Theory. the Abstract Principles Embodied in Music and the Sounds of Which It Consists

Home , Conductus, Discant, Franco of Cologne, Jerome of Moravia, Johannes de Muris, Marchetto da Padova

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, musica 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. Lindny, ed. (Oxford: Clarendon Pr, 19 1 1); SR, pp. 93- 100 (chaps. 15-23 complete); CCT 12 (1980): 1 1 -20. Book 3 treats music as a branch of mathematics (with arithmetic, géometry, astron- omy) and outlines three parts of music-harmonics, rhythmics, metrics (chap. 18), then divides sounds into harmonic (vocal), organic (wind instruments), and rhythmic (string and mrcussion instruments) (chaps. 19-22). Book 6 diKusses singing in the Of- âce. Aurelian of Réôme (:. 840- boj.Musica disciplina (The Discipline of Music). GS 1:28 - 63; Lawrence Gushee, ed., CSM 2 1 ( 1975); Joseph Ponte, trans., CCT 3 (1968). Chapters 1 - 7 discuss general topics (drawn from Boçthius, Cassiodoms, and Isidore), chaps. 8- 18 examine the dght modes, chap. 19 explains recitation formulas, and chap. 20 lists classes of plainsong. Hucbald (ca. 8no-g3oj.Deharmonica institutione (The Principles ot Harmony). GS 1:104- 25; Yves Chalier, ed., trans. Fr., çtLa Musica d'Hucbald de Saint-Amand'' (Ph.D. diss., Paris IV, 1973); Warren Babb, trans., YTS 3 ( 1978): 3-46. Synthesizes Greater Perfect System and tetrachords with eight modes and Byzantine *noeane syllables. Considers intervals, consonances and dissonances, tones and semitones three times-whhout musical notation, with neumatic notation and tetrachords, and with the Greater Perfect System plus Greek letter names. Anonymous (ca. 900). *Musica enchiriadis (Music Handbook). GS 1:152- 73; CS 2:74- 78; Hans Schmid, ed. (Munich: Verlag der Bayerischen Aka- demie der Wissenschaften, 198 1); Ernst Waeltner, ed., trans. Ger., chaps. 13- 18, Die Lehre von Or- ganum bis zur Mitte des 11. Jahrhunderts (Tutzing: Schneider, 1975), pp. 2- 19; Léonie Rosenstiel, trans., CCF7 (1976); RFSC, pp. 13- 15. Also related texts: Scolica enchiriadis, GS 1:173- 2 12; NR, pp. 126- 38. Commemoratio brevis, GS 1:2 13-29; Ter- ence Bailey, ed. (Ottawa: U of Ottawa Pr, 1979). Alia musica, GS 1:125- 52) Jacques Chailley, èd. (Paris: Centre de documentation universitaireg 1965); Ed- mund B. Heard, ed. and trans. (Ph.D. dlss., Univ. of Wis., 1966). Examines intervals, hexachords, modes, parallèl and free organum. Gives gamut built only from disjunct tetrachords in Daseian notation. Anonymous (ca. 1000). Dialogus de musica (Dia- logue Concerning Music). GS 1:252-64; P. Bohn, trans. Ger., MfMg 12 (1880): 23- 34, 39-48; SR, pp. 103- 16; Michel Huglo, çtL'auteur du tDialogue sur la Musique' attribué à Odon,'' RdM 55 ( 1969): 1 19 - 7 1. Starts with division of monochord (chaps. 1 - 2) and consonances (chaps. 4- 5), then gives lengthy account of eight modes as deâned by hnal (chaps. 6- 18). Uses I6-note gamut (G-a'). Formerly attrib- uted to Odo of Cluny. Guido of Arezzo (ca. 99 1 -after 1033). Micrologus (Little Discourse), 1025-26 or 1028 - 32. GS 2:2-

