HARMONY.

Harmony is derived from the classical Greek harmonia (meaning a joint between the planks of a ship or a joining of those planks). From the beginning, the term was also used in its current metaphorical sense, that of a combination of parts or related things to form a consistent whole or an agreement.

Harmony in Ancient Greek Writings on Music

In ancient Greek writings on the subject of music, harmony (also known as "harmonics") was the study of the formation of melody. This study began with the elements of melody—the individual notes—and continued with the specification of appropriate ways in which pairs of notes, a higher and a lower, could be combined successively into melodic intervals. (The simultaneous combination of notes was not a part of classical Greek musical practice.) These melodic intervals were in turn combined into a variety of complex scalar systems, the defining structures of complete melodies. In general terms, classical Greek harmonics falls into two traditions: the Aristoxenian and the Pythagorean.

Aristoxenian harmonics.

To Aristoxenus (c. 375–300 b.c.e.), a prolific writer on a variety of philosophical and historical subjects and the son of a musician, harmonics was the study of music as we hear it. Its task was to arrive at an understanding of the musical sounds that the human ear hears as pleasing or melodic through a systematic analysis of the perceived phenomena. The definition of "melodic" must concern itself only with the sounded elements of music—the notes, described exactly as they are heard by the ear, namely, as different pitches on a melodic continuum; furthermore, the general rules that govern melodic structure must not be derived from any abstract principles. However, Aristoxenus, to support his phenomenalist argument for the existence of certain melodic combinations of notes that have an "affinity" with one another, makes an analogy to a related property of speech: "And yet the order which relates to the melodic and unmelodic is similar to that concerned with the combination of letters in speech: from a given set of letters a syllable is not generated in just any way, but in some ways and not in others" (Barker, p. 153).

Pythagorean harmonics.

By contrast, Pythagorean harmonics, the set of beliefs about music attributed to the contemporary followers of Pythagoras of Samos (c. 580–c. 500 b.c.e.) and his intellectual heirs in the fourth and third centuries b.c.e., was centered on the discovery that the fundamental melodic intervals of the octave, the fifth, and the fourth could be produced by the lengths of the two sections of a stretched string in the simple and elegant mathematical ratios of 2:1, 3:2, and 4:3, respectively. The Pythagoreans took this musical discovery as an affirmation of their belief in the mathematical nature of reality and argued that certain musical intervals are pleasing to the ear because of their underlying structure, not for any reason having to do with musical sound considered only as an audible phenomenon. Within Pythagorean harmonics, all subsequent combinations of tones into scalar systems were generated from these basic ratios, including 9:8, the mathematical ratio associated with a musical whole tone. To the Pythagoreans, these scalar constructions—the Pythagorean system of intonation—were musical embodiments of a cosmic scalar relationship among the planets governed by their distances from the earth and their revolutionary speed: the harmony of the spheres. (This idea may have had Mesopotamian roots; see Kilmer, "Mesopotamia.")

Plato's Harmonic Cosmology

None of the writings produced by Pythagoras or his contemporaries are extant (see Burkert for a discussion of the authenticity of information about fifth-and sixth-century b.c.e. Pythagoreans), but the impact that the simplicity and exactness of the Pythagorean proportions had on the philosopher Plato (428–348 or 347 b.c.e.) is revealed most clearly in his late dialogue Timaeus, in which Timaeus, a man trained in Pythagorean doctrine, describes the origin and nature of the physical world. Central to this cosmological drama is the Demiurge, a kind of primary arranger, who begins with formless matter in a primitive state of chaos. He proceeds by using a fixed set of numbers to construct the soul of the world as a mixture of metaphysical oppositions (indivisible and divisible existence, indivisible and divisible sameness and difference). Successive lengths of primary material are mixed in the ratios 2:1, 3:2, 4:3, and 9:8, that is, exactly the Pythagorean harmonic ratios (Timaeus 35–36, pp. 64–73). Thus the world's soul is constructed as a harmony of opposites permeated by number in which the formative principles of Platonic cosmology are identical to those of Pythagorean harmonic theory. In a following section, Plato turns to the construction of the physical universe as an "eternal image, moving according to number" (Timaeus 37D, pp. 76–77). The seven celestial bodies—the Moon, Mercury, Venus, the Sun, Mars, Saturn, and Jupiter—are created and placed in orbits about the earth determined by these same harmonic ratios. According to this model, the sun is located at the midpoint of a seven-note scale of revolving bodies; the whole thing is contained within an outer starry sphere that sets the limits of the universe. In his Republic, written some thirty years before the Timaeus, Plato had used striking imagery rather than mathematical relationships to describe his harmonic universe of planetary spheres: "And on the upper surface of each circle is a siren, who goes round with them, hymning a single tone or note. Together they form the concord of a single harmony [musical scale]" ("The Myth of Er," Republic 617B, pp. 502–505).

