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Aldwell_ifc_se.indd 1 2/1/10 2:39 PM EDITION IV Harmony & Voice Leading 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 Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd i 2/12/10 6:58:35 AM Harmony & Voice Leading, Fourth Edition Edward Aldwell, Carl Schachter, and Allen Cadwallader Publisher: Clark G. Baxter Assistant Editor: Nell Pepper Editorial Assistant: Ashley Bargende Senior Media Editor: Wendy Constantine © 2011, 2008, 2003 Schirmer, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means, graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. Marketing Manager: Mark Haynes Marketing Coordinat; or: Josh Hendrick Marketing Communications Manager: Heather Baxley Art Director: Faith Brosnan Content Project Manager: Georgia Young For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Further permissions questions can be e-mailed to email@example.com. Senior Manufacturing Buyer: Justin Palmiero Permissions Acquisitions Manager: Roberta Broyer Production Service/Compositor: Graphic World Inc. Library of Congress Control Number: 2009942240 ISBN-13: 978-0-495-18975-6 ISBN-10: 0-495-18975-8 Text Designer: Glenna Collett Cover Designer: Pehlke Design Cover Image: The Art Archive/Bach House Leipzig/Alfredo Dagli Orti. From the autograph score of the Invention in F major by J. S. Bach Schirmer 20 Channel Center Street Boston, MA 02210 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Schirmer, visit www.cengage.com/Schirmer. Purchase any of our products at your local college store or at our preferred online store www.CengageBrain.com. Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd ii 2/12/10 6:58:38 AM In Memoriam Edward Aldwell (January 30, 1938–May 28, 2006) Edward Aldwell's untimely death occurred during the planning stages for this Fourth Edition but before actual work had begun. Nevertheless, 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. He was best known as an interpreter of Bach, and indeed he performed most of Bach's keyboard music and recorded it extensively. But he was also a marvelous player of Mozart, Beethoven, Schubert, Chopin, Brahms, Fauré, Hindemith, and others. His interest in theory and analysis developed out of a desire to hear and understand music better so that he, and those he taught, would learn to perform better. He had little (if indeed any) interest in theoretical speculation for its own sake, but he was convinced that a comprehensive grounding in counterpoint, thoroughbass, and related disciplines would lead to more insightful—and even to more technically secure—performances. Edward was a great teacher both of theory and piano; he was also an outstanding chamber-music 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. Carl Schachter Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd iii 2/12/10 6:58:38 AM Contents Preface to the Fourth Edition Preface to the Third Edition PART I xi xiii The Primary Materials and Procedures 1 Key, Scales, and Modes 1 3 Tonal Relationships; Major Keys 4 Minor Keys; Modes; Tonality 14 2 Intervals 21 Recognizing and Constructing Intervals The Overtone Series 24 Consonance and Dissonance 26 Intervals in a Key 30 3 Rhythm and Meter 35 Rhythmic Organization 35 Rhythm and Dissonance Treatment 4 Triads and Seventh Chords Triads 47 Seventh Chords 56 Texture and Structure 5 43 46 59 Introduction to Counterpoint Species Counterpoint Cantus Firmus 64 First Species 65 Second Species 69 Third Species 74 Fourth Species 78 Fifth Species 85 Exercises 91 21 61 62 v Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd v 2/12/10 6:58:38 AM vi Contents 6 Procedures of Four-Part Writing 93 Chord Construction 93 Counterpoint in Chorale Textures: Voice Leading Points for Review 110 Exercises 111 PART II 103 I-V-I and Its Elaborations 7 I, V, and V 7 113 115 Tonic and Dominant 116 I-V-I in Four Parts 118 The Dominant Seventh 122 Points for Review 128 Exercises 129 8 I6, V 6, and VII6 131 I6 and V6 132 VII6 (Leading-Tone Triad) Points for Review 142 Exercises 143 9 Inversions of V 7 138 147 V 65 , V 43 , and V 42 147 Contrapuntal Expansions of Tonic and Dominant Points for Review 159 Exercises 159 10 Leading to V: IV, II, and II6 154 161 Intermediate Harmonies 162 IV and II in Contrapuntal Progressions 169 Expansions of II and IV 172 Harmonic Syntax; Rhythmic Implications 175 Points for Review 178 Exercises 179 11 The Cadential 64 181 An Intensification of V 181 Points for Review 191 Exercises 191 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd vi 2/12/10 6:58:39 AM vii Contents 12 VI and IV 6 194 Uses of VI 195 Uses of IV 6 200 Points for Review 205 Exercises 205 13 Supertonic and Subdominant Seventh Chords 208 Supertonic Seventh Chords 209 Subdominant Seventh Chords 221 Points for Review 224 Exercises 225 14 Other Uses of IV, IV 6, and VI 229 IV and IV 6 230 VI 236 Summary of Cadences 242 Points for Review 243 Exercises 243 15 V as a Key Area 245 Tonicization and Modulation Applications to Written Work Points for Review 263 Exercises 263 16 III and VII 246 260 265 Uses of III 266 Uses of VII 275 Points for Review Exercises 280 280 5 , 6 , and 6 Techniques 3 3 4 PART III 17 283 5 -Chord Techniques 285 3 Progressions by 5ths and 3rds 285 Contrapuntal Chord Functions 290 V as a Minor Triad 296 Points for Review 299 Exercises 299 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd vii 2/12/10 6:58:39 AM viii Contents 18 Diatonic Sequences 301 Compositional Functions 302 Sequences with Descending 5ths 306 Sequences with Ascending 5ths 310 Sequences Using the Ascending 5-6 Technique 312 Sequences Using the Descending 5-6 Technique (Falling in 3rds) Less Frequent Sequential Patterns 319 Sequences Moving to Tonicized V 321 Sequences in Minor 323 Points for Review 326 Exercises 328 19 6 -Chord Techniques 330 3 6 Chords in Parallel Motion 331 3 337 Other Uses of 63 Chords Points for Review 343 Exercises 343 20 6 -Chord Techniques 346 4 Dissonant 64 Chords 348 Special Treatment of Cadential 64 Chords 361 Consonant 64 Chords Some Special Cases 363 Points for Review 365 Exercises 365 PART IV Elements of Figuration 21 Melodic Figuration 316 356 369 371 Chordal Skips (Arpeggios) 373 Passing and Neighboring Tones 376 Points for Review 388 Exercises 388 22 Rhythmic Figuration 392 Suspensions 392 Anticipations 410 The Pedal Point 414 Points for Review 416 Exercises 417 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd viii 2/12/10 6:58:39 AM ix Contents PART V Dissonance and Chromaticism I 23 Leading-Tone Seventh Chords 419 421 The Diminished Seventh Chord 422 The Half-Diminished Seventh Chord 429 Points for Review 432 Exercises 433 24 Mixture 435 Combining Modes 436 Points for Review 447 Exercises 447 25 Remaining Uses of Seventh Chords 449 Seventh Chords in Sequence 450 Expanded Treatment of Seventh Chords Apparent Seventh Chords 460 Points for Review 465 Exercises 465 26 Applied V and VII 467 Applied Chords 468 Chords Applied to V 472 Other Applied Chords 477 Applied Chords in Sequence Points for Review 492 Exercises 492 27 Diatonic Modulation 483 494 Modulatory Techniques 495 Modulation, Large-Scale Motion, and Form Points for Review 509 Exercises 509 PART VI 456 501 Dissonance and Chromaticism II 28 Seventh Chords with Added Dissonance Ninths 517 "Elevenths" and "Thirteenths" Points for Review 533 Exercises 533 515 517 528 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd ix 2/12/10 6:58:39 AM x Contents 29 The Phrygian II (Neapolitan) 536 A Chord Leading to V 537 Other Uses of ∫II 549 Chromatic Notation 552 Points for Review 556 Exercises 556 30 Augmented Sixth Chords 559 A Chromatic Preparation for V 560 Approaching Augmented Sixth Chords 565 Details of Voice Leading 572 Augmented Sixths and Modulation 575 "Inversions" of Augmented Sixth Chords 577 Motion to Applied Dominants