24., Joseph Smits van Waesberghe, ed., CSM 4 (1955)) Warren Babb, trans., YTS 3 (1978): 49- 83., RFSC, pp. 15- 16. Also Alia regulae, GS 2:34-42., Regulae rhythmicae, GS2..2ô - 34; Epistola de Iknoto cantu, G5*2:43- 50., SR, pp. 12 1 -25. Guido expands the gamut of the Dialogus to 2 1 steps (chap. 2), considers the monochord (chaps. 7 - 13), modes by am- bitus and ânal (chap. 14), melodic design (chaps. 15- 17), and free organum (especially the occursuè (chaps. 18 - 20). The Epistola gives six solmization syllables- ut, re, mi, fa, sol, Ia -while Alia regulae and Regulae rhythmicae introduce staF notation. Neither the Guidonian hand nor the system of natu- ral, hard, and soft hexachords appears in Guido's extant works. Johannes Aëighemensis (:. ca. 1 100). De musica (On Music). GS 2:230-65., Joseph Smits van Waes- Grghe, ed., CSM 1 ( 1950)., Warren Babb, trans., YTS 3(1978): 87 - I9O',RFJC, pp. 17 - 18. Modeled on the Micrologus. Covers general dehnitions (chaps. 1 -4), the monochord (chaps. 5- 7), intervals (chaps. 8 -9), modes (chap. 10), chant (chaps. 14- 19), solmization syllables (chap. 20), notation (chap. 2 1), and free organum (chap. 22). Prefers contraryto parallel motion in organum. Chapters 23- 27 are reserved for a top- ary. Bibl.: Gà, vols. 1 - 2. Edmond de Coussemaker, Histoire de I'harmonie au moyen âge (Paris, 1852; R: Hildesheim: Olms, 1966). Adrien de La Fage, Essais de diphtérographie musicale (Paris, 1864). CS, vols. 1 -2. Hans Peter Gysin, Studien zum Vokabular der Musiktheorie im Mittelalter (Zurich: A KöMer, 1959). RISM, B/III. GilYrt Reaney, çtThe Question of Authorship in the Medieval Treatises on Music,'' MD 18 ( 1964): 7- 17. JoKph Smits van Waes- Grghe, Musikerziehung: Lehre und Theorie, Musikge- Khichte in Bildern 3, pt. 3 (Gipzig: œutxhef Verlag fûr Musik, 1969). Michel Huglo, Les tonaires.. Inventaire, ana- Iyse, comparaison (Paris: Société française de musicologie, 197 1). F. AlYrto Gallo, çtphilological Works on Musical Treatixs of the Middle Ages,'' AM 44 (1972): 78- 101. uwrence Gushee, ççouestions of Genre in Medieval Trea- tises on Musics'' Schrade, 1973, pp. 365-433. Ernst Waeltner, Die Lehre von Organum bis zur Mitte des 11. Jahrhunderts (Tutzing: Schneider, 1975). Mathias Bielitz, Musik und Grammatik.. Studien zur mittelalterichen Mu- Jïk/àe/rïctMunich: Katzbichler, 1977). Michael Markovits, Das Tonsystem der abendlândischen Musik im#ûhen Mit- telaltèr (Befn: P Haupt, 1977). Andrew Hughes, Medieval Music, rev. ed. (Toronto: U of Toronto Pr, 1980). Sarah Fuller, ttTheoretical Foundations of Early Organum Theory,'iWM53 (198 1): 52- 84. Charles M. Atkinson, çtThe Parapteres.. Nothi or Not?'' MQ 68 ( 1982): 32 - 59. Calvin M. Bower, review of YTS 3 in JAMS 35 ( 1982): 157 -67. Hans Heinrich ERebrecht, ttDie Mehrstimmigkeitslehre von ihren Anfângen bis zum 12. Jahrhundert,'' in Ge- schichte der Musiktheorie, vol. 5, ed. Frieder Zaminer (Darmstadt: Wissenschaftliche Buchgesellschaû, 1984), pp. *-87. 111. Representative theorists 1200- 1400. Anonymous (0. early 13th century). Dlàcantu positio vulgaris (General Types of Discant). CS 1:94 - 7; Simon M. Cserba, ed., Hieronymous de Moravia Regensburg: F Pustet, 1935), pp. 189- 94) Janet napp, trans., JMT6 (1962): 200-207;RFJC, p. 19. short treatise on discant rules and six rhythmic nodes. Also dehnes sound, ligatures, consonance, Itc., and lists various polyphonic forms-e.g., or- Ianum, motet, conductus, hocket. Transmitted by rerome of Moravia. Anonymous IV (:. ca. 1270- 80). De mensuris et iiscantu (Concerning Mensuration and Discant). CS I :327 -65; Frits Reckow, ed., with commentary, 4./Aw, Beihefte 4- 5 (1967); Luther Dittmer, trans., VTT 1 ( 1957)., RFJC, pp. 20-2 1; Jeremy Yudkin, rd. and trans. (Ph.D. diss., Stanford Unlv., 1982). Lengthy synopsis of rhythmic modes, consonance Ind dissonance, discant and organum, plus réfer- Inces to specihc