Neoplatonic Speculative Harmony

At some point Plato's sirens were replaced by Muses, possibly in the lost commentary on the Republic by the Neoplatonist Porphyry (c. 234–c. 305). A unique passage transmitting the "musical" version of Plato's image is found in the first book of Martianus Capella's fifth-century Neoplatonic treatise, De nuptiis Philologiae et Mercurii (On the marriage of Philology and Mercury). Martianus describes the Muses arriving at their celestial locations:

The upper planets and the sevenfold spheres produced together the clear harmonies of a certain sweet melody in a sound even more pleasant than usual, undoubtedly because they knew that the Muses were approaching. Passing through all the spheres one by one, each Muse stopped when she recognized the pitch that was familiar to her. Urania occupied the most distant sphere of the starry universe, which was carried along resonating an acute clear tone. Polymnia took the circle of Saturn; Euterpe that of Jupiter. Erato, once she had entered the sphere, sang the pitch of Mars. Melpone took the middle orbit where the sun makes the sky beautiful with his flaming light. Terpsichore was united with the gold of Venus. Calliope took possession of the sphere of Mercury, and Clio the innermost circle … on the moon, which resonated a deep pitch in a harsher tone. (Martianus Capella, De nuptiis, book 1, pp. 12–13)

Later in the treatise, Martianus describes Philology's traveling from earth through the heavenly bodies to the outermost sphere, giving the length of each leg of her journey in terms of a specific musical interval and indicating that the entire musical distance is equal to six whole tones, or an octave. Thus he presents an explicit working-out of the Platonic cosmological parallel between the ordering of the universe and the harmonic organization of a musical scale.

Continuation of the Pythagorean-Platonic Tradition in Music

Pythagorean–Platonic musical mathematics was transmitted to scholars during the Latin Middle Ages principally through the following texts: Calcidius's late-fourth-century Latin translation of most of the Timaeus (up to Timaeus 53c) and his accompanying commentary on that text; Macrobius's fifth-century commentary on Cicero's Somnium Scipionis (Scipio's dream); Martianus Capella's De nuptiis philologiae et Mercurii, quoted above; book 2 of Cassiodorus's sixth-century Institutiones divinarum et humanum litterarum (Institutions); book 3 of Isidore of Seville's sixth-century Etymologiae (Etymologies); and finally the extremely important work of Boethius (c. 480–c. 524), De institutione musica (The fundaments of music), a learned statement of the Neopythagorean and Neoplatonic traditions in music probably based on the work of the Greek writers Nicomachus of Gerasa (fl. c. 100 c.e.) and Ptolemy (2nd century c.e.). After affirming the correctness of Plato's assertion that the soul of the universe is joined according to musical concord (Boethius, book 1, chapter 1), Boethius formulates an evocative triadic version of the tradition (Boethius, book 1, chapter 2): musica mun-dana (cosmic music), the principle of the universe controlling planetary motion, seasons, and elemental combinations; musica humana (human music), the integrating force between body and soul; and musica instrumentalis (instrumental music), the music produced by string, wind, and percussion instruments.

By about the middle of the ninth century, during the Carolingian era, writers on the subject of music began to produce treatises that contained large excerpts from the writers cited above, but that attempted to place the Pythagorean–Platonic harmonic tradition within a Christian framework and adapt it to the current need to codify and regulate the performance of liturgical chant within the mass and the holy office. For example, Aurelian of Réôme in his Musica disciplina (The discipline of music), written around the middle of the ninth century, justifies the classification of liturgical chant into eight different modes by referring to the seven planetary orbits plus the outer starry sphere (now referred to as the Zodiac), exactly as laid out in Plato. Referring to his Latin authorities (Boethius, Cassiodorus, Isidore), he writes that "the whole theory of the art of music consists of numbers … Music has to do with numbers that are abstract, mobile, and in proportion" (Aurelian, Musica disciplina, chapter 8, pp. 22–23).

