Harmony and voice leading 5th edition edward aldwell

Harmony and Voice Leading 5th Edition Edward Aldwell

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HARMONY & VOICE LEADING 5th Edition

Edward Aldwell

The Curtis Institute of Music Mannes College of Music

Carl Schachter Mannes College of Music

The Juilliard School

Allen Cadwallader Oberlin Conservatory of Music

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Harmony & Voice Leading, Fifth Edition

Edward Aldwell, Carl Schachter, Allen Cadwallader

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Preface

PART I THE PRIMARY MATERIALS AND PROCEDURES 1

1

PART VI DISSONANCE AND CHROMATICISM II

Equal Divisions of the Octave 619

Points for Review 629

Exercises 630

33 Chromaticism in Larger Contexts 632

New Modulatory Techniques 633

Chromatic Tonal Areas 641

Points for Review 652

Exercises 652

Appendix I Keyboard Progressions 656

Appendix II Score Reduction 678

Appendix III

Roman Numerals and Registers 682

Index of Musical Examples 690

Subject Index 694

Dedication

In Memoriam Edward Aldwell

(January 30, 1938–May 28, 2006)

Edward Aldwell’s untimely death occurred durinsg the planning stages for the Fourth Edition but before actual work had begun. His ideas about music and its teaching are evident on every page of this new edition. Edward was a complete musician. He was primarily a pianist, and a wonderful one. Edward was a great teacher of both theory and piano; he was also an outstanding chambermusic coach. His performance students benefited from his analytical insights, and his theory students benefited from his ability to derive implications for performance from theoretical observations—and from the way he translated these implications into sound in his beautiful demonstrations at the piano. My work with Edward on Harmony & Voice Leading was a genuine and close collaboration; whatever might be good in the book is due equally to both of us. Our friendship lasted more than forty years, and his death is a grievous loss to me and to the countless students and colleagues who gained so much from his artistry and wisdom.

Preface

The Fifth Edition of Harmony & Voice Leading retains the approach and goals of its four predecessors. It offers a thorough and comprehensive course of study in harmony in the music of the eighteenth and nineteenth centuries. At the same time, it emphasizes the linear aspects of music as much as the harmonic, with relationships of line to line and line to chord receiving as much attention as relationships among chords. Large-scale progressions—both harmonic and linear—are introduced at an early stage so that students can gain an understanding of the connection between detail and broad, inclusive plan in a musical composition. They learn that “harmony” is not merely the progression from one chord to the next and that “voice leading” is much more than the way two consecutive chords are connected.

In preparing this new edition, we have reviewed the entire text to improve our manner of presentation, our examples from the literature, our use of terminology, and our exercises. We hope that these changes will make the book more effective for both teacher and student. Among the new features are the following: We have incorporated the upper- and lower-case system of Roman numeral analysis to facilitate students in the recognition of chord quality. Furthermore, we have informally introduced Schenker’s notion of linear progressions, an important concept in later, more advanced work in tonal analysis.

The book is suitable either for a self-contained course in harmony or for an integrated program combining harmony with other aspects of music. Harmony & Voice Leading touches on many of these aspects, including rhythm, melody, counterpoint, and form. It can function, therefore, as the basic text for an integrated program and can serve as a convenient point of departure for systematic work in the other areas, with or without a supplementary text. Many theory programs have returned to the study of species counterpoint, usually at an early stage. This book would combine very well with work in species counterpoint; such a combination would provide an excellent basis for the understanding of tonal music. But counterpoint need not precede or accompany work in Harmony & Voice Leading; this is a completely self-contained and self-sufficient text.

In most theory programs, instruction in harmony or counterpoint usually follows a review of fundamentals: scales, key signatures, intervals, and so forth. This initial phase can pose difficult problems for instructors. Students vary

widely—even wildly—in the quality of their previous training. And even those with a reasonably secure grasp of the fundamentals seldom understand the significance of the material they have learned by rote. The first three units of Harmony & Voice Leading attempt to deal with these problems. They offer both a review of the fundamental materials and a glimpse—a first glimpse for most students—of their significance for musical structure. Thus, these opening units attempt to provide both a practical and a conceptual basis for the students’ later work. For students deficient in their knowledge of fundamentals, we have provided a large number of written drills in the accompanying workbooks, as well as a smaller group in the text itself. Better prepared students will not need to devote much time to these drills, but they will profit from reading through the first three units and from classroom discussion of their contents.