two-, three-, and four-part composi- rions. Preserves essential historical data about Notre Dame School. Johannes de Garlandia (:. ca. 1240). De mensura- qili musica (Concerning Measured Music). CS I :175- 82; Erich Reimer, ed., with commentary, 4fMw, Beiheûe 10- 1 1 ( 1972); Stanley H. Birnbaum, rrans., CCT9 ( 1978); Rudolf Rasch, zd.tlohannes de Sarlandia, IMM Musicological Studles 20 (New fork: Institute of Mediaeval Music, 1969); RFSC, p. 20. Describes three types of organum -discant, co- lula, organum-and focuses on the notation of rhythmic modes both in pure form and in combina- rion in diFerent voices. Franco of Cologne (f1. ca. 1250). Ars cantus men- ïurabilis (The Art of Measured Singing). GS 3:1 - 16., TS 1:1 17 - 36; Friedrich Gennrich, ed., Magistri Branconis Ars cantus mensurabilis (Darmstadt: Dennrich, 1957); Gilbert Reaney and André Gilles, ,d:., CSM 18 ( 1974); SR, pp. 139 - 59 (complete). Detailed treatise on mensural notation, discant, con- ionance and dissonance, copula, organum: and âocket. Provided basis for many later theonsts- ,.g., Marchetto da Padova, Jehan des Murs, Jacques Je Liège. Jerome of Moravia (;. 1272- 1304). Tractatus de musica (Treatise on Music). CS 1:1 -94., Simon M. Oserba, ed., Hieronymous de Moravia (Regensburg: F Pustet, 1935);AFJC, pp. 2 1 -22. Encyclopedic sur- zey of dehnitions, gamut, solmization, intervals, rodes, plainsong, etc. Quotes identiâed extracts rrom other writers and transmits four complete dis- zant tçzaiszs- Discantuspositio vulgaris, Johannes âe Garlandia, Franco of Cologne, and Petrus de Pi- zardia CCS 1:94- 154). Johannes de Grocheo (0. 1300). Ars musice (The drt of Music). Facs., ed., trans. Ger. with commen- Lary, Ernst RohloF, Die Quellenhandschréten zum Vusiktraktat des Johannes de Grocheio (Leipzig: Deutscher Verlag fur Musik, 1972)., trans. Eng. A1- àert Seay, CCT 1 (1967)., RFSC, pp. 23-24. Dis- :usses music in use in Paris, dividing it into musica 97; Simon M. Cserba, ed., Hieronymous de Moravia (Regensburg: F Pustet, 1935), pp. 189-94) Janet Knapp, trans., JMT6 (1962): 200-207;RFJC, p. 19. Short treatise on discant rules and six rhythmic modes. Also dehnes sound, ligatures, consonance, etc., and lists various polyphonic forms-e.g., organum, motet, conductus, hocket. Transmitted by Jerome of Moravia. Anonymous IV (:. ca. 1270-80). De mensuris et discantu (Concerning Mensuration and Discant). CS 1:327 -65; Frits Reckow, ed., with commentary, W./Aw, Beihefte 4- 5 (1967); Luther Dittmer, trans., MTT 1 ( 1957)., RFJC, pp. 20-2 1; Jeremy Yudkin, ed. and trans. (Ph.D. diss., Stanford Unlv., 1982). Lengthy synopsis of rhythmic modes, consonance and dissonance, discant and organum, plus réfer- ences to specihc two-, three-, and four-part composi- tions. Preserves essential historical data about Notre Dame School. Johannes de Garlandia (:. ca. 1240). De mensurabili musica (Concerning Measured Music). CS 1:175- 82; Erich Reimer, ed., with commentary, WfMw, Beiheûe 10- 1 1 ( 1972); Stanley H. Birnbaum, trans., CCT9 ( 1978); Rudolf Rasch, zd.tlohannes de Garlandia, IMM Musicological Studles 20 (New York: Institute of Mediaeval Music, 1969); RFSC, p. 20. Describes three types of organum -discant, copula, organum-and focuses on the notation of rhythmic modes both in pure form and in combina- tion in diFerent voices. Franco of Cologne (f1. ca. 1250). Ars cantus mensurabilis (The Art of Measured Singing). GS 3:1 - 16., CS 1:1 17 - 36; Friedrich Gennrich, ed., Magistri Franconis Ars cantus mensurabilis (Darmstadt: Gennrich, 1957); Gilbert Reaney and André Gilles, edg., CSM 18 ( 1974); SR, pp. 139 - 59 (complete). Detailed treatise on mensural notation, discant, consonance and dissonance, copula, organum: and hocket. Provided basis for many later theonsts- e.g., Marchetto da Padova, Jehan des Murs, Jacques de Liège. Jerome of Moravia (;. 