UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 THEORY AND PRACTICE: RAMEAU S FUNDAMENTAL BASS APPLIED TO THE CONTEMPORARY FRENCH OVERTURE A thesis submitted to the Division of Research and Advanced Studies Of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF MUSIC In the Department of Composition, Musicology, and Theory Of the Cincinnati-Conservatory of Music June, 2004 by Tamara A. McKinney B.M., Belmont University, 1994 Committee Chair: Dr. Robert Zierolf

3 ABSTRACT Music theorists at the turn of the eighteenth-century focused much of their energy introducing new theoretical methods for musical composition. As a result the field of music theory grew into several important areas. Jean-Philippe Rameau ( ) first presented many of the ideas that form our modern analytical practice. In Traité de l harmonie, Rameau established a system of harmonic principles currently in use today. These include: constructing chords in a series of thirds, the formation of the major triad from the overtone series, the relationship of chords to a tonal center, the theory of chordal inversions, and the theory of fundamental bass. Although Rameau first intended his theory of fundamental bass to be used as a pedagogical tool, the function of the fundamental bass developed into revealing the foundation or root of each chord. The purpose of the chordal root was to serve as an early theoretical and analytical perception later interpreted by Roman numerals. The purpose of this Thesis will be to present an analysis of Rameau s fundamental bass theory as it is applied to the music of the French overture style. In addition to Rameau s overtures, some by Lully and Telemann were chosen to allow comparison both chronologically and nationally. Pieces selected include Alceste, Cadmus et Hermione, and Amadis by Lully; Hippolyte et Arcie, La Princesse de Navarre, and Castor et Pollux by Rameau; and Suite in D, Overture in D minor, and Orchestral Suite in F minor by Telemann. Rameau s theory of fundamental bass was a theoretical revolution. It provided a new model to indicate not only the origin of harmonies, but how these harmonies progressed in music over real time. His theory also verified the interpretation of a succession of harmonies to be a process of motion. The essence of this Thesis is to demonstrate that the fundamental bass theory developed by Rameau

4 is a practical description of music from that time and beyond.

5

6 ACKNOWLEDGEMENTS To my family: for their unfailing love and support. To Dr. Robert Zierolf: my eternal gratitude for giving the opportunity to come to CCM and to be apart of such a wonderful institution. To the other the faculty and staff at CCM for making this an experience I will never forget.

7 CONTENTS Introduction vii Chapter 1. Theorists A. Theorists before Rameau. 1. Johann Joseph Fux. 2. Johann David Heinichen. B. Rameau's contributions. 2. Theory of fundamental bass Analysis of Corelli's Sonatas Fundamental bass analysis of overtures by Lully, Rameau, and Telemann Conclusions Bibliography Appendices vi

8 INTRODUCTION Seventeenth-century theorists in France contributed to the wide variety of European theoretical literature. They were concerned with many theoretical and compositional issues such as: an ornamentation classification system for performers; the reconstruction of theory manuals to include all levels of pedagogy; establishing clear definitions of all theoretical terminology; the documentation of early music history; and distinguishing between specific theories of thoroughbass, counterpoint, and a formal organization of music. 1 By the late seventeenth century French music theorists faced a substantial amount of diverse theoretical concepts concerning music theory and composition. For example, major/minor keys overlapped with different modal systems. Empirical thoroughbass theorists constructed chords but could not associate them with a limited number of pitches. Other theorists insisted upon explaining the fundamentals of music by acoustical science, which conflicted with those theorists still using numerical ratios. By the turn of the eighteenth century, theorists focused on introducing even more new theoretical methods for musical composition. As a result, the field of music theory grew into several important areas. For example, in Gradus ad Parnassum (1725), Johann Joseph Fux ( ) codified a species approach to counterpoint. 1 Albert Cohen, 17 th -Century Music Theory: France, Journal of Music Theory 16 (1972), vii

9 Johann David Heinichen ( ) provided information on the thoroughbass method in his treatise Der General-Bass in der Composition, (1728). These two traditions prepared the way for Jean-Philippe Rameau ( ), who first presented many of the ideas that form our modern analytical practice. In his Traité de l'harmonie (1722) and other treatises that followed, Rameau established a system of harmonic principles then in use. These included: constructing chords in a series of thirds; the formation of the major triad from the overtone series; the relationship of chords to a tonal center; the theory of chordal inversions; and the theory of fundamental bass. Rameau first intended his theory of fundamental bass to be used as a pedagogical tool to simplify the instruction of composition and thoroughbass. The fundamental bass reveals the foundation or root of each chord. Identification of the root was necessary to determine chordal progressions. These chordal roots were not intended to be performed; instead, their purpose was to serve as a theoretical and analytical perception later interpreted by Roman numerals. Although Rameau revised his theory of fundamental bass, he did not extensively apply this theory to his own music. Theorists did, however, applied fundamental bass to music years later. The purpose of this thesis will be to present a test case study and analysis of Rameau's fundamental bass theory as it is applied to the music of the French overture styles. French overtures included a variety of harmonic and rhythmic styles, homophonic and polyphonic textures, viii