The length of time needed to work through the book will vary, depending on the students’ preparation, the number of class hours devoted to harmony, and the thoroughness of the course. We have tried to treat the subject as comprehensively as possible, knowing that different teachers may have very different ideas about which aspects of the study to emphasize. Some may choose to skim over certain units or sections within units, while others may devote a fair amount of time to those very units or sections. We hope that our comprehensive approach will give teachers the possibility of designing a course that best fits their students’ needs. We know that some schools that have adopted the book go through it in three semesters and that others take as long as three years (together, of course, with other aspects of music, notably counterpoint). Most schools seem to accommodate it within a two-year theory sequence. The text and the two workbooks contain far more exercise material than could be covered in any single course. Instructors can thus choose the number and type of exercises that best meet the needs of their particular class. The remaining exercises will provide valuable material for classroom demonstration, exams, and review.

The order in which important materials and procedures are presented differs from that found in any other text. After a discussion of chord vocabulary, chord construction, and voice leading, the fundamental harmonic relationship between tonic and dominant is introduced, and the discussion then proceeds quickly to the most frequent linear expansions of tonic harmony. Confining students’ work in these initial stages to a single harmonic relationship and to a number of closely related contrapuntal ones makes it much easier for them to hear what they are doing than if they are confronted immediately with seven root-position chords, each with a different sound and function. In subsequent units, students learn new usages a few at a time in a way that relates to and expands on the techniques they have already mastered. This order of presentation also makes it possible to show examples from the literature at a much earlier stage than in other approaches—and without including usages that students have not yet learned. Thus, they develop their ability to hear in a logical and orderly fashion, and they can begin their analysis of music of the highest quality much sooner than in other approaches. The book’s order of presentation also makes it possible to pursue a number of fundamental concepts, such as tonic-dominant relationship, voice exchange, and 5-6 technique, by starting with their simplest manifestations and gradually revealing more complex developments and ramifications. By relating new

material to large inclusive ideas, rather than simply piling rule upon rule, we hope to help students to begin thinking about music in productive ways that will benefit their analysis, writing, and performing.

Although Harmony & Voice Leading probably covers more material than any comparable text, it does not require an inordinate amount of time to complete. Nonetheless, this book offers no shortcuts. There are no shortcuts in learning music theory—especially in the development of writing skills. If twenty-first-century students wonder why they need to master such skills—why they need to take the time to learn a musical language spoken by composers of the past—they can be reminded that they are learning to form the musical equivalents of simple sentences and paragraphs. The purpose is not to learn to write “like” Mozart or Brahms, but to understand the language the great composers spoke with such matchless eloquence, the language that embodies some of the greatest achievements of the human spirit.

In the initial years of the twenty-first century, no one can minimize the importance of a thorough study of twentieth-century music. But we believe that to combine in a single text an intensive study of tonal harmony with an introduction to twentieth-century techniques would fail to do justice to either subject. For one thing, some of the simplest and most fundamental principles of earlier music—the functioning and even identity of intervals, for example—become radically altered in twentieth-century usage, so that it is impossible to proceed directly from one kind of music to the other. And the twentieth century has seen the development of compositional styles that sometimes differ from one another so profoundly as to amount to different languages. To deal adequately with this disparate and often complex material requires a separate text.

Many readers will realize that this book reflects the theoretical and analytic approach of Heinrich Schenker, an approach many musicians recognize as embodying unique and profound insights into tonal music. Harmony & Voice Leading is not a text in Schenkerian analysis—no knowledge of it is presupposed for either instructor or student—but the book will provide a valuable preparation for the later study of Schenkerian analysis. We believe that a solid foundation in harmony and voice leading is an indispensable prerequisite for learning Schenker’s approach; without it, students have no secure basis for the analytic judgments they are called upon to make.

The Fifth Edition features MindTap, which offers several challenging and interesting features, including playable musical examples, flashcards, ReadSpeaker, and opportunities for instructors to add their own teaching materials to the learning path.

Students are now able to use their computers as an adjunct to the text and may listen to examples while reading the explanations and studying the scores.

Students may access MindTap using a passcode either bundled with their text or purchased online at www.cengagebrain.com.

Your MindTap for Harmony & Voice Leading, Fifth Edition, includes:

● The eBook for Harmony & Voice Leading , Fifth Edition, including the narrative, examples, and exercises as well as the appendices.

● All the audio examples, as part of the eBook.

● Guided exercises, which show in stages the solution for one exercise from the core book’s Units 7–16 and 18–33.

● All workbook exercises in PDF format.

Finally, we have developed for instructors a series of guidelines that highlight the primary topics of each unit; we hope the suggested strategies will be of use in developing class presentations.

Acknowledgments

We are grateful to several individuals who have helped us bring the Fifth Edition to fruition. Karen Bottge offered many fine suggestions for the new edition of the workbooks and applied her keen editorial skills in proofreading the manuscript and page proofs. We also express our sincere gratitude to Sue Gleason Wade, our first editor at Cengage, for her frequent support and consultation, sometimes given on a daily basis. Revising a book of this scope is no easy task, and Sue graciously shared her keen editorial skills on every aspect of the project, making our work easier and more enjoyable.

