The Principles and Practice of Modal Counterpoint.pdf

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The Principles and Practice

of Modal Counterpoint

Covering modal music from Gregorian chant through the seventeenth century, The Principles and Practice of Modal Counterpoint is a comprehensive textbook combining stylistic composition, theory and analysis, music history, and performance. By supplementing a modified species approach with a wealth of complete musical examples and historical information, this textbook thoroughly joins principle with practice, providing a truly immersive experience in the study of modal counterpoint and familiarizing students with modal repertoire.

Features:

• A balanced approach to learning counterpoint combining technique, style, and composition • Synthesizes the species approach and also the direct approach in an original way

• Includes many complete musical scores

• Provides self-tests and exercises throughout each chapter

Douglass M. Green (1926–1999) was a founding member of the Society for Music Theory. He last taught at the University of Texas at Austin, where he was Professor of Music Theory until his death. Widely known as an expert in the music of Debussy and Berg, Green was the author of many articles and books on musical form and harmony, including the seminal analysis text Form in Tonal Music. He won several honors throughout his lifetime, including appointment as a Fulbright Scholar to Italy, the ASCAP-Deems Taylor Award, and the E.W. Doty Professorship of Fine Arts at UT-Austin. Green’s counterpoint classes remain legendary among his students.

Evan Jones is Associate Professor and Coordinator of Music Theory and Composition at the Florida State University College of Music. He has received a Sproull Fellowship from the University of Rochester, a Doctoral Fellowship from the Social Sciences and Humanities Research Council of Canada, and the Alfred Mann Dissertation Prize from the Eastman School. He has published research on music by Lassus, Quantz, Schubert, and Xenakis in peer-reviewed journals and essay collections, and has edited a two-volume collection of essays on twentieth-century string quartets.

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The Principles and Practice

of Modal Counterpoint

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by Routledge

270 Madison Avenue, New York, NY 10016 Simultaneously published in the UK

by Routledge

2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN

Routledge is an imprint of the Taylor & Francis Group, an informa business

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registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging in Publication Data

Green, Douglass M. (Douglass Marshall), 1926–1999.

The principles and practice of modal counterpoint / Douglass Green and Evan Jones. p. cm.

Includes bibliographical references and index. 1. Counterpoint. I. Jones, Evan. II. Title. MT55.G814P75 2011 781.2'86–dc22 2010005907 ISBN13: 978–0–415–87821–0 (hbk) ISBN13: 978–0–415–98865–0 (pbk) ISBN13: 978–0–203–84655–1 (ebk)

This edition published in the Taylor & Francis e-Library, 2011.

To purchase your own copy of this or any of Taylor & Francis or Routledge’s

collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.

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Foreword

When Douglass M. Green passed away in 1999, his magnum opus, The Principles and Practice of Counterpoint, was almost completely finished, with only the final chapter of the second volume left unwritten. The present volume is the first of two. Throughout his long and distinguished career as a theorist, composer, organist, and church musician, counterpoint was always his primary interest. He began work on Principles after moving to The University of Texas at Austin in 1977; early iterations of the text were used by hundreds of Doug’s students at UT-Austin and Indiana University (where he taught during a sabbatical), many of whom have gone on to teaching in their own counterpoint classrooms, and making do with faded photocopies of Doug’s text. I first made Doug’s acquaintance as one of those students; later (after getting through those courses successfully!), I summoned up the courage to ask his daughter out on a date, and eventually became his son-in-law.

Doug’s heirs turned to me, as a family member with expertise in the field, after his death to shepherd the book through the publication process. Extensive discussions about Principles with his friends, col-leagues, and former students made it clear that, while Doug’s innovative approach to contrapuntal pedagogy was as fresh and valid as ever, the prose of the text would require some editing for a more contemporary audience. This was a difficult prospect for all of us who loved Doug and were reluctant to change his final work. On the other hand, I strongly felt that the best testament to Doug’s memory would be the publication and continuation of Principles as a living text, meeting the evolving needs of con-temporary counterpoint teachers and students. For these reasons, when Constance Ditzel at Routledge expressed an interest in an updated version of Principles, I suggested that we find a co-author who respected Doug’s pedagogical aims, but would not feel constrained by his memory, someone who was an active counterpoint teacher with a sympathetic but independent pedagogical approach. I believe we’ve found this person in Evan Jones.

On behalf of Doug’s family, I’d like to thank Evan Jones, Constance Ditzel, and Routledge for intro-ducing his work to new generations of counterpoint students, and for keeping alive the memory of an esteemed and beloved teacher, mentor, and friend.

Jonathan C. Santore, Ph.D. Professor of Music Theory and Composition and Chair

Department of Music, Theatre, and Dance Plymouth State University (New Hampshire)

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Preface

It was an honor to be asked to prepare Douglass Green’s two-volume manuscript, which he collectively entitled The Principles and Practice of Counterpoint, for publication. I was immediately impressed by the breadth and depth of the present volume, extending from Gregorian chant through seventeenth-century developments, and including a great number of complete pieces of music. Professor Green’s incisive stylistic observations and his unique pairing of “principles and practice” provide an expert introduction to musical developments over quite a lengthy historical span.

FEATURES

Species Counterpoint

Combining the study of stylistic composition, theory and analysis, music history, and performance, a course in modal counterpoint is without doubt a multifaceted endeavor. Students face the challenge of composing not only “correct” but also musically convincing examples of a centuries-old style, and teachers face the challenge of finding an efficient way of transmitting the necessary knowledge and assessing compositional attempts. The long-standing popularity of Johann Joseph Fux’s “species” approach to modal counterpoint indicates its success in this regard. Fux’s eighteenth-century treatise, Gradus ad Parnassum, established the species approach as the dominant instructional paradigm for over two hundred years. Species counterpoint sets a series of developmental stages for budding contrapuntists to encounter in turn, gradually attaining a more and more elaborate compositional result.

Complete Musical Works

Unfortunately, many species-oriented books (starting with Fux’s own, of course) have neglected to provide a substantial quantity of complete musical works by modal composers, perhaps expecting too much of the method itself and underestimating the importance of studying such issues as text setting. A number of recent textbooks have modified or abandoned the progressive organization of the species approach, aiming instead to extract stylistic rules with reference directly to the musical literature, but have sacrificed something of the pedagogical clarity of Fux’s method. In supplementing a modified species approach with a wealth of complete musical examples and historical information, the present volume keeps the promise of its title, joining principle with practice in an unusually thorough and sensitive way.

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Interwoven Organization

The particular ordering of components in this volume has been chosen so as to achieve the maximum connection between historical, theoretical, and practical dimensions. I share Professor Green’s conviction that Fux’s third species (four notes against one) should be delayed until after fourth species (syncopes); the introduction of smaller note values thus coincides with the study of melody and rhythm in Palestrina, Lassus, and Victoria, beginning in Chapter 14. Professor Green was less certain about the ideal ordering of material before that point in this volume. From certain anomalies of example numbering in Professor Green’s manuscript, it was apparent that at one point he had begun with all the species-based chapters in a row, and had later decided to begin with all of the historical chapters. Neither of these, however, was a perfect solution.