and Nondominant Chords German Sixth and Dominant Seventh 583 Points for Review 585 Exercises 586 31 Other Chromatic Chords 580 589 Advanced Uses of Mixture 590 Augmented Triads 596 Altered Dominant Seventh Chords 601 Common-Tone Diminished Seventh Chords 602 Other Chromatic Embellishing Chords 606 Points for Review 607 Exercises 608 32 Chromatic Voice-Leading Techniques 610 Chromaticism Based on Parallel Motion 611 Chromaticism Based on Contrary Motion 623 Equal Divisions of the Octave 631 Points for Review 641 Exercises 642 33 Chromaticism in Larger Contexts 645 New Modulatory Techniques 646 Chromatic Tonal Areas 655 Points for Review 666 Exercises 667 Appendix I Keyboard Progressions 670 Appendix II Score Reduction Appendix III Roman Numerals and Registers; Explanatory Tables and Charts 692 Index of Musical Examples Subject Index 712 696 709 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd x 2/12/10 6:58:39 AM Preface to the Fourth Edition The Fourth Edition of Harmony & Voice Leading amplifies the approach to previous editions in significant ways. Unit 5 presents an introduction to two-voice species counterpoint, which we hope will benefit programs that do not have dedicated counterpoint components and established source materials. We are aware that many schools already include counterpoint in the theory curriculum. For such programs, this unit may serve as an ancillary reference, which students may consult during work with the initial topics in harmony (Units 7–11) when they are beginning to learn how "chords" function contrapuntally within broader "harmonies." We also suggest that students refer back to Unit 5 when studying the numerous formations in Units 21 and 22 (Melodic Figuration and Rhythmic Figuration). In some programs, the study of tonal theory is limited to three semesters, certainly not an ideal situation. We nevertheless recommend devoting at least a full year to the study of counterpoint, proceeding simultaneously with the presentation of topics in harmony. One strategy, assuming three classes per week, involves beginning first species with Unit 2 (Intervals), second species with Unit 6 (Procedures of Four-Part Writing), and third species with Unit 9 (Inversions of V7); this pacing is designed so that fourth species coincides with the study of the cadential 64 in Unit 11 (a "chord" understood as arising through contrapuntal elements over dominant harmony). Fifth species is then used not only to review passing tones, neighbor notes, and suspensions but also to help students learn to write rhythmically balanced lines. The Fourth Edition is accompanied by a Premium Website, which contains various resources for both students and instructors. The sound files of examples from the Third Edition remain available in CD format and may be accessed from individual web pages (the site also contains 14 examples new to the Fourth Edition). Students are therefore 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. For most units, we have provided a multistage solution for one of the exercises (drawn primarily from the text), which students may use as models for homework assignments. The site presents many of the exercises as PDFs, permitting students and instructors to download and print individual pages. 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. We are grateful to several individuals who have helped us bring the Fourth 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 of the text; Don Giller prepared the examples for the guided assignments of the Premium Website with his usual artistry and expertise. We also offer our appreciation to Miriam Kartch and Olli Väisälä for pointing out numerous mistakes in the Third Edition, some of them quite obscure. We also exxi Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xi 2/12/10 6:58:40 AM xii Preface to the Fourth Edition tend special thanks to the Oberlin Conservatory staff and students who performed and recorded the new examples for the Fourth Edition of the text: Paul Eachus, production engineer; Simone Gheller, organ; Scott Cuellar, piano; Brendan Shea, violin; Jing Qiao, violin; Dillard Cheek, viola; and Zizai Ning, cello. We are grateful to the following reviewers of the Fourth Edition: Reginald Bain, University of South Carolina–Columbia; David Damschroder, University of Minnesota–Twin Cities; Stacey Davis, University of Texas–San Antonio; John Paul Ito, Lawrence University; Igor Karaca, Oklahoma State University–Stillwater; Anthony Kelley, Duke University; Donna Menhart, Hartt School of Music/ University of Hartford; and Cindy Moyer, Humboldt State University. We wish to thank the staff of Schirmer/Cengage Learning, especially Publisher Clark Baxter, Assistant Editor Nell Pepper, Project Manager Georgia Young, Senior Media Editor Wendy Constantine, and Editorial Assistant Ashley Bargende. Finally, we wish to thank the staff of Graphic World Inc., especially Dan Fitzgerald. Carl Schachter Allen Cadwallader Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xii 2/12/10 6:58:40 AM Preface to the Third Edition The Third Edition of Harmony and Voice Leading retains the approach and goals of its two 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: More exercises of a quasi-compositional nature, mostly consisting of writing phrases and phrase groups. In the workbooks, a few of these exercises include the setting of texts Greatly expanded treatment of phrase rhythm and phrase organization. Among the concepts and terms introduced are hypermeasure, the Schoenbergian sentence, and Fortspinnung Emphasizing the most important concepts and techniques through clearer visual presentation Presenting the descending bass tetrachord 8̂-7̂-6̂-5̂ as a basic device (Units 12, 16, 19, and 26) More precise treatment of chord succession, under the rubric "harmonic syntax" Instructions for making simple score reductions (Appendix 2) Clearer and more consistent terminology and symbols throughout More guidance in working out the exercises Perhaps most significantly, an audio component. A set of CDs is now available separately, with performances of the excerpts from the literature in the textbook (not the workbooks) 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 and xiii Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xiii 2/12/10 6:58:40 AM xiv Preface to the Third Edition 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 and 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 and 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 workbook 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 (Units 4 and 6), 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 * The exercises for Units 1–4 appear on the Premium Website. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xiv 2/12/10 6:58:40 AM Preface to the Third Edition xv 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 and 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 these opening 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 and 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. In preparing this Third Edition, we profited from the advice of many colleagues, students, and friends; and we wish to thank all of them, including the Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xv 2/12/10 6:58:40 AM xvi Preface to the Third Edition large number we are not able to list here. In particular, we wish to thank our colleagues David Gagné and William Rothstein, who drew upon their great theoretical expertise and long familiarity with the previous editions and made particularly valuable suggestions. We also owe special thanks to Professor John Hanson of the Eastman School, who, on his own generous initiative, sent us a large number of comments that were most helpful to us in the revision of the workbooks. And we thank Jean Aldwell for countless acts of support and forbearance. We owe a great debt of gratitude to Eric Wen of the Curtis Institute of Music, who produced the accompanying CDs. He is an expert musician and an expert record producer, and we were fortunate in being able to draw upon his varied skills. We are very grateful to the following reviewers of the Third Edition: Emelyne Bingham, Vanderbilt University; Ellon Carpenter, Arizona State University; David Damschroder, University of Minnesota; David Gagné, Aaron Copland School of Music/Queens College/City University of New York; David Kopp, University of Washington– Seattle; David Pereira, University of California at Berkeley; William Rothstein, Aaron Copland School of Music/Queens College/City University of New York; and Deborah Stein, New England Conservatory. We wish to thank the staff of Wadsworth Group for their enthusiastic support—especially publisher Clark Baxter, Senior Developmental Editor Sharon Adams-Poore, and Project Managers Dianne Toop and Erica Silverstein. Elsa Peterson was a most helpful editor in the early stages of production. And finally, we wish to thank the staff of Buuji, Inc., especially Sara Dovre Wudali and Scott Rohr, for their valiant efforts in producing this book. Edward Aldwell Carl Schachter Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_00_FM_i-xvi.indd xvi 2/12/10 6:58:41 AM PART I The Primary Materials and Procedures Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 1 2/4/10 12:17:28 PM UNIT 1 Key, Scales, and Modes 1-1 Mozart, Piano Sonata, K. 545, I 3 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 3 2/4/10 12:17:29 PM 4 Unit 1 Key, Scales, and Modes 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. 1-2 Example 1-1, most prominent tones Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 4 2/4/10 12:17:32 PM Tonal Relationships; Major Keys 5 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. 1-3 Mozart, Piano Sonata, K. 545, I 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 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 5 2/4/10 12:17:35 PM 6 Unit 1 Key, Scales, and Modes 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: 1̂ 2̂ 3̂ 4̂ 5̂ 6̂ 7̂ Tonic Supertonic Mediant Subdominant Dominant Submediant 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. 1 Throughout 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. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 6 2/4/10 12:17:38 PM Tonal Relationships; Major Keys 7 1-5 registers 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 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 3̂ and 4̂ and between 7̂ and 8̂; all the others are whole steps. 1-6 major scale 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 3̂ and 4̂ and 7̂ and 8̂. 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. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 7 2/4/10 12:17:39 PM 8 Unit 1 Key, Scales, and Modes 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 (1̂, 3̂, and 5̂). 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 1̂-3̂-5̂ 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 rela- Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 8 2/4/10 12:17:41 PM Tonal Relationships; Major Keys 9 tionships 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 1̂, 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 1̂, that they tend to move to this stable, central tone. However, 3̂ and 5̂ can also function as stable tones, though they are less stable than 1̂. 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 1̂. Motion to 3̂ or 5̂ will not have the same finality as motion to 1̂. Many melodies begin on 3̂ or 5̂ rather than on 1̂. 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 1̂ does not serve as the initial melodic tone. 9. Passing Tones; Neighboring Tones. Example 1-8 contains a diagram of the Cmajor scale. The stable tones, 1̂, 3̂, 5̂, and 8̂, are shown as whole notes; the more active tones are written with black noteheads. 1-8 stable and active tones As the diagram indicates, the active tones lead from one stable tone to another: up from 1̂ to 3̂, 3̂ to 5̂, and 5̂ to 8̂; 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 1̂ with 3̂ and 3̂ with 5̂ but that two passing tones are needed to connect 5̂ with 8̂. 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 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 9 2/4/10 12:17:41 PM 10 Unit 1 Key, Scales, and Modes 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 1̂ and 3̂; with the melody's prominent 5̂ (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 8̂ down to 5̂ (bar 1) and the second from 5̂ to 1̂ (bars 1–2). In listening to the first segment, we experience no stability until the line comes to rest on 5̂; the transitional, "passing" character of 7̂ and 6̂ is very evident. In the second segment, 4̂ and 2̂ receive more rhythmic emphasis than 3̂ and 1̂, 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 1̂ up to 5̂, which is embellished with its upper neighbor, 6̂. This upward motion is balanced by a line that moves down, but only as far as 3̂, not 1̂. Note that bar 2 contains the motion 5̂-4̂-3̂ 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 5̂ in the Handel moving to the 4̂ 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) Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 10 2/4/10 12:17:43 PM Tonal Relationships; Major Keys 11 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, 4̂ tends to move more readily to 3̂ than to 5̂, the other possible goal tone. And 7̂ is very strongly attracted to 8̂; in fact, the term leading tone refers to the active way in which 7̂ "leads into" 8̂. The half steps are very well situated in major; the instability of 7̂ and 4̂ helps to strengthen 1̂ and 3̂ 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. 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 (1̂-3̂-5̂ 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 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 11 2/4/10 12:17:44 PM 12 Unit 1 Key, Scales, and Modes 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 3̂ and 4̂; the music would no longer be in F major. 1-13 Mozart, Piano Sonata, K. 545, I 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 3̂ and 4̂ and 7̂ and 8̂. 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) Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 12 2/4/10 12:17:47 PM Tonal Relationships; Major Keys 13 (b) 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 Fπ of bars 9 and 10 to the G of bar 11; it functions, therefore, as a kind of passing tone. 1-15 chromatic 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 Fπ-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 (Fπ-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 Aπ-A∫-G. Chromaticism sometimes involves the Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 13 2/4/10 12:17:49 PM 14 Unit 1 Key, Scales, and Modes use of double sharps and double flats. A chromatic passing tone between F∏ and G∏, for example, would be FV ; 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∫, Aπ 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 Cπ. 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. 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). 1̂ and 5̂ are the same in both chords; only 3̂ varies (the B∫ is closer to G than is the Bπ); the 3rd G-B∫, therefore, is smaller than the 3rd G-Bπ. Minor and major simply mean smaller and larger. A minor key is a key containing a small or minor 3rd between 1̂ and 3̂; a major key is a key containing a large or major 3rd between 1̂ and 3̂. 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. 