10 and a contrast between tempi. Overtures in France were divided into two parts: a slow section in duple meter with dotted rhythms followed by a faster fugal section often in triple or compound meter. At the end a return to the opening material is often found, allowing this form to be a representative type of binary or rounded binary form with harmonically contrasting parts. The French overture was established as the standard type in France during the reign of Louis XIV and quickly spread to Germany and England. In addition to Rameau s overtures, those by Lully and Telemann, representing France and Germany respectively, were chosen for this study to allow comparison both chronologically and nationally. Pieces selected include Alceste, Cadmus et Hermione, and Amadis by Lully; Hippolyte et Aricie, La Princesse de Navarre, and Castor et Pollux by Rameau; overtures from Suite in D, Overture in D minor, and Orchestral Suite in F minor by Telemann. Rameau s theory of fundamental bass was the most important theoretical revolution of the eighteenth century. It provided a new model to indicate not only the origin of harmonies but how these harmonies progressed in music over time. His theory also verified the interpretation of a succession of harmonies in music to be a process of motion. The essence of this thesis is to examine how the fundamental bass theory developed by Rameau is a practical description of music from that time. xi

11 CHAPTER I Johann Joseph Fux ( ) and Johann David Heinichen ( ) contributed vital information on species counterpoint and thoroughbass methods, respectively, during the early eighteenth century, thus providing a foundation for Rameau and his theory of fundamental bass. Fux, an Austrian composer and theorist, decided on a career in music during his early childhood. By 1698 he was appointed court composer for Emperor Leopold I. During his lifetime Fux held a variety of positions including Vice-Kapellmeister, Kapellmeister, and Principal Court Kapellmeister for three Habsburg emperors. Fux wrote secular and sacred works including trio sonatas, masses, Te Deum settings, oratorios, and operas. His Gradus ad Parnassum (1725) was quickly regarded as one of the most important counterpoint manuals and remains so today. Written in Latin, Gradus was translated into German, Italian, French, and English, and influenced musicians including Beethoven, Haydn, Mozart, and Schubert. This treatise was constructed in the dialogue style, in which Fux disguised himself as the pupil Josephus. Fux s greatest influence, Palestrina, was the wise teacher Aloysius. Throughout the treatise Aloysius introduced Josephus to counterpoint principles using the species approach along with exercises for him to master. Josephus worked through the exercises and remained under constant surveillance from Aloysius, who corrected his exercises and answered his questions. Joel Lester, in his book entitled Compositional Theory in the Eighteenth Century, gave a detailed description of Fux s treatise. Fux began the treatise by listing 1

12 2 the voice-leading rules for five types or species of two-part writing and consonances needed for species counterpoint. First species created counterpoint using whole notes above or below a cantus firmus in whole notes. Second species used half notes against the cantus firmus. Third species used quarter notes; fourth species featured suspensions; and fifth species combined the possibilities from the preceding species. These concepts were then introduced in three- and four-part writing followed by imitation, fugue, and invertible counterpoint. Although Fux illustrated his points through exercises, he gave only the information needed to know to complete that specific exercise, which resulted in some confusion for readers. In one exercise Aloysius corrected the leap F-B to F-C. However, the leap F-A was changed to F-B-flat by Josephus two exercises later. Why was B-flat mentioned here and not in the previous exercise? With the first exercise in the E mode and the second exercise in the F mode, B-flat was more appropriate in the second exercise. Fux chose not to explain this situation because it consisted of a complicated discussion concerning contrapuntal and modal issues. 2 It is clear that Fux wanted the student to face the reality of compositional problems and to learn to make choices. Gradus endured due to Fux s understanding of basic voice-leading principles by which a student can master the compositional styles of the eighteenth and nineteenth centuries. Within this pedagogical framework, Fux recognized the prima practica traditions passed on from Zarlino. Although he was against innovations and new ways of thinking established by the Enlightenment in the eighteenth century, his treatise educated students 2 Joel Lester, Compositional Theory in the Eighteenth Century (Cambridge: Harvard University Press, 1992), 33.

13 3 in counterpoint and composition. Fux s work was the culmination of Germanic Baroque music in Austria and became the foundation of Viennese Classicism. Johann David Heinichen, German composer and theorist, began his music career by composing and conducting sacred music. After completing a law degree from Leipzig University, Heinichen traveled to Venice and Rome, where his reputation as a composer grew. This enabled him to win the post of Kapellmeister in 1717 to the court in Dresden. By the beginning of the eighteenth century, Heinichen had composed in every form and established himself as a major theorist. His contemporaries regarded him as the Rameau of Germany. 3 While in Venice Heinichen wrote his theoretical masterpiece Der General- Bass in der Composition (1728). This treatise consisted of 960 pages with musical examples and is divided into two sections: Part One covered thorough-bass principles; Part Two explained harpsichord accompaniment. As Lester explains, thoroughbass writers, including Johannes David Heinichen ( ), Friderich Erhard Niedt ( ), Francois Campion ( ), and C.P.E. Bach ( ), clarified how to read figured bass symbols with the bass becoming the foundation of all harmony and chords, and each explained procedures to follow for solving unfigured bass lines. Their treatises revealed views toward composition that valued an improvisational approach more than the traditional contrapuntal methods. 4 3 George Buelow, Heinichen s Treatment of Dissonance, Journal of Music Theory 6 (1962), Lester, 52.