Instead, I have arrived at an interwoven ordering of historical chapters and species-based chapters that reinforces certain topics and skills from different perspectives along the way. The introduction to modal melody and plainsong in Chapter 1 is followed by a consideration of melodic line from a Fuxian perspective in Chapter 2; the survey of medieval polyphony in two and three parts is followed by an introduction to Fuxian first species in two and three parts; and the evolving nature of consonance and dissonance in the music of early Renaissance is complemented by Fux’s presentation of dissonance treatment in second and fourth species.

Even though Fux approached these issues from an eighteenth-century point of view, and was primarily focused on the style of Palestrina rather than the music of earlier centuries, this volume integrates the gradual introduction of the species and the intervening historical and analytical chapters so that they reinforce each other very fruitfully, and can be pursued in tandem.

TO THE STUDENT

The study of the rules and norms of modal music necessarily exists in a symbiotic relationship—indeed, in a kind of counterpoint—with the study of real musical literature. Students will of course become acquainted with the strictures that are discussed and illustrated, but should also aim to achieve a more intuitive awareness of the expressive, affective character of the music being studied. This is only possible if the musical repertoire can be experienced through listening and performance. Even when considering the most technical, abstract details of the style, it is also crucial to recognize a connection to the particulars of compositional practice and to the musical effect that results. I believe this volume strikes this balance in a useful and accessible manner, and provides a truly immersive experience of modal music.

Although most musicians are more familiar with eighteenth- and nineteenth-century classical music than with earlier styles, the study of modal counterpoint provides a direct path to the understanding and appreciation of many different eras of modal music. The traditions and practices that developed toward well-loved examples of sacred sixteenth-century vocal music may initially strike the beginning student as somewhat foreign in sound, and will probably prove difficult to imitate compositionally at first. Following the pluralistic program of study in this volume, however, will bring the student into a closer relationship with modal repertoire, will certainly enhance an awareness of many of its aspects, and will equip the student to compose in a stylistically faithful manner. Further, since the composers of modal music established ideas of consonance and dissonance, harmonic sonority, melodic gesture, and textual treatment that persisted through later centuries, the rewards of studying this subject are similarly long-range in scope.

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ACKNOWLEDGMENTS

I have many people to thank in connection with this project. First, I would like to thank Constance Ditzel (at Routledge) and Jonathan Santore for selecting me to co-author this volume and its companion, and for providing immeasurable assistance along the way. This volume has also benefitted greatly at various stages from the help of Mhairi Baxter and Denny Tek at Routledge, Maggie Lindsey-Jones, Emma Wood, and others at Keystroke, and John Banks. Musical examples were expertly realized by Chris Burton and Jeff Yunek. I would like to thank all my colleagues at the Florida State University College of Music for their support, and all the students who have taken my counterpoint classes over the years. My parents, Peter and Helen Jones, and my parents-in-law, David and Barbara Ferguson, have been extremely supportive and generous during my work on this project. I am also grateful to my wife, Kim, for her patience, her selflessness, and for the pleasure of her companionship.

Finally, I owe a debt that can never be repaid to my esteemed co-author, with whom it has been my very great privilege to collaborate in the manner that we have.

Evan Jones July 2010

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Chapter 1 Modes and Monophony

1.1 Authentic and Plagal Melodies in Folksong

Beginning on a low C (C3 for men, C4 for women),1_{sing the melody of “Home on the Range” (it starts}

“Oh, give me a home”). You will be in the key of F and you will end on the note F. Starting again from the F, in the same key, sing the melody of “On Top of Old Smokey.” You probably noticed that the first one was easy to sing while the second one was uncomfortably high. It would seem more natural for the second tune to be transposed down a perfect fourth or a perfect fifth to begin on low C (C3 for men, C4 for women) or even low Bb (Bb2 for men, Bb3 for women). Now try “Home on the Range” beginning on F4 (F3 for men). Unless you have an unusually high voice, you will wish you were in a lower key.

The easiest singing range for the untrained voice is about an octave, C3–C4 for men, an octave above that for women, C4–C5. Most people can go fairly easily a third below and a second above this octave. By putting “Home on the Range” in the key of F and “On Top of Old Smokey” in the key of C, all these songs can be comfortably sung by anyone.

Folksongs generally have a range of about an octave, rarely more than a tenth. In order to accommodate folksongs to a singable range, the accompanist has to be able to play in at least two keys, a fourth or fifth apart. Most folksongs can be placed into one of two categories: those whose prevailing ranges lie within the octave formed by the tonic notes and those whose prevailing ranges lie within the octave formed by the dominant notes. The former is referred to as an authentic range, or simply an “authentic melody,” and the latter as a plagal range, or a “plagal melody” (Example 1-1).

In relation to the keynote, a melody with a plagal range lies lower than does a melody with an authentic range, for the tonic note lies approximately in the middle of its compass.2

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EXERCISE 1.1

Sing as many of the following songs as you know. Which are plagal and which authentic? 1. “I’ve Been Working on the Railroad” (“The Eyes of Texas”)

2. “Camptown Races”

3. “When Johnny Comes Marching Home” 4. “Alouette”

5. “On Top of Old Smokey” 6. “The Streets of Laredo” 7. “Old Folks at Home”

8. “Oh Dear, What Can the Matter Be?” 9. “Doe, a Deer”

10. “Greensleeves” (or “What Child Is This”)

1.2 Scales and Modes

A thorough discussion of scales and modes, their origins, their tunings and temperaments, their functions and uses throughout music history would be an enormous undertaking. For our purposes here it is enough to describe the chief modes and their scales without inquiring closely into the chain of events that led to their development in the first place.

The notes of plainsong can be abstracted to form diatonic scales. A diatonic scale has one and only one note for each letter name, seven different notes in all. The notes of each diatonic scale can be rearranged to form an uninterrupted seven-note segment of the cycle of perfect fifths. Starting arbitrarily on Bb, the cycle of fifths is shown in Example 1-2 along with the possibilities for various diatonic scales. (Some “fifths” are shown as fourths, in order to keep the pitches within a narrower range on the staff.)

Ionian (major) = notes 2–8, beginning on C Dorian = notes 2–8, beginning on D

Phrygian = notes 2–8, beginning on E Lydian = notes 2–8, beginning on F Mixolydian = notes 2–8, beginning on G Aeolian (minor) = notes 2–8, beginning on A

Transposed Ionian (major) = notes 1–7, beginning on F Transposed Dorian = notes 1–7, beginning on G Transposed Phrygian = notes 1–7, beginning on A Transposed Lydian = notes 1–7, beginning on Bb

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Transposed Mixolydian = notes 1–7, beginning on C Transposed Aeolian (minor) = notes 1–7, beginning on D

Other major scales = notes 3–9, beginning on G; notes 4–10, beginning on D; notes 5–11, beginning on A; etc.

