Invertible Counterpoint Research Articles

Fuga and invertible counterpoint in Byrd’s Cantiones sacrae (1589): some preliminary observations

Abstract This article aims to shed light on aspects of Byrd’s contrapuntal technique—particularly in relation to the crafting of fuga (or imitation)—by subjecting to close analysis a small sample from his Cantiones sacrae (1589), and in so doing offers in effect a trailer for the study of the collection as a whole. A number of issues are raised which indicate some of the directions in which future research in this area might be taken. These include the extent to which material is, or appears to be, reused by Byrd within a motet or across different motets, or indeed whether it may be borrowed from another composer, although this is a subordinate consideration in relation to Byrd’s reuse of his own material. While passages may appear to resemble each other—because they use similar subjects—sometimes it is a superficial likeness only, and when one delves below the surface one finds that the fuga is constructed quite differently. Byrd’s use of fuga is often far from straightforward, and leads one to conjecture that at times there was likely to have been some kind of pre-compositional planning: duos with inbuilt cadence formulas arise, and at the same time complexes of triple (invertible) counterpoint are introduced that take the procedures to another level of intricacy.

The Fundamenta compositionis Jean Kuhnaus 1703. Edition, Translation, and Commentary

Until now, scholars have generally doubted whether Johann Kuhnau, J. S. Bach’s predecessor as Thomaskantor in Leipzig, was indeed the author of the manuscript treatise, Fundamenta Compositionis Jean Kuhnaus 1703. Nevertheless the Fundamenta appears to originate from the Thomasschule around 1700, suggesting that it may stem from Kuhnau’s broader circle. Moreover, recent research has revealed concordances between the Fundamenta and an anonymous manuscript titled Kurtze Verfasung (“Brief Instruction”), in addition to a number of other sources. Thus, regardless of who compiled the Fundamenta, this manuscript is significant because it reveals the pedagogical priorities, or “fundamentals” (according to the title), that often lie buried in more discursive, theoretical treatises. Therefore the ultimate value of the Fundamenta is not its originality, but rather its explication of topics such as consonance and dissonance, modes, clausulae, cadences, invertible counterpoint, and fugue in a highly condensed, practical manner with numerous musical examples. The Fundamenta is thus an excellent resource for historically informed analysis and composition at the turn of the eighteenth century, before thoroughbass accompaniment displaced vocal polyphony as the dominant pedagogical paradigm in compositional instruction. Part one of this article explores the Fundamenta’s provenance in detail; part two provides a transcription and English translation.

Zelenka and the Combinative Impulse: Contrapuntal Techniques in the Miserere in D Minor, ZWV 56

Commissioned in 1722 for the Holy Week services at the Dresden Hofkapelle, Zelenka’s Miserere in D minor (ZWV 56) demonstrates virtuosic handling of a wide range of contrapuntal techniques. Zelenka drew upon his deep knowledge of past musical traditions, honed through his earlier studies in Vienna with Fux and his circle, to provide individual sections of the Miserere with systematic explorations of freely imitative, canonic and fugal writing as well as subtle demonstrations of invertible counterpoint, retrograde motion and melodic inversion. The result is a work that offers insight into Zelenka’s abilities to exploit the combinative potential of his motivic materials in settings that maintain expressive as well as technical originality. The present study adopts a range of analytical techniques to uncover the inner workings of Zelenka’s counterpoint, thereby providing greater understanding of this composer’s versatility and accomplishments.