We are grateful to the following reviewers of the Fifth Edition: Matthew Bribitzer-Stull, University of Minnesota, Twin Cities; Stephen Emmons, Angelo State University; Ellie Hisama, Columbia University; Timothy Howard, California State University, Northridge; Jennifer Iverson, University of Iowa; Kyle Jenkins, Georgia State University; Andrey Kasparov, Old Dominion University; Francis Massinon, Austin Peay State University; Edward Smaldone, City University of New York, Queens College; and Heather Worden, Broome Community College.

PART I

The Primary Materials and Procedures

Key, Scales, and Modes

1-1 Mozart, Piano Sonata, K. 545, I

Listen to the audio of all the unit’s examples.

Practice the workbook’s exercises, available in PDF.

Tonal Relationships; Major Keys

1. Key. We’ll begin by considering the opening of Mozart’s familiar Sonata in C major, K. 545 (Example 1-1). The piece obviously contains many tones besides C. Why, then, do we call it a “sonata in C major,” or say that “it’s written in the key of C”? Most people would answer that music is “in a key” when its tones relate to one central tone—the one that has the same name as the key—and when the functions of the other tones result from the ways in which they relate to the central one. According to this answer, the Mozart Sonata is in C because C is the central tone; we hear the other tones as subordinate to C. (Why it’s not simply in C but in C major, we’ll discuss presently.)

This explanation of key is certainly correct as far as it goes, but it tells us little about the kinds of relationships that exist between the central tone and the others. (A definition of chess as “a game played on a board by two people, each with sixteen pieces” would be correct in the same way. But it wouldn’t help anyone to learn to play chess.) Let’s now look more closely at these relationships.

2. The Tonic. We call the central tone of a key the tonic. In Example 1-2, which shows the most prominent tones of the Mozart, both hands begin on the tonic, C. The left hand stays around C for most of bars 1–4 and moves on to C as the lowest point in the downward motion F-E-D-C, bars 5–8. The right-hand part does not return to C after the opening bars, but its subsequent course points to C as its eventual goal. In bars 3 and 4, the melody moves from the high A down as far as E. The sixteenth-note scales that follow repeat, in varied form, the melodic line A-G-F-E but then carry it one step further, to D (bar 9). In listening to the melody, we are led to expect it to finish on C, to complete the circle by ending where it began. But it doesn’t—not yet, at any rate. Instead the D is taken up again in bars 11 and 12; the first part of the piece closes without having arrived at its melodic goal.

And, in fact, C’s function as a goal is not fulfilled until almost the end of the piece (Example 1-3). Generalizing from the Mozart, we can state that the tonic, the central tone of the key, forms the point of departure from which the other tones move and the goal to which they are directed. As in bars 1–12, the music does not always reach its goal at the moment we expect it to; by ending a part of the piece in a state of suspense, a composer can enhance the feeling of finality at the very end.

3. Scales. In Example 1-1, Mozart uses only some of the tones that the piano keyboard can produce. Almost all the sounds in these twelve bars result from playing the white keys; of the nearly 200 notes, the only exceptions are two C ’s (bar 9) and one F (bar 10). And if we were to look at other pieces in C major, we would find similar tonal materials. For the most part the pieces would contain the tones C, D, E, F, G, A, and B, and any other tones would play a subordinate role.

When all the tones that belong to a key occur in consecutive order, each one next to those closest to it in pitch, the result is a scale (Latin scala, “steps, staircase, or ladder”). In bars 5–8 of the Mozart, C-major scales occur beginning on A, G, F, and E. The basic form of a scale, however, is the one that

begins and ends on the tonic. A scale in this basic form can be thought of as a symbol of, or abstraction from, the natural flow of music—at least of music that is written “in a key.” For such a scale begins on the tonic as its point of departure and concludes on the tonic as its goal (Example 1-4).

1-4 scale degrees in C1

The capped numbers above the notes in Example 1-4 indicate scale degrees (sometimes called scale steps) and will be used for this purpose throughout the book. In addition to numbers, the following traditional names are used so often for the scale degrees that you should memorize them:

ˆ 1. Tonic

2. Supertonic

3. Mediant

4. Subdominant

5. Dominant

6. Submediant

7. Leading tone

4. The Octave. The beginning and ending tones of Example 1-4 are both C, but they are not one and the same tone. The last tone sounds considerably “higher” in pitch than the first. Yet, despite this marked difference in register, the sounds of the two C’s are very similar; that’s why we call them both by the same letter name. When two tones are separated by an octave (Latin octava, “eighth”) they are equivalents—that is, they are variants of the same sound. This phenomenon of octave equivalence is one of the most important aspects of pitch organization in music. In technical writing about music, it is frequently helpful to indicate the register in which a tone occurs. Example 1-5 shows how this can be done.

