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Journal of Seventeenth-Century Music

Journal of Seventeenth-Century Music

Volume 12 (2006) No. 1

Columbus’s Egg: Andreas Werckmeister’s Teachings on Contrapuntal Improvisation in Harmonologia musica (1702)

Michael R. Dodds*


In his treatise Harmonologia musica, Andreas Werckmeister (1645–1706) devotes special attention to keyboard improvisation of fugue and invertible counterpoint. The essence of Werckmeister’s approach to this most challenging type of improvisation consists of sequential imitative subjects and pervasive doubling in thirds. Ingenious in their own right, Werckmeister’s teachings on contrapuntal improvisation also shed light on the organ works of his friend Dieterich Buxtehude, who contributed to the treatise two poems honoring Werckmeister. For modern performers recovering the lost art of contrapuntal improvisation, Werckmeister’s treatise furnishes an invaluable guide.

1. Introduction

2. The Friendship of Werckmeister and Buxtehude

3. Werckmeister’s Harmonologia musica

4. Columbus’s Egg: Stepwise Sequential Subjects

5. Buxtehude’s Use of Sequential Subjects

6. Parallel Thirds as a Key to Improvising Invertible Counterpoint

7. Conclusions



1. Introduction

1.1 Improvisation of fugue has long ranked as a supreme manifestation of the organist’s art. The fame accorded to the likes of Frescobaldi, Froberger, Kerll, Scheidemann, Weckman, Pachelbel, Buxtehude, Bach, and Handel arose in large measure from their legendary ex tempore performances. Their improvisations have long since echoed into silence, but they are not all that has faded with time: so too have many of the musical techniques and strategies with which these artists approached their art. Fortunately, important clues about improvisational practices survive in composers’ written music, eye-witness accounts, the instruments they played, and contemporaneous music theory treatises. This last source of clues can prove problematic, for while Baroque treatises abound on such topics as thoroughbass or counterpoint, few address keyboard improvisation with the sophistication of a Renaissance treatise like Tomàs de Santa María’s Libro llamado arte de tañer fantasias (Valladolid, 1565) or bear a demonstrable relationship with a major organist-composer, especially in a way that sheds light on that composer’s musical style.

1.2 In this respect the Harmonologia musica of Andreas Werckmeister constitutes an exceptional bequest.1 Werckmeister’s ingenious yet simple teachings on fugue, canon and invertible counterpoint merit special mention in their own right, but they also shed light on the improvisational techniques of his contemporaries, most notably Dieterich Buxtehude, organist of the Marienkirche in Lübeck. In many of his works in improvisatory genres, including his praeambula and chorale-based compositions, Buxtehude employs the very techniques described by his friend Werckmeister. At issue is not any question of influence: notwithstanding Werckmeister’s avowals of originality, it seems more likely it was he, an innovative theorist but mediocre composer, who learned from the Lübeck master than vice versa. Indeed, by the time Werckmeister published Harmonologia musica in 1702, the sexagenarian Buxtehude had already composed most of his organ works. Rather, the striking parallels between Werckmeister’s teachings and Buxtehude’s music merit attention because they constitute mutually supporting evidence about contrapuntal improvisation in the Baroque era.

1.3 If to Baroque minds invertible counterpoint and canon exemplified some of the most arcane expressions of musical science, then for composers of fugue these procedures also provided indispensable tools. Mastery of invertible counterpoint enables composers to create fugue subjects and countersubjects that can be rotated through any voice of the texture.2 Knowledge of canon is crucial for constructing strettos, the most contrapuntally rigorous sections of many Baroque organ fugues, in which a subject is stated in densely overlapping imitative entries. What may seem remarkable to modern minds is that fugue was not limited to worked-out, written-down composition: in the Baroque era church organists were expected to be able to improvise fugues, just as public speakers were expected to possess the skill of ex tempore oration. Thus, these contrapuntal techniques also belong to the theory and practice of musical rhetoric in general and Figurenlehre in particular. For recovering the lost art of contrapuntal improvisation—an aspect of keyboard performance practice as deserving of mastery by modern early-music performers as thoroughbass and ornamentation—Werckmeister’s Harmonologia musica furnishes an invaluable guide.3

