are possible.


A polychord combines two or more chords into a more complex sonority.  The crucial qualification for a polychord is that the listener is able to perceive that separate harmonic entities are being juxtaposed.

Polychords usually result with the combination of tertian triads or 7th chords, but anything is possible.

In order to be heard as a polychord, the individual sonorities that make up the polychord must be separated by some means, such as register or timbre.

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