The last truly original and complete statement of Pythagorean-Platonic speculative harmony was given in the Renaissance by the German astronomer and writer on music Johannes Kepler (1571–1630), who in his treatise Harmonice mundi (The harmony of the world) posits a world created by God in accordance with the archetypal harmonies represented in the principal musical consonances. The tradition, however, is clearly implicit in subsequent works, such as Die Welt als Wille und Vorstellung (The world as will and representation) by Arthur Schopenhauer (1788–1860 ), and explicit in the work of the twentieth-century Swiss aesthetician Hans Kayser (1891–1964). Kayser's principal contribution was a harmonic theory, informed by a close reading of Kepler's Harmonice mundi, according to which the measurement of the intervallic properties of sound could also serve as an exact measurement of feeling. Finally, late-twentieth-century "superstring theory," a cosmic physical theory that, in principle, is capable of describing all physical phenomena, is clearly Pythagorean in essence and scope:

Music has long since provided the metaphors of choice for those puzzling over questions of cosmic concern. From the ancient Pythagorean "music of the spheres" to the "harmonies of nature" that have guided inquiry through the ages we have collectively sought the song of nature in the gentle wanderings of celestial bodies and the riotous fulminations of subatomic particles. With the discovery of superstring theory, musical metaphors take on a startling reality, for the theory suggests that the microscopic landscape is suffused with tiny strings whose vibrational patterns orchestrate the evolution of the cosmos. The winds of change, according to superstring theory, gust through an aeolian universe. (Greene, p. 135)

Harmony as the Organizing Principle of Western Music

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.

See also Composition, Musical ; Mathematics ; Musicology ; Platonism ; Pythagoreanism .

bibliography

PRIMARY SOURCES

Aurelian of Réôme. Musica disciplina. Translated by Joseph Ponte. Colorado Springs: Colorado College Music Press, 1968.

Boethius. De institutione musica. Translated, with introduction and notes, by Calvin M. Bower. Edited by Claude V. Palisca. New Haven: Yale University Press, 1989.

Martianus Capella. De nuptiis Philologiae et Mercurii. Edited by James Willis. Leipzig: B. G. Teubner, 1983.

Plato. Republic, Books 6–10. Translated by Paul Shorey. Loeb Classical Library no. 276 (Plato, vol. VI). Cambridge, Mass., and London: Harvard University Press, 1935. Reprint, 2000.

Plato. Timaeus. Translated by R. G. Bury. In Loeb Classical Library no. 234 (Plato, vol. IX). Cambridge, Mass. and London: Harvard University Press, 1942. Reprint, 1989.

SECONDARY SOURCES

Barker, Andrew, ed. Greek Musical Writings. Vol. 2: Harmonic and Acoustic Theory. Cambridge, U.K., and New York: Cambridge University Press, 1990.

Burkert, Walter. Lore and Science in Ancient Pythagoreanism. Translated by E. L. Minar. Cambridge, Mass.: Harvard University Press, 1972.

Godwin, Joscelyn, ed. Harmony of the Spheres: A Sourcebook of the Pythagorean Tradition in Music. Rochester, Vt.: Inner Traditions International, 1993.

Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York and London: W. W. Norton, 1999.

Kilmer, Anne Draffkorn. "Mesopotamia." In The New Grove Dictionary of Music and Musicians, edited by Stanley Sadie and John Tyrrell, vol. 16, pp. 480–487. London: Macmillan, 2001.

Krumhansl, Carol L. Cognitive Foundations of Musical Pitch. Oxford Psychology Series No. 17. New York and Oxford: Oxford University Press, 1990.

Mathiesen, Thomas J. Apollo's Lyre: Greek Music and Music Theory in Antiquity and the Middle Ages. Lincoln: University of Nebraska Press, 1999.

Nattiez, Jean–Jacques. Music and Discourse: Toward a Semiology of Music. Translated by Carolyn Abbate. Princeton: Princeton University Press, 1990.