14 Nevertheless, Der General-Bass was the most accomplished treatise on thoroughbass 4 methods. This document transcended the other Baroque treatises with numerous examples of thoroughbass techniques. Heinichen, as well as other thoroughbass writers following the Zarlino tradition, built harmonies above the bass and explained rootposition triads and consonant chords followed by dissonant chords. Root-position chords were defined as a fifth and third over the bass and did not need figures in order to be recognized. Heinichen further encouraged the performer to play root position chords in three different right-hand positions, which he called the three principal chords (drey Haupt-Accorde). 5 Heinichen then discussed other consonant chords, such as 6/3, and the principle of inversion. Lester suggested that Heinichen s addition of theories on inversion after 1728 may have been influenced by Rameau. The second part of this treatise reflected Heinichen s tonal and harmonic views. Lester broke down his discussion of unfigured basses and lists his three methods to assist performers. His first showed how to construct chords based solely on the intervals between the solo and bass parts. His second method, which consisted of standardized intervals regardless of key, was supplemented by his third method, is labeled Special Rules. These Special Rules enumerated the position of miscellaneous chords within particular keys. They specify the necessary notes in major and minor keys and their modulations. Heinichen covered all 24 major and minor keys by using a circle that revealed their relationships as we see in Example Ibid., Lester, 78.

15 5 Example 1: Buelow, 275. Heinichen s twenty-four major and minor keys using a circle revealing their relationships. One moved in either direction on the circle to an opposite neighboring position or to a skipped position. Heinichen explained that major keys modulated to their third, fifth, and sixth while minor keys modulated to their third, fourth, fifth, and seventh. 7 In Der General-Bass, Heinichen expanded the six Special Rules into eight General Rules discussed in the theatrical or operatic style. Due to its free treatment of dissonances, Heinichen chose theatrical style rather than church or chamber styles. On page 587 of Der General-Bass Heinichen explained why he wanted to regulate dissonant treatment: 7 Ibid., Normally [in the theatrical style] no chord or progression can be considered correct that is not followed by a correct resolution of the dissonance, whether it occurs before or after the inversion of harmonies, in the upper, middle, or lowest part. If the chord passes this test, it is fundamental; when it does not, it is incorrect and without a very important reason to the contrary can not be allowed. 8 8 George Buelow, Heinichen s Treatment of Dissonance, 219, quoting Johann David Heinichen, Der General-Bass in der Composition (Dresden, 1728), 587.

16 6 Heinichen demonstrated the resolutions of dissonances by dividing them into eight categories called General Rules. These General Rules expanded the six Special Rules and covered many major and minor situations. Heinichen used a solo cantata by Alessandro Scarlatti to demonstrate these rules as Example 2 illustrated. Here Heinichen used scale-step rules for the accurate chord choice over the first eight notes (see Example 2). 9 Example 2: Lester, 79; Heinichen 1728; Buelow 1966a, p , mm The realization is according to Buelow, page 282. Lester's comments paraphrase Heinichen s text. The first note implies the key of B minor; but the cadence (B to E) indicate E minor with an incorrect signature. Conclusion: never judge the key from the first note but rather from the first cadence. Each B, as scale-step 5, carries a major third. The C in measure 1 carries a sixth, being the sixth degree skipping to the fourth degree. 9 George Buelow, Thorough-bass Accompaniment According to Johann David Heinichen, 2 nd ed. (Los Angeles: University of California Press. Ann Arbor; UMI Research Press, 1986),

17 The note D#, the leading tone, carries a sixth. The note G, scale-step 3, carries a sixth. 7 The note F#, as scale step 2, could have a major sixth. But when 2 immediately precedes a cadence, it is more beautiful to retain a seventh over it. The B's at the cadence could have 5/4-#; 6-5/4-#; or just a major third. Remember, this cadence pattern because it is very common. 10 Lester explained that Heinichen s choice of chords was based on the appropriate harmony for each scale degree. Scale-step and cadence patterns were of primary importance in this example. Der General-Bass contained a wealth of knowledge concerning thorough-bass practice and composition. Heinichen wanted to organize harmonic principles using the theatrical style of music to teach his students the art of composition using thoroughbass methods. One of his most important contributions was clarifying the use of dissonances in the theatrical style. Unlike other Baroque theorists, Heinichen freely admits that he can not classify every harmonic situation into one of his eight categories. Heinichen s treatise earned him the praise and respect of his contemporaries and a permanent place among the most influential theorists of the Baroque era. In 1722 Rameau s treatise, Traité de l'harmonie, contributed a new vision of harmony. As a music theorist Rameau presented music as a scientific process with harmonic principles provided by nature. Thomas Christensen, in his book entitled 10 Joel Lester, Compositional Theory in the Eighteenth Century, (Cambridge: Harvard University Press, 1992), 79-80, quoting George Buelow, Thorough-bass Accompaniment According to Johann David Heinichen, 2 nd ed. (1986), 282.