Other minor scales = notes 3–9, beginning on E; notes 4–10, beginning on B; notes 5–11, beginning on F#; etc.

You may already have guessed that a scale in itself gives you neither key nor mode. One note of the scale must be selected as the tonic (or final) and the other notes arranged in ascending or descending order if you are to speak of the “scale of C major” or the “scale of Bb Lydian.” We can take any one of the seven-note segments from the cycle of fifths and, selecting each of the seven seven-notes in turn, construct the scales of seven different keys and modes (see Example 1-3).

In each instance the half-steps occur between different scale degrees. Thus, each one is a different mode, with no cases of the same mode being transposed. The names are those that have come to be attached to these particular modes through tradition.3

Although the mode on B has been given the name Locrian (among others) it is not a mode that has ever really been in use, at least not until the twentieth century. It has always seemed defective because its fifth degree forms the interval of a diminished fifth with the tonic. Thus it has no true dominant, nor does it have a consonant tonic triad. Folksongs tend to divide the octave into two unequal parts consisting of a perfect fifth and a perfect fourth, but the Locrian mode does not lend itself to this kind of division. The lower fifth outlines the diminished fifth and the upper fourth outlines the tritone.4

The Lydian mode, though accepted by theorists as a viable mode, tended to be shunned by composers until the twentieth century, at least in its pure form. The tritone between the first and fourth scale degrees sounds so rough that composers have regularly lowered the latter by a half-step to form a perfect fourth. But to do this consistently turns the Lydian mode into a transposed version of Ionian, the major mode. It

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is only when scale-degree4 is kept well away from scale-degree ^ ^1 that a satisfactory pure Lydian mode can be maintained.

1.3 Plainsong

Medieval theorists found that they could accommodate most of the plainsong repertory in eight categories. Separating the authentic ranges from the plagal ranges gives two large groups. Within each of these groups there are four modes: Dorian, Phrygian, Lydian, and Mixolydian.5 _{The names indicate the authentic}

group. The plagal group uses the same four names with the prefix “hypo-.” During the Middle Ages, as in modern times when dealing with plainsong, it was more usual to denote the modes by numbers than by names. Odd numbers refer to the authentic modes and even numbers to the plagal modes.

SELF-TEST 1.1

Fill in the blanks.

1. The range of a plagal melody is approximately from scale-degree ______ to scale-degree ______. 2. “Joy to the World” is an example of ________________ melody.

3. The Phrygian scale is distinct from the other modes in that the distance from scale-degree ^1 to scale-degree ^2 is a ___________-step rather than a ___________-step.

4. Using only the white keys of the piano, the diatonic scale beginning on G is called ________________ and the one beginning on D is ________________.

5. The scale with half-steps between scale-degrees ^3 and ^4 and between ^7 and 8 is the _______________.^ This scale is also called ________________.

6. The Aeolian (natural minor) scale has half-steps between scale-degrees ______ and ______ , and between scale-degrees ______ and ______.

7. The scale that is similar to major except for the raised scale-degree ^4 is called ________________. 8. The scale that is similar to minor except for the raised scale-degree ______ is called ________________. 9. The difference between major (Ionian) and Mixolydian is in scale-degree ______.

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Psalms and other scriptures tended to be chanted mainly on a single pitch. This pitch is called the reciting tone. In the table above it can be seen that the reciting tone is always F, A, or C except in Mode 7 where it is D. It would have been unthinkable to recite on a B, since in relation to the other notes of the scale B would necessarily be pulling against the F, a tritone away. In the authentic modes, then, the reciting tone is a perfect fifth above the final unless that perfect fifth would put it on a B. The reciting tones of the plagal modes are a third below those of the corresponding authentic modes, except in the case of Mode 8 where C replaces B.6_{The range of Mode 4 violates the pattern of the other plagal modes for the same}

reason. All the other modes are shown with a range of a fourth below the final to a fifth above the final, but the typical range of Mode 4 extends from C to C instead of B to B.

EXERCISE 1.2

Examples 1–4 through 1–16 comprise a small anthology of plainsong in every mode. Sing through these several times each, trying to get a feel for the quality of each mode.7_{It is important to keep in mind that the} notated pitches are not absolute but only relative. Music was not written with various key signatures but only in “white” notes, although B could be either natural or flatted. You may sing these, then, at any comfortable pitch level. As you sing, take note of the melodic intervals and ask yourself these questions:

1. Do the melodic intervals include leaps of thirds, fourths, or fifths? 2. Do the examples include leaps of octaves and/or sixths?

3. Are the gaps produced by leaps invariably followed by melodic motion in the opposite direction? 4. Are there two successive leaps in the same direction?

Also take note of other characteristics.

1. In a large number of chants it is easy to perceive that the melody revolves around the reciting tone of the mode, even though the music is by no means a simple case of reciting a psalm or scripture reading on a single pitch. In many others either the note around which the music revolves is not the reciting tone of the mode or else the music does not concentrate on any one note.

2. Notice how the final cadence is approached. Is it by step from above or below, or is it approached by leap? 3. Finally, think about the form of the piece. Does it divide into clear-cut sections? Does one section seem to be based on one of the authentic modes while another section has the plagal version of the same mode?

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EXAMPLE 1-4

MODE 1: From the Ordinary of the Massa

Dorian (transposed Aeolian)

Lord, have mercy. Christ, have mercy.

a_{Up to the asterisk the music is sung by a solo voice (cantor or priest). The choir joins in at that}

point.

EXAMPLE 1-5

MODE 1: Ambrosian hymn (Second Vespers of Christmas)

Dorian

Christ, the Redeemer of all, from the Father, the only one of the Father,

who alone ineffably was born before the beginning.

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EXAMPLE 1-6

MODE 1: Antiphon (Second Vespers of Christmas)

Dorian (transposed Aeolian)

Today Christ is born: today the Savior has appeared: today on earth the angels sing and the archangels rejoice:

today the just exult, saying: Glory to God in the Highest,

Alleluia.

EXAMPLE 1-7

MODE 2: Introit (First Mass of Christmas)

Hypodorian

The Lord said to me, “You are my Son; today I have begotten you.” (Psalm) Why do the nations rage so furiously, and the people think vain things?

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EXAMPLE 1-8

MODE 3: Hymn (feast of Corpus Christi)

Phrygian

Let my tongue tell the mystery of the glorious Body and of the precious Blood which, for the ransom of the world, the fruit of a noble womb,

the King of the nations, has shed.

EXAMPLE 1-9

MODE 4: Antiphon (Second Vespers of Trinity Sunday)

Hypophrygian

Let praise to God the Father, and to His equal, the Son, and with unceasing zeal to thee, Holy Spirit,

resound from our mouth through all ages.

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EXAMPLE 1-10

MODE 5: Antiphon (first Sunday of Advent at Vespers)

Lydian

Behold, the Lord will come, and all his saints with him:

and there will be on that day a great light. Alleluia.

