Advances in Music Theory during the Scientific Revolution: The Cartesians and Modern Music Theory

Pre-Baroque Developments
In addition to the great artistic merit and beauty of the music of the Renaissance and Baroque periods, this music was also the test bed for the evolution in the late Baroque of a coherent theory of triadic tonality.

The Florentine Camerata's innovation of recitativo was far more than forward strides toward a more emotive music, it represented a stage toward a chord-based rather than linear- based music. It also represented the culmination of performance practice and theoretical evolution that had begun as early as fifteenth century.

The intial step toward a cohesive triadic tonality, of course, was the embrace by continental composers of the English view that the vertical intervals of the third and sixth were consonant. The addition of a fourth voice to the musical texture of the motet and mass movement in the fifteenth cnetury also contributed a critical voicing to the cadence, what we call today root motion from the dominant to the tonic chord (V-I). This motion has become a signal of closure in the cadence of Western music, as well as the determinant of harmonic motion when the interval occurs in the bass within a harmonic progression.

The signal function seems to have been established fairly early in major keys, and the fact may be verified in cadential treatment in the major-key bass frameworks that emerged in the Renaissance. The minor-key bass frameworks were another matter, however, and nearly a century and a half passed from the first appearance of the folia family of minor-key bass frameworks until musicians arrived at a universal and modern definition of the dominant harmony in minor keys. In minor key bass frameworks, one always finds VII-i and V-i juxtaposed, as if the Renaissance ear could not quite decide which chord really fulled the function of the dominant chord. VII-i ultimately evolved into the V-vi "deceptive" cadence in the major key, and V-i became the standard modern cadential treatment in minor keys. The lack of clear definition of function of these chords, however, lingered well into the Baroque period.

A second performance practice from the fifteenth century contributed siginificantly to dominant-chord definition as well as modern triadic theory. Singers began to routinely apply musica ficta to "smooth-out" the melody at the final cadence. Here singers heard that the upward motion of a half- step between the seventh and eighth (tonic) scale degree created more effective closure than whole step motion between the minor or lowered seventh scale degree and the tonic. This alteration of the seventh scale degree of the various modes played out in the bass framework, the minor seventh degree being applied in the progression VII-i and the raised, major seventh degree being applied, in the same progression, in V-i. Hence the minor key bass framework presented the two possibilities in close proximity, and the Renaissance and early Baroque "ear," over time, made the final decision regarding desirability. The passamezzo antico, one of the fundamental bass frameworks, followed the progression:

i-VII-i-V-i (or III)-i-V-i

Musica ficta also had profound effects upon the Church modes, the second critical part of this evolution, ultimately reducing the Church modes to the two modern scales, major and minor.

The first step came in the work of Hemricus Glareanus (1448-1563). In his Dodecachordon (1547), Glarean cited twelve Church modes instead of the traditional eight, and here he added two new modes. These new modes were the Aeolian (based on the note A) and Ionian (based on C). Respectively, these modes are the modern natural minor and natural major scales. In this instance, the scale discovered by Pythagoras (c 500 B.C.E) in his division of the monochord and used for centuries in Western music education became a scale that was actually used in real music. In his Institutioni harmoniche (1558) a decade later, Giosefo Zarlino (1517-1590) reordered the modes modes, which had begun on the note D, to begin on C, laying the foundation for modern concepts of the C scale as the model for all other scales and the starting point in music study.

The impact of singer's application of musica ficta, along with the changing concepts already noted, drove Western music toward a two-scale system. The Phrygian and Locrian modes, both minor modes which are characterized by half-step intervals between the first and second degrees, fell from use and were ultimately discredited by theorists. The remaining Church modes already divided into major (Ionian, Lydian, and Mixolydian) and minor types (Dorian, Aeolian). As noted, the Ionian and Aeolian modes already shared the same intervallic patterns as the modern major and minor scales, respectively. Lowering the raised fourth degree of the Lydian turned it into a major scale, as did raising the lowered seventh degree of the Mixolydian scale. Similarly, raising the seventh degree of the Aeloian mode yielded the modern harmonic minor scale, the scale that permits the change of the v chord in minor to the modern dominant chord V, and raising the seventh degree in the Dorian mode creates the modern melodic minor scale.