18 Rameau and Musical Thought in the Enlightenment, described his early years as a composer. 8 Legend has it that Rameau s formal musical education did not include theory. However, in his treatise Démonstration du principe de l'harmonie (Paris, 1750), Rameau explained how his attention became occupied with theory: drawn since my youth by a mathematical instinct to the study of an art for which I found myself destined, and which has singularly occupied me my entire life. 11 At the age of seven or eight Rameau explained how he sensed that the tritone should be resolved by the sixth and I made this into a rule. 12 Christensen, on page 30 of his book Rameau and Musical Thought in the Enlightenment, summarized the state of music theory in France during the 17 th century. Theorists such as Marin Mersenne ( ) and Rene Descartes ( ), influenced by the musica theorica tradition, were mainly focused on the interrelationships between the origin and nature of musical material such as pitches, intervals, modes, prolations, and tunings. Discovering solutions for such problems as mode classifications, the mathematical generation of intervals, and the evaluation of tuning systems at first eluded them. Nevertheless, these theorists contributed to advancements in revising consonant hierarchy, temperament, and modal theory, and with their work established a turning point in the way of thinking about the musica theorica tradition. The musica practica or musica attiva tradition largely dominated the second half 11 Thomas Christensen, Rameau and Musical Thought in the Enlightenment, (New York: Cambridge University Press, 1993), 22, quoting Jean-Philippe Rameau, Démonstration du principe de l'harmonie (Paris, 1750), Ibid., 23, quoting Jean-Philippe Rameau, Génération harmonique (Paris, 1737), 223.

19 of the seventeenth century. Branching away from the philosophical, French music theory 9 became more practical. Texts by Guillaume Nivers ( ), Etienne Loulie ( ), Marc-Antoine Charpentier ( ), and Michel L Affilard ( ), ranged from singing treatises to thorough-bass primers, methods for transpositions, and dictions. Rameau bridged the gap between these two traditions. He acted both as a speculative theorist, by offering explanations, and as a practical theorist determined to produce results. His basse fondamentale principle was both theoretical and practical: theoretical in the sense of finding the origin of musical material and used as a practical way of describing music to musicians. Rameau, unlike seventeenth-century theorists, united the two traditions, speculative theory and practice, into one. 13 The information on the following four pages is derived from Joan Ferris s article, The Evolution of Rameau s Harmonic Theories, pages Ferris s article chronologically summarized Rameau s theoretical treatises published throughout his career. Rameau s theoretical system was based on several fundamental principles found in his first and most provocative work, the Traité de l'harmonie Reduite a ses Principles Naturels, which first appeared in This treatise presented a theoretical system founded upon the senario, the first six partials of a vibrating string. Rameau illustrated how intervals and chords in their fundamental positions or inversions and chord successions were generated from the senario. A more comprehensive study of his harmonic theories from this treatise will be presented in chapter two. Four years later, after becoming acquainted with the acoustical studies of Marin Mersenne ( ), Rameau wrote his next treatise, the Nouveau Systême de musique 13 Ibid., 30.

20 théorique (1726). From these studies Rameau observed how overtones originating from the 10 harmonic divisions of a string are also produced by a resonating body. Other advancements in this treatise included labeling the fourth scale degree subdominant and defining its role pertaining to the theory of double-employment. Dissertation sur les differentes métodes d'accompagnement (1732), his next treatise, was a practical description of how fundamental harmony can explain the basic mechanics of music. Although the theoretical concepts found in Nouveau Systême are not improved upon, Rameau constantly practiced and expanded his theoretical viewpoints. In 1737, Génération harmonique was written, which focused on his latest experiment. He thought he had discovered how similar properties of the resonating body were achieved by the covibration of strings with lengths that are multiples of the original pulsating string. Thirteen years later, in Démonstration du principle de l'harmonie (1750) Rameau realized this experiment found in Génération and obliterated the idea of the fundamental being the lowest tone of a partial series. The pulsating string's length caused only portions of the long string to vibrate in conjunction with the unison of the pulsating string. This discovery led to the demise of the subdominant. Rameau then theorized on the origins of the subdominant as the product of nature and art. In 1752 he wrote his next treatise, Nouvelles réflexions de M. Rameau sun la Démonstration du principle de l'harmonie, which explained new experimental discoveries. Rameau tried to discover a natural relationship between music and architecture, and he experimented using brass instruments, proving that the first six partials are in tune. Rameau further advanced his study of the subdominant and concluded that it was out of tune due to its location in the partial series, which in turn

21 proved the superiority of the tonic and dominant over the subdominant. 11 Observations sun notre instinct pour la musique (1754) expanded upon his concept of melody arising from harmony and how harmony was second nature to man. Rameau then described the physical effects such as the effect on the ear and the emotional effect harmony creates. Code de musique pratique (1760) was a practical treatise divided into two parts. In part one Rameau sorted through ambiguous points and presented an accurate statement of his theories. He addressed conflicting issues such as the construction of chords by added thirds versus the harmonic generation of chords. The second half of this treatise articulated Rameau's theories for the last time. He dismissed his ideas concerning undertones and that chords came into existence as a product of art. Confirming his original theory of one single source generating everything in music, Rameau attempted to construct a sensible system of music. In Origine der Sciences, published in 1762, Rameau examined the philosophical and emotional effects of the tonic sound, melodic inflection, and fundamental harmony on people. As a theorist, Rameau remained loyal to several primary concepts. He insisted that harmony is the foundation for all music. Contrary to other views on harmony, he claimed that it does not occur at random but is firmly established by concrete evidence. After studying the senario, the division of a string with the fundamental sound depicted as unity, he studied the harmonics produced by a resonating body and established the idea of a single musical sound containing partials within itself. 14 Throughout his works Rameau reduced music to a science using 14 Joan Ferris, The Evolution of Rameau's Harmonic Theories, Journal of Music Theory 3, (1959),