1Throughout the examples, the exercises, and the workbook, capital letters are used for major keys and lowercase letters are used for minor keys. Thus, G and g indicate the keys of G major and G minor, respectively. (Names of individual pitches, in contrast, are always indicated with a capital letter.)

5. Major Scales; Whole Steps and Half Steps. If we play a white-key scale from C to C on the piano, we can easily see that there is a black key between most of the adjacent white keys—between C and D, D and E, F and G, and so on. However, no black key appears between E and F or between B and C. The distance between one tone of a scale and the next is usually called a step. The scale from C to C contains two kinds of steps: small ones between E and F and between B and C, larger ones between the other adjacent tones. The small ones occur where there is no intervening black key; the larger ones occur where there is a black key. (See Example 1-6.)

We call the smaller steps half steps (or semitones) and the larger ones whole steps (or whole tones). The half steps occur between ˆ 3 and ˆ 4 and between ˆ 7 and ˆ 8; all the others are whole steps.

A scale with half steps and whole steps arranged in this order is called a major scale. Only the major scale has half steps between ˆ 3 and ˆ 4 and ˆ 7 and ˆ 8. Any piece whose tones form the same pattern of whole and half steps, starting with the tonic, is a piece in a major key.

The major scale is one kind of diatonic scale. All diatonic scales contain five whole steps and two half steps within the octave, but each of the different types of diatonic scale has the half steps in different places. From the time of the ancient Greeks through the nineteenth century, most Western art music was based on diatonic scales. Other kinds of scales are used in some Western folk music, music of non-Western cultures, and much twentieth-century music.

6. Intervals. Example l-7a shows the tones that begin each of the first four bars of the Mozart Sonata in both the left-hand and the right-hand parts. We call the relationship between two tones heard in a single context an interval. Intervals formed by simultaneously sounding tones are called vertical (because they are written one above the other). Intervals formed by tones that sound one after the other are called horizontal (Example l-7b). The terms harmonic and melodic are often used instead of vertical and horizontal.

1-7 (a) vertical intervals (b) horizontal intervals

We can describe intervals, whether horizontal or vertical, by ordinal numbers arrived at by counting letter names up from the lower to the higher tone, or down from the higher to the lower. Thus C up to G is a 5th, because it spans five letter names, C, D, E, F, and G. From B to C is a 2nd, because it spans two letter names. From G to the next G above is an octave (not an “eighth,” though it has the same meaning as “octave”). Finding the numerical size of an interval does not identify it completely. For example, B-C and C-D are both 2nds. Yet C-D (a whole step) is larger than B-C (a half step). Later on we will specify intervals more exactly; for now, it is enough to be able to determine the numerical size.

7. Chords; Triads. Compare the first and last bars of the Mozart (Examples 1-1 and 1-3). Both bars contain the same three tones (with octave duplications); the tones are C, E, and G ( ˆ 1, ˆ 3, and ˆ 5). These three tones are very closely associated, the basis of their association being membership in the same chord. A chord is a group of three or more tones that make sense when played or sung all at the same time. In essence a chord is a vertical unit; the simplest and most basic way to present it is as a block chord, with all the tones sounding at once (as in the last bar of the Mozart, second beat). But a composer can also present the tones one after the other, as Mozart does in bar 1. Because our ear and memory can group the tones into a unit, we still hear a chord. But not a block chord; it is a broken chord or arpeggio.

The chord C-E-G contains three tones; the upper two form the intervals of a 5th and a 3rd from C, the lowest. A three-tone chord formed in this way is a triad. The triad is the basic chord in Western music from the fifteenth through the nineteenth centuries. All other chords are derived from it. In every key, the triad ˆ 1ˆ 3ˆ 5 has the tonic as its lowest tone. The lowest tone, called the root, functions as the basis of the chord, so we call this triad the tonic triad or tonic chord.

8. Active Tones; Stable Tones. Among the many mysterious powers of music is its ability to suggest motion. In listening to a piece of music, we do not hear a succession of static tones; rather, we hear tonal motions, one tone moving to another. In part, this impression comes from rhythm, for musical rhythm has

close relationships to some of the physical activities—walking, for instance— that form our primary experience of motion. But the impression of motion also arises from tonal organization. We have already seen that ˆ 1, the tonic, functions as the goal to which the other tones are directed. (And musical motion is essentially directed motion, motion to a goal.) We might say that all the other scale degrees, in different ways, are active in the direction of ˆ 1, that they tend to move to this stable, central tone. However, ˆ 3 and ˆ 5 can also function as stable tones, though they are less stable than ˆ 1. They can serve as goals to which other, still more active tones can move because they are members of the tonic triad and thus closely associated with ˆ 1. Motion to ˆ 3 or ˆ 5 will not have the same finality as motion to ˆ 1.