The Contributions of Pre-Cartesian Theorists
The use of chords and the recognition of their identity and function within a larger system represents a huge step. Other theoretical tenets had to be in place before a heirarchy could emerge. Johannes Lippius (1585-1612) furnished critical information in his identification in Synopsi musicae novae of the trias harmonia as the most important component of music. In his concept of the triad, the composition shifted from the tenor voice, the voice which in earlier times contained the cantus firmus, to the bass. He recognized, especially in the progressive Italian music of his day, that the bass was the foundation of the music and that it determined which notes could sound above it. Other advances attributable to Lippius are the first use of the word "scale" and his model of the scale as a modern seven-note entity, and his description of possible voicings of the notes of the chord above the bass. He did note recognize the invertibility of the chord, but did recognize that the spacing of the tones of a chord had no impact on their meaning or identity.

The Cartesians
Rene Descartes (1596-1650) was the inventor of analytical geometry, but his most important contribution came in his advocacy of deductive rather than inductive reasoning. His most famous postulate, ego cogito, "I think, therefore I am," distilled his point to its essence, and was the starting point from which he worked outward. This position, of course, stood at the opposite polarity from scientific method.

Descartes contributed significantly to music theory. His most important studies involved the physical nature of sound. He was among the earliest theorist-scientists to attempt to define the relationship between the physical production of sound and the psychological perception of it. The study profoundly influenced later German philosophy of affections, the capability of music to irrationally elicit emotions in the listener. An observation of animal behavior later became the theory of conditioned response.

His other musical studies explain the existence of certain intervals as "residues" or "shadows." In Descartes' system, the division of the octave yields the interval of fifth (three and one-half steps), but the "residue" is the other interval that is created, the fourth (two and one-half steps). The division of the fifth yields a major third (two whole steps), but the residue or the other interval, is the minor third (one and a half steps). Descartes system of octave division offered an important and practical, if not scientific, alternative to Pythagoras' definition of the notes in the octave. Pythagoras divided the string in ratios to find the primary intervals, and then located the remaining notes in the octave by adding and subtracting the ratios of the primary divisions.

Pere Marin Mersenne (1588-1648) was an important theorist who reconciled the accomplishments of the Renaissance and the important new questions of the Baroque. Although he was a priest, he strongly believed in the reason of man and encouraged the development of scientific method in relation to music. He maintained an extensive international correspondence with the leading thinkers of the day including Galilei, Descartes, and Hobbs.

In Harmonie universelle, Mersenne laid the foundation of acoustic physics in his empirical observations on the nature of sound. He correctly identified sound as vibration or pure motion rather than substance. He also was the first to describe correctly the method of sound transmission and formulated the rules governing vibrating strings based on variables such as length, diameter, tension, and mass. Mersenne was also among the first to observe and identify upper partials and their relationship to a fundamental note (see Saveur below). His studies laid the foundations for the study of the speed of sound, resonance and echo, and the character of a vibrating column of air as in the tube of a wind or brass instrument.

Joseph Saveur (1683-1764) was a French mathematician, but not a musician, who presented his findings to the Academie des sciences his Memoires (1701). Starting with Mersenne's assessment of sound as pure vibration, Saveur identified and described the nature of the vibrating string including its "loops" (wave) and "nodes" (stationary points). He devised a formula to predict the behavior of a vibrating string within one percent. Saveur's application of logarithms to measure the octave superseded in accuracy all previous systems from Pythagoras to the present and enabled measurement of intervals "to a speck." His work founded the discipline of acoustics, which he named, and he contributed to the study the terms "harmonic" and the aforementioned loop and node. Saveur was among the first to anticipate the need for standard tuning of musical instruments. Using purely mechanical means, he extended the work of Mersenne and Newton by identifying more than 270 upper partials.

An upper partial, harmonic, or overtone is a note that rings faintly above a note that is sounded. The note sounded, called the fundamental, actually consists of many notes that are produced with it but do not overpower it.

In the overtone series, the first note to ring above the fundamental C note is the C note an octave higher. Above that rings a fifth, the G note, another octave C note, the third, the E note, and another fifth, the G note. Significantly more notes ring faintly above the C-C-G-C-E-G series cited, but do so out of tune. The further from the fundamental, the fainter the note becomes.

The significance of overtones is threefold. The strong presence or absence of certain overtones imbues the various instruments and human voices with their timbre, that is, the quality of sound that permits identification of the source. Secondly, the series cited also describes the notes that are necessary to the major triad. The chord, then, is a natural phenomenon. Third, Pythagoras' first two divisions of the string, the octave and the fifth, are found as the first, second, third, and fifth harmonics. The octave and the fifth constitute the strongest harmonics and suggest more than a coincidence that all music systems use the octave and fifth as their fundamental octave division and their fundamental underpinnings.