2. The Friendship of Werckmeister and Buxtehude

2.1 The friendship between Werckmeister and Buxtehude is well documented. The Lübeck master contributed two congratulatory poems to Harmonologia musica, praising Werckmeister highly and calling him his “hochgeschätztem Freunde” (highly treasured friend). The longer of the two poems reads as follows:

Wer ein Kunst-Werck recht betrachtet, Es nicht unerkannt verachtet, Redet frey ohn’ arge List, Christlich, wie es billig ist; Kömmt es denn auch auf die Proben, Muss das Werck den Meister loben. Er mein Freund! hat wol erwogen, In dem Buch, und ausgezogen, So der Kunst erspriesslich sey, Treulich und ohn Heucheley, Er ist auch Werckmeister worden, Rühmlich in der Musen-Orden. Dieses wolte dem Herrn AUTHORI als seinem Hochgeschätztem Freunde glückwünschend zurufen, sein Ergebenster Dieterich Buxtehude Organ. an der Haupt-Kirchen zu St. Marien in Lübeck.4 [Whoever views a work of art properly, Does not disdain it anonymously, Speaks freely without arrant cunning, In a Christian manner, as is right, For when it comes to the test, The work must praise the master. He, my friend, has considered well In the book, and excerpted What is useful to art, Honestly and unfeignedly. He has also become workmaster, Praiseworthy in the order of the muses. (The undersigned) wanted to exclaim this to congratulate the author, as his highly treasured friend, his most devoted Dieterich Buxtehude Organist at the principal church of St. Mary in Lübeck.]5

2.2 We also know that Buxtehude wrote letters to Werckmeister, for Johann Gottfried Walther wrote to Heinrich Bokemeyer in 1729 that Werckmeister (with whom Walther studied) had once “honored me … with some letters and keyboard works by Buxtehude,” some of them in the Lübeck master’s own hand.6 Buxtehude and Werckmeister would certainly have had much to discuss in their correspondence, for as church organists they shared many interests, including not only learned counterpoint but also organ construction and tuning. From 1664 Werckmeister served as organist at churches in various Thuringian towns, including Hasselfelde, Elbingerode, and Quedlinburg; in 1696 he became organist of St. Martin’s church in Halberstadt, where he served until his death in 1706. Well known in his day as an organ examiner, his most enduring fame derives from the circulating temperaments he propounded in a series of treatises beginning with the Orgel-Probe of 1681. Thus, by the time Buxtehude wrote his congratulatory poems for Werckmeister in late 1701, he may well have known of Werckmeister—if not known him personally—for as many as twenty years.7

3. Werckmeister’s Harmonologia musica

3.1 Werckmeister’s many writings on music reveal a curious mixture of Medieval thought and the avant-garde. He regarded the study of music as a speculative, mathematical science, yet justified his innovations in musical temperament and counterpoint on the pragmatic and progressive grounds that a creative artist should not be fettered by arbitrary constraints.8 These contrasting strains of thought are united in the title of Harmonologia musica:

HARMONOLOGIA MUSICA Oder Kurtze Anleitung zur Musicalischen Composition. Wie man vermittels der Regeln und Anmerckungen bey den General-Bass einen Contrapunctum simplicem mit sonderbahrem Vortheil durch drey Sätze oder Griffe Componiren, und extempore spielen: auch dadurch im Clavier und Composition weiter zu schreiten und zu variiren Gelegenheit nehmen könne: Benebst einen Unterricht, wie man einen gedoppelten Contrapunct und mancherley Canones oder Fugas Ligatas, durch sonderbahre Griffe und Vortheile setzen und einrichten möge, aus denen Mathemathischen und Musicalischen Gründen aufgesetzet und zum drucke herausgegeben Durch Andream Werckmeistern, Benicosteinensem Cheruscum, p.t. Organisten in der Haupt-Pfarr-Kirche zu St. Martini in Halberstadt. [HARMONOLOGIA MUSICA or Brief Introduction to Musical Composition. How one can compose and improvise simplex counterpoint using the rules and symbols of thoroughbass with the special aid of three chords or hand positions, and through this have the opportunity to advance and to make variations at the keyboard and in composition; together with instruction on how one may make or arrange double counterpoint and all sorts of canons, or fugas ligatas, through special devices and advantageous techniques, all on the basis of mathematical and musical foundations; brought to press and published by Andreas Werckmeister of Benneckenstein in Thuringia, currently organist in the main parish church of St. Martin in Halberstadt.]