Many melodies begin on ˆ 3 or ˆ 5 rather than on ˆ 1. If these melodies are harmonized, the tonic will almost always appear in the lowest part. Thus, the music will still move from a tonic at the beginning to a tonic as the final goal even if ˆ 1 does not serve as the initial melodic tone.

9. Passing Tones; Neighboring Tones. Example 1-8 contains a diagram of the C-major scale. The stable tones, ˆ 1, ˆ 3, ˆ 5, and ˆ 8, are shown as whole notes; the more active tones are written with black noteheads.

As the diagram indicates, the active tones lead from one stable tone to another: up from ˆ 1 to ˆ 3, ˆ 3 to ˆ 5, and ˆ 5 to ˆ 8; down in the reverse order. A tone that forms a stepwise connection between two stable tones is called a passing tone (abbreviation, P); the term clearly conveys the transitional character of these tones. We can readily hear this transitional character if we play the scale in the right hand while holding the tonic triad in the left. Note that a single passing tone connects ˆ 1 with ˆ 3 and ˆ 3 with ˆ 5, but that two passing tones are needed to connect ˆ 5 with ˆ 8.

Motion along the scale—that is, motion with passing tones—is by no means the only type of melodic progression, though it is the basic type. Example 1-9 shows another important possibility. Here the active tones decorate a single stable tone rather than move from one to another. A tone that moves by step away from and back to a stable tone is called a neighboring tone, or simply neighbor (N). Sometimes it is helpful to specify the direction of a neighboring tone by referring to it as an upper or a lower neighbor (UN or LN).

1-9 neighboring tones

In Unit 5 we will examine passing and neighboring tones in greater detail. For now, look at the two excerpts of Example 1-10, which will introduce how these tones work in a musical composition. In the Brahms, the accompaniment sustains ˆ 1 and ˆ 3; with the melody’s prominent ˆ 5 (opening upbeat and long notes in bars 1 and 2), a complete tonic chord seems to form the background of the whole line. Against this background, two segments of the descending E scale create melodic motion: the first one leads from ˆ 8 down to ˆ 5 (bar 1) and the second from ˆ 5 to ˆ 1 (bars 1–2). In listening to the first segment, we experience no stability until the line comes to rest on ˆ 5; the transitional, “passing” character of ˆ 7 and ˆ 6 is very evident. In the second segment, ˆ 4 and ˆ 2 receive more rhythmic emphasis than ˆ 3 and ˆ 1, but the dissonances they form against the “background” tonic direct them strongly toward the more stable tones.

In the Messiah excerpt, the scalar motion leads from ˆ 1 up to ˆ 5, which is embellished with its upper neighbor, ˆ 6. This upward motion is balanced by a line that moves down, but only as far as ˆ 3, not ˆ 1. Note that bar 2 contains the motion ˆ 5ˆ 4ˆ 3 on two different levels. The main note of the first beat, G, moves on to F and E on beats 2 and 3, but a smaller version of the same line fills out beat 1. In music, as in language, we perceive connections between elements that are not right next to one another. We hear ˆ 5 in the Handel moving to the ˆ 4 of beat 2 despite the two notes in between, just as we connect the subject and verb of a sentence even if they are separated by many words.

1-10

(a) Brahms, Intermezzo, Op. 117/I

(b) Handel, Pastoral Symphony (from Messiah)

10. Half Steps as Melodic Intensifiers. When an active tone, P or N, lies a half step from the stable tone to which it is attracted, its motion to the goal tone has a particularly intense character. The closeness in pitch between the two tones draws the active tone into the gravitational field, as it were, of the stable one and enhances the attractional power of the latter. In major, therefore, ˆ 4 tends to move more readily to ˆ 3 than to ˆ 5, the other possible goal tone. And ˆ 7 is very strongly attracted to ˆ 8; in fact, the term leading tone refers to the active way in which ˆ 7 “leads into” ˆ 8. The half steps are very well situated in major; the instability of ˆ 7 and ˆ 4 helps to strengthen ˆ 1 and ˆ 3 and leads to a clear definition of the key. Play the right-hand part of Example 1-1, extending the duration of B (bar 2) and F (bar 4). Notice how urgently the ear demands a continuation to C and E, respectively.

11. Incomplete Neighbors; Double Neighbors. Sometimes a neighboring tone will connect with only one statement of the stable tone rather than two; the neighboring tone will move either to the stable tone or away from it, but not both. In the melodic fragment shown in Example 1-11, the stable tones are A, C , and E ( ˆ 1ˆ 3ˆ 5 of A major). The G , D, B, and F are active tones that precede or follow (but not both) one of the main tones. We use the term incomplete neighbor (IN) to denote neighboring tones connected with one rather than two main tones.