Rameau: the Father of Modern Music Theory
Jean-Philippe Rameau (1683-1716) wore many hats in his lifetime and left lasting contributions in the various areas of endeavor. In addition to being one of France's most important composers of opera and an important member of the French clavecin school, his Traite de l'harmonie (1722) established the modern understanding of music theory. His work explained how the notes of the octave could generate all the intervals and chords, and how they interconnected. He described the formulation of the triad as a natural phenomenon as it was laid out but not recognized in the works of Mersenne. Later exposure to the work of Saveur gave him a stronger argument to support his theories.

Rameau was the first to recognize the possibility of chord inversion, that is, that, from lowest to highest, C-E-G and E-G-C are the same chord but with differently ordered notes. Earlier theorists interpreted the combinations as being different harmonies!

Rameau identified the functions of chords, dividing them into two categories. He recognized the function of the tonic chord, which he named as such, as the chord of repose, and that the chord had a special relationship to the chord built on the fifth scale degree. This chord he called the dominant, identifying it as the chord within the key of greatest tension and urgency to resolve.

Rameau took the idea of the "dominant" chord somewhat further than just a description of the relationship of the V and I chords, and noted that in progression, the most potent bass motion among chords involved the same interval as the motion of the bass of the dominant chord to the bass of the tonic chord, that is, the leap upward of the interval of a fourth or the leap downward of the interval of the fifth. His observations explained the relationships of chords within the harmonic sequence (i.e. i-iv-VII-III-VI-ii-V or the chords Am-Dm-G-C-F-B half- diminshed-V). Here the roots each move as if in a dominant-tonic relationship although the chord qualities are not correct for true V-I motion. The result is a very aurally satisfying terraced chain of consecutive roots, each pair occurring at the next lower pitch level. The dominant-tonic relationship breaks down, of course, between the F and B half-diminished chords since the interval between the roots is a tritone. It is worth noting that this relationship of roots is common enough in other progressions. The progression I-IV exmplifies the point, and the common closing harmonic formula of ii-V-I represents the same root motion.

The combination of Rameau's recognition of harmonic invertibility and his concept of root motion led to another startling breakthrough. Rameau recognized that chords in a progression are actually determined by the motion of the chord roots regardless of whether or not the root is in the bass or even stated in the chord. The root he called the basse fundamentale, and observed root motion controlled the direction and cohesion of the progression. In other words, the progression C-F-G-C is recognizable regardless of the vertical ordering of the notes of each chord. The chord roots C-F-G-C must appear somewhere in each of the chords and form the scaffold for the music.

Rameau's understanding of chord formulation also embraced the addition of a third above to form the 7 chord or below a chord to form the 9, 11, and 13 chords. The concept is employed by modern jazz players in formulating chords that yield richer sound without disrupting basic harmonic function.

Rameau's treatment of the IV chord, however, differed somewhat from the modern approach. The addition a seventh to the chord and regarding it as a "dominant" would have not have explained the function of the chord. The tone a fourth above or a fifth below the root of the IV chord is the root of the vii chord, a chord that cannot be used as a satisfactory target of resolution since it is a diminshed chord. Instead, Rameau added a sixth, calling it the chord of the sixt ajoute. In modern terms the sixt ajoute is a first-inversion ii7 chord. Its advantage is two-fold. First, in the correct motion of voices between the IV and V chords, the only possible four-voice treatment is that the root moves upward and all other voices move downward to the nearest chord tones. The procedure does not yield a wide range of musical options. Composers often avoided the situation by using a three-voice texture and, in keyboard and lute music, composers simply broke the rules, often resulting in the cardinal sin of parallel fifths. The sixt ajoute, however, permitted the composer freedom of voice motion. In my observation, the sixt ajoute is actually a reflection of nearly two centuries of practical usage . The application of the first-inversion ii chord permits the bass to ascend stepwise from the fourth to the fifth scale degree, smoother voice-leading, freedom of motion of the upper voices, and the preparation of a 4-3 suspension over the V chord.

Although Rameau is responsible for assigning the names such as tonic, dominant, and subdominant, which identify both the chord and its function, the assignment of numbers to each chord remained for a German priest, Abbe Volger, a short time later. In Vogler's system, the chord built on the first degree is given the label "I." The chord built on the second degree is called "ii." The case indicates chord quality, that is, whether the chord is major or minor. Upper case identifies major. Both nomenclatures remain in use today on an equal footing. The advantage of each system is that the musician can recognize and label the relationships he hears in music without knowing the key and without having his instrument in hand. Hence he can understand, learn, or improvise over the music more quickly. Hearing in this way constitutes the strongest and most important skills of the jazz musician and the studio musician, both of whom are afforded little time to develop their parts.