3.2 The three Sätze (chords) or Griffe (hand positions) Werckmeister advertises refer to three ways of voicing triads when each hand plays in thirds—an essential element of Werckmeister’s approach to invertible counterpoint. While resting on “mathematical and musical foundations,” Werckmeister’s contrapuntal strategies succeed—especially in improvisation—through their relation to elements of keyboard technique that even comparative novices will have mastered. Indeed, it is to such novices that Werckmeister addresses Harmonologia, as he reminds the reader from time to time: “Ich sage aber noch einmahl, dass ich vor exercirte, hochverständige, und virtuose Leute, auch vor die unverständigen Lästerer, nichts wil geschrieben haben, allein, nur vor die Incipienten, und Music-Liebende” (But I say yet again that I have not desired to write anything for experienced, erudite, and virtuosic people, nor for uncomprehending slanderers, but only for beginners and music-lovers.) 9 “Incipienten” here denotes not rank beginners (the book does not after all address such fundamentals as scales, intervals, and key signatures), but rather musical apprentices who possess basic knowledge and are ready to advance further. In the preface, Werckmeister indicates that he originally conceived this treatise as a clarification and extension of his Nothwendige Anmerkungen vom General-Bass (1698) for the purpose of introducing the fundamentals of composition.10 Thus in the main body of the treatise he addresses such topics as thoroughbass, the modes, transposition, and the requisite traits of the church organist—traits that include the ability to improvise fugues and chorale settings.

4. Columbus’s Egg: Stepwise Sequential Subjects

4.1 Werckmeister's most original ideas in Harmonologia musica concern the improvisation of invertible counterpoint and canon. He addresses this topic not in the main body of the treatise but the supplement:

In dieser Methode der Composition wird mir es gehen, wie dem Columbo, welcher das Ey mit sonderlichen Vorthel auff die Spitze gestellet, und stehend gemacht; Diese Kunst hatte vorher von den Anwesenden keiner gewust, nachdem aber, wie er den Vorthel entdecket, sagten Sie: O! Diese Kunst können wir auch? Also werden meine Verfolger und Missgönner auch sagen, wann Sie diese Griffe und Vortheile dieser Composition sehen werden, da sie doch vorher immer [recte nimmer] darauff gedacht haben; Ich kan wohl mit guten Gewissen sagen, dass ich im keinen Autore, diese Methode und bequeme Vortheile gesehen, auch niemahls in Discursen davon gehöret; Denn viel Musici sind heimlich und rahr mit ihren Wissenschafften.11

[In this method of composition I will fare like Columbus, who by special means placed an egg on its point and made it stand. Until then none among the knowledgeable had known this technique, but after he had revealed how he did it, they said: “Oh, we can do this too!” Thus too will speak my persecutors and detractors when they see the devices and means of this manner of composition, since they indeed had never thought of it before. I can affirm with good conscience that I have never seen this method and easy means in any author, nor have I heard of it in discourse; for many musicians are secretive and reticent with their knowledge.]12

4.2 Likening the fashioning of double counterpoint to the challenge of standing an egg on end, Werckmeister proposes a solution as simple and ingenious as the one found by the great explorer. Columbus reputedly stood the egg on end simply by making a slight indentation in its shell. For his part, Werckmeister relies not upon the sorts of complicated rules or numeric inversion tables usually provided in counterpoint treatises (although he does include them), but upon simple, sequential melodic patterns. These formulas can be combined easily in imitation at all sorts of pitch and time intervals, including simultaneous doubling in thirds, thus producing nearly automatic conformance with the stringent rules of invertible counterpoint. Although conceived in terms of an underlying note-against-note framework, Werckmeister’s sequential formulas can easily be expanded and embellished with diminutions and rhythmic adjustments. The basic patterns in their ascending forms are shown in Example 1.13