1-11 Mozart, Piano Concerto, K. 488, I

Also derived from the neighbor is a four-note group consisting of a stable tone, both the upper and the lower neighbor (in either order), and a return to the stable tone. Two neighbors occurring together are called a double neighbor (DN), as in Example 1-12 (we will examine double neighbors again in Unit 5).

1-12 Haydn, Symphony No. 98, II

12. Transposition; Key Signatures. In bar 42 of the Mozart movement with which we began this unit, the opening idea returns; the technical name for such a return in a sonata movement is recapitulation. Usually a recapitulation is in the same key as the beginning of the movement, but most exceptionally, Mozart does not begin his recapitulation in C. Instead of C, F functions as the central tone; the music has moved to F major. Example 1-13 quotes the opening few bars of this F-major recapitulation; observe that every time a B occurs, it is modified by a flat. A moment’s reflection (and, perhaps, a glance at the keyboard) will show why the B is needed. Without it, there would be a whole step between ˆ 3 and ˆ 4; the music would no longer be in F major.

Putting a piece (or section) of music into another key is called transposing it. If we transpose a piece from C to any other major key, we have to use flats or sharps to preserve the half steps between ˆ 3 and ˆ 4, and ˆ 7 and ˆ 8. These sharps or flats are gathered together into a key signature that occurs at the beginning of each line of music. Sometimes a change of key within a piece is accompanied by a new key signature, but very often, as in the Mozart, the necessary flats, sharps, naturals, and so on, occur in the body of the music as accidentals, like the flats before the B’s in Example 1-13.

Example 1-14 shows the signatures of the fifteen major keys. Note that the keys with sharps move up in 5ths; the tonic of each new key lies a 5th above the preceding tonic. Also note that the keys with flats do just the opposite—they move down in 5ths.

1-14 major key signatures

(a)

If you have not already done so, memorize these key signatures immediately. Not doing so will cause you unnecessary difficulties in studying music theory. Note that memorizing them means being able to recall them instantly and automatically.

13. Chromaticism; Chromatic Half Steps. In the recapitulation of the Mozart, the use of an accidental—B —results from a change of key to F major. However, accidentals do not always signal a change of key; in fact, they usually do not. Very often they occur when a composer wishes to emphasize a scale degree by means of the melodically intense half-step progression. In Example 1-1, the F in bar 10 (left hand, last tone) intensifies the G that follows. Example 1-15 shows the specific function of this F by leaving out some of the less important tones and simplifying the use of registers. It reveals that the F leads from the F of bars 9 and 10 to the G of bar 11; it functions, therefore, as a kind of passing tone.

The use of tones that normally do not belong to a key is called chromaticism; Mozart’s F , therefore, is a chromatic passing tone. Chromatic elements embellish a basically diatonic substructure; the term chromatic (Greek chroma, “color”) clearly conveys the decorative character of these tones. As Example 1-15 indicates, the use of chromatic tones creates the possibility for a new kind of half step, the chromatic half step. The half step F -F involves two tones with the same letter name, whereas the diatonic half step (for example, B-C) involves two tones with adjacent letter names. Chromatic passing tones divide a whole step into a chromatic half step plus a diatonic one (F -F -G). The chromatic half step normally comes first; the chromatic passing tone uses the same letter name as the preceding diatonic tone. Thus, a chromatic passing tone from A down to G would be A ; the melodic progression, therefore, would be A -A -G. Chromaticism sometimes involves the use of double sharps and double flats. A chromatic passing tone between F and G , for example, would be F ; one between B and A would be B .

Not every chromatic tone produces a chromatic half step. The C ’s in bar 9 of Example 1-1 do not. They intensify the motion to D through the half-step progression, but they lie a whole step above the preceding tone, B.

14. Enharmonics. On the piano, we depress the same key to produce C and D , A and B , and so on. To facilitate playing in all keys and to make possible an extensive use of chromaticism, keyboard instruments are tuned to the equally tempered scale, a scale that divides the octave into twelve equal semitones. Tempered tuning eliminates the minute differences in pitch between, say, G and A or B and C . Two tones with different names but the same pitch (or, in nontempered tuning, almost the same pitch) are called enharmonic equivalents.

Minor Keys; Modes; Tonality

15. Minor Keys. Example 1-16 is the beginning of a variation movement by Handel. The key signature contains two flats, but this composition is clearly not in B major. The lowest part begins and ends on G; the highest begins on D and ends on G; the opening chord contains the tones G, B , and D. All of this points to G as the tonic and to G-B -D as the tonic triad. And, in fact, the piece is in the key of G, but G minor, not G major.

