Theory. The abstract principles embodied in music and the sounds of which it consists. With respect to Western music, theory has traditionally encompassed the properties of single sounds, pitch, duration, timbre and those of collections of sounds: acoustics, tuning and temperaments, intervals, consonance and dissonance, scales, modes, melody, harmony, counterpoint, rhythm, meter, form, and analysis. Today the term also refers specifically to the teaching of the fundamentals or rudiments of music- e.g, elementary harmony and counterpoint, general musicianship, ear training, solfege. Non-Western musical cultures with rich traditions of explicit formulation and study of music theory include those of East Asia and south Asia. 1. General. In its most general sense, the term theory refers to the contemplation rather than the practice of music. Following Aristotle's division of knowledge (episteme) into praktikë, poiëtikë, and theörëtikê (Metaphysics 1025b-26a), writings about music before I 600 were usually classified as musica theorica musica speculativa, musica contemplativa, musica arithmeticaj, musica practica çmusica at- tivaj, or, particularly after the Musica of Nikolaus Listenius ( 1537), musicapoetica. In this scheme, mu- sica theorica dealt with the arithmetic foundations of music-i.e., music as one of the four mathematical arts Isee Quadrivium). Such discussions typically in- cluded detailed accounts of the gamut and Greater Perfect System, interval ratios, and consonance and dissonance. Musica practica then applied these ideas to musical composltions. Though the distinctions were blurred, especially in the early Middle Ages, musica practica was often subdivided into two branches: musicaplana, which dealt with plainsong, Solmization, mode, and even intervals, etc., and Musica mensurabilis (musica /g/rl/ïll, which explained mensural notion, Discant, and counterpoint. Musica poetica considered more general aspects of musical composition, usually in relation to rhetorical models and afective content Isee Allèctions, doctrine of Figures, doctrine of Rhetoric). Although Aristotle's threefold plan lived on through the 17th and 18th centuries (e.g., in the dictionaries of Janovka, Brossard, Walther, Rousseau), the scope of musical thought grew in several imgon tant areas: through the crystallization of major - minor tonality and the recognition of triadic tonal harmony (Lippius, Werkmeister, Rameau); through the codification of species counterpoint (Zacconi, Dinlta, Berardi, Fuxl; through the use of the thoroughbass method (Agazzari, Penna, Heinichen); through the emergence of the science of acoustics (Descartes, Mersenne, Sauveur); and through the 11. owering of formal analysis toward the end of the 18th century Riepel, Koch, Kollmann, Momigny). During the 19th and 20th centuries, theorists further codihed the structural functions of tonality (Fétis, Riemann, Schoenberg, and, above all, Schenker), thereby rehning the techniques of tonal analysis. Some of the new analytic methods of the 20th century have been prompted by dramatic changes in musical style and others by attempts to adopt the methods of other disciplines, such as linguistics and mathematics. These new methods include the use of ordered and unordered set theories (Babbitt, Lewin, Forte), information theory (Hitler), generative gram- mars (Lerdahl, Jackendoff, Keiler), structuralism and Semiotics (Nattiez, Ruwet), as well as theories of new musical resources (Héba, Yasser, Cowell, Bu- soni, Xenakis, and others). Besides specifying a general area of study, theory is also used in the narrow sense to mean a comprehensive and consistent set of statements that attempt to explain a given musical phenomenon. Prior to the 20th century, little progress was made in the study of theory construction (metatheoa) after the work of classical writers such as Aristoxenus and the Pythagoreans. But since 1900, music theorists have used the many advances in contemporary logic and philosophy of science to reâne their thinking about theoretical discovery. Several attempts have recently been made to present musical systems as axiomatic theories. In axiomatic theories, statements are of two types-axioms or given statements (primitives, postulates) and theorems or derived statements. For a theorem to Y valid, it must follow from the axioms according to a given set of rules for deductive inference. Axiomatic theories may be presented either informally in everyday language orïbrmally, using the notation of symbolic logic. As they stand: axiomatic theories make no claim about the material truth of individual statements - the only concern is that theorems follow the axioms by logic alone. In this sense, the theories are unintemreted. Like scientiâc theories, however, music theories