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Handbook for Composition and Analysis
Pitch Interval, Consonance, and
As we learned in Sound and
Its Notation, music has both vertical and horizontal dimensions.
It consists of both pitches sounded simultaneously (harmony) and pitches
sounded in series (melody). The intervals that separate pitches (harmonically
or melodically) provide those pitches with either a sense of stability
The study of harmony is the study of these two states, and of the
imaginative and dramatic control of the relationship between them. We begin
that study by identifying and classifying the different aspects of each.
of Pitch Intervals
A pitch interval (hereafter simply called and "interval") is
the distance between two notes. We calculate that distance by counting
the lines and spaces that separate the two notes on the staff. Alternatively,
we can calculate the distance by counting the number of half steps that
separate the notes on the keyboard. The former measurement of an interval
is its ordinal or diatonic size. The latter measurement its
absolute size. The absolute size of an interval determines that
interval's quality. We refer to an interval by both its diatonic
name and its quality.
The Ordinal or Diatonic
We determine the ordinal size of an interval by counting, inclusively,
the number of lines and spaces that separate the two notes involved. This,
in effect, measures the number of white keys that make up the interval.
The white keys of the piano form what we call a diatonic collection.
Thus, we frequently call the ordinal size of an interval its diatonic
size. (The diatonic collection is discussed in detail in Chapter 3
and Appendix K.)
The Absolute Size
We measure the absolute size of an interval by counting the number of half
steps between the bottom and top notes.
When you calculate
the absolute size of an interval, count the distances between successive
piano keys, not the keys themselves.
Remember that the absolute size of an interval determines that interval's
THE DIATONIC QUALITIES
Within a diatonic collection (the white keys of the piano, for instance),
each diatonic interval smaller than an octave comes in two (absolute) sizes.
For instance, of the seven thirds found between white keys, three are large
(four half steps) and four are small (three half steps).
The larger thirds are called major thirds. The smaller thirds
are called minor thirds.
With one exception, diatonic fourths (and fifths) come in a single size--five
half steps (fourths) and seven half steps (fifths). These are called perfect
fourths and fifths.
A diatonic interval can be of a type that is major or minor or
of a type that is perfect. It cannot be of both types.
THE CHROMATIC QUALITIES
One diatonic fourth (F-B) and one diatonic fifth (B-F) are not perfect.
Six half steps span the fourth F-B, but a perfect fourth is only five half
steps. A fourth that is one half step larger than a perfect fourth is an
The fifth B-F spans six half steps as well, but a perfect fifth
is seven half steps. This B-F fifth, then, is a half step smaller
than perfect. Such a fifth is called a diminished fifth. The interval
of six half steps (however it is spelled) is a tritone.
Diminished Intervals. If the
absolute size of a diatonic interval is one half step less than
the minor or perfect interval of that diatonic type, we call it diminished.
If it is two half steps less, we call it doubly diminished.
Augmented Intervals. If the
absolute size of a diatonic interval is one half step more than
a major or perfect interval of that diatonic type, we call it augmented.
If it is two half steps more, we call it doubly augmented.
Appendix E catalogues interval
names, qualities, and sies.
SIMPLE AND COMPOUND INTERVALS
Intervals of an octave or less are simple intervals. Intervals that
exceed the octave are compound intervals. A compound second (that
is, the interval of an octave plus a second) is a ninth. A compound
minor second is a minor ninth, a compound major second
is a major ninth, and so on.
To invert an interval we take the lower pitch of the interval and
make it the higher one. Alternatively, we can take the higher pitch of
the interval and make it the lower one. To invert an interval, follow these
steps in this order.
First, find the inverted diatonic name by subtracting
the original interval's diatonic size from nine.
Second, find the inverted quality by converting the original
quality as follows:
A minor interval inverts into a major interval (and vice versa).
Third, find the inverted absolute size by subtracting the
absolute size of the original interval from twelve.
A perfect interval inverts into another perfect interval.
A diminished interval inverts into an augmented interval (and vice
Remember: You must first
find the diatonic size of the inverted interval. For example, a third always
inverts into some kind of sixth, no matter what the absolute size.
Appendix F illustrates these
The Acoustic Foundations of Consonance and Dissonance
When we hear a note played by some instrument, we hear not only that primary
pitch, or fundamental, but a series of other pitches as well. We
call these subsidiary pitches overtones. Their frequencies are whole-number
multiples of the fundamental frequency. (That is, the first overtone is
twice the frequency of the fundamental, the second is three times that
of the fundamental, and so on.) As the overtones ascend above the fundamental,
the interval between successive overtones gets smaller as the ratio between
their frequenies becomes more complex.
Informally, musicians sometimes call overtones harmonics or partials.
This is not quite right. When these terms are used properly, the first
overtone refers to the second harmonic or partial. That is, the
first harmonic is the fundamental itself (see example 2-6).
The fundamental and the first five overtones above it form what is sometimes
called the chord of nature. In what sense this object is "natural"
is open to question. However, we do find in it the prototype for each of
the traditionally consonant intervals. (Gutwein's
students: study this link> Appendix P:
the harmonic series)
In large part, this book concerns the relation between dissonance and consonance.
We will continually reevaluate these terms as we go along. For now, think
of consonance as a state of stability and rest, and dissonance as a state
of instability or motion. Disregard the colloquial usage that associates
consonance with acoustic pleasure and dissonance with acoustic pain.
CONSONANCE AND DISSONANCE
We call consonant all perfect intervals, as well as all major and
minor intervals that do not contain adjacent pitch classes. Of the
consonant intervals, we call those with the least complex interval ratios
perfect consonances. We call the remainder imperfect consonances.