22 mathematics. Although he was immersed in the intricacies of his theoretical methods, he 12 exalted musical experience over then, thereby threatening the stability of his theoretical system by looking for other explanations. As a product of the Enlightenment, Rameau endlessly searched for faults and ways to improve his theories in pursuit of establishing universal harmonic principles. At age 39 Rameau wrote his first and monumental treatise, Traité de l'harmonie. One of the most important concepts Rameau articulated was the basse fondamentale, the fundamental bass. In his book Rameau and Musical Thought in the Enlightenment, Thomas Christensen gave a brief synopsis of Rameau s theories, which are as follows: Rameau established the structure of music as harmonic in nature and as generated from a single fundamental source. This source began as the string divisions of the monochord and in later works was the physical base of the corps sonore, a vibrating body generating the harmonic upper partials. The ratios and proportions produced by the monochord and the corps sonore enabled Rameau to summarize harmonies (chords). He also categorized all chord-root motion as the cadential formula of a dissonant seventh chord resolving to a consonant triad, thus proving how chord successions and ratios mirror each other. Dissonant progressions resembled the basic fundamental bass progression of a dissonant seventh chord resolving down by a perfect fifth to a consonant triad by new ideas such as the generative fundamental, inversional identity, and supposition. Therefore, this acoustical source accounted for musical syntax, terminology, and components such as melody, counterpoint, and rhythm of all tonal music. In his subsequent treatises Rameau eagerly sought to prove his theories worthy enough to be

23 seen as a scientific system. 13 Rameau s theories were often compared by his contemporaries to Newton s and Descartes. Newton s Philsophiae Naturalis Principia Mathematica (1687) scientifically unified theories from his predecessors, Galileo and Kepler, which proved how earthly material bodies and planetary motion behaved identically. By presenting a theoretical system using mathematical principles Rameau united dissonant treatment rules and thoroughbass chordal formulations. Rameau often found himself called the Newton of Music by his colleagues. 15 Rene Descartes ( ), in his Discours de la methode, changed the way of thinking when searching for the truth on any subject. Descartes method dismissed previously existing facts regardless of how long they existed. After clearing your mind of all preconceptions, one investigated the subject, formed principles, then tested these principles against a methodological foundation based upon mathematics. Rameau relentlessly searched for theories to replace the unorganized quantity of principles of counterpoint and thoroughbass. In the Preface to the Traité, Rameau expressed this view: Music is a science which should have certain rules; these rules should be drawn from an evident principle; and this principle can not really be known to us without the aid of mathematics. 16 Rameau was very clear in stating his beliefs that a single source, the monochord and corps sonore combined with mathematics, produced an entire theoretical system of music. Jean Benjamin de Laborde, music lexicographer, historian, and 15 Christensen, Thomas Christensen, Rameau and Musical Thought in the Enlightenment (New York: Cambridge University Press, 1993), 11-12, quoting Jean-Philippe Rameau, Treatise on Harmony, trans. by Philip Gossett. (New York: Dover Publications, Inc., 1971), xx.

24 contemporary of Rameau, compared him correctly to Newton and Descartes. 14 One can say that Rameau was both Descartes and Newton, Since he did for music what these two great men together did for philosophy. Like Newton, he began with what existed in practice in order to find the principle. And like Descartes, he began with nature herself (that is to say, the phenomenon known as the corps sonore) in order to deduce along with all its consequences the principles and individual rules. By his efforts [music theory] has become elevated to a practical science whose mechanical operations are at once the most plausible and simple. 17 Regardless of what label his colleagues bestowed upon him, Traité de l'harmonie is an incredible accomplishment. The information on the following three pages has been paraphrased from Christensen s book, pages 24-29, Rameau and Musical Thought in the Enlightenment, where he gave a brief history of Rameau s sketches and professors before he published his Traité in Rene Suaudeau, a former professor of music at the Ecole Nationale De Musique in Clermont-Ferrand, described Rameau's original exercise, sketches, and notes from his second Clermont residence for the Traité in a monograph. The original notes for this treatise were lost. According to Suaudeau s study, he established the main ideas from the Clermont notes. Rameau borrowed ideas then shaped them to appropriately reinforce his own. The fundamental bass idea can be traced from new individual sources in the seventeenth century. His intention for the fundamental bass was to serve as a teaching tool. Using thoroughbass theory for his background, Rameau began by establishing the 17 Thomas Christensen, Thomas Christensen, Rameau and Musical Thought in the Enlightenment (New York: Cambridge University Press, 1993), 18, quoting Jean Benjamin de Laborde, Essai sur la musique ancienne et moderne, 4vols. (Paris, 1780), III,