1-16 Handel, Passacaglia (from Harpsichord Suite No. 7)

Why this piece is in minor becomes very clear if we compare its tonic triad with the tonic triad of G major (Example 1-17). 1-17

ˆ 1 and ˆ 5 are the same in both chords; only ˆ 3 varies (the B is closer to G than is the B ); the 3rd G-B , therefore, is smaller than the 3rd G-B . Minor and major simply mean smaller and larger. A minor key is a key containing a small or minor 3rd between ˆ 1 and ˆ 3; a major key is a key containing a large or major 3rd between ˆ 1 and ˆ 3. There are other significant differences between major and minor, but the contrast in sound between the two kinds of 3rds marks the fundamental distinction between them.

16. The Natural Form of Minor. Example l-18a shows the beginning of a later variation from the Handel Passacaglia. The right-hand part contains descending scales that follow the key signature exactly; no chromatic alterations occur. Example 1-18b is a diagram of the scale Handel uses, showing its stable and active tones, as well as the location of its two half steps. The scale in this diagram is the natural (or pure) minor scale.

1-18

(a) Handel, Passacaglia

(b) natural minor scale

The contrast with major is striking. The minor 3rd between ˆ 1 and ˆ 3 lends its characteristic color to the scale. The half steps between ˆ 2 and ˆ 3, and ˆ 5 and ˆ 6, create an intensity in the motions from ˆ 2 to ˆ 3 and from ˆ 6 to ˆ 5 quite different from the corresponding progressions in major. Finally—and very significantly—the whole step between ˆ 7 and ˆ 8 fails to lead into the tonic with the same conviction as in major. For this reason, the term leading tone is not used to indicate the 7th degree of the minor scale in its natural form. We use the term subtonic instead.

When the minor scale descends (as in Example l-18a), the lack of a leading tone does not present a problem, for ˆ 7 leads away from ˆ 8 rather than toward it. But when the scale ascends, the whole step between subtonic and tonic can constitute a real defect because ˆ 8 does not sound like a goal; its power to act as the central tone of the key is impaired. For this reason, ˆ 7 in minor must be raised to create the necessary half step whenever it moves to ˆ 8 as a goal, or whenever the composer wishes to suggest such a motion, even if it is not immediately fulfilled. That is why Handel uses F instead of F in bars 3 and 4 of Example 1-16.

17. The Harmonic Form of Minor. Raising the 7th degree but leaving the others unaltered produces the scale shown in Example 1-19. This scale is called the harmonic minor, for many important chord progressions use the tones it contains. However, one characteristic of this scale makes it unsuitable for normal melodic progression. The interval between ˆ 6 and ˆ 7 is larger than a whole step; it is equivalent, in fact, to a step and a half. This larger interval creates a gap in the continuity of the scale that can destroy melodic flow. As Example 1-19 shows, the harmonic minor has three half steps: between ˆ 2 and ˆ 3, ˆ 5 and ˆ 6, and ˆ 7 and ˆ 8. In the keys of G minor, D minor, and A minor, raising ˆ 7 requires a double sharp; in A minor, for example, the leading tone is G .

1-19 harmonic minor scale

18. The Melodic Form of Minor. If we raise ˆ 6 as well as ˆ 7, we gain a leading tone, but without creating an awkwardly large interval between ˆ 6 and ˆ 7. In a melodic line in minor, therefore, if ˆ 6 comes before the leading tone (raised ˆ 7), it too will be raised. Note, for example, the E in bar 4 of Example 1-16, used instead of the E called for by the key signature.

The minor scale that raises ˆ 6 and ˆ 7 ascending is called the melodic minor scale (Example 1-20a). Example 1-20b, still from the Handel Passacaglia, illustrates its use in a composition. Note that it contains two half steps, between ˆ 2 and ˆ 3, and ˆ 7 and ˆ 8.

1-20 (a) melodic minor scale

(b) Handel, Passacaglia

Because ˆ 6 and ˆ 7 are raised so they lead convincingly to ˆ 8, the raised forms of these degrees will normally occur only when the scale goes up. The descending form of the melodic minor, therefore, reverts to the natural form, with the accidentals for ˆ 6 and ˆ 7 canceled.

19. The Three Forms of Minor. Beginning students sometimes have the misconception that the three forms of minor constitute three independent and unrelated scales. Actually, the harmonic and melodic forms are variants of the natural minor scale. The fact that the key signature always corresponds to the natural minor indicates that this is the basic form of the scale.

Most compositions in minor will contain elements of all three forms of the scale. Some successions of chords will come from the natural form (Example 1-16, bars 1 and 2); others will come from the harmonic (Example 1-16, bars 3 and 4). Melodic lines that ascend ˆ 6ˆ 7ˆ 8 tend to use the ascending melodic scale; descending lines tend to use the descending melodic (or natural) form (Examples 1-18 and 1-20).