4.3 Strettos using sequential subjects can be found in compositions dating almost to the earliest rhythmically notated polyphony, but Werckmeister seems to be the only German Protestant theorist of the Baroque or Renaissance eras explicitly to describe and recommend sequential subjects in the context of contrapuntal imitation, and especially in improvisation. His own claim never before to have seen the technique in print may well be true, but he was not in fact the first theorist to describe such patterns. The Portuguese theorist Vincenzo Lusitano, for example, addresses sequential subjects in his Introduttione facilissima et novissima from 1553, listing more than fifty possible canonic combinations, mostly for two or three voices.14 Lusitano and several subsequent Iberian and Italian authors address canonic sequences primarily in relation to cantus firmus settings, whether worked out or improvised. Tomàs de Santa María, for example, extensively explores note-against-note sequential formulas in his Arte de tañer fantasia of 1565, emphasizing their voice-leading advantages in cantus firmus settings rather than their canonic and inversional possibilities.15 Orazio Tigrini briefly treats invertible canonic sequences composed or improvised against artificial cantus firmi; in his Compendio della musica of 1588, he acknowledges his debt to Lusitano.16 Pedro Cerone likewise commends their use against cantus firmi in Melopeo y maestro from 1613.17 In connection with artificial cantus firmi, Lodovico Zacconi addresses sequential canons in Prattica di musica, including how to embellish subjects with diminutions, but he does not address their invertibility as fully as Werckmeister, nor direct his comments to improvisation as specifically as Lusitano and Santa María.18 For all these theorists, however, sequential subjects provide not so much a stimulus to fugal invention as a useful set of voice-leading patterns to be applied to segments of a cantus firmus.

4.4 North of the Alps, the Vienna court organist Alessandro Poglietti includes numerous examples of elaborated canonic sequences in two and three voices in a 1676 manuscript treatise for organists.19 In a similar vein, Spiridion à Monte Carmelo demonstrates free sequential imitation in his Nova instructio pro pulsandis organis, spinettis, manuchordiis etc.20 Spiridion, a well-traveled Carmelite priest active mostly around Bamberg, advertises his treatise as an easy, teach-yourself method for improvising toccatas, versets, and other free liturgical keyboard genres; his intended market seems to have been monastic organists. Spiridion’s manual, which like Poglietti’s consists more of examples than text, draws inspiration from the improvisatory techniques of Frescobaldi, whose works provide many of its examples.21 The essence of Spiridion’s improvisation method consists of elaborating simple figured-bass patterns called “cadentiae.” Since many of these patterns are sequential, they lend themselves to free contrapuntal imitation in the manner described by Werckmeister. In a series of examples, Spiridion demonstrates how sequential bass patterns may be realized in thoroughbass as well as elaborated in both imitative and non-imitative styles.22 Spiridion also demonstrates how a stretto based on a sequential subject may be varied using melodic and textural inversion as well as retrograde motion.23 Spiridion’s approach anticipates Werckmeister’s in relying on simple voice-leading patterns rooted in thoroughbass technique rather than complex contrapuntal rules. Unlike Werckmeister, however, Spiridion provides no formal instruction on counterpoint, canon, or fugue; his teachings are less sophisticated and less thoroughly worked out than Werckmeister’s. It is possible, notwithstanding Werckmeister’s protestations of originality, that he may have known Spiridion’s treatise directly or indirectly.

4.5 That such old-style theorists as Zarlino, Calvisius, and Artusi do not endorse sequential subjects reflects not a lack of awareness of their properties, but rather the strict style’s eschewal of extended sequences, which contradict the Renaissance ideal of varietas.24 Not so in the Baroque, of course: for many composers the sequence furnished a veritable stock-in-trade. None other than Claudio Monteverdi ingeniously exploited sequential imitation in many works, including the “Dixit Dominus” from the 1610 Vespers, in which each falso bordone section gives way to sequential imitative flourishes of tremendous vitality. A 1655 portrait of the Cremonese organist Tarquinio Merula represents him, pen in hand, in the act of composing a six-voice canon featuring what appears to be a falling-third, rising-second sequential subject.25 Closer to Werckmeister’s ambit, awareness of the combinatorial properties of sequential subjects may be a factor in the story told to Johann Nikolaus Forkel by C.P.E. Bach about his father, J.S. Bach (who may well have known Werckmeister’s contrapuntal teachings, either directly or through Buxtehude, Walther, or any number of other channels):

Bey Anhörung einer starck besetzten u. vielstimmigen Fuge, wuste er bald, nach den ersten Eintritten der Thematum, vorherzusagen, was für contrapuncktische Künste möglich anzubringen wären u. was der Componist auch von Rechtswegen anbringen müste, u. bey solcher Gelegenheit, wenn ich bey ihm stand, u. er seine Vermuthungen gegen mich geäussert hatte, freute er sich u. stiess mich an, als seine Erwartungen eintrafen.26