25 15 necessary consonant framework of music as being the major and minor triads. Then he further clarified the three fundamental chord types: the accord parfait (consonant triad); the accord de grande-sixte (added-sixth chord); and the accord de 7e de dominante (seventh chord). For example, F-major and G-major triads, according to Rameau, complemented and reinforced C major as the tonic. The addition of dissonances of the F- and G-major triads, he illuminated their function and classified them as representing one of the three fundamental chord types. He labeled the bottom note of each of these chords, what we call today the root, as the basse fondamentale. He also discussed how the fundamental bass of the chord remained the same despite the inversion (renversement) in which a chord appears. Rameau s Clermont notes offered explanations of ninth and eleventh chords by suppositions, discussions of addedsixth chords, and the concept of double employment (double emploi). Furthermore, Rameau accomplished two very important tasks in these Clermont notes. First, he diminished the multitude of figured-bass signatures to a few fundamental types; second, he revealed how composition is much easier by following his chord succession rules. These principles in the Clermont notes, according to Suaudeau, were his most important ideas which lead to the development of his scientific theory of music found in Traité. 18 Traité de l'harmonie was the first treatise to recognize the needed separation of harmony from counterpoint as a compositional tool. In this treatise Rameau narrowly defined the subject of harmony in four books. Book 1, titled On the Relationship between Harmonic Ratios and Proportions outlined the underlying framework for 18 Christensen,

26 chords, ratios, proportions, and their existing relationships with each other. He demonstrated 16 how chords are generated from a single source, that being the monochord. Through string divisions Rameau used the monochord to produce all of the ratios needed to build any chord. Next he labeled the consonant triad and the dissonant seventh chord as the two fundamental chord types, with all other chords originating from these two. He emphasized the important role the monochord plays in providing the foundation for those chords that control all harmony and music. In Book 2, On the Nature and Properties of chords and Everything which May Be used to Make Music Perfect, Rameau introduced the fundamental bass (basse fondamentale). This book interpreted the chords found in Book 1 using the fundamental bass. Rameau s most significant discovery was how the monochord and the fundamental bass produce similar interval ratios, which controlled both chord construction and succession. Book 3, Principles of Composition and Book 4, Principles of Accompaniment, provided a detailed discussion of the text found in Book 2. Rameau claimed that composition and accompanying became easier if one used the fundamental bass. Even after a brief survey of Traité it became clear that Rameau spent many years investigating and fine-tuning his theories. This treatise presented his ideas in the most inventive and creative form. It contained a substantial amount of his most important and fundamental principles including chord generation and inversion, the fundamental bass, supposition, and the obvious relationship between melody and harmony. Although these ideas were presented with little organization or structure, Traité consequentially was his most important treatise because it represented his debut into the theoretical community. This same treatise, however, was criticized because many of the individual

27 17 elements originated from sources other than Rameau. Christensen highlighted the fact that the ideas leading up to the development of the fundamental bass were found in seventeenthcentury theory, including speculative and practical texts. Many thoroughbass and compositional treatises of the seventeenth-century provided informal guidelines that Rameau could use as a starting point for the fundamental bass. 19 Even though Traité was hailed as the first complete theoretical treatise on harmony, Rameau may not have cared to admit his degree of dependency upon his predecessors. At the beginning of the seventeenth century, the triad rapidly gained prominence in many treatises. Theorists such as Joachim Burmeister ( ) and Johannes Lippius ( ) faithfully followed Zarlino and labeled the major triad as the most concrete and stable structure in music. They based their belief on Zarlino s arithmetic division of the octave and fifth in his senario. By mid-seventeenth century, French theorists were divided by different styles. For instance, some theorists leaned towards the simplification of triads with even more emphasis on tonal characteristics. Opposing this was the style of exploiting dissonances in the name of expression, which permitted a subtle breakdown of tonality. French music theorists displayed characteristics from both of these styles in their thoroughbass treatises. Thoroughbass practice became popular in France by the mid-seventeenth century. If the thoroughbass practice had not transformed French triadic theory, Rameau might never had discovered the fundamental bass. Thoroughbass led Rameau to the realization of harmony as having been broken down into temporal units by a dominating bass line. In Nouveau systême, Rameau wrote: the shortest and surest means for becoming 19 Christensen, 43.

28 18 properly sensitive to harmony is by accompanying on the harpsichord or organ, since one will always hear a most regular succession of full harmonies. 20 Through accompaniment he heard the bass line producing harmonic successions continually. As a result Rameau focused on accompaniment as the most important method to illustrate his fundamental bass theory. Moreover, he staunchly believed the complexities of accompaniment were greatly reduced and simplified by using the fundamental bass. Before we continue our study of the fundamental bass, a brief discussion of Rameau's harmonic and triadic theories found in Books 1 and 2 of his Traité will be presented here. Rameau begins Book 1 of his treatise by quoting this passage from Descartes Compendium Musicae: That all the consonances are determined by the first six numbers; for the sounds produced by the whole string and its different divisions correspond to the notes C,c,g,c,e,g,(if C be taken to represent the sound produced by the entire string) in which, if the Octave c be added, all the consonances will be found; for this reason all the force of harmony has been attributed to number. That the origin and degrees of perfection of these consonances are determined by the order in which the numbers arise. Thus the Octave is the most perfect consonance; after it comes the Fifth, which is not so perfect as the Octave, then the Fourth, and so on. That the sounds which arise form these divisions of the string give, when heard together, the most perfect harmony that one can imagine. That all these sounds are generated for the whole string, or from its parts; but just as numbers must be related to Unity, which is the source of numbers, so must the different divisions of the string be related to the entire string in which they are contained; and the 20 Thomas Christensen, Rameau and Musical Thought in the Enlightenment, (New York: Cambridge University Press, 1993), 51 quoting Jean-Philippe Rameau, Nouveau systême (Paris, 1726), 91.Christensen, 43.