A Taxonomy for Teaching Music Theory: J. S. Bach and Lessons in Invertible Counterpoint

A Taxonomy for Teaching Music Theory: J. S. Bach and Lessons in Invertible Counterpoint

A Taxonomy for Teaching Music Theory: J. S. Bach and Lessons in Invertible Counterpoint

Many aspects of J. S. Bach's pedagogical tradition remain relevant to current thinking in undergraduate music theory instruction. This essay draws inspiration from Bach as a teacher, particularly his systematic presentation of ideas and his preference for teaching from compositions rather than abstract exercises. Part 1 explores the potential for applying Bloom's Taxonomy to music theory instruction. Building on recent applications of the taxonomy to music education, particularly work by Deborah Rifkin and Philip Stoecker, I propose a revised taxonomy for teaching undergraduate written theory. This approach provides a great deal of flexibility in the classroom since it can be applied to any theory topic. To demonstrate its effectiveness, Part 2 presents a series of detailed activities for teaching invertible counterpoint following the six categories of the taxonomy. Each activity includes excerpts either from Bach's Well-Tempered Clavier or two-part inventions.

Counterpoint and Partimento: Methods of Teaching Composition in Late Eighteenth-Century Naples by Peter van Tour

Counterpoint and Partimento: Methods of Teaching Composition in Late Eighteenth-Century Naples. By Peter van Tour. (Acta Universitatis Upsaliensis; Studia Musicologica Upsaliensia, no. 25.) Uppsala: Uppsala Universitet, 2015. [318 p. ISBN 9789155491970. $82.50.] Music examples, companion Web sites, tables, appendices, bibliography, index.The compositional, cognitive, and pedagogical history of eighteenth-century music and musicians has experienced a groundswell in contemporary scholarship. Still in its nascence a few short decades ago, several important studies have propelled these challenging topics to a position of prominence in the field (Robert O. Gjerdingen, Music in the Galant Style [New York: Oxford University Press, 2007]; Rosa Cafiero, Early Reception of Neapolitan Partimento Theory in France: A Survey, Journal of Music Theory 51, no. 1 [2007]: 137-59; and Giorgio Sanguinetti, The Art of Partimento: History, Theory, and Practice [New York: Oxford University Press, 2012]; to name but a few). As the de facto nucleus of eighteenth-century musical training, Naples and its population of teachers and composers have garnered particular attention. Unlike their French and German contemporaries, who penned lengthy treatises, Neapolitan musicians left precious little prose or explication con- cerning their work for posterity; an eighteenth-century Neapolitan treatise often contains little more than opaque examples and exercises in composition or improvisation, rendering a rich historiography exceedingly difficult.A portion of this historiography is precisely what Peter van Tour constructs in his 2015 dissertation, Counterpoint and Partimento: Methods of Teaching Composition in Late Eighteenth-Century Naples, published by Acta Universitatis Upsalienis as number 25 of the series Studia Musicologica Upsaliensia. This important volume elucidates complex relationships among a web of eighteenth-century primary sources and historical accounts, many of which have been previously misunderstood, misattributed, or given little attention. The clarification and nuance that Counterpoint and Partimento provides is essential to a deep understanding of the methods and materials of Neapolitan musical training.As Giorgio Sanguinetti admits in The Art of Partimento, the very act of defining is a tricky endeavor, given that Italian treatises rarely provide satisfactory definitions themselves. The effective working definition that he proposes is potential musical works; that is, while a partimento is most often a bass line, it also contains complex implications for voice leading, melody, and even imitation (Sanguinetti, pp. 167, 11). Van Tour defines his primary terms (counterpoint and partimento) as follows:By 'counterpoint' I mean the process of conceiving independent melodic progressions according to principles fixed beforehand, both written and improvised, in two or more parts, freely or against pre-existing material, such as a cantus firmus, or a soggetto cavato, a partimento bass, a melody, the hexachordal scale, etc. 'Partimento' is understood as a notational device, commonly written on a single staff in the F clef, either figured or unfigured, applied both in playing and in writing activities, and used for developing skills in the art of accompaniment, improvisation, diminution and counterpoint. (p. 19)In light of its various purposes and deployments, van Tour traces the essential paths toward partimento, including practical exercises at the keyboard, diminution, and acquiring contrapuntal skills such as invertible counterpoint, imitation, and fugue (p. 20).As he situates his research in the context of existing scholarship, van Tour argues that previous studies of various methods and materials utilized at Neapolitan conservatories have, in some cases, reached insufficient depth. In particular, he addresses Jesse Rosenberg's 1995 dissertation, in which Rosenberg tackles the specific characteristics of the two rival Neapolitan schools: one following the teachings of Francesco Durante (1684-1755) and the other following the teachings of Leonardo Leo (1694-1744) (Jesse Rosenberg, Experimental Music of Pietro Raimondi [Ph. …

