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Traite de l' Harmonie By Jean Philippe Rameau

Jean Philippe Rameau was one of the foremost musicians of his day. His theories of the generation of chords from thirds, chord inversions and the fundamental bass, and the function of diatonic chords in major or minor keys, were considered great revelations in the science of music. Rameau sought to derive the fundamental principals of harmony from the laws of acoustics. While Rameau's theories were not without their critics, they formed the backbone of the traditional methods of analysis for the music of the period between 1600 and 1900.

Jean Philippe Rameau was born in Dijon France in 1683. A conscientious artist, he died in 1764 in Paris a prosperous and distinguished composer and music theorist. Attacked first as an innovator, then as a reactionary, he proposed a theory of harmony based on acoustical phenomena. Rameau attempted to ascertain the "natural principles" that govern all music whether melodic or harmonic, and to demonstrate that these natural principals reside within the musical sounds themselves.

Rameau's theories are based on the theoretical works of Zarlino, and Rene Descartes extensively, but his influences can be traced back to several centuries before Christ, and the Greeks discoveries that the consonances of the Octave, Fifth, and Fourth could be expressed in ratios 1:2, 2:3, 3:4. Usually attributed to Pythagoras, this is one of the greatest achievements in musical theory, and led directly to the fundamental principles in the theories of Zarlino, Rene Descartes, and Rameau. The ratios represent the division of a string of a monochord, represented by the first number, compared to the division of the string of a second monochord, represented by the second number. The intervals resulting from the harmonic or melodic sounding of the two notes in the above mentioned ratios, produce the consonant intervals that formed the basis of the musical system of the Greeks. Both Zarlino and Descartes based their theories on the assumption that all the consonance's, that is, the constituent parts of harmony, arose not arbitrarily, but from a certain mathematical principle called the "senario" or the arithmetical progression 1 : 2 : 3 : 4 : 5 : 6. From the senario Rameau develops his theory of the fundamental note in chords, chord inversion and chord succession.

Before Rameau's treatise was published, it was widely accepted that each of the three chords ceg, egc, gce, had a different fundamental note C, E, G, respectively. Using the senario as a starting point, Rameau determined that the fundamental note of the three previous chords had the same fundamental note C. After the publication of his Traite de l'harmonie Rameau discovered, (to his astonishment) that his principle was not only of a mathematical basis, but a natural phenomena.

Traite de l'harmonie is divided into four Books, the first deals with chords, ratios, and proportions and their relationships to one another, the second, with the fundamental bass and the properties of chords,. The third book is on the principals of composition and the fourth is on the principles of accompaniment. These four books contain the essential principles of Rameau's theories, although still in the embryonic stage, and with the confirmation of the natural existence of the overtone series confirming the Zarlino's senario, which was Rameau' starting point, Rameau gained international acclaim.

Partly due to the difficult nature of the subject, the somewhat bulky treatise is often written in a pell-mell confusion with little order or arrangement, and is often difficult, obscure and diffuse. Further, Rameau did not proceed far in the development of his theories before he encountered serious difficulties. His dissatisfaction of some of the features of his theories is proved by the fact that he frequently changed his mind concerning the origin of the subdominant chord, the origin of the minor harmony, the generation of chords, of the fundamental bass in thirds, and the relationship of the major and minor modes.

The chief criticism of Rameau's theories however, dealt with the use of the term "natural". Unclear as to its exact meaning, phrases like "derived directly from nature" and "harmony is a natural effect" lead to criticisms by some including Berlioz and Fetis, concluding that in respect to music and harmony, the ear is the sole judge. However, while it is true that the ear is the final judge on the consonant or dissonant quality of intervals, it is not true that the composer is free to create or dispose of a consonance or dissonance at will. At best, a composer can control and manipulate the intervals, he can never create them, to this end Rameau's principles attempt to enlighten us as to the nature of the intervals.

Following the examples set by Zarlino and Descartes, Rameau determined that the sounds produced by the whole string and its different divisions correspond to the notes C, c, g, c', e', g', and "that the origin and degrees of perfection of these consonance's are determined by the order in which the numbers (of the senario) arise". The octave is the replica and is considered most perfect, after the octave is the fifth, which is not as perfect as the octave, then the fourth and so on. All these sounds arise from the division of the whole string or its parts, and must be considered as being generated from the first or fundamental sound. The octave is not regarded as really differing from the fundamental sound although the fundamental sound has greater importance attached to it. By identifying the octave as a replica of the fundamental tone, he considers it the inversion of the fundamental tone. Rameau extrapolates from the principle of inversion, and applies it to the other intervals. The fourth arises as the inversion of the fifth, the major sixth arises as the inversion of the minor third, and the minor sixth is the inversion of the major third.