29 sounds arising from these divisions must be considered as being generated from the first or fundamental sound, which is therefore the source and foundation of all other sounds. The harmony therefore resulting from the consonant intervals produced by the entire string and its divisions is not perfect unless this fundamental sound is heard below the other sounds; for this sound must appear as the principle or source of these consonances, and of the harmony which they form; it is the base and foundation Shirlaw explained how important the octave is to Rameau s theories. By defining an interval as being the difference or distance between a lower and an upper sound, the octave, with the 2:1 ratio located between the first and second partials, was Rameau's most important interval. As a result of being the most perfect consonance, the octave now functioned as the outer perimeter for all other intervals to measure against. By understanding the mathematical and physical principles of the octave, it provided him with the foundation of his harmonic theories: harmonic generation (generation harmonique), harmonic inversion (renversement), and fundamental bass (basse fondamentale). Rameau then concluded by saying the octave, being the most perfect consonant interval, was a replica or repetition of this sound. 22 Referring back now to Joan Ferris s article, The Evolution of Rameau s Harmonic Theories, Ferris showed how the concept of the octave allows Rameau to develop his theory of inversion for both intervals and chords. This first distinguishable interval to be heard after the octave, which later was established by Rameau as the backbone of harmony, was the fifth (3:2). Rameau then illustrated his principle of 21 Matthew Shirlaw, The Theory of Harmony, (London: Novello & Company, Limited, 1955), 66, quoting Jean-Philippe Rameau, Traité, (1725) Bk. I, Ch Matthew Shirlaw, The Theory of Harmony (New York, 1969), 66.

30 inversions by explaining the interval of the fourth. As the second replica of the fundamental, 20 the fourth (4:3) was located by finding the difference between the fourth partial and the fifth. It was seen as merely as a by-product of the fifth. Rameau continued along these same lines by using the principle of inversion to explain the minor sixth and the major third, and the major sixth and its inversion the minor third. The minor third, however, plagued Rameau throughout his career. He could not justify it because it can not be traced directly back to the fundamental or a replica of the fundamental. 23 In Traité, he simply dismissed this problem by saying the principle of the minor third seems to be different from that of the major third. 24 Consequently Rameau's three essential consonances were the fifth and the two thirds. His secondary consonances were the fourth and the two sixths. Rameau stated that primary consonances cannot be seen as the inversion of secondary consonances; likewise, secondary consonances were totally defined by the primary consonances. However, Rameau had tremendous difficulty defining dissonances. He began by establishing the major tone (9:8) by subtracting the fourth (4:3) from the fifth (3:2). By using mathematical concepts such as adding, cubing, and squaring, Rameau constructed other dissonant intervals. He devised other methods of establishing dissonant intervals such as chromatically altering consonances. One dissonant interval, the seventh, gave Rameau much concern. He wrestled with the possibility of the seventh actually 23 Ferris Joan Ferris, The Evolution of Rameau s Harmonic Theories, Journal of Music Theory 3, (1959): 234, quoting Jean-Philippe Rameau, Traité, trans. By Philip Gossett, (New York: Dover Publications, Inc., 1971), 13.

31 21 being classified as an interval. Later he abandoned the labeling of the seventh as a dissonance interval and renamed it a fundamental interval in connection with his fundamental bass concept. 25 For Rameau the presence of the fundamental sound in the bottom part of the interval was not necessary because it was still considered to be the generating tone or root. He had now clearly established both consonant and dissonant intervals from the senario. Rameau continued by listing the characteristics that define a chord: 1. The chord is not to exceed the range of an octave. 2. The foundation of all chords is the fifth (is the most significant harmonic component of music). 3. Chord construction can be determined by the major or minor thirds. 26 Rameau limited chords to only two basic types: the consonant accord parfait, or perfect chord, and the dissonant accord de la septieme, or seventh chord. The perfect chord, also known as the major triad, was the only chord that directly comes from the senario. He reduced all chords to either a perfect chord, a seventh chord, an inversion of either of these, or a specific type of seventh chord produced by supposition or by the addition of a sixth. When Rameau encountered chords without a perfect fifth, such as augmented or diminished chords, he viewed them as being incomplete. Chords such as ninth or eleventh chords were explained by supposition (accords par supposition). Rameau s intentions when discussing the basic chord types was to clarify and reduce thoroughbass harmonies. Throughout this treatise Rameau focused on thoroughbass figures and methods. His theory of inversion was designed to show every 25 Shirlaw, Joan Ferris, The Evolution of Rameau s Harmonic Theories, Journal of Music Theory 3, (1959): 235, quoting, quoting Jean-Philippe Rameau, Treatise on Harmony, trans. by Philip Gossett. (New York: Dover Publications, Inc., 1971), 32.