1-16 Handel, Passacaglia (from Harpsichord Suite No. 7) Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 14 2/4/10 12:17:50 PM Minor Keys; Modes; Tonality 15 1-17 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 1̂ and 3̂ lends its characteristic color to the scale. The half steps between 2̂ and 3̂ and 5̂ and 6̂ create an intensity in the motions from 2̂ to 3̂ and from 6̂ to 5̂ quite different from the corresponding progressions in major. Finally—and very significantly—the whole step between 7̂ and 8̂ 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 seventh 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 7̂ leads away from 8̂ rather than toward it. But when the scale ascends, the whole step between subtonic and tonic can constitute a real defect because 8̂ does not sound like a goal; its power to act as the central tone of the key is impaired. For this reason, 7̂ in minor must be raised to create the necessary half step whenever it moves to 8̂ 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 Fπ in bars 3 and 4 of Example 1-16. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 15 2/4/10 12:17:52 PM 16 Unit 1 Key, Scales, and Modes 17. The Harmonic Form of Minor. Raising the seventh 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 6̂ and 7̂ 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 2̂ and 3̂, 5̂ and 6̂, and 7̂ and 8̂. In the keys of G∏ minor, D∏ minor, and A∏ minor, raising 7̂ requires a double sharp; in A∏ minor, for example, the leading tone is GV . 1-19 harmonic minor scale 18. The Melodic Form of Minor. If we raise 6̂ as well as 7̂, we gain a leading tone, but without creating an awkwardly large interval between 6̂ and 7̂. In a melodic line in minor, therefore, if 6̂ comes before the leading tone (raised 7̂), it too will be raised. Note, for example, the Eπ in bar 4 of Example 1-16, used instead of the E∫ called for by the key signature. The minor scale that raises 6̂ and 7̂ 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 2̂ and 3̂ and 7̂ and 8̂. 1-20 (a) melodic minor scale (b) Handel, Passacaglia Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 16 2/4/10 12:17:53 PM Minor Keys; Modes; Tonality 17 Because 6̂ and 7̂ are raised so they lead convincingly to 8̂, 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 6̂ and 7̂ 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 from the harmonic (Example 1-16, bars 3 and 4). Melodic lines that ascend 6̂-7̂-8̂ 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 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 17 2/4/10 12:17:55 PM 18 Unit 1 Key, Scales, and Modes 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 6̂ in the relative major. For example, the tonic of G∏ minor is 6̂ in B major; the key signature of G∏ minor, therefore, contains five sharps. Conversely, in minor the tonic of the relative major is 3̂. For example, the tonic of B∫ major is 3̂ in G minor. 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 6̂ and 7̂ 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. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 18 2/4/10 12:17:57 PM Minor Keys; Modes; Tonality 1-22 19 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 twentiethcentury 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. For one thing, the association of major or minor mode and emotion is a very old one; it goes back at least 400 years. Writing in 1558, Gioseffo Zarlino, the greatest theorist of the late Renaissance, remarks that melodies (and modes) featuring a major 3rd above the central tone sound cheerful and that those with Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 19 2/4/10 12:17:57 PM 20 Unit 1 Key, Scales, and Modes a minor 3rd sound sad.2 Any cultural tradition that has persisted for so long takes on a certain importance even if it is based on nothing more than custom. That the great composers of the eighteenth and nineteenth centuries believed in this association is evident to anyone who studies their songs and other music they composed to texts. And (as you will see in Unit 2) there is a strong possibility that the emotional connotations of major and minor may reflect more than habit or conditioning—that they may arise out of qualities inherent in tonal relationships. Exercises for Units 1–4 can be found on the Premium Website. 2 Gioseffo Zarlino, The Art of Counterpoint, translated by Guy A. Marco and Claude V. Palisca (New Haven, Conn.: Yale University Press, 1968), pp. 21–23. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_01_Ch01_001-020.indd 20 2/4/10 12:17:59 PM UNIT 2 Intervals 2-1 Verdi, Recordare (from the Requiem, Dies Irae) (a) horizontal intervals (b) vertical intervals Recognizing and Constructing Intervals 1. Numerical Size and Quality. As you should remember from Unit 1 (Section 6), intervals exist in two dimensions, horizontal (melodic), as in Example 2-1a, and vertical (harmonic), as in 2-1b. And you will recall that the numerical size of an interval depends on the number of letter names the two tones span. But numerical size alone is not enough for the complete identification of an interval. For example, the intervals from C down to G∏, Gπ, and G∫ are all 4ths, for they all have the same letter names. Yet the three intervals are slightly different in size and very different in sound or quality. Thus, the complete identification of an interval, whether horizontal (melodic) or vertical (harmonic), depends on both its numerical size and its quality. 21 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 21 2/4/10 12:21:02 PM 22 Unit 2 Intervals Example 2-1b shows the intervals between the two melodic lines labeled both by size and by quality. Intervals come in five qualities: Major (M) Minor (m) Perfect (P) Augmented (A) Diminished (d) (Occasionally one encounters doubly augmented or doubly diminished intervals.) For purposes of classification, intervals divide into two groups: Group 1: Unisons, 4ths, 5ths, and octaves Group 2: 2nds, 3rds, 6ths, and 7ths The intervals belonging to the first group are basically perfect; they are never major or minor. The intervals belonging to the second group are basically major or minor; they are never perfect. Thus, musicians never speak of a "major 5th" or a "perfect 6th." Intervals of either group are sometimes augmented or diminished. If the upper tone of the interval belongs to the major scale of the lower tone, the interval is perfect. If the interval is a chromatic half step larger than perfect, it is augmented; if it is a chromatic half step smaller than perfect, it is diminished (Example 2-2). Group 1 2-2 intervals: group 1 Group 2 If the upper tone belongs to the major scale of the lower tone, the interval is major. If the interval is a chromatic half step larger than major, it is augmented. If it is a chromatic half step smaller than major, it is minor. And if it is a chromatic half step smaller than minor, it is diminished (Example 2-3). 2-3 intervals: group 2 Identifying and building intervals is easy if the lower tone normally begins a major scale and if the interval is built up from the lower tone. It is slightly more difficult if the lower tone is the tonic of an improbable major scale or if the inter- Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 22 2/4/10 12:21:03 PM Recognizing and Constructing Intervals 23 PROBLEM 1: IDENTIFY THE FOLLOWING INTERVAL Few of us are familiar with the exotic key of B∏ major, so we'll disregard the sharp for a moment. From Bπ to A is a minor 7th. (B to A∏ would be a major 7th because A∏ belongs to the B-major scale. B-Aπ is a chromatic half step smaller than major; therefore, it is minor.) If B-A is a minor 7th, then B∏-A must be a diminished 7th because raising the lower tone makes the interval a chromatic half step smaller than minor. PROBLEM 2: WRITE AN AUGMENTED 6TH BELOW D. To solve this problem, we must first find the letter name of the lower tone. In this case, it is F, because F-D forms a 6th. From F to D would be a major 6th because D fits into the F-major scale. An augmented 6th is a chromatic half step larger. We cannot change the given tone, D, so we can enlarge the interval only by lowering the F. The correct answer, therefore, is F∫. Note: Be careful! When you raise the lower tone of an interval, it becomes smaller; conversely, lowering the bottom note results in a larger interval. val must be built down from the upper tone. The two problems above show the way to proceed. 