Nolan, Catherine. "Music Theory and Mathematics." In The Cambridge History of Western Music Theory, edited by Thomas Christensen, pp. 272–304. Cambridge, U.K., and New York: Cambridge University Press, 2002.

Piston, Walter. Harmony. 5th ed. Revised and expanded by Mark De Voto. New York and London: W. W. Norton, 1987.

Stephenson, Bruce. The Music of the Heavens: Kepler's Harmonic Astronomy. Princeton: Princeton University Press, 1994.

Blair Sullivan

harmony. The simultaneous sounding (i.e. combination) of notes, giving what is known as vertical mus., contrasted with horizontal mus. (counterpoint). Composers, in much the greater proportion of their mus., maintain in their minds some melody which ranks as the principal one, and which they intend the listener to recognize as such, whilst other melodies which are combined with it, or chords with which it is acc., rank as subsidiary. The word chord may be defined as any combination of notes simultaneously perf., and even when the main process in the composer's mind is a weaving together of melodic strands he has to keep before him this combinational element, both as regards the notes thus sounded together and the suitability of one combination to follow and precede the adjacent combination.

At different periods composers have given more attention to one or the other of the two aspects of their work: (a) the weaving together of melodic strands and (b) the chords thus brought into existence from point to point.

The former aspect of the result is the contrapuntal element (see counterpoint) and the latter the harmonic element. In less elaborate mus. (as, for instance, a simple song with pf. acc.) the contrapuntal element may be unimportant or even non-existent. Counterpoint necessarily implies also harmony, but harmony does not necessarily imply counterpoint.

Over a long period the resources of harmony may be said to have widened: new combinations introduced by composers of pioneering spirit have been condemned by unaccustomed ears as ugly, have then gradually come to be accepted as commonplace, and have been succeeded in their turn by other experimental combinations. The following definitions concern traditional and basic harmonic procedures:(a) DIATONIC HARMONY: harmony which confines itself to the major or minor key in force at the moment. CHROMATIC HARMONY: harmony which employs notes extraneous to the major or minor key in force at the moment.(b) OPEN HARMONY: harmony in which the notes of the chords are more or less widely spread. CLOSE HARMONY: harmony in which the notes of the chords lie near together.(c) PROGRESSION: the motion of one note to another note or one chord to another chord.(d) TRIAD: a note with its 3rd and 5th (e.g. C–E–G). COMMON CHORD: a triad of which the 5th is perfect. MAJOR COMMON CHORD: a common chord of which the 3rd is major. MINOR COMMON CHORD: a common chord of which the 3rd is minor. AUGMENTED TRIAD: a triad of which the 5th is augmented. DIMINISHED TRIAD: a triad of which the 5th is diminished.(e) ROOT of a chord: that note from which it originates (e.g., in the common chord C–E–G we have C as the root, to which are added the 3rd and 5th). INVERSION of a chord: the removal of the root from the bass to an upper part. FIRST INVERSION: that in which the 3rd becomes the bass (e.g. E–G–C or E–C–G). SECOND INVERSION: that in which the 5th becomes the bass (e.g. G–E–C or G–C–E). THIRD INVERSION: in a 4-note chord that inversion in which the fourth note becomes the bass (e.g., in the chord G–B–D–F the form of it that consists of F–G–B–D or F–B–G–D, etc.). FUNDAMENTAL BASS: an imaginary bass of a passage, consisting not of its actual bass notes but of the roots of its chords, i.e. the bass of its chords when uninverted.(f) CONCORD: a chord satisfactory in itself (or an interval that can be so described; or a note which forms a part of such an interval or chord). CONSONANCE: the same as concord. DISCORD: a chord which is restless, requiring to be followed in a particular way if its presence is to be justified by the ear (or the note or interval responsible for producing this effect). See, for instance, the examples given under dominant (seventh) and diminished (seventh). DISSONANCE: the same as discord. RESOLUTION: the satisfactory following of a discordant chord (or the satisfactory following of the discordant note in such a chord). SUSPENSION: a form of discord arising from the holding over of a note in one chord as a momentary (discordant) part of the combination which follows, it being then resolved by falling a degree to a note which forms a real part of the second chord. DOUBLE SUSPENSION: the same as the last with 2 notes held over.(g) ANTICIPATION: the sounding of a note of a chord before the rest of the chord is sounded. RETARDATION: the same as a suspension but resolved by rising a degree.PREPARATION: the sounding in one chord of a concordant note which is to remain (in the same ‘part’) in the next chord as a discordant note. (This applies both to fundamental discords and suspensions.)UNPREPARED SUSPENSION: a contradiction in terms meaning an effect similar to that of suspension but without ‘preparation’.FUNDAMENTAL DISCORD: a discordant chord of which the discordant note forms a real part of the chord, i.e. not a mere suspension, anticipation, or retardation. Or the said discordant note itself (e.g. dominant seventh, diminished seventh, etc.).PASSING NOTE: a connecting note in one of the melodic parts (not forming a part of the chord which it follows or precedes).(h) FALSE RELATION: the appearance of a note with the same letter-name in different parts (or ‘voices’) of contiguous chords, in one case inflected (sharp or flat) and in the other uninflected.(i) PEDAL (or ‘point d'orgue’): the device of holding on a bass note (usually tonic or dominant) through a passage including some chords of which it does not form a part. INVERTED PEDAL: the same as the above but with the held note in an upper part. DOUBLE PEDAL: a pedal in which two notes are held (generally tonic and dominant).