EXAMPLE 1-11

MODE 5: From the Ordinary of the Mass

Lydian (transposed Ionian)

Lamb of God who takest away the sins of the world, have mercy upon us.

EXAMPLE 1-12

MODE 6: Hymn (Seven Sorrows of the Blessed Virgin Mary, Second Vespers)

Hypolydian

The grieving mother stood in tears next to the cross where her Son was hanging.

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EXAMPLE 1-13

MODE 6: Introit (Mass for the Dead)

Hypolydian

Grant them eternal rest, O Lord, and let perpetual light shine upon them.

EXAMPLE 1-14

MODE 7: Communion (Mass on the feast of the Sacred Heart of Jesus)

Mixolydian

One of the soldiers pierced his side with a spear, and at once there came out blood and water.

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EXAMPLE 1-15

Before Sunday Mass

Mixolydian

Thou shalt sprinkle me with hyssop, O Lord, and I shall be cleansed;

thou shalt wash me,

and I shall be made whiter than snow. (Psalm) Have mercy on me, O God, according to thy great loving-kindness.

EXAMPLE 1-16

MODE 8: Hymn (Whitsunday at Second Vespers)

Hypomixolydian

Come, Creator Spirit, visit thy minds: fill with celestial grace the hearts

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1.4 The Problem of Ionian and Aeolian

You may be wondering why the medieval theorists recognized only four different modes with their authentic and plagal versions. What about Aeolian and Ionian modes such as seem to occur in Example 1-4, 1-6, and 1-11?

There are two possible answers to this question. First, because of the problem of the tritone with F, it was always possible to use Bb instead of B-natural. But the use of Bb can alter the character of a mode. In Example 1-4 and much of Example 1-6, for instance, the stress on F calls for Bb, to avoid the F–B tritone. It would not be strictly necessary to recognize the mode on A (Aeolian) as a separate entity, since music with Aeolian characteristics could be easily explained as Dorian-with-Bb transposed up a perfect fifth. Similarly, when Lydian used Bb, as it often did, it did not have to be explained as a different mode (Ionian) since it could be sufficiently understood as Lydian-with-Bb, the flats being there for the purpose of correcting the tritone.

Another answer to the question has to do with the medieval concept of the organization of the modes. They were not thought of as scales of seven different pitch classes spread throughout an octave. Rather, each mode was considered to consist of a perfect fifth and a perfect fourth placed conjunctly—that is, the same pitch served as the upper extreme of the perfect fifth and the lower extreme of the perfect fourth, or vice versa. Authentic modes had the perfect fifth on the bottom, plagal modes the perfect fourth. Each fifth and each fourth was characterized by the placement of the intervals between their extremes. Thus each mode had its particular type of perfect fifth and its particular type of perfect fourth, as demonstrated in Example 1-17. Notice the location of the half-steps. In the first four modes the half-step placement in the fifth corresponds to that in the fourth.

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It is true that Dorian and Mixolydian share the same type of fourth. Since a fourth offers only three possible placements of the half-step, it is inevitable that two of the four fourths would be alike. But their fifths are unique. The Aeolian and Ionian modes on the other hand have neither unique fourths nor unique fifths. Aeolian is a mixture of Dorian and Phrygian, Ionian a mixture of Mixolydian and Lydian. (See Example 1-18.) Not until the Swiss theorist Heinrich Glarean published his treatise Dodecachordon in 1547 were Aeolian and Ionian accepted as modes in themselves.8

EXAMPLE 1-18

SELF-TEST 1.2 Mode numbers

Reciting tone Final Fill in the mode number (1–8)

A F Mode no. _____ A E Mode no. _____ A D Mode no. _____ F D Mode no. _____ C F Mode no. _____ C G Mode no. _____ C E Mode no. _____ D G Mode no. _____

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EXERCISE 1.3

Answer the following questions about each of the four pieces given. 1. What is its mode?

2. Can the music be said to revolve around the traditional reciting tone of its mode? 3. Which is more prevalent, motion by step or motion by leap?

4. Generally speaking, does the motion turn downwards after an upward melodic leap, and vice versa? 5. How are the cadences approached, by stepwise motion or leap? From above or below?

A. Antiphon in honor of the Blessed Virgin Mary

Hail Mary, full of grace, the Lord is with thee.

Blessed art thou among women and blessed is the fruit of thy womb, Jesus. Holy Mary, Mother of God, pray for us sinners

now and in the hour of our death. Amen.

SELF-TEST 1.3 Fourths and fifths

1. The Dorian fifth has half-steps between its ______________ and ______________ notes. It appears not only in Dorian mode, but also in ______________ and Hypo______________.

2. The fourth with a half-step between the first and second notes is called the ______________ fourth. It appears in the ______________, the Hypo______________, and the ______________ modes.

3. Aeolian mode consists of a ______________ fifth and a ______________ fourth. 4. Ionian mode consists of a ______________ fifth and a ______________ fourth.

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B. Hymn for Vespers, First Sunday of Advent

Nurturing Creator of the stars, eternal light of believers,

Jesus, Redeemer of all, hear the suppliants’ prayers.

C. Hymn before Benediction (Stanza 5 of Example 1-8) Spanish chant

Therefore, prostrate, we revere such a great sacrament: and the old teaching yields to the new rite:

let a supply of faith surpass the insufficiency of the senses.

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D. Hymn of St. Thomas Aquinas in honor of the Blessed Sacrament

Godhead here in hiding, whom I do adore

Masked by these bare shadows, shape and nothing more, See, Lord, at thy service low lies here a heart

Lost, all lost in wonder at the God thou art.

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Chapter 2 The Single Line

2.1 Species Counterpoint

During the sixteenth century a number of treatises dealing with counterpoint were written. Several authors, among them Gioseffo Zarlino and Thomas Morley, used an approach in which the simplest steps were presented first, then modifications and difficulties were gradually introduced one by one. Characteristic of this approach was the presentation to the student of a given unchangeable melody, called the cantus firmus (“fixed song”), against which the student wrote counterpoints. At first only one note was placed against each note of the cantus firmus—note against note, or in Latin, punctus contra punctum. After this step was mastered, two notes were placed against one, and so forth in various combinations called species.

This pedagogical approach, known today as “species counterpoint,” was thoroughly systematized by Johann Joseph Fux in his famous Latin treatise Gradus ad Parnassum. His book is known to have been admired by J. S. Bach, whose pupil Lorenz Mizler translated it into German. It became the basis of contrapuntal study for virtually all eighteenth-century composers, including Haydn and Mozart. When Mozart taught counterpoint to his English pupil Thomas Attwood, Fux’s treatise was the basis of their lessons. Haydn and Albrechtsberger both had recourse to it when teaching their most famous pupil, Ludwig van Beethoven.