(Later, we will discuss in what sense one consonance is more "perfect"
THE PERFECT CONSONANCES
The perfect unison, the perfect fourth, the perfect fifth, and the perfect
octave are all perfect intervals. They are also perfect consonances.
THE IMPERFECT CONSONANCES
Major and minor thirds and sixths are imperfect consonances. (Major and
minor seconds and sevenths, since they contain adjacent pitch classes,
are not consonant at all, but dissonant.)
If you combine any three pitch classes so that none is a step from another,
you have created a triad. A triad contains no adjacent pitch
Origin. Many theorists derive
the triad from the "hord of nature." Many others question the adequacy
and others the accuracy of this derivation.
Structure. The triad consists
of three pitch classes: the root, third, and fifth. The root
of a triad is that pitch class standing respectively a third and a fifth
below the other two pitch classes of the triad. The third of a triad
is that pitch class standing a third above the root of the triad. The fifth
of a triad is that pitch class standing a fifth above the root of the triad.
Thus, with the root at the bottom of a triad, the other two pitch classes
stand a third and a fifth above that root. The kinds of thirds and fifths
that make up the triad determine the quality of the triad.
Qualities. A triad can be either
major, minor, diminished, or augmented. A major triad
has a major third between the root and third and a perfect fifth
between root and fifth. A minor triad has a minor third between
the root and the third. A perfect fifth spans the distance from root to
A diminished triad has a minor third between the root and third
and a diminished fifth between root and fifth. An augmented triad
has a major third between the root and third. As a result, an augmented
fifth spans the distance between root and fifth.
Appendix G provides a graphic
synopsis of triad structure.
DISSONANCE AND CONSONANCE
Major and minor triads are consonant since they contain only consonant
intervals. Diminished and augmented triads, however, contain fifths that
are not consonant (that is, not perfect). Accordingly, augmented and diminished
triads are dissonant.
All sevenths and diminished and augmented intervals are considered
dissonant. Whether we play the pitches of a seventh or an augmented
or diminished interval simultaneously (harmonically) or successively (melodically),
they remain unstable.
Major and minor secondsare, however, more ambiguous. If we express their
pitches harmonically, they are dissonant; but if we express them melodically
they are consonant.
HARMONIC AND LINEAR DISSONANCE
Because triads contain only nonadjacent pitch classes, the distance between
them is always some kind of skip. Melodies, however, move mainly by step
between adjacent pitch classes. Thus, in a harmonic context, major and
minor seconds behave as dissonances. In a melodic context, they behave
Two intervals of the same absolute size but of two different diatonic
sizes are enharmonically equivalent.
G-sharp-B spans a minor third--an interval of three half steps. Although
A-flat-B spans an augmented second, the absolute size is the same three
half steps. The distinction is not trivial, however. By the definitions
given above, that third is an imperfect consonance and the augmented second
is a dissonance, even though each is the same absolute size.
TO THE TRIAD
Major and minor triads contain only consonant intervals. A major or minor
triad contains no dissonant intervals--that is, no augmented or diminished
intervals and no adjacent pitch classes (seconds or sevenths). We can define
"dissonance" circularly, then, as any interval not present in either
a major or a minor triad.
COMPELLED TO MOTION
Major and minor triads shape and control the harmonic or vertical aspect
of music. These triads are consonant--that is, stable. What characterizes
the music we love, however, is a sense of motion, of dramatic arrivals
and departures. Dissonance provides this sense of motion and drama.
THE PASSING NATURE OF
Consonance is both a point of departure and a goal. Dissonance is neither;
it is unstable. Dissonance takes us from one place (consonance) to another.
In fact, "good" harmony is nothing more (or less) than the imaginative
and dramatic use of dissonance.
Intervals have both a diatonic name and a quality. To identify an interval
fully, we need both designations. We can invert an interval by placing
he bottom pitch on top or the top pitch on the bottom. Intervals are either
consonant or dissonant. Acoustics suggests an origin for the consonant
Three pitch classes chosen so that none is adjacent to another make
up a triad. We call the pitch classes of a triad the root, the third, and
the fifth. We consider those triads with perfect fifths consonant and all
Any interval not contained in a consonant triad is a harmonic dissonance.
Major and minor seconds, though harmonically dissonant, are melodically
Aldwell, Edward, and Carl Schachter. Harmony and Voice Leading.
2d ed. 2 vols. New York: Harcourt Brace Jovanovich, 1989. Chapters 1-2.
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Bamberger, Jeanne Shapiro, and Howard Brofsky.
The Art of Listening: Developing Music Perception. 5th ed. New York:
Harper & Row, 1988. Chapters 1-3.
Fux, Johann Joseph. The Study of Counterpoint from Johann Fux's "Gradus
ad Parnassum." Translated and edited by Alfred Mann. New York: Norton,
1965. Chapter 1.
Hall, Donald E. Musical Acoustics. 2d ed. Pacific Grove, CA:
Brooks/Cole, 1991. Chapter 1.
Mitchell, William J. Elementary Harmony. 2d ed. Englewood Cliffs,
NJ: Prentice-Hall, 1948. Chapter 1.
Piston, Walter. Harmony. 5th ed. Revised and expanded by Mark
DeVoto. New York: Norton, 1987. Chapter 1.
Schenker, Heinrich. Harmony. Edited by Oswald Jonas. Translated
by Elisabeth Mann Borgese. Chicago: University of Chicago Press. 1954.
Schoenberg, Arold. Theory of Harmony. Translated by Roy E. Carter.
Berkeley: University of California Press, 1983. Chapter 1.
Westergaard, Peter. An Introduction to Tonal
Theory. New York: Norton, 1975. Chapter 1.
Williams, Edgar W. Harmony and Voice Leading. New York: HarperCollins,
1992. Chapter 2.