2. Compound Intervals. Compound intervals are those larger than an octave. According to the principle of octave equivalence (see Unit 1, Section 4), compound intervals are functionally the same as the corresponding simple ones. As Example 2-4 demonstrates, a 12th is simply an expanded 5th, a 15th is an expanded octave, and so forth. And such intervals are almost always called 5ths and octaves rather than 12ths and 15ths. The only compound intervals whose names we need for our present purposes are the 9th and the 10th. 2-4 compound intervals 3. Interval Inversion. We invert an interval of an octave or less by bringing the lower tone up an octave or the upper tone down an octave while leaving the other tone in place (Example 2-5); with compound intervals one of the tones would have to be displaced by two or more octaves. The numerical size of an interval plus that of its inversion adds up to 9. Thus, the inversion of a unison is an octave (1 plus 8 equals 9), the inversion of a 3rd is a 6th (3 plus 6 equals 9), and so on. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 23 2/4/10 12:21:03 PM 24 Unit 2 Intervals 2-5 inverted intervals The inversion of a perfect interval is also perfect. Inversion changes the other qualities to their opposites: Major Minor Augmented Diminished becomes becomes becomes becomes minor major diminished augmented Because interval inversion results from the octave displacement of one of the interval's tones, an interval and its inversion form a related pair; this relationship is another consequence of octave equivalence. The Overtone Series 4. Composite Sounds and Overtones. Most musical tones are composite sounds. Their pitch results from the frequency with which the sounding body vibrates. (The sounding body may be a string, as on a violin, or an air column, as inside an oboe; and so forth.) As it vibrates, the sounding body divides itself into segments that vibrate independently. The vibration of the segments produces overtones. Normally we are not conscious of these overtones for they and the fundamental tone (the pitch we hear) blend into a single sound. But if you have a good musical ear, you can easily train yourself to hear overtones, especially when the fundamental tone is in a low register. Overtones help to determine the timbre (or tone color) of the various instruments; they make possible the playing of harmonics on string instruments and the technique of overblowing on wind instruments. The intensity (loudness) of the different overtones will vary depending on the instrument and on how it is played, but almost all musical sounds of any pitch contain the same group of overtones; we call this group the overtone series. Example 2-6 shows the series from great C (review Example 1-5) through the 16th tone (or partial). The series continues infinitely, the intervals between successive partials becoming smaller and smaller. But the higher partials are so weak that they lose any musical significance. Note that the partials are numbered from the fundamental, which is the first partial. All the other partials, or overtones, are literally "over" the fundamental. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 24 2/4/10 12:21:04 PM The Overtone Series 25 2-6 the overtone series 5. The Overtone Series and the Tonal System. We can see from Example 2-6 that the overtone series contains elements that coincide with some of the most important materials of the tonal system. Between the fundamental and its upper partials we find: The perfect octave (partials 2, 4, 8, and 16) The perfect 5th (partials 3, 6, and 12) The major 3rd (partials 5 and 10) The major triad (partials 1–6) However, the tonal system does not make use of all the sounds that occur in the overtone series. The partials shown by black noteheads in Example 2-6 (7, 11, 13, and 14) do not form part of major or minor scales, and their notation in the example is only an approximation of their true pitch. Furthermore, some of the important elements of tonal music—the minor triad, for example—do not relate directly to the overtone series. The significance of the overtone series for the theory of tonal music is a matter of controversy. In the past, many theorists went to absurd extremes in their attempt to use the series as a "scientific" basis for music, contorting it in various ways to extract a minor triad from it and making it the basis for arbitrary "rules" of composition—rules that no great composer has ever followed. And even where the series and the tonal system correspond very closely—as with the major triad— there is no proof that the acoustical relationship causes the musical one. Nowadays most musicians would maintain that the foundations for music theory should lie in the works of great composers, not in the laboratories of acousticians. But the following characteristics can be observed in the works of the great composers of tonal music: 1. The major triad functions as the most stable chord. From the Renaissance on, composers showed a marked preference for the major triad as final chord, even in modes containing the minor 3rd. This tendency was strongest in the earlier stages of triadic music, but it never died out altogether. 2. In major–minor tonality, the major mode is normally the positive, happy, bright one, and the minor is the negative, sad, dark one. This again points to the greater stability of the major triad. 3. Two tones a 5th apart are in a particularly close relation. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 25 2/4/10 12:21:04 PM 26 Unit 2 Intervals 4. The most stable intervals are the octave, 5th, and 3rd (the major 3rd more so than the minor). 5. A triad is generated from its root, or lowest tone, much as overtones are generated from the fundamental. All these characteristics are at least compatible with the view that some of the most important features of tonality give expression to relationships that are inherent in a single musical tone. As noted before, it is impossible to demonstrate a causal connection between the overtone series and these aspects of tonal music. But if it is a coincidence, it is a most remarkable one. Consonance and Dissonance 6. Stable and Unstable Intervals. Some intervals produce the impression of stability; others, the effect of activity or tension. We call the stable intervals consonances or consonant intervals; the unstable ones are dissonant. The consonant intervals are the following: The perfect unison The perfect octave The perfect 5th The perfect 4th (sometimes) Major and minor 3rds Major and minor 6ths The dissonant intervals are the following: All 2nds All 7ths All augmented and diminished intervals The perfect 4th (sometimes) For the moment, in discussing consonance and dissonance, we will concentrate on vertical intervals, those whose tones sound simultaneously. Melodic intervals can also be characterized as consonant or dissonant, as we will discuss in later units. 7. The Consonant Intervals. In major–minor tonality, the consonant intervals are the unison and octave, plus all the intervals that make up major and minor triads. The most stable triadic intervals are those that lie between the lowest tone (root) of a triad and one of the upper tones; these are the perfect 5th, the major 3rd, and the minor 3rd. The remaining consonances—the major 6th, the minor 6th, and the perfect 4th—result from the inversion of the more stable ones. Example 2-7 illustrates the consonant intervals in an order that proceeds from the more to the less stable. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 26 2/4/10 12:21:06 PM Consonance and Dissonance 2-7 27 consonant intervals The unison and octave are the most stable of all the consonances; in the unison the two tones agree completely, and in the octave they differ only in register. The lack of tension in these intervals is reflected in the tendency of composers to end pieces on unisons or octaves. Next comes the perfect 5th. In music based on the triad, the 5th is uniquely important, for its upper tone defines the lower one as the root of a chord. Thus if we hear the bare 5th F-C, we understand F as the root, for F-C occurs in no triads except F major and F minor, in both of which F is the root. And because our feeling for key rests partly on the stability of the tones of the tonic triad, the 5th between 1̂ and 5̂ plays a most significant role in defining the key. Composers have tended to treat the major triad as more stable than the minor. The major 3rd, therefore, which characterizes the major triad, is a more stable interval than the minor 3rd, which characterizes the minor triad. Both 3rds are more active intervals than the 5th. Still more active are the major and minor 6ths, inversions of the 3rds. Differences in stability between the two kinds of 6th are not particularly significant. Their fluid character is reflected in the fact that they are not used to end pieces except for special and unusual effects. The perfect 4th—the only interval that is sometimes consonant and sometimes dissonant—is in a category of its own and will be discussed in Section 11. 8. Perfect and Imperfect Consonances. We call unisons, octaves, and 5ths perfect consonances; we call major and minor 3rds and 6ths imperfect consonances. In two-part textures (music containing two melodic lines), composers prefer the more stable perfect consonances for important points of articulation—beginnings and endings of phrases, sections, or pieces (we will examine strict two-part writing in Unit 5). Because of their less stable, more fluid character, the imperfect consonances normally predominate in places where the music moves from one point of articulation to another. In textures of more than two parts, imperfect consonances tend to occur between the highest and the lowest parts (the most prominent lines), except at points of articulation. 9. The Dissonant Intervals. Unlike the consonances, all of which form part of major or minor triads and therefore function as chordal elements, dissonant intervals between two parts arise out of melodic activity in one or both of the parts. In Unit 1, Section 9, we saw that passing tones move by step from one stable tone to another and that neighboring tones arise from the stepwise decoration of a single tone. In Example 2-8, the passing and neighboring tones in one part create dissonant intervals between the two parts. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 27 2/4/10 12:21:06 PM 28 Unit 2 Intervals 2-8 melodic origin of dissonant intervals All the dissonant intervals in Example 2-8 arise from stepwise motion. This is a fundamental characteristic of dissonance treatment in tonal music. Approaching and leaving the dissonance by step ensures a close connection between it and the surrounding consonances. The stepwise connection channels the tension and energy of the dissonant intervals so that dissonance becomes a powerful force for musical direction. On the other hand, isolated dissonances—those without a close connection to consonances—run the risk of creating tensions that serve no musical purpose because they lead to no goals. We will continue to examine the dissonant intervals—and how they relate to perfect and imperfect consonances—in greater detail in Unit 5. 10. Dissonance and Activity. In Unit 1, we saw that 2̂, 4̂, 6̂, and 7̂ function as active tones tending to move to 1̂, 3̂, and 5̂. The division of scale degrees into stable and active tones relates directly to the phenomenon of consonance and dissonance because the active tones are those that form dissonances with one or more tones of the tonic chord, whereas 1̂, 3̂, and 5̂ (the tonic chord) are all consonant with each other. The simplest and most basic use of consonance and dissonance, therefore, would be 1̂, 3̂, and 5̂ as consonances and the other scale degrees as dissonances against the other part or parts (Example 2-9). 2-9 4^, 2^, and 7^ form dissonances For composers to restrict themselves to the simplest possibilities, however, would be far too limiting. A most important compositional resource, therefore, is stabilizing the normally active tones by giving them the support of consonant intervals; at the same time, normally stable tones may become unstable by appearing as dissonances (Example 2-10). Note that 4̂, 2̂, and 7̂, the active tones stabilized by consonant support, do not altogether lose their active character, as we can ascertain by playing the example and stopping on one of those tones. The music does not sound at rest until it arrives at the final 1̂. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 28 2/4/10 12:21:06 PM Consonance and Dissonance 29 2-10 4^, 2^, and 7^ given consonant support 11. The Perfect 4th. In the early stages of medieval polyphony, the perfect consonances formed the basis for music composition. Not only unisons, octaves, and 5ths, but perfect 4ths as well, functioned as stable intervals. Over the course of several centuries, composers experimented with the possibilities made available through the use of 3rds and 6ths; the most important of these possibilities were the complete triads, major and minor, that became the basis for later music. Using complete triads effected a fundamental change in musical structure; one consequence of this change threatened the consonant status of the 4th. Once the 3rd became a pervasive element in musical texture, many situations arose in which the 4th sounded less like an inversion of the 5th—and thus a more or less stable interval—than like an active interval gravitating to the 3rd. In such situations, the 4th takes on the character of a dissonance (Example 2-11). 2-11 dissonant 4ths However, if the 4th occurs in close proximity to the 5th of which it is an inversion, it sounds perfectly stable and consonant; it has no tendency to move to a 3rd. The same is true in situations where the 4th appears in an arpeggiated triad. Example 2-12 illustrates the 4th as a stable, consonant interval. 2-12 consonant 4ths Unlike any other interval, therefore, the 4th is sometimes consonant, sometimes dissonant. It is consonant whenever the context shows it to function as an inverted 5th; otherwise, it is dissonant. In simple textures, the 4th is mostly dissonant when it occurs in a two-part setting or between the lowest part and one of the upper ones in a setting of more than two parts. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 29 2/4/10 12:21:07 PM 30 Unit 2 Intervals Intervals in a Key 12. Intervals between Scale Degrees. The special character of any scale degree depends partly on the intervals it forms with the other degrees. In major, for example, 3̂, 6̂, and 7̂ form a major 3rd, major 6th, and major 7th above 1̂. In minor, the corresponding intervals are all minor. These differences in intervallic quality help give a different character to the tonic note in the two modes (Example 2-13). 2-13 intervals in major and minor keys In any diatonic mode, the group of intervals formed by any scale degree sounding against all the others is unique. Each group will differ from all the other groups by two or more intervals. In Example 2-14, we show the intervals that contain 3̂ in G major (2-14a) and those that contain 6̂ (2-14b). The two groups of intervals are almost the same, but not quite. 