Triple Counterpoint and Six‐Four Chords in Bach's Sinfonia in F Minor

ABSTRACTThe Sinfonia (Three‐Part Invention) in F minor stands out among J. S. Bach's masterworks of triple counterpoint because of its bold departures from conventional practices of dissonance treatment and harmonic syntax with respect to six‐four chords, licence that Bach takes to accommodate thematic material in the bass. This essay explores several interrelated musical components central to the extraordinary nature of the piece: the polyphonic structure of the subject and the transferred resolution of its most prominent dissonance; the above‐mentioned departures from conventional treatment of six‐four chords; the altered linear and harmonic meanings of the subject, first countersubject and second countersubject in the context of modulation via auxiliary cadence; Bach's use of particular permutations to mark with perfect authentic cadences both keys and sections of form; and his use of certain permutations to create a prolongational structure on a deep middleground level. The essay also briefly explores the views of the eighteenth‐century theorists Johann Philipp Kirnberger and Friedrich Wilhelm Marpurg regarding six‐four chords in the context of invertible counterpoint; Bach's employment of melodic and cadential formulas common in eighteenth‐century fugue and partimento; and the descriptions of those cadential formulas by several seventeenth‐ and eighteenth‐century theorists.

Sergei Taneev’s Vertical-Shifting Counterpoint

In his 1909 treatisePodvizhnoi kontrapunkt strogogo pis’ma(Movable counterpoint in the strict style), Sergei Taneev reconceived invertible counterpoint as a type of “vertical-shifting” counterpoint, in which the voices move higher or lower in pitch but do not necessarily cross. The conditions on vertical shifting at all intervals operate according to the same mathematical principles; Taneev’s system allows these conditions to be efficiently calculated. This article introduces Taneev’s terminology and method in order to demonstrate how his work generalizes, refines, and extends the insights of prior theorists. I apply the method analytically to investigate shifts at uncommon intervals in one of Taneev’s fugues, and I extend the method to account for Taneev’s compositional use of counterpointalla riversa, or “reversible” counterpoint, in which the voices are flipped upside-down in melodic inversion.

The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening. By Steven G. Laitz

The Complete Musician is a comprehensive theoretical training text for undergraduates. The main volume, containing the principal text and exercises, with a disc of recorded examples, is accompanied by a workbook of supplementary exercises with its own CD. The main volume has four appendices: these either offer further material on basic subjects for less experienced students or they extend various sections of the text, with such topics as intervals, sequences, and invertible counterpoint. Appendix 3, subtitled ‘The Motive’ (pp. 768–803), provides a useful analogue to Part 7, on ternary, rondo, and sonata forms. The book was first published in 2004, a second edition appearing in 2008. Under present review is the third edition. This includes a number of revisions listed in the Preface (pp. xv–xvii) such as: a reduced number of chapters; more material on musical fundamentals; solutions to certain exercises in Appendix 6; many new examples from the repertory, and a new companion website (www.oup.com/us/laitz).

The Theory and Practice of Integrating Invertible Counterpoint into Classical Theme-Types

This essay investigates the use of invertible counterpoint within conventional theme-types encountered within Classical-style works. Whereas recent research has focused on how this contrapuntal technique provides a means of creating continuity, this article expands the purview by elucidating how it works in concert with theme-types, forms regulated by cadences (devices that arguably suppress continuity). A basic model provides the theoretical framework with which to analyze excerpts from a selection of works by Beethoven, Haydn, and Mozart. The analyses reveal that some theme-types are more successful than others at integrating invertible counterpoint into their formal design. Moreover, the analyses uncover the purpose of using invertible counterpoint within theme-types.