32 chord in its primary, or fundamental, form, proving how chords may be inverted using the 22 same process as when inverting intervals. Shirlaw summarized Rameau s inversion theory as follows: In the major harmony (as c-e-g), which is represented by the numbers 4:5:6, if we place 4 an Octave higher we obtain the first inversion of the harmony, that is, a chord of the Sixth (e-g-c), represented by the numbers 5:6:8. If in the same way we place 5 an Octave higher, we obtain the second inversion of the harmony, a chord of the Fourth and Sixth (g-c-e), represented by the numbers 6:8:10. We can not however here carry the process of inversion further, for if we place 6 an Octave higher, we get a chord represented by the numbers 8:10:12. But this proportion is the same as 4:5:6, and indeed represents the original harmony itself. The first chord is called Perfect; the two chords derived from it are called Imperfect; for in the case of these derived chords the fundamental sound, c, is not in this bass; it is transposed, and represented by another sound, namely its Octave. 27 Rameau has thus demonstrated the intervals generated over the fundamental, that being the perfect fifth and major third, and he justified the perfect fourth and minor sixth as being complements of the fundamental. Furthermore, Rameau tried in vain to assign a relationship between the minor third and major sixth parallel to that of the perfect fifth and major third: Since all intervals were generated by the octave and begin and end there, so should the minor third. It should not be found indirectly, between the major third and the fifth, but related directly to the fundamental sound or its octave. 28 He 27 Matthew Shirlaw, The Theory of Harmony (London: Novello & Company, Limited, 1955), 68, quoting Jean-Philippe Rameau, Treatise on Harmony, trans. by Philip Gossett. (New York: Dover Publications, Inc., 1971), Ibid, 67, quoting Jean-Philippe Rameau, Treatise on Harmony, trans. by Philip Gossett. (New York: Dover Publications, Inc., 1971), Bk. I, Ch. 3, Article 5.

33 failed to illustrate the fundamental generating the minor third and as a result diminished the 23 possibilities of major and minor chords being related. Although Rameau could not justify the origin or inversion of the minor third, his inversion principle alone transformed the role of the interval or chord without nullifying its harmonic use or foundation. The theory of inversion allowed him to arrange and categorize consonant and dissonant intervals. In fact, Rameau's principles of harmonic generation, fundamental bass, and the inversion of chords were very closely interrelated. Furthermore, the inversion of intervals or chords could not exist without having been previously founded in theoretical concepts such as harmonic generation and the fundamental bass. Shirlaw pointed out the obvious lack of appreciation for Rameau's theories. Many musicians and music theorists alike refused to acknowledge a purely acoustical phenomena science as the basis of harmony. 29 Whether we as musicians and theorists acknowledge it or not, we have accepted Rameau's principles of harmonic generation and the fundamental bass when we utilize his theory of inversion. 29 Shirlaw, 75.

34 CHAPTER II Rameau s fundamental bass principle, found in the second book of Traité, was his best attempt to simplify harmonic theory. Two leaders in the theoretical concepts of Rameau s fundamental bass are Matthew Shirlaw and Thomas Christensen. The information presented in this chapter has been compiled from Thomas Christensen s book, Rameau and Musical Thought in the Enlightenment, and Matthew Shirlaw s book, The Theory of Harmony. The fundamental bass concept actually distinguished two distinct basses for the perfect chord: an actual sounding bass note found in the basso continuo and the fundamental bass. The fundamental bass or fundamental note (bassefondamentale, son fondamental), known as the root, was used by Rameau to justify harmony as a real science controlled by laws of harmonic succession. This fundamental bass provided the harmonic principles overseeing the progressions of one harmony to another. The fundamental bass was the foundation of the harmony and the sole determining factor on which harmonic succession depends. As Shirlaw pointed out, Rameau emphasized his view by saying: Zarlino has compared the bass to the earth, which serves as a foundation for all other elements. It is called the bass of the harmony, because it is the basis and foundation of it. If the foundation were to fail, that would be as if the earth were to fail: all the beautiful order of Nature would fall into ruin; every piece of music would be filled with dissonance and confusion. 24

35 25 When one wishes to compose a bass, it is necessary to proceed by movements somewhat slow and separate. The higher parts may move more quickly and in diatonic [conjunct] progression. 30 The fundamental bass principle was based upon the mathematical division of the monochord. Rameau states: the string with its divisions furnishes us with a perfect harmony, the bass of this harmony resulting from the entire string, which is the source and foundation of all the other sounds. 31 Although the idea of the monochord can be traced back to both Zarlino and Descartes, Christensen explained how Rameau utilizes the monochord in different ways. The main function of the monochord in the past was to measure intervals. Rameau s dilemma was producing a combination of intervals resulting in the chords he wanted. With his theory of the son fondamental or fundamental sound, he attempted to prove how intervals are generated by separate sounds. Chords as a result were produced by the combination of intervals with the same fundamental sounds. The main purpose of the monochord was for the monochord string itself to generate intervals and chords that later functioned as the most natural resource of his harmonic principles. 32 Accepting the monochord or senario as the basis for his theories, Rameau must solve the problem of generating dissonant chords. The following information was 30 Matthew Shirlaw, The Theory of Harmony (London: Novello & Company, Limited, 1955), 98-99, quoting Jean-Philippe Rameau, Traité, (1725), Bk. II, Ch. I. 31 Ibid., Christensen,