3̂ forms a minor 2nd above and major 7th below, whereas 6̂ forms a major 2nd above and a minor 7th below. Every scale degree generates a unique collection of intervals, which gives each tone of a diatonic scale its own distinctive character. Using the tones that belong to major, natural minor, and other diatonic scales, we can produce the following intervals: perfect unisons and octaves, perfect and diminished 5ths, perfect and augmented 4ths, major and minor 3rds and 6ths, and major and minor 2nds and 7ths. No other intervals can be produced by combining these tones. 2-14 (a) intervals from 3^ (b) intervals from 6^ Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 30 2/4/10 12:21:08 PM Intervals in a Key 31 13. The Diminished 5th and the Augmented 4th in Major. Among the intervals found in major and in natural minor are six perfect 5ths and six perfect 4ths (inversions of the 5ths). But there is only one diminished 5th and only one augmented 4th. In major, the diminished 5th occurs between 7̂ and 4̂; the augmented 4th, between 4̂ and 7̂. In Unit 1, we saw that 4̂ and 7̂ gravitate to the stable tones 3̂ and 1̂ because of the particularly intense character of the half-step relationship. When 4̂ and 7̂ occur at the same time, their melodic tendencies remain; in fact, they are considerably enhanced by the tension of the dissonant interval they form. The motion of a dissonant interval to the consonance that acts as its goal is called a resolution. The diminished 5th resolves by moving in to a 3rd; the augmented 4th resolves by moving out to a 6th (Example 2-15). 2-15 resolving d5 and A4 The resolution of the diminished 5th and augmented 4th to 1̂ and 3̂ creates a strong drive toward the tonic triad and helps orient the listener as to the position of the tonic; for this reason we call it a key-defining progression. The key-defining function of these intervals is connected with the fact that any particular diminished 5th or augmented 4th occurs in only one major key. Thus, the minor 2nd E-F occurs in two major keys: C and F; the major 3rd C-E occurs in three: C, F, and G. But the diminished 5th B-F (and the augmented 4th F-B), unlike any other interval, occurs in one major key alone—the key of C. Traditionally, the augmented 4th is called the tritone, which means three whole steps, thus: F-G, G-A, and A-B. (Strictly speaking, the diminished 5th is not a tritone, for it contains not three whole steps but a diatonic half step, two whole steps, and another diatonic half step: B-C, C-D, D-E, and E-F. However, for convenience, the term tritone is often used to mean the diminished 5th as well as the augmented 4th.) 14. The Diminished 5th and the Augmented 4th in Minor. In the natural minor, the diminished 5th lies between 2̂ and 6̂, the augmented 4th between 6̂ and 2̂. They resolve to 3̂ and 5̂, expressed as a 3rd (resolution of diminished 5th) or as a 6th (resolution of augmented 4th). Although 3̂ and 5̂ are members of the tonic triad, these resolutions do not define the key nearly as successfully as do the corresponding resolutions to 1̂ and 3̂ in major. When 3̂ and 5̂ are heard without 1̂, 3̂ tends to be heard as the root of a triad. Thus, the progression shown in Example 2-16 suggests E∫ as tonic rather than C. It is partly because of this implication that the minor mode tends to gravitate to its mediant degree (or relative major). Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 31 2/4/10 12:21:09 PM 32 Unit 2 Intervals 2-16 does not define c Raising 7̂ in the harmonic and melodic minor creates an "artificial" tritone between 4̂ and 7̂ that resolves to 1̂ and 3̂ as in major. The use of this tritone (or diminished 5th) lends to minor the clear definition of the key that occurs naturally in major (Example 2-17). 2-17 defines c The raised 6̂ of the ascending melodic minor scale creates another tritone, in this case with 3̂. This tritone occurs much less often than the other two and has no significant influence on key definition. 15. The Diminished 7th and the Augmented 2nd. The interval between raised 7̂ and natural 6̂ in the harmonic minor is a diminished 7th; inverted, it becomes an augmented 2nd. Like all diminished and augmented intervals, these are dissonant. As Example 2-18 indicates, they resolve to 1̂ and 5̂. The diminished 7th is the more useful of the two intervals because it resolves to a 5th. The interval of the 4th, to which the augmented 2nd resolves, is itself often dissonant. Therefore, the augmented 2nd cannot occur very freely; as a rule, it is used in those situations where the 4th to which it resolves is consonant, as shown in Example 2-12. 2-18 resolving d7 and A2 (a) (b) (c) This pair of dissonant intervals has a very strong key-defining power. The resolution to 1̂ and 5̂ unmistakably points out the location of the tonic. Furthermore, among the intervals in major and minor scales, the diminished 7th and augmented 2nd appear only between raised 7̂ and natural 6̂. Thus, the diminished 7th C∏-B∫, for example, immediately points to D as tonic, for no other tonic can generate this particular interval. Because of its powers of key definition, the diminished Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 32 2/4/10 12:21:10 PM Intervals in a Key 33 7th often appears in major as a consequence of mixture (see Unit 1, Section 22). We can bring the diminished 7th C∏-B∫ into the key of D major by introducing B∫, 6̂ of the parallel minor (Example 2-18c). 16. The Remaining Intervals. We have already mentioned most of the intervals that are significant in the study of music theory. Of those not yet mentioned, one pair, the augmented 5th and diminished 4th, occurs in the inflected forms of minor; these intervals arise from the combination of 3̂ and raised 7̂ (Example 2-19). 2-19 A5 and d4 Another pair, the diminished 3rd and augmented 6th, is the product of chromaticism. These intervals normally come about as a consequence of raising 4̂ in minor; the intervals between raised 4̂ and natural 6̂ are the diminished 3rd and augmented 6th. As Example 2-20 indicates, raised 4̂ functions as a lower neighbor to 5̂ or as a chromatic passing tone leading from 4̂ to 5̂. 2-20 (a) A6 and d3 (b) (c) Chromaticism sometimes produces other intervals, but they are of less significance. The diminished octave and augmented 3rd of Example 2-21 are formed by melodic ornamentation in one of the parts; the intervals are mere by-products of this ornamentation. 2-21 (a) intervals as by-products (b) Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 33 2/4/10 12:21:10 PM 34 Unit 2 Intervals 17. Enharmonically Equivalent Intervals. The use of enharmonically equivalent tones makes it possible to construct two intervals of different size and quality, but whose tones have the same pitch in tempered tuning (see also Unit 1, Section 14). Example 2-22a shows some of these enharmonically equivalent intervals. In isolation, an interval is indistinguishable from its enharmonic equivalent. In context, however, the two can sound very different indeed (2-22b). 2-22 enharmonic equivalents (a) (b) Exercises for Units 1–4 can be found on the Premium Website. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_02_Ch02_021-034.indd 34 2/4/10 12:21:12 PM UNIT 3 Rhythm and Meter 3-1 Mozart, Piano Concerto, K. 467, II Rhythmic Organization 1. The Beat. Music moves in time; musical rhythm organizes the flow of time. This organization involves many factors, the most important being duration, accent, and grouping. The basic unit of duration is the pulse or beat. A beat is a span of time that recurs regularly; a succession of beats divides the flow of time into equal segments.