The New Grove Dictionary of Music and Musicians * * * Theory, theorists. Teoria,teoricos. Theory is now understood as principally the study of the structure of music. This can be divided into melody, rhythm, counterpoint, harmony and form, but these elements are difficult to distinguish from each other and to separate from their contexts. At a more fundamental level theory includes considerations of tonal systems, scales, tuning, intervals, consonance, dissonance, durational proportions and the acoustics of pitch systems. A body of theory exists also about other aspects of music, such as composition, performance, orchestration, ornamentation, improvisation and electronic sound production. (There are separate articles on most of these subjects, but for more detailed treatment of the most fundamental of them see in particular Acoustics; Analysis; Counterpoint; Harmony; Improvisation; Melody; Mode; Notation; Rhythm.)The Western art music tradition is remarkable for the quantity and scope of its theory. The Byzantine, Arabic, Hebrew, Chinese and Indian traditions are also notable in possessing significant bodies of theoretical literature. Recently there has also been some theoretical treatment of jazz and other genres of popular music. This article, however, will deal exclusively with the Western art music tradition. (For these other traditions see particularly Arab music; China, II; India, III; Iran, II; Japan, I; Jewish music, III; see also Byzantine chant, 17; Greece, I; Mode, V; Jazz; Popular music.)1. Introduction.Treatises as disparate as De institutione musica (c500) by Boethius, Larte del contraponto ridotta in tavole (15869) by Giovanni Maria Artusi, Larmonico pratico al cimbalo (1708) by Francesco Gasparini, and Der freie Satz (1935) by Heinrich Schenker are all commonly subsumed under the category of thought called music theory. Yet these four books have little in common. That of Boethius was totally divorced from the music of his time and probably not intended to be read by musicians or composers. In it a student of the liberal arts sums up the speculations about music of a number of Greek authors, mainly from the 2nd century. Artusis book was a text for the training of musicians and composers in counterpoint as practised and taught by his, then older, generation. Gasparinis is a manual for harpsichordists on the art of accompanying from a thoroughbass. Schenkers expounds some fundamental hypotheses about masterpieces of 18th- and 19th-century music through an analysis of their tonal and harmonic content.Even allowing for the span of time encompassing them from about 500 to 1935 and the changing practice of music, the absence of any significant overlap in these four books, whether of content, purpose or intended audience, demonstrates at once the diffuseness and richness of the concept of theory. The term can be given an inclusive or an exclusive definition; in the one case it will embrace all of these works, in the other only one or two of them. It is useful to begin with an attempt at an inclusive definition.2. Definitions.The word theory itself has broad implications. Its Greek root theoria is the noun form of the verb theoreo, meaning to inspect, look at, behold, observe, contemplate, consider. A theoros is a spectator, as at