The value of the method is indicated in the title “Steps to Parnassus” (Mount Parnassus is the mythical home of the muses). The student is asked to take one step at a time. Difficulties are isolated from each other so that one does not have too much to contend with at any given moment. But the method has a drawback, too, for if followed systematically through all five species, first in two voices, then in three, then in four and more, the time normally allotted for the study of counterpoint in the usual college curriculum will long have gone by. Only artificially contrived counterpoint will have been dealt with and the music of the great composers bypassed. For this reason, we will use the species approach only up until Chapter 13 of this book, for the purpose of introducing and practicing only the most essential concepts of coun-terpoint.

Although species counterpoint is intended to acquaint the student with contrapuntal principles rather than with any particular historical style, it was developed at a time when Palestrina’s manner was looked on as the epitome of the strict sacred vocal style. The practice of species counterpoint, then, is an excellent introduction to composition according to the aesthetic of the Roman School of the late sixteenth century. This music was done almost entirely in strict diatonicism either with no key signature or with a signature of one flat. Accidentals are strictly limited:

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1. When no key signature is present one may, under certain circumstances to be explained later, alter a B-natural to a Bb. With a signature of one flat, one may alter an E-B-natural to an Eb.

2. When cadencing on D, G, or A, one should create a leading-tone by raising the seventh degree of the scale. To cadence on D, use C# instead of C-natural. Similarly, to cadence on G or A, use F#or G# respectively.

3. When ascending through the leading-tone one may raise the sixth degree of the scale in order to reach the seventh degree by step (e.g., F#–G#–A).

In summary, the only accidentals available in this style are Bb, C#, F#, and G#. With a key signature of one flat, Eb is also available, but G#is unavailable. It is important to remember that accidentals may not be used arbitrarily. Sharps are used only to produce a leading-tone for a cadence. Flats are used only to avoid the tritone (explained in 2.3 below).

With this limited spectrum of pitch classes, exercises can be done in any of the standard church modes. The modes were discussed in detail in Chapter 1. For now, we need to remember especially the following:

1. The Dorian mode beginning on D uses the scale D–E–F–G–A–B–C–D with no signature. It can be transposed up a perfect fourth by using a key signature of one flat and beginning the scale on G: thus, G–A–Bb–C–D–E–F–G.

2. The Phrygian mode beginning on E uses the scale E–F–G–A–B–C–D–E. It can be transposed up a perfect fourth by using a key signature of one flat and beginning the scale on A: A–Bb–C–D–E– F–G–A. The Phrygian mode differs from the others in that it never uses a raised seventh degree of the scale in order to cadence. The move into the tonic from below is by whole step, D-natural to E or, if transposed, G-natural to A.

3. The Lydian mode beginning on F uses, like the other modes, only those pitches with unaltered letter names (white keys of the piano). Transposed to Bb, it has a signature of one flat.

4. Mixolydian uses the “white notes” on G. Transposed up a fourth, its scale is C–D–E–F–G–A–Bb–C. 5. Aeolian is the same as natural minor on A; transposed up to D, it uses one flat in the signature. 6. Ionian is the same as major on C; transposed up to F, it uses one flat in the signature.

In modal terminology the note on which the scale is built is known as the final. In Dorian, the final is D; in Phrygian it is E, and so on. The word “final” is analogous to the word “tonic” as used in the analysis of tonal music.

2.2 The Melodic Line

Sing the cantus firmus in Example 2-1, play it, then sing it again.1_{As you perform this and the following}

melodies, try to notice which specific intervals occur melodically. When there is a leap, how is it approached and left?

EXAMPLE 2-1

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The eleven pitches comprise seven stepwise motions and three leaps (notes 1–2, 4–5, and 6–7). Although other melodies may have a greater proportion of leaps, on the whole this ratio is about normal— two-thirds stepwise motion and one-third leaps. This is not surprising when we remember that a melody is not simply a number of notes sung or played one after another. Rather, it has something about it that makes these notes hang together in such a way that we can refer to them in the singular: we say “a melody,” not “some notes.” One of the qualities that brings about this coherence is the fact that most of the notes in the melody move to other notes that are close by.

There is a second fact about Fux’s cantus firmus (hereafter referred to as C.F.) that plays a part in producing a coherent quality: the melody has a single climax. This is note 7, higher than the other pitches in the melody. But the simple fact that it is the highest pitch is not enough to cause it to serve as a climax. Rather, it is due to notes 2 and 5, which prepare the climax by leading the ear upward, and notes 8 through 11, which bring about a gentle descent.2_{In other words, a high or low note cannot act as a focal point if}

it is isolated from the other notes of the melody. Rather, such isolated notes will simply not seem to be a part of the line. Sing and play the “bad” examples given in Example 2-2. The notes marked X are isolated. Only by altering the notes or adding others can the melodies of Example 2-2 be made satisfactory.

In the case of melody (a) in Example 2-3 the zenith is not led up to as it was in Fux’s C.F., Example 2-1. Nevertheless, the climactic C5 is well enough incorporated into the line because of the descending stepwise motion, which fills in the gap created by the leap from note 3 to note 4.3_{In the case of melody}

(b) in Example 2-3, the second note (B3), though highest, sounds less like the climax than part of a circular motion around the pitch G3 (notes 1 and 4). Rather, it is the low D3 (note 5) that seems to be the focal point of the melodic line. A negative climax like this is called a nadir.

In melody (a) of Example 2-3, the gap between notes 3 and 4 was filled in by stepwise motion in the opposite direction. A similar means of filling-in occurred in melody (b) after the gap produced by notes 4 and 5. But motion in the opposite direction serving to fill in a gap does not necessarily have to be by an immediate step. It is enough that the gap is compensated for in some way. Melody (c) inserts an extra note

EXAMPLE 2-2

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(4a), which moves by leap in the opposite direction. Melody (d) reverses notes 6 and 7, yet the gap between notes 4 and 5 is satisfactorily filled in. Sing-play-sing all the melodies of Example 2-3 several times.

Do the same with the melodies in Example 2-4. Which are satisfactory? Whenever one seems unac-ceptable, try to determine precisely what makes it so. DO NOT READ THE SUBSEQUENT PARAGRAPH UNTIL YOU HAVE DONE THIS.

You probably noticed that melody (a) of Example 2-4 was extremely static, owing to the back-and-forth motion of notes 2 through 6. This motion also means there is no single climax, neither zenith nor nadir. Melodies (b) and (c) are satisfactory. In the latter, notice how note 2 is brought into the mainstream by notes 9 and 12. These three notes together form an ascending scale (G3–A3–B3) which is fulfilled by the final C4. Melody (d) would be acceptable if only it began with note 2. But the first note, A2, is isolated from the rest of the line. Note 8 is raised, of course, in order that it may move by whole step into the leading tone, as in the melodic minor scale.

2.3 Melodic Intervals

Here are more C.F.s to be sung, then played, then sung again (do not neglect this triple performance!) (Example 2-5).