1272- 1304). Tractatus de musica (Treatise on Music). CS 1:1 -94., Simon M. Cserba, ed., Hieronymous de Moravia (Regensburg: F Pustet, 1935);AFJC, pp. 2 1 -22. Encyclopedic survey of dehnitions, gamut, solmization, intervals, modes, plainsong, etc. Quotes identiâed extracts from other writers and transmits four complete discant tçzaiszs- Discantuspositio vulgaris, Johannes de Garlandia, Franco of Cologne, and Petrus de Pi- cardia CCS 1:94- 154). Johannes de Grocheo (0. 1300). Ars musice (The AM of Music). Facs., ed., trans. Ger. with commentary, Ernst RohloF, Die Quellenhandschréten zum Musiktraktat des Johannes de Grocheio (Leipzig: Deutscher Verlag fur Musik, 1972)., trans. Eng. A1- bert Seay, CCT 1 (1967)., RFSC, pp. 23-24. Dis- cusses music in use in Paris, dividing it into musica vulgarlà, musica mensurabilis, and musica ecclesias- tica, and describes specihc forms of Ocred and secu- lar music at length. Walter Odington (0. 1298- 1316). Summa de spe- t culatione musicae (Highçst Smculations about t Music). CS 1:182 -250; Frederick F. Hammond, ed., t CSM 14 ( 1970); Jay A. HuF, trans., pt. 6, MSD 31 ( 1973)., RFSC, pp. 22 -23. Comprehensive treatise in t six parts: pts. 1 -4 deal with speculative topics, in- ê cluding arithmetic proportions. intervals, and divi- 1 . t 6 : sions of the monochord; pt. 5 provides a tonary, p . surveys mensural notation and discusses rondellus, conductus, copula, motet, and hocket. l Marchetto da Padova (f1. 1305-26). Lucidarium . in arte musicae planae (Explanation of the Art of 1 P Plainsong), 1309- 18. GS 3:65- 121; Jan W. Her- linger, The Lucidarium of Marchetto of Padua.. W J Critical Edition, Translation, and Commentary J (Chicago: U of Chicago Pr, 1985); id., ççMarchetto's 4 Division of the Whole Wonz,''JAMS 34 ( 198 1): 193- t ê 2 16. Pomerium arte musicae mensuratae (Orchard . of the Art of Measured Music), 1318 -26., GS 3:12 1 - 4 87., Giuseppe Vecchi, ed., CSM 6 ( 196 1); SR, pp. 4 160- 7 1; RFSC, pp. 24-25. Lucidarium outlines the 1 arithmetic aspects of interval ratios, modes, and 1 chant construction, while the Pomerium codihes 1 Trecento Italian mensural notation. Pomerium pre- cedes and is probably independent of the Ars nova k tradition. Jehan des Murs (ca. 1300-ca. 1350). Notitia artis 4 musice (Knowledge of the AM of Music) or Ars nove d musice (The Art of New Music), 132 1. Bk. 1, ttmusica theorica,'' GS 3:312, 256- 57, 313- 15; bk. 2, d ttmusica practica,'' GS 3:292 - 30 1 (the whole con- ' densed as Compendium musice practice, ca. 1322, E GS 3:301 -6); Ulrich Michels, ed., CSM 17 (1972); ' SR, pp. 172 -79. Also Musica speculativa, GS ï 3:249 - 83; Libellus cantus mensurabilis, CS 3:46- 58., be/nning of Ars contrapuncti, CS 3:59-68. See Ulrich Michels, Die Musiktraktate des Johannes de Muris, W-/Mw, Beihefte 8 ( 1970). tçMusica theorica'' treats tuning, propolions, etc., and ttmusica practica'' conforms with Vitry's ideas on duple and triple notation. Philippe de Vitry ( 129 1 - 136 1). Ars nova (The New Art), ca. 1322 -23. CS 3:13-22; Gilbert Reaney, André Gilles, Jean Maillard, eds., trans. Fr., CSM8 ( 1964)., Leon