From Wagner onwards the resources of harmony have been enormously extended, and those used by composers of the present day often submit to no rules whatever, being purely empirical, or justified by rules of the particular composer's own devising. Among contemp. practices are:

Bitonality—in which two contrapuntal strands or ‘parts’ proceed in different keys.

Polytonality—in which the different contrapuntal strands, or ‘parts’, proceed in more than one key.

Atonality—in which no principle of key is observed.

Microtonality—in which scales are used having smaller intervals than the semitone.

In the 20th cent. greater freedom in the treatment of the above procedures has developed, together with a much wider application of dissonance. Chords of 7th, 9th, 11th, and 13th are treated as primary chords, and there has been a return to the use of pentatonic scales, medieval modes, and the whole-tone scale. A prin. revolution c.1910 was the abandonment of the triad as the prin. and fundamental consonance. Composers such as Bartók, Stravinsky, Schoenberg, and Webern widened the mus. spectrum of tone-colour by showing that any combination of notes could be used as a basic unresolved chord. The tritone has been used as the cause of harmonic tensions in place of tonic-dominant relationships. Another 20th-cent. harmonic feature is the ‘layering’ of sound, each layer following different principles of organization. Milhaud produces bitonal passages from two layers in different tonalities.

Since 1950 much mus. has been comp. in which harmony has hardly any place, for example in some of the serial works of Boulez and Stockhausen. Where non-pitched sounds are used, harmony no longer exists and its place is taken by overtones, densities, and other concomitants of ‘clusters’, etc.

In amplification of this entry see added sixth, augmented, consecutive, counterpoint, and chromatic intervals.

har·mo·ny / ˈhärmənē/ • n. (pl. -nies) 1. the combination of simultaneously sounded musical notes to produce chords and chord progressions having a pleasing effect: four-part harmony in the barbershop style | the note played on the fourth beat anticipates the harmony of the following bar. ∎  the study or composition of musical harmony. ∎  the quality of forming a pleasing and consistent whole: delightful cities where old and new blend in harmony. ∎  an arrangement of the four Gospels, or of any parallel narratives, that presents a single continuous narrative text.2. agreement or concord: man and machine in perfect harmony.PHRASES: harmony of the spheressee sphere.

harmony In music, structure of chords and the relationships between them. The diatonic scale (from one C to the next on a piano, for example) is the basis of chord construction, and a harmonic progression from one chord to the next is defined by the key. The tonic, dominant, and subdominant chords are the primary chords of a key (C, G, and F chords in the key of C). Composers, especially of the 18th and 19th centuries, follow specific rules of harmony. See also scale

harmony †melody, music XIV; (mus.) combination of notes to make chords; agreement, accord XVI. — (O)F. harmonie — L. harmonia — Gr. harmoníā joint, agreement, concord, f. *harmo- of harmós joint, harmózein fit together.
So harmonic XVI. — L. — Gr. harmonikós. harmonica first applied (1762) by B. Franklin to a developed form of musical glasses; fem. sg. or n. pl. (used sb.) of L. harmonicus. harmonious XVI. — (O)F. harmonium XIX. — F. harmonize XV (rare before XVII).