EXAMPLE 2-4

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In performing these and the previous C.F., you may have made some or all of the following observations: 1. Species counterpoint is strictly diatonic. No chromatic half-steps appear and the only accidentals are those necessary to create a leading tone or to avoid an augmented second in approaching the leading tone from below. Two half-steps in succession must be avoided: for instance, Bb–A–G#.

2. Occasionally a note is repeated, but rarely more than once.

3. Only easily sung melodic intervals are employed. No leaps of augmented or diminished intervals, no leaps larger than a perfect fifth and minor sixth except for the octave. Minor sixths occur only very occa-sionally and always in ascending, never descending, motion. Major sixths are entirely avoided.

4. Melodies end on the final of the mode. The final is approached by step either from above or from below. Although in these examples all melodies also begin on the final, this is not a strict rule. If not the final, the first note will probably be the fifth degree of the scale.

5. Approach to the cadential ^2 is usually by step from above. When the final is approached by step from above, that is, from scale-degree ^2, this note is itself approached by step or by a descending third. In other words, the last three notes will be ^3–^2–^1, ^1–^2–^1, or ^4–^2–^1. Of these, ^3–^2–^1 is the most common. 6. A melodic leap must be compensated for in one way or another. Either the note immediately preceding or the note immediately following a leap should move in the opposite direction to the leap itself. That is, leaps must be either approached or left by motion in the opposite direction. While leaps may be both approached and left by movement in the opposite direction, this contour is not a requirement except in the case of large leaps—the ascending minor sixth and the octave, ascending or descending (see item 7 below). The compensating movement in the opposite direction does not necessarily have to be by step. The rule for leaps of a major or minor third, perfect fourth, or perfect fifth may be put this way: when a leap occurs it is either at the bottom of a line in a single direction or at the top, not somewhere in the middle.

7. Large leaps are both approached and left by contrary motion. The ascending minor sixth and the octave are considered large leaps. For the sake of balance in the line, the notes both preceding and fol-lowing must lie within the gap produced by these leaps. (See Example 2-5(b), notes 6–8; Example 2-5(c), notes 4–6; and Example 2-5(e), notes 2–5 and 7–11.)

8. Occasionally double leaps occur. Two successive leaps in the same direction occur twice in Example 2-5(a), notes 4–6 and 8–10. In this case the double leap must be both preceded and followed by motion in the opposite direction. Moreover, the double leap itself must outline a major or minor triad (not diminished or augmented). Or, the two leaps may outline an octave—a perfect fifth as the lower leap, a perfect fourth as the upper leap (as in Example 2-5(d), notes 1–3). A general principle regarding double leaps is that the smaller leap is never the lower one.

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9. A high or low note is not isolated by register from the other notes, but is incorporated into the line by means of notes a step or a third away.

There is one more observation very important to an acceptable melodic line in species counterpoint. This has to do with the avoidance of the stressed tritone. In everyday language the word “tritone” is often used to mean either the augmented fourth or the diminished fifth. When correctly used, however, the term applies only to the augmented fourth, which indeed comprises three (whole) tones: hence, “tri-tone.” The diminished fifth, on the other hand, is made up of two half-steps, one at either end, and two whole tones. Of course the total number of semitones in either case is six, but the effect in tonal and modal music of a tritone is very different from that of the diminished fifth. In illustration, sing, play, and sing the melodies given in Example 2-6. DO NOT READ FURTHER UNTIL YOU HAVE DONE THIS.

SELF-TEST 2.1

The following are intended to help you reinforce this information. Directly above each melody, write “good” or “bad” with the numbers 1 through 7 of the observation that it illustrates or ignores.

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You probably noticed a certain difficulty in singing note 5 of melody (b), or at least in singing it exactly in tune. There is a slight roughness here since note 5 is the lower end of a tritone begun with note 2. Although notes 5–7 in the same melody also produce a tritone, the fact that the melodic line continues on upward to the C4 (note 8) smoothes out this roughness so that the latter part of the melody is acceptable. Similarly notes 2–4 in melody (c) cause a difficulty. On the other hand, melody (a) offers no problems whatever, notes 2–6 comprising not a tritone but a diminished fifth. We can, then, formulate Observation No. 10.

10. The last three or four notes of a passage in a single direction should not outline a tritone (an augmented fourth).

Principle

In species counterpoint the single melodic line is simple and coherent.

Practice

Simplicity of line is produced by limiting the pitches to those occurring in the diatonic scales, by stressing stepwise motion, and by ending with a stepwise approach to the tonic.

Coherence is produced by compensating for leaps by subsequent contrary motion, and by providing a clear focal point, zenith or nadir.

EXERCISE 2.1

Review Section 2.2 (“The melodic line”). Then write twelve melodies in whole notes, very carefully following Observations 1–10 as listed in Section 2.3. Use (at least) two different clefs. The melodies should be in various modes, some transposed (one flat in key signature) and some not. Always begin on scale-degree ^1 or ^5 and end on ^1.

Sing-play-sing each one before deciding you are satisfied with it. Remember, your best friend is your eraser.

SELF-TEST 2.2

The following should help to reinforce the observation regarding the tritone. Write “good” or “bad” directly above each melody. If the melody is “bad,” indicate which notes comprise the offensive tritone.

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Chapter 3 Counterpoint During

the Middle Ages

A thorough account of counterpoint from the tenth century through the sixteenth century must inevitably deal with virtually all part music written during this period of more than five hundred years. Such accounts are presented to varying degrees of detail in music history books and musical dictionaries. Our purpose here is to highlight some of the aspects of counterpoint from that period, especially aspects that have a bearing on the contrapuntal writing of later periods.

3.1 Early Organum

About the end of the ninth century, theorists began to describe a way of singing portions of plainsong in more than one part. There is reason to believe that the practice of singing secular music in at least two parts had, for a long time, been practiced in some places, but we have no authoritative written accounts of such activities. A segment of plainsong would be sung by one voice, the vox principalis, while another voice, the vox organalis, would double it at the fourth or fifth below. Perhaps two higher voices would double the principal and organal voices at the upper octave. Much of the piece would move along this way in parallel fourths and/or fifths. The voices generally began with a unison and spread out through a second or third to the fourth or fifth and reversed the process for cadences. In his treatise entitled Micrologus (c. 1025), Guido of Arezzo talks about these approaches to the cadential unison, showing that theorists of the time were interested in how two-part cadences could be achieved. Example 3-1 is a piece found in a handbook of music, the Musica Enchiriadis, an anonymous work dating from around 900.

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In addition to the dominating parallel motion, it includes oblique motion at the beginning of phrases and two or three instances of contrary motion. By the next century, contrary motion in organum was being both practiced by composers and advocated by some theorists.

