Harmony

In a significant departure from its original meaning in Greek music theory as the melodic or horizontal combination of two different notes, the term harmony, beginning with the two-voice polyphony of the Middle Ages, came to refer to the simultaneous or vertical combination of two or more notes, as well as the horizontal or linear relationships between the complex sounds thus produced. The primary subjects of harmonic analysis are relationships between notes, properties of chords, consonance and dissonance, and tonality and key. Since the Middle Ages, the study of harmony has developed in two basic areas: speculative or theoretical harmony and practical harmony.

Speculative or theoretical harmony.

The theoretical study of harmony developed directly from ancient mathematical speculations on the foundation and structure of music, in particular, Pythagorean, Platonic, and Neoplatonic theory. At least through the seventeenth century, mathematical relationships were generally considered to be the formative principles of musical phenomena. Then in the eighteenth century, Jean-Philippe Rameau (1683–1764), in a series of treatises inspired by the acoustical research of Joseph Sauveur (1653–1716), broke with the Pythagorean–Platonic tradition and attempted to discover a strict scientific basis for musical sound through a consideration of the physical principles observable in the natural harmonic series, consisting of a principle tone generated by a vibrating body (corps sonore) at a particular frequency and the integral multiples of that tone that are known as its harmonics or overtones. Although this "natural" theory—which can be viewed as a continuation of the Aristoxenian or phenomenalist tradition of harmonic analysis—was intuitively more appealing as a basis for the generation of musical intervals and chords than the proportional divisions of a stretched string offered by the purely mathematical tradition, Rameau's analysis failed to provide a complete systematic explanation of all commonly used chords and chordal progressions.

In the mid-nineteenth century, Moritz Hauptmann (1792–1868), in Die Natur der Harmonie und der Metrik (The nature of harmony and meter), turned away from both the mathematical tradition and the type of physical explanation proposed by Rameau to argue—probably following Hegelian philosophical doctrine—that the universal principles underlying music must be identical to those of human thought: unity, opposition, and reunion or higher unity. In recent years, in addition to theoretical investigations of the physical properties of musical sound, empirical studies of musical perception and cognition have produced information of a comparative nature about different musical styles and cultures in an effort to demonstrate the existence of universally perceived harmonic properties, such as scalar organization of tones and tonal centers (see, for example, Krumhansl, Cognitive Foundations).

Practical harmony.

The study of practical harmony, rather than having the aim of producing speculative theories about musical phenomena, is intended to educate musical practicians: composers, performers, educators, and amateurs. Its history largely coincides with that of harmonic tonality, the Western music of the "common practice" period, approximately 1600 through 1910. From the beginning, practical harmony, or harmonic practice, has included topics such as rules for the composition of counterpoint; techniques of improvisation; and the vertical and horizontal analysis of chordal structures and chordal progressions, the so-called Roman numeral analysis of tonal music. Throughout much of the twentieth century, following the nineteenth-century consolidation of the canon of music within the Western "classical" tradition (that is, "art music" rather than "popular music"), textbooks on harmonic practice tended to concentrate on the analysis of the musical structures at work in the compositions of Bach, Beethoven, Brahms, and other composers of this repertory (see, for example, Piston, Harmony). This approach, however, produces somewhat chaotic results when applied to the extremely chromatic, and therefore not strictly tonal, music of the late nineteenth century. Musicologist Jean-Jacques Nattiez, for example, cites thirty-three different functional harmonic analyses of one particular chord—the famous "Tristan chord," F–B–D–G—which occurs in the opening measure of the prelude to Richard Wagner's opera Tristan und Isolde (Nattiez, chapter 9). A significant late-twentieth-century development has been the loosening of the distinction between classical and popular music and the inclusion of largely diatonic, rather than chromatic, musical idioms—such as folk music, jazz, blues, and rock and roll—within contemporary textbooks on harmonic practice.