COMMON-PRACTICE TONALITY: A Handbook for Composition and Analysis
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# Pitch Interval, Consonance, and Dissonance

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 or instability.

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.

Measurement 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 Size

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 quality.

#### 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 augmented fourth.

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.

### Inverting Pitch Intervals

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).

A perfect interval inverts into another perfect interval.

A diminished interval inverts into an augmented interval (and vice versa).

Third, find the inverted absolute size by subtracting the absolute size of the original interval from twelve.

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 inversional relationships.

### 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)

## CONSONANCE AND DISSONANCE

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

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" than another.)

#### 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 classes.

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 fifth.

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.

### Dissonance

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 as consonances.

#### ENHARMONIC EQUIVALENCE

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.

#### DISSONANCE RELATED 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.

#### DISSONANCE 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 DISSONANCE

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.

## SUMMARY:

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 intervals.

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 others dissonant.

Any interval not contained in a consonant triad is a harmonic dissonance. Major and minor seconds, though harmonically dissonant, are melodically consonant.

Aldwell, Edward, and Carl Schachter. Harmony and Voice Leading. 2d ed. 2 vols. New York: Harcourt Brace Jovanovich, 1989. Chapters 1-2.

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. Division I.

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.

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