* We are aware of the beat even if it is not always expressed in the music. Thus, in the Mozart Piano Concerto passage of Example 3-1, the quarter note takes the beat. But the music does not move only—or even mainly—in quarter notes. Beats * Actually, the segments are approximately, rather than strictly, equal, for in performance, slight deviations from exact measurement are the rule rather than the exception. And the word "beat" is often used to denote the point in time where the span begins, rather than the span itself. 35 Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_03_Ch03_035-045.indd 35 2/4/10 12:21:32 PM 36 Unit 3 Rhythm and Meter combine into half notes; they divide into eighths, triplets, and sixteenths; and additional time values are produced by dots, double dots, and ties. In the Mozart, we relate these other values to the quarter note as basic unit, so the beat persists as a background against which we hear the varied rhythms of the piece. The simplest way both to divide and to combine beats is by twos (Example 3-2). Thus, a quarter note divides into two eighths, an eighth into two sixteenths, a sixteenth into two thirty-seconds, and so on. Similarly, two quarters combine into a half note and two half notes into a whole note. The division and combination of rests follow exactly the same principle. 3-2 The use of dots permits more complex divisions and combinations (Example 3-3). A dot following a note or rest adds half its value to it; a second or third dot adds on half the value of the preceding dot. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_03_Ch03_035-045.indd 36 2/4/10 12:21:32 PM Rhythmic Organization 37 3-3 A division of the beat into three (triplets) is indicated by the numeral 3 above or below the notes. Other divisions can also be indicated through the use of the appropriate numerals. Example 3-4 shows some possibilities. 3-4 Using ties makes it possible to combine notes across a bar line and to create durations that cannot be achieved through note values or dots (Example 3-5). Sometimes two tied notes will replace a single dotted note to make for easier reading or a clearer expression of the rhythmic structure. 3-5 (a) (b) (c) Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_03_Ch03_035-045.indd 37 2/4/10 12:21:35 PM 38 Unit 3 Rhythm and Meter 2. Tempo. The musical term for the pace of a composition is tempo (Italian, time). In music whose rhythms are based on beats, the impression of a quick, moderate, or slow tempo comes from the pace of the beats, not from the speed of the quickest notes. A slow tempo such as an Adagio or Largo may contain passages in, say, 64th notes that move very quickly indeed. Yet, we do not hear a change to a quicker tempo; as long as the beats move slowly, the tempo remains slow. The pace of the beats relates to tonal movement, to the frequency with which chords change, or to the rate of motion of the main tones of the melody. The fact that music moves at different tempos relates to an important aspect of our notational system: note values indicate relative rather than absolute durations. A quarter note that occupies a beat in a Largo lasts much longer than a quarter note that gets the beat in a Presto. But no matter what the tempo, a quarter note equals two eighths or four sixteenths. 3. Accent. Accent means emphasis. A note that receives more emphasis than the ones surrounding it is heard as accented. Accents often arise in performance when a note is stressed by being played more loudly than those around it or when the performer emphasizes the beginning (attack) of the note. Other kinds of accents are, so to speak, built into the composition itself. In general, long notes attract accents because their long duration creates an emphasis (these are often referred to as agogic accents). Unusually high or low notes come across more strongly than do those in a normal register. Dissonant or chromatic elements, because of the tensions they create, tend to sound accented compared with consonant or diatonic elements. 4. Meter and Metrical Accent. A repetitive pattern that combines accented and unaccented beats is called meter. Usually we speak of strong beats and weak beats to distinguish beats with and without accents. If the first of every two beats is strong, the meter is duple; if the first of every three is strong, the meter is triple. Quadruple meter (derived from duple) has a primary emphasis on the first beat and a weaker emphasis on the third beat of four. In normal musical notation, the bar line appears just before the strong beat; the accent that falls on the first beat of the bar is called the metrical accent. The inner organization of a divided beat mirrors in miniature the metrical organization of a measure. The beginning of a beat is stronger than the subdivisions that follow it. Within a divided quarter note, for example, the accent will fall on the first of two eighths, the first of three triplet eighths, or the first of four sixteenths. If the tempo is slow, the third of four sixteenths may get a subsidiary accent, just like the third beat in a bar of quadruple meter. 5. Time Signatures. Composers indicate meter by means of time signatures placed at the beginning of a piece after the key signature and at any subsequent point where the meter changes. The time signature contains two numbers, one above the other. The lower number normally indicates the note value of the beat; the upper one indicates the number of beats per measure. Most often the quarter note gets the beat. Therefore duple, triple, and quadruple meter usually have the time signatures 24 , 34 , and 44 (or its equivalent, c). However, composers can suggest the character and, sometimes, the tempo of a piece by using another note value, usually a half note or an eighth, for the beat. Time signatures like 22 (| ), 28 , 32 , 38 , 42 , and 48 occur frequently. Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 89756_03_Ch03_035-045.indd 38 2/4/10 12:21:38 PM Rhythmic Organization 39 Some meters contain accentual patterning on more than one level. This is especially true of the so-called compound meters, those with beats grouped in 6 6 multiples of three ( 68 , 98 , 12 8 , 4 , and so on). In a bar of 8 time, for instance, the first eighth note of each three receives an accent; the strong eighth notes, therefore, are the first and fourth. At the same time, a larger pattern of half bars is superimposed on this one; of the two dotted quarters in the bar, the first is the stronger. If the tempo is slow, we hear six beats in the bar arranged in two groups of three beats each; the beginning of the first group is stronger than the beginning of the second. If the tempo is quick, however, the 68 has only two real beats; it sounds exactly like 24 with triplet subdivisions. Meters containing five or seven beats are frequent in twentieth-century music and occur occasionally in earlier music. These meters often result from the com3 , 5 , 4 , and bination of duple and triple meter. Changing meters (such as 28 , 38 , 16 8 8 so on) also occur in a good deal of twentieth-century music—much less often in music of the nineteenth century. 6. Rhythmic Accent versus Metrical Accent. Very often a composer underscores the metrical accent by making it coincide with some other kind of emphasis. In Example 3-1, long notes appear at the beginnings of bars 1, 2, 4, and 5. At these points, the rhythmic accents caused by longer note values coincide with the metrically strong beats. In general, the simplest and most natural kinds of rhythm are those whose emphases fit into the metrical pattern. Sometimes, however, a rhythmic emphasis contradicts the meter. The presence of a rhythmic accent at a metrically weak place is called syncopation. Syncopations arise in various ways; for our purposes the most important are those caused by a note that begins on a weak beat (or part of the beat) and is held through the next strong beat (or part). Because the beginning of a note is heard as stronger than its conti