Rameau takes his principle further by applying it to chords. The major chord is represented by the numbers 4 : 5 : 6, which correspond to the division of the strings of three monochords sounding simultaniously. The string the first monochord is divided four times, the string of the second is divided five times and the string of the third monochord is divided six times. The intervals produced combine to form a major triad represented by the numbers 4 : 5 : 6. If we place the 4 an octave higher we obtain the first inversion of the chord, represented by the numbers 5 : 6 : 8 . If we place the 5 an octave higher we obtain the second inversion represented by the numbers 6 : 8 : 10. The first chord is perfect, the second two are imperfect.

Rameau has difficulty in explaining the minor third though. After telling us that the only intervals generated from the fundamental are the fifth and major third, he tells us that the minor third must be generated from the harmonic division of the fifth! Unlike the octave, fifth, and major third, the minor third and major sixth cannot be considered as derived from the fundamental, and the major sixth cannot be considered a derived interval until the minor third is established as a fundamental.

The minor chord too, is of more difficulty. The senario clearly provides him with the major triad, it does not provide him with the minor. Rameau generates the minor third indirectly from the fundamental note; the fifth being composed of two thirds, the only difference being in the "different dispositions of the third which together make up the fifth". "This makes no difference in the character of the fifth, which always has a third on one side or the other". Rameau makes a brief mention of Zarlinos explanation of the minor chord, and determines that the minor chord must be generated from the fundamental note, or "this third could never alter its position within the triad". But he finds himself totally unable to give any rational account of the origin of the minor chord. Nonetheless, Rameau considers himself at liberty to place the minor third as well as the major third wherever it suits him. He remarks "the fifth and thirds not only divide the principle chords, they also compose them, whether by their squares or their the generation of chords by means of not only added thirds but also the squaring of thirds. It is not only the thirds that are manipulated in this way, for example, we are told that the square of the fourth produces a seventh, and the square of the fifth is a ninth". It is the addition of thirds that allows Rameau to form "perfect" chords by adding one third to another, and "dissonant" chords by adding three or four thirds together.

Rameau established a principle that chords must not exceed the compass of an octave. Therefore, in order to account for chords of the ninth and the eleventh, he invents a theory of chords by supposition. In it Rameau supposes that the chord of the ninth is a seventh chord with a "supposed" bass a third below its fundamental bass, and that the chord of the eleventh is a seventh chord with a "supposed" bass a fifth below it's fundamental bass.

The apparent contradiction in the two theories of chord generation are never reconciled. To further confuse matters, Rameau rejects the diminished and augmented triads as not being fundamental chords, yet there is nothing in his new theory of harmonic generation by means of added thirds, that indicates that the addition of major thirds as in the augmented chord, or the addition of minor thirds as in the diminished triad, is any less fundamental than the chords formed by the addition of a major and a minor third. The attempts to reconcile the two theories inevitably lead Rameau to the grossest of absurdities. In abandoning his first principle of harmonic generation, he has had to give up his theories of the fundamental bass and the inversion of chords. Without the fundamental bass Rameau is left without a harmonic foundation, from which he can build an entire system of harmony. According to Rameau, a real science of harmony must explain harmonic progressions, not just chords as single entities, and discover the underlying principles that govern all harmonic and melodic progressions. It is the bass that is the foundation of all the other elements, if the bass fails it would be as if the earth had failed. This principle Rameau claims, cannot be overstated, and it receives confirmation from the arithmetical division of the string. Rameau uses Zarlino's model and determines that even in cases where the bass is not present, it is understood. Rameau further tells us that the most "perfect" progression of the fundamental bass is to descend a fifth. Because the fifth is the first interval generated from the division of a string, the "perfection" in the descending fifth as in the perfect cadence is due to the closeness of the fifth to the fundamental sound. The lower is understood as the fundamental sound, and the higher understood as the dependent sound.

In books II, III, and IV, Rameau puts forth a number of observations respecting the nature and function of chords. The Dominant Seventh chord is of such importance to Rameau that he gives it the full name Dominant-Tonic, because this chord is most naturally followed by the Tonic. Rameau deduces that "harmonic succession is nothing but a chain of Tonics and Dominants". Without the dissonance of the Dominant Seventh chord, the key cannot be properly determined. If the seventh of the dominant chord is not present, then the cadence is not perfect. Here is the source of the doctrine of "Tonality" that has been widely disseminated and is still taught today. Our modern music has been determined by the need to resolve two dissonant notes of the Dominant Seventh chord. Rameau further concluded that the Subdominant chord represents the first inversion of the Supertonic chord with an added seventh, or the "grande sixte", although the seventh cannot be added without flagrantly violating Rameau's own rules for the preparation and resolution of discords. In 1726 Rameau published Nouveau Systeme de Musique Theorique where his ideas underwent some development, but the principles set fourth four years earlier in Traite de l' Harmonie have produced results of a real and lasting value to the science of harmony and the craft of composition.

Jean Philippe Rameau's theories of the fundamental bass, the tertian system of chord generation, the principle of chord inversion, and the function of diatonic harmony, were considered great revalations in the science of music. Rameau sought to derive the undamental principals of harmony from the laws of acoustics and while Rameau's theories were not without their critics, they became the basic method for the traditional methods for the analysis of music that influenced generations of musicians.