In the twelfth century in certain places in Spain and France, another type of organum was being practiced, which has come to be called florid or melismatic organum. The organal voice, rather than moving note-against-note with the principal voice, sings a melisma in an improvisatory manner above the slower-moving notes of the plainsong. Thus what had originally been the chief tune, the plainsong itself, was now more of a foundation made up of a series of long sustained notes acting as a support for the fantastic arabesques in the upper voice (Example 3-2).1

Since the manuscripts, found in the monastery of St. Martial in south-central France and the monastery of Santiago de Compostela in northwestern Spain, are by no means consistently clear as to precisely where the simultaneities occur, we cannot draw hard-and-fast conclusions regarding consonance and dissonance treatment in melismatic organum. What is interesting for us in this study is the texture of this music— two melodic lines performed simultaneously but maximally contrasted to each other.

By the late twelfth century, composers may have felt the need for a notation by which they could indi-cate the rhythms of the two voices more precisely. Over a period of years the practice of notating by means of rhythmic modes came into being. Each of the six modes provided a basic rhythmic pattern similar to those found in poetry, and means were worked out by which slight variations of these patterns could be notated. Example 3-3 is based on the first rhythmic mode, long-short, here transcribed as a quarter-note followed by an eighth-note.

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As might be expected, phrase endings formed the interval of an octave or perfect fifth. Rarely one might find a perfect fourth. More often than not, phrases also began with one of these perfect consonances, but they did not do so consistently. They might even begin with dissonances. Leonin’s Gaude Maria (Example 3-3) begins its first two phrases (not shown) with a minor seventh and a major ninth, respectively, both moving immediately into an octave. In twelfth-century counterpoint there was often a great deal of parallel motion, particularly parallel perfect fifths.

3.2 Voice Interchange

The anonymous work given as Example 3-4 is a fascinating motet from the thirteenth century. The text itself is amusing. It interrupts the word “Alleluya” by inserting other words between its second and third syllables, at each repetition adding length to this interruption. Finally, as a kind of coda, we hear the word in its normal form.

EXAMPLE 3-3

From Gaude Maria

Leonin

EXAMPLE 3-4

Alle, psallite

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Alle- sing with -luya. Alle- noisily sing with -luya

Alle- to God with a full heart sing with -luya Alleluya.

The tenor sings a bit of melody, which may or may not be taken from plainsong, three measures in length. Above this a second voice, the duplum, sings a counterpoint, virtually note-against-note. Above this a third voice, the triplum, sings a slightly more florid counterpoint. These three measures are imme-diately repeated, the sole difference being that the duplum sings what the triplum previously had, and the triplum sings what the duplum previously had. This type of voice interchange, also known by the German

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term Stimmtausch, is common in medieval music. The exchange is at the identical pitch level so that the repetition is essentially the same, although of course there will be some difference in timbre between the two singers.

The second phrase, extended to four measures, is similarly repeated with voice interchange, as is the third phrase, five measures in length. Two voices always rest at the end of a phrase while the other voice artfully bridges the gap between phrases. Each phrase ends with a perfect fifth above a diminished third, the final of the mode. Harmonically, the piece is quite static. Such is not always the case, but many thirteenth-century pieces tend in this direction. The famous rota from thirteenth-century England, Sumer is icumin in, makes elaborate use of voice interchange while being even more harmonically static than the motet Alle, psallite. Its correct performance takes six voices, four to sing the round itself, and two for the ostinato pes (“foot”) which supports it. Example 3-5(a) shows this pes; Example 3-5(b) shows the essential harmony of the entire piece.

3.3 Music Without a Plainsong Basis

Not all sacred music used plainsong as a foundation. One of the chief musical types in the thirteenth century was the conductus, a piece normally composed of newly invented melodies. The prevailing, though not the only, texture for the conductus was basically note-against-note, so much so that musicologists today speak of such texture as “conductus style.” Example 3-6 is the beginning of a hymn in conductus style for two voices.

Born for us today of the Virgin Mary . . .

Mark the harmonic intervals, noticing where consonances and where dissonances occur. Is there any voice crossing? Do all types of motion appear? Are there parallel octaves, unisons, and/or fifths? What is the rhythmic relationship between the voices?

EXAMPLE 3-5

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Example 3-7 is the first stanza of the complete conductus Orientis partibus. It is a three-voice setting of the Song of the Ass, a tune that existed in various versions and may still be familiar today as the Christmas carol “The Friendly Beasts.” This conductus was performed as part of a ceremony celebrating the flight of the Holy Family into Egypt, during which a young girl dressed as the Virgin Mary rode a donkey into the church. It is known to have been performed regularly in France at Beauvais and Sens, and probably also in Madrid. The tune is in the lowest voice.

Out from lands of Orient was the ass divinely sent; strong and very fair was he, bearing burdens gallantly,

Heigh, Sir Ass, oh heigh!

(Translated by Henry C. Greene)

Perform the piece several times by singing the tune and playing the upper voices. Then ask yourself the same questions asked in reference to Example 3-6. If you did this carefully, you probably made some or all of the following observations:

1. The chief consonant sonority is the perfect fifth, which begins and ends each phrase and dominates the strong beats within the phrase.

2. Voice crossing is quite common, even below the lowest voice.

3. Parallel perfect fifths, perfect octaves, and perfect unisons are all to be found. 4. The upper voices are slightly more rhythmically active than the lowest voice.

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Secular music, of course, did not use plainsong as a basis. The instrumental dance illustrated in Example 3-8 uses the first rhythmic mode in both voices. Since it is a dance it has a regular metric structure that transcribes into modern notation as four-measure phrases. The second phrase is similar to the first except that its ending is altered to conclude on the final of the Lydian mode, while the first phrase ended on the second scale degree. Thus the two phrases together form what we would call today the antecedent and consequent of a period. In the thirteenth century, the era of this dance, the phrase that begins the period was called “open” and the phrase that ended it was “closed.”

The remainder of the piece consists of four more “periods” on the same tune but with varying counter-points. Beginning with the third “period” the tune is transferred to the upper voice and, consistent with the lack of key signature in that voice, transposed up a perfect fifth.2_{The last time, the theme itself is}

varied.

The style of the estampie is similar to that of a two-part conductus. The intervallic structure between the two voices is grounded solidly in the sonority of the perfect fifth. Not only do most phrases begin and end with the perfect fifth, but so do most measures begin with it. Dissonances tend to be a step away.

Play the estampie fragment several times, listening carefully to the effect of the Bb3 in the lower voice and the B3 in the upper voice. Although the B-natural occurs but once in the first period (m. 7) it gives a very distinctive sound to the passage. It leads to the upper voice cadence on C as a leading-tone to scale-degree ^

5 of the mode, while the E3 in the lower voice acts as leading-tone to scale-degree ^1. The quality of this double leading-tone became very prominent in later thirteenth-century music and permeates the sound of fourteenth-century music, occurring in a much more obvious way than in this estampie (see Chapter 6).

The trouvère Adam de la Halle, to whom our Example 3-9 is attributed, composed a number of ron-deaux in three parts that are fairly typical of thirteenth-century conductus style. Notice how the two upper voices consist, for the most part, of parallel fourths in opposition to the lowest voice.

