MUS 312 Form & Analysis
L. The Vertical Dimension: Chords &
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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
- Secundal chords (also tone clusters)
- Tertian chords (including 9ths, 11ths, 13ths, and
- 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
Finally, the possibility of juxtaposing two or more aurally
*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
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
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.
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.
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 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.
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
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
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."
Any chord whose members could be obtained from a single whole-tone
scale is a whole-tone chord. A number of such chords are
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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