"A Fallacious Concept": Invertible Counterpoint at the Twelfth within the Ursatz

"A Fallacious Concept": Invertible Counterpoint at the Twelfth within the Ursatz

The Reinstatement of Polyphony in Musical Construction: Fugal Finales in Haydn's Op. 20 String Quartets

The title, borrowed from Paul Henry Lang's description of Haydn's op. 20 string quartets in Music in Western Civilization, characterizes Haydn's endeavor to create more independent partwriting in the string quartet. First, Haydn's fugal practice is noteworthy particularly for the construction of the fugal exposition and his treatment of multiple subjects, the question of what constitutes a regular countersubject, and the treatment of redundant entries. Second, the chief strategy in these movements is the invention of invertible counterpoint in three voices. Haydn writes a double fugue (with a regular countersubject), as well as a triple and quadruple fugue, in which the principal issue is the ability of each subject (including the double fugue's countersubject) to serve as any voice—top, middle or bottom—in a texture of invertible counterpoint. The expertise he attained with these works then allows him to exploit the technique in later quartets, principally in the development sections of sonata-form movements. There, he uses invertible counterpoint to establish the independence of each voice, and to create longer passages unarticulated by cadences, sections distinct from the more clearly articulated periodic expositions and recapitulations. The three fugal finales of op. 20, therefore, constitute Haydn's advanced study not so much in fugal procedure as in the practice of invertible counterpoint.

Theories of fugue from the age of Josquin to the age of Bach

Part 1 Fugal theory of the Renaissance and early Baroque: fugue in the High Renaissance fugue at the end of the Renaissance, I - Italy and the Netherlands fugue at the end of the Renaissance, II - Germany German theory during the Thirty Years War - fugue in Latin School music texts. Part 2 The makings of the modern fugue - Italy and Germany in the mid-17th century: Italian influence on German fugal theory, 1640-1680 instrumental fugue at the emergence of fugal structure in the third quarter of the 17th century the Hamburg circle of theorists - invertible counterpoint, the countersubject, and permutation fugue. Part 3 German fugal theory of the mature Baroque, 1680-1740: fugal theory, 1680-1710 fugal theory in German lexicographic texts fugal theory, 1710-1740 - Mattheson and Fux.

A Lesson from Lassus: Form in the Duos of 1577

This article demonstrates how contrapuntal techniques contribute to the formal structure of Lassus's twenty-four famous duos. Three types of contrapuntal events are identified in each piece: time interval of imitation, fuga (type of melodic variation), and “blocks” (contrapuntal combinations, usually varied by transposition or invertible counterpoint). Occurrences of these events are mapped onto time lines for all twenty-four pieces. The time lines suggest conclusions regarding Lassus's normative formal procedures. These procedures can in some cases be related to text and, in others, taken as suggestions for expressive performance.

Some Group Properties of Triple Counterpoint and Their Influence on Compositions by J. S. Bach

Invertible counterpoints in tonal music are a special and unique class of thematic repetition. Like unvaried reprises, recapitulations, and transpositional sequences, invertible counterpoints reuse thematic material in different parts of a musical form. Unlike these other procedures, which preserve interrelationships among the several melodic lines, invertible counterpoint alters these interrelationships so that the melodic components are rearranged with respect to each other upon repetition. For this reason invertible counterpoint can be a more subtle and covert means of thematic repetition than the other procedures mentioned above, and therefore can become a powerful way to unify musical structure. In the hands of J. S. Bach, who recognized new and unexplored possibilities for invertible counterpoints, it is used so often and with such imagination that the formal economies of his compositions are often extraordinarily simple in relation to the beautifully complex sound of the music. In the pages following, I will examine both the general properties of triply invertible counterpoints (triple counterpoint) and the use of this compositional technique in some of Bach's music. The reader might well ask why a study of Bach's use of double counterpoint is not prerequisite. The answer lies in the different abstract structure of these two types of invertible counterpoint and the effect these differences have on compositional procedure.