require empiric interpretation; such interpretation usually involves the analysis of specific pieces or repertories. The problem is that analytic methods are always heavily theory laden. It may not be possible: therefore, to propose neutral tests for the comparison of rival theories. Though the study of music theory has a long tradition, many questions about the construction, selection, transmission, and genres of music theories remain unanswered. Bibl.: Bibliographies. James B. Coover, ttMusic Theory in Translation: A Bibliography,'' JMTD ( 1959j: 70-96; supp., JMT 13 ( 1969): 230 -48. JoKph Smits van Waeserghe, Pieter Fischer, and Christian Mnnq, eds., The Fâe/?'y of Music#om the Carolingian Era up to 1400, RISM, B/IIl/ 1 - 2 (Mynich-Duiserg: Henle, 196 1 - 68). François Hsure, ed., Ecrits imprimés concernant la musique, RJSM, B/VI/ 1 - 2 (Munich-Duisberg: Henle, 197 1). David Russell Wil- liams, a4. Bibliography of the History of Music Theory, 2nd ed. (Faimorq N.Y.: Rochester Music Pubs, 197 1). Periodicals. Die Reihe, 8 vols. (Austrian ed., 1955-62; trans. Eng., 1958 -68). Journal ofMusic Theory ( 1957 - ). Perspectives of New Music ( 1962 - ). The Music Forum ( 1967 - ). ZeitschrsfûrMusiktheorieç 1970-78). In Fàet)r.p Only ( 1975- ). avusic Theory Spectrum ( 1979 -). Music Analysis ( 1982-). Musiktheorie ( 1986 -). General. Hugo memann, Gèschichte der Musiktheorie im IX. -XIX. Jahrhundert (Leipzig, 1898; 2nd ed., Berlin: M Hesse, 192 1, R: Hildesheim: Olms, 196 1); trans. Eng., bks. 1 -2 by Raymond H. Hae (Lincoln: U of Nebr Pr, 1962) and bk. 3 by William Mickelsen (Lincoln: U of Nebr Pr, 1977). Oliver Strunk, Source Readings in Music History (New York: Nolon, 1950). Gustave Reex, Fourscore Clas- sics ofMusic Literature (NiW York: Lieral Arts Pr, 1957). David Kmehenbuehl, Norman Phelps, Howard Mumhy, Gordon Binkerd, and Roert Melcher, ççThe Professional Music Theorist-His Habits and Training: A Forum,'' JMT4 (1960): 62 -84. FranciKö Jok Vôn Tello, Estudios de historia de Ia teorîa musical (Maddd: CSIC, 1962). Mi- chael KaKsler, CQA Sketch of the U% of FormnlizM Imn- guages forthe Asxrtion of Music,''##M 1/2 ( 1963): 83-94. Milton Babbitt, tt-l'he Structure and Fundion of Music Theory,'' CMS 5 (1965): 49-60. David Vwin, ççBehind the Beyond,'' PNM 712 ( 1969): 59 -69. Benjamin Boretz, ttMeta- variations: Studies in the Foundations of Musical Thought,''##M8/1 (1969): 1-74; 8/2 (1970): 49- 1 1 1; 9/1 (1970): 23-42; 9/2 and 10/1 (1971): 232-70; 11/1 (1972): 146-223; 1 1/2 (1973): 156-203. Thomas Clifton, ççsome Compadsons Ytween Intuitive and Scientiâc Dexriptions of Music,'' JMT 19 (1975): 66- 1 10. Richmond Browne, Vernon L. Kliewer, Peter Westergnnrd, Carl E. Schachter, Carlton Gamer, and Allen Forte, çtMusic Theory: The Art, the Profession, and the Futurey'' CMS 17 (1977): 135-62. John Rahn, ttAsmcts of Musical Explanation,'' PNM 17/2 (1979): 204-24. Harold S. PoweD, ççunguage Models and Musical Analysis,'' Ethno 24 (1980): 1 -60. Richmond Browne, ed., Music Theory. Special Topics (New York: Acad Pr, 198 1). 11, Representative theorists 350- 1200. Augustine (354-430). De musica (On Music), 387-9 1. PL 32:108 1 - 1 194; Giuseppe Vecchi, ed., Praecepta artis musicae; collecta ex Iibris sex Aurelii Augustini ''De Musica, '' Reale accademia delle scienze dell'1stituto di Bologna, Classe di scienze morali, Memorie, ser. 5, vol. 1 (Bologna, 1950), py. 93- 153; R. Catesby Taliaferro, trans., On Music, ln Writings ofsaint Augustine, pt. 2, The Fathers ofthe Church: ,4 New Translation, vol. 4 (New York: Fa- thers of the Church, 1947). Deânes music as the ççart of measuring we11.'' Books 1 - 5 consider musical rhythm and quàntitative poetic meter and may have been impetus behind modal rhythm. Boethius (ca. 480-ca. 524). De institutione musica (The Principles of Music). PL 63:1 167 - 1300; Gott- fried Friedlein, ed., W. M. T. S. Boetii. De institutione arithmetica Iibri duo. De institutione musica Iibri quinque. (Leipzig, 1867; R: Frankful: Minerva, 1966)., Calvin M. Bower, trans., téBoethius, Theprin- clè/c.î ofMusic'' (Ph.D. diss., George Peabody Co1- lege, 1967); SR, pp. 79-86; RFSC, pp. 12- 13. See ë Calvin Bower, çt-f'he Modes of Boethius,'' JM 3 ( 1984): 252 -63. Principal text for the transmission of Greek theory to the Middle Ages. Bks. 1 -4 proba- ' bly translated from Nicomachus (with material from Euclid (?), Sectio canonis in bk. 4) and bk. 5 from '. molemy. Covers harmonics, arithmetic proportions, ' tones and semitones, comma, notation, tuning, and : modes. Divides music into musica mundana, musica : humana, and musica instrumentalis (bk. 1). d Isidore of Seville (ca. 559 -636). Etymologiarum I sive originum Iibri xx, bk. 3, chaps. 15-23, bk. 6, chap. 19. PL 82:163-69, 251 -60., GS 1:20- 25;Wa1alace M.