A rondeau consists of two parts, A and B, both parts being performed according to the scheme AB aA ab AB. Capital letters indicate identity of the poetic line as well as the musical. In Tant con je vivrai, B consists of three phrases, each three measures in length, and all closely related motivically. Example 3-9(b) shows a summary outline of the four cadences in this piece. In the first two cadences, two voices move stepwise into the cadential note and its fifth, while the third voice jumps to the octave of the cadential

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note. The last two cadences are forerunners of what became known as the clausula vera, the “true” or “authentic” cadence, in which two voices move by step—one up and one down—into the cadential note. In the thirteenth and fourteenth centuries the third voice normally ascends by half-step into the fifth above the cadential note. This is the double leading-tone cadence.

EXAMPLE 3-9 (a)

Rondeau: Tant con je vivrai

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Tant con je vivrai is also typical of much thirteenth- and fourteenth-century music in that no particular mode is made manifest. The overall tonal organization is not without logic, however, as is shown in Example 3-10. The first part moves from an 8

5on A down to an 85on G. The second part begins again on

A and moves through G to conclude on F.

The feeling of conclusion at m. 14 is not due to the melodic lines having reached a “tonic.” It is more likely due to their having simultaneously arrived at a low point. The listener senses an affinity with the natural law of gravity.3

EXAMPLE 3-10 EXAMPLE 3-9 (b)

SELF-TEST 3.1

Circle the correct words or figures.

1. In early organum the vox principalis sang a segment of plainsong that sounded above / below the vox organalis.

2. In the late twelfth / thirteenth century composers such as Leonin organized their music rhyth-mically by the use of rhythmic modes.

3. In the motet Alle, psallite cum luya the German term Stimmtausch refers to the insertion of text

within a word / the repetition of material in a new voice.

4. Sumer is icumin in is a conductus / rondeau / canon.

5. It is true / false that all music of the Middle Ages was built on a plainsong basis. 6. Orientis partibus is a conductus / rondeau / canon.

7. The estampie is an instrumental dance with a texture similar to florid organum / a rota / a

conductus.

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Chapter 4 First Species in Two Voices

4.1 Harmonic Intervals: Consonance and Dissonance

For two reasons the definition of consonance and dissonance1_{has changed over the years: (1) from time}

to time opinions as to which intervals are consonant and which dissonant have varied; and (2) the precise meaning of the terms themselves has not had a uniform explanation from writers on music.2_{For instance,}

in 1597 Thomas Morley defined a consonant sound as one which “delights” the ear, a dissonant sound as one which “offends.”3_{Many other writers, especially those writing before the nineteenth century, also}

speak of “pleasantness” and “unpleasantness” as aspects of consonance and dissonance respectively. In our own time, writers often describe the difference in terms of relative stability or instability. A con-sonance sounds relaxed, not particularly demanding further motion: it is more or less stable. A discon-sonance sounds tense, demanding resolution to a more stable sound. A slightly different point of view is that a dissonance is a sound that needs explanation—in order to understand it the listener must depend on its resolution to a consonance.

To complicate matters still further, intervals that are generally acknowledged to be consonant are not nec-essarily perceived to be equally so. Thirds and sixths are both considered consonant intervals, yet the third is stable enough to end a piece of music while its inversion, the sixth, is not. And the perfect fourth has, over the years, been in the ambiguous position of being consonant in some contexts and dissonant in others.

Until the eighteenth century theorists regarded as most consonant those intervals whose ratio was made up of the smallest numbers. For example, if one vibrating string produces the note C2, a string half its length will produce C3, an octave higher. The ratio of the two pitches’ frequencies is 2:1, and the resulting octave is a perfect consonance. If the ratio between two frequencies is 3:2, a perfect fifth results. It is called a perfect consonance, but is slightly less stable than the octave. The ratio 4:3 gives the perfect fourth, which is the inversion of the perfect fifth and less stable than the fifth. The imperfect consonances are given by the other ratios: 5:4 and 6:5 result in the major and minor third respectively, and their inversions are the minor and major sixths, ratios 8:5 and 5:3.4_{Sixths are less stable than thirds. It might appear, then, that}

the “stability” of an interval is in direct proportion to the ratio that produces it—that is to say, the ratio made up of two adjacent lower numbers gives a more consonant interval than does a ratio deriving from higher numbers or from numbers not adjacent. Thus, the perfect fifth (3:2) is more stable than the major third (5:4), but the major third is more stable than the major sixth (5:3).

It was during the seventeenth century that the laws of vibrations of strings were discovered. When a string vibrates it does so not only in its full length but also in halves, thirds, quarters, and so on. Not only

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is the fundamental pitch of the string produced, but also overtones in increasingly smaller intervals. The same is true of the column of air in an organ pipe or a wind instrument. Thus the early theorists with their consonances based on small-numbered ratios were justified since the first—and therefore most prominent —overtones correspond exactly to these ratios. A pitch produced by a musical instrument is a complex sound, made up of various pitches. We speak of the fundamental as the first partial, the first overtone as the second partial, and so on. (See Example 4-1.)

The first six partials correspond to the pitch classes of a major triad, but beyond that some of the partials do not correspond very closely with the pitches used in Western music. The seventh and eleventh partials, for instance, are perceptibly lower than the notated Bb4 and F#5 in Example 4-1 (hence the minus signs above them). Moreover, the overtones from the seventh partial upwards are too close together to be perceived as consonances, for they are notes of adjacency—a whole step or smaller. Therefore tradition has it that the first six partials along with their octave replicas give us our consonances, as shown in Example 4-2.5_{The fifth and eighth partials produce the minor sixth, the third and fifth partials the major sixth.}

Consonances, then, are the perfect intervals and major and minor thirds and sixths. But it is not quite so simple as this, for there is a problem regarding the perfect fourth. Beginning in the fourteenth century, composers have tended to treat the perfect fourth as a consonance only when it appears between two upper voices. When the lower of the two notes comprising the perfect fourth is the lowest-sounding voice, this interval has been perceived as so unstable as to be for all intents and purposes a dissonant interval, and has been used as such by composers. Possible reasons for this rather odd phenomenon are demonstrated in Example 4-3.

EXAMPLE 4-1 THE OVERTONE SERIES

The Principles and Practice of Modal Counterpoint.pdf

The Principles and Practice

of Modal Counterpoint

The Principles and Practice

of Modal Counterpoint

Contents

Foreword

Preface

FEATURES

TO THE STUDENT

ACKNOWLEDGMENTS

Chapter 1

Modes and Monophony

1.1 Authentic and Plagal Melodies in Folksong

1.2 Scales and Modes

1.3 Plainsong

1.4 The Problem of Ionian and Aeolian

Chapter 2

The Single Line

2.1 Species Counterpoint

2.2 The Melodic Line

2.3 Melodic Intervals

Chapter 3

Counterpoint During

the Middle Ages

3.1 Early Organum

3.2 Voice Interchange

3.3 Music Without a Plainsong Basis

Chapter 4

First Species in Two Voices

4.1 Harmonic Intervals: Consonance and Dissonance