Plantinga, trans., JMT5 ( 196 1): 204- 23*, RFSC, pp. 25 -26. Following a description of intervals, gamut, hexachords, musica hcta, etc., Vitry explains duple and triple notation at four levels - maximodus modus, tempus, prolatio-and in- ? troduws four tlme signatures. Jacques de Liège (ca. 1260-aher 1330). Speculum musice (The Mirror of Music), not before 1330. CS 2:193-433 (bks. 6, 7),. Roger Bragard, ed., complete in 7 vols., CSMDI 1955-73); SR, pp. 180-90;RFJC, pp. 27 - 28., F. Joseph Smith, Iacobi Leodiensis Spe- culum musicae, IMM Musicological Studies 13, 22, 42 (Brooklyn: Institute of Mediaeval Music, 1966-). Largest extant treatise on music from the Middle Ages (7 bks., 518 chaps.). Books 1 - 5 mostly devoted to intervals, consonances and dissonances, proportions, etc.; bks. 6 - 7 survey modes, 13th- to I4th-century polyphony, and mensural notation. Bibl.: GS, vol. 3. Edmond de CousKmaker, Traités iné- dits sur Ia musique du moyen âge (Li11e, 1865-69). CS, vol. 3. Richard Crocker, ttDiscant, Countemoint, and Har- mony,'' JAMS 15 (1962): 1 -21. Id., tçl-lermann's Major '' JAMS 25 (1972): 19-37. See also bibl. for 11 above. Sixth, IV. Representative theorists 1400- 1600. Prokocimus de Beldemandis (ca. 1380- 1428). Tractatus musicae speculative (Treatise on Sxcula- tive Music), 1425. D. RaFaello Baralli and Luigi Torri eds., RIM 20 (1913): 707 -62. Also Tractatus : practlce cantus mensurabilis, CS 3:200-28. Contra- punctus, CS 3:193-99; Jan Herlinger, ed. and trans. (Lincoln: U of Nebr Pr, 1984). Tractatus practice cantus mensurabilià ad modum ytalicorum, CS 3:228-48; Jay A. Hufll ed. and trans., MSD 29 ( 1972); RFSC, p. 30. Includes a polemic against Mar- chetto's Lucidarium; rejects Marchetto's division of the tone into flve dieses and restores Pythagorean ratios. Stresses advantages of Italian trecento notation over French ars nova. Ugolino of Orvieto (ca. 1380 - 1457). Declaratio musice disclàline (Declaration of the Discipline of Music), 1430- 35. Albert Seay, ed., 3 vols., CSM 7 ( 1959-62); id., ttugolino of Orvieto, Theorist and Composer,'' MD 9 (1955): 1 1 1 -66, 1 1 (1957): 126- 33; Andrew Hughes, trans., chap. 34, MSD 27 (1972), pp. 2 1 - 39; RFSC, p. 31. Treatise in two sec- tions: bks. 1 - 3 consider practical problems of mode, intervals, chant, countemoint, and mensural notation; bks. 4 - 5 are more speculative and deal with the arithmetic foundations of music. Johannes Tinctoris (ca. 1435- 151 1?). Liber de arte contrapuncti (The AM of Countemoint), 1477. CS 4:76 - 153; Albert Seay, ed. and trans., M5>D 5 (196 1); SR, pp. 197 -99; RFJC, pp. 33- 34. Also Liber de natura etproprietate tonorum, C54:16 -4 1; CCTZ ( 1967; 2nd ed., 1976). Proportionale musices, CN4:153-77; CCT 10 (1979); SR, pp. 193-96. Ex- positio manus, CS 4:1 - 16; Albert Seay, trans., JMT 9 (1965): 194-232. See CSAf 22 (1975). Taken to- gether, Tinctoris's 12 treatises cover the spectrum of 1sth-century music theory. Liber de arte contrapuncti has prologue and three books. Book 1 deânes countemoint and gives the 22 consonances of note- against-note style; bk. 2 describes 27 dissonances and false concords; and bk. 3 lists 8 general rules for countemoint. Bartolomeo Ramos de Pareia (ca. 1440-9 1 or thereafter). Musica practica (Practical Music), Bolo- ' gna, 1482. Facs., BMB sez. 2, 3 (1969); Johannes Wolf, ed. (Leipzig: Breitkopf & Hàrtel, 1901)) SR, pp. 200- 204) RFSC, pp. 36 - 37. Text in three parts with prologue and epilogue. One of the most original of all