20. Key Signatures in Minor. Like C major, A minor has neither sharps nor flats. As we move up in 5ths from A, we must add one sharp each time to the key signature to preserve the correct pattern of whole steps and half steps. As we move down in 5ths, we add flats. Example 1-21 shows the signatures for the fifteen minor keys, which you should memorize.

1-21 minor key signatures

(a)

(b)

21. Relative Major and Minor. The terms relative major and relative minor are often used to denote a major key with the same signature as a given minor one, and vice versa. Thus, C major would be the relative major of A minor, and D minor would be the relative minor of F major. These terms sometimes confuse

students, who might think that F major and D minor, for instance, are the same key. Nothing could be more misleading; F major and D minor have different tonics; therefore, they are different keys.

Knowing the relative major can help you learn the minor key signatures. Remember that the tonic of the minor key is ˆ 6 in the relative major. For example, the tonic of G minor is ˆ 6 in B major; the key signature of G minor, therefore, contains five sharps. Conversely, in minor the tonic of the relative major is ˆ 3. For example, in G minor, ˆ 3 is the tonic of the relative major—B major.

22. Parallel Major and Minor; Mixture. Major and minor keys with different signatures but with the same tone as tonic are called parallel. G minor would be the parallel minor of G major. Actually, parallel major and minor keys are much more closely related than are relative majors and minors. In G minor, as in G major, tonal activity is directed to the same goal, to G. In many compositions, elements from major and minor occur in very close proximity; in such cases, we speak of a mixture of major and minor. Using raised ˆ 6 and ˆ 7 in minor constitutes one kind of mixture, for these tones are the same as the corresponding ones in the parallel major.

Strictly speaking, the parallel minor keys of D , G , and C major would be D , G , and C minor. But because these keys would require one, two, or three double flats in their signatures, they are never used except for brief passages without a change of key signature. Instead, the corresponding minor keys with sharp signatures—C , F , and B minor—will occur. Thus, in Chopin’s Prelude in D major, the extended middle section, which moves to the parallel minor, is written in C minor, with a signature of four sharps. Similarly, the parallel major keys of G , D , and A minor are written as A , E , and B major.

23. Modes; the Diatonic Order. Writers on music often refer to major and minor as modes. If we build scales starting on each of the white keys of the piano as a tonic and use only white keys for the other tones, the result will be seven scales, each with a different pattern of five whole steps and two half steps within its octave. We will have created seven different tonal systems, for the different arrangement of whole and half steps creates a different tonal structure in each of these scales.

The seven “white-key” scales constitute segments of the diatonic order, the pattern of whole and half steps that has given rise to most of the tonal systems of Western music. Like major and minor, these segments are known as modes. The seven patterns are shown in Example 1-22, together with their traditional names. Some of these modes had great importance in music before the eighteenth century, but some did not. The Locrian mode was scarcely more than a theoretical possibility, and the Lydian, at least in polyphonic music, made such regular use of B as to be indistinguishable from Ionian (or major). Much great music was composed in the Dorian, Phrygian, and Mixolydian systems, and to understand early music, you must certainly investigate the way the modes were used. General information appears in any standard history of music and in some counterpoint texts.

1-22 seven diatonic modes

In the music we are dealing with in this book—the music from the Baroque through the Romantic periods—there are only two modes of any importance: major and minor. In this music, elements from some of the other modes— especially Phrygian—will sometimes appear. But they do so, for the most part, in a larger context of major or minor.

24. Tonality. Many musicians and writers use the term tonal to describe any piece or type of music organized around a central tone. And the principle of organization would be called tonality. Under these broad definitions of tonal and tonality, many—indeed, most—kinds of music would be tonal: music in major and minor keys, modal music, much non-Western music, and a good deal of twentieth-century music. The presence of a tonal center is an important common feature of these different kinds of music. But how the other tones function with respect to the central tone may vary considerably. Since the music we will deal with in this book is based, for the most part, on major and minor, the term major–minor tonality is the most accurate, though it is fairly unwieldy. So, we will sometimes use the words tonal and tonality in a narrower sense as an abbreviated form of major–minor tonality.

25. The Contrast between Major and Minor. Many people feel that music in a major key is “happy” and that music in a minor key is “sad.” Sophisticated musicians often question this association, believing that it is a purely arbitrary one based on nothing except, perhaps, habit. It is true that the emotional character of a piece depends on many factors in combination. Light and even comical pieces—some of Mendelssohn’s scherzos, for instance—are in minor. And some very solemn pieces are in major, for example the “Dead March” in Handel’s Saul. But it is a mistake to ignore the likelihood that choice of mode is one of the factors that determine the character of a piece. And sometimes it may be the most important factor.