[When he listened to a rich and many-voiced fugue, he could soon say, after the first entries of the subjects, what contrapuntal devices it would be possible to apply, and which of them the composer by rights ought to apply, and on such occasions, when I was standing next to him, and he had voiced his surmises to me, he would joyfully nudge me when his expectations were fulfilled.]27

4.6 Closely related to the use of stepwise sequential subjects is extensive doubling in thirds, sixths, and tenths. If this improvisational strategy is less original from a theoretical point of view (Zarlino viewed the practice with ambivalence, particularly when used overzealously in contrapunto à mente),28 it nonetheless often appears in North German organ music. It is also found in the canonic works of the Hamburg school of learned counterpoint, a cohort of musicians active in Hamburg and Lübeck in the early 1670s that in addition to Buxtehude included Christoph Bernhard, Johann Adam Reincken, Johann Theile, and probably Matthias Weckman.29 As in the canons of these composers, Werckmeister’s sequential subjects and the practice of doubling in thirds are closely intertwined, since in any stretto in three or more voices in simplex counterpoint that employs one of Werckmeister’s typical subjects, at least two voices will be moving in parallel thirds, sixths, or tenths—easily invertible intervals. (Parallel thirds and tenths are invertible at the octave and twelfth, while parallel sixths are invertible at the octave. The octave and the twelfth are the intervals of inversion most employed by fugue composers.)

4.7 In spite of invertible counterpoint’s traditionally arcane and mathematical nature, Werckmeister repeatedly reminds the reader that he is attempting to address it only in the most simple, practical terms, both because of his intended audience (relative beginners) and because precepts to be applied in improvisation must be simple. It is typical of Werckmeister’s simplified approach in this treatise that he begins his discussion of double counterpoint not with a list of rules, as do Zarlino and most of his Italian successors (although Werckmeister does eventually present such a list),30 nor with numeric inversion tables—as do Angelo Berardi, Johann Josef Fux, Johann Gottfried Walther, and many later writers—but with the sequential melodic patterns referred to earlier. Few composers with recourse to pen and paper would restrict themselves to these plain sequential patterns, but for beginning improvisers they provide sure-fire formulas, and for experienced musicians they can furnish the underlying shape of more elaborate subjects. Far from presenting sequential subjects as mere templates, Werckmeister considers them a stimulus to invention, full of endless possibilities for variety. In characteristic fashion he writes that “die Inventiones gehen nach dem Menschlichen Verstande in infinitum, dieselben kan man keinen einflüssen, und sind vielmehr Gottes Gaben, die er einen jeden mittheilet, nach dem Maasse wie er will” (The possible inventions surpass human comprehension into infinity; they cannot be contrived, and are rather the gifts of God, who apportions them to each in the measure that He wills).31 And elsewhere he marvels that

In Summa, es ist das Werck so reich, dass man sich hoch darüber verwundern muss, und müste ein Klotz oder Atheiste (ach die armen verstockten Hertzen) seyn, der hierüber den Schöpfer zu preisen, nicht solte bewogen werden. Und dieses ist doch nur ein schlechter Contrapunct, wie? wann nun die mannigfaltigen Variationes und Figuren darzu kommen.32

[In sum, the technique is so rich that one must wonder greatly at it and be a blockhead or an atheist (oh, the poor impenitent hearts!) not to be moved to praise the Creator because of it. And these examples are but plain counterpoint—what then of the manifold variations and figures that can be added to this?]

4.8 Werckmeister works out numerous possible contrapuntal combinations for melodic motion per secundas and per tertias before leaving the reader to realize the remaining types of melodic motion. These combinations begin very simply. Melodic motion ascending per secundas Werckmeister combines in interlocking imitation as shown in Example 2; he combines the same subject differently in three voices with syncopations and a 5–6 voice-leading pattern in Example 3. Melodic motion ascending per tertias Werckmeister works out in stretto with answers at the fifth as shown in Example 4; the bass and soprano exhibit an underlying pattern of parallel tenths, while the tenor and alto move in parallel thirds with one another and in contrary motion with the outer voices. In these as in most of his examples, Werckmeister presents his sequential subjects in the tightest of stretto, often per arsin et thesin with respect to rhythm.33 This conforms to his precept that invertible counterpoint should be conceived initially in a simplex framework. Werckmeister’s note-against-note canons also mirror one of the main “practical” applications of canon, namely the strettos of organ fugues.34 The close relation between canon and fugue is implicit in Werckmeister’s terminology: he calls canon fuga ligata (strict fugue), while fugue he terms fuga soluta (free fugue).35

4.9 Having demonstrated the combinatorial properties of sequential subjects, Werckmeister next demonstrates how to embellish them with diminutions.36 He transforms a voice-leading paradigm based on a sequential subject doubled in thirds (Example 5a) into a simplex canon or stretto with entries on C and G (Example 5b; soprano and alto are inverted at the octave). He then enlivens this underlying framework with diminutions (Example 5c).37

4.10 Werckmeister’s presentation of plain subjects in stretto, followed by a demonstration of how they may be elaborated with diminutions or syncopations, provides a procedural model for the composer or improviser who intends to present a subject both in an “open” point of imitation and in stretto. To ensure the subject’s suitability for both contrapuntal contexts, a composer with pen in hand must first work out the unadorned subject in stretto or fuga ligata before composing the first point of imitation or fuga soluta. In the case of extemporaneous composition, the improviser should either choose a subject whose combinatorial properties are already familiar, or, if handed a subject (as would be the case in a seventeenth- or eighteenth-century organ audition), quickly identify the ways it can be combined with itself (and possibly a countersubject) before beginning to play.38 If employing a sequential subject of the sort Werckmeister proposes, the improviser will likely wish to embellish it with diminutions or other rhythmic alterations, or even modify the sequential profile (while honoring in the breach the underlying voice-leading paradigm) in order to avoid too predictable a contour.

4.11 Werckmeister’s contrapuntal teachings accord well with notions of musical rhetoric that by his day were already firmly established in theory and practice. The three rhetorical tasks of inventio, dispositio, and elaboratio (also termed elocutio or decoratio) are all aided by his approach. As he himself makes clear, stepwise sequential patterns provide a stimulus to the invention (inventio) of subjects rich in combinatorial properties. These combinatorial possibilities in turn suggest the arrangement of the work’s parts, or dispositio. Contrapuntal elaboratio—including textural and melodic inversion, combination of subjects at various pitch and time intervals, fragmentation, and stretto—lies at the heart of the Figurenlehre propounded by the likes of Joachim Burmeister, Athanasius Kircher, Christoph Bernhard, and Johann Mattheson. Putting this into practice, the composer of a North German organ praeambulum might begin the exordium by foreshadowing the subject in a free, toccata-like style. In the main body of the work the composer would then present the subject in an open point of imitation before exploring various contrapuntal elaborations, culminating in a climactic stretto before the final cadential section.39

5. Buxtehude’s Use of Sequential Subjects

5.1 Buxtehude seems to follow the rhetorical procedure just described in his Praeambulum in A Minor, BuxWV 158.40 This fairly typical praeambulum exemplifies the technique of generating an entire improvisatory composition from a single melodic idea—in this case a sequential subject of the sort addressed by Werckmeister. The head of the fugue subject, introduced at m. 17, displays a clearly sequential profile of falling thirds and rising seconds (Example 6a). Buxtehude has already adumbrated this melodic idea in highly ornamented fashion in the piece’s toccata-like exordium (Example 6b). As soon as the answer enters in m. 18, he abandons the subject’s melodic sequence in order to avoid parallel thirds: though suitable for thicker textures later on, their use here would undesirably limit the independence of the answer. After completing the brief fugal exposition, at m. 32 he presents the subject in stretto per arsin et thesin (generating alternate thirds and sixths in both the soprano-alto and tenor-bass pairs), with a slight rhythmic alteration to ensure that the stretto will be heard as such (Example 6c). The countersubject that appears in the left hand at the pickup to m. 38 (against the subject in the alto, introduced one beat earlier) is derived from the subject through a combination of diminutions and rhythmic displacement. (The first note of the sequence has now become an eighth-note anacrusis, a rhythm that seems to develop from the stretto at m. 32.) Since the underlying contours of subject and countersubject are the same, their combination results in a pattern of alternating thirds and sixths similar to that of the stretto at m. 32. Non-parallel thirds and sixths, as Werckmeister duly observes, are invertible at both the octave (as in the second half of m. 39) and tenth (as in the first half of m. 42).

5.2 At the change to 6/4 at m. 60 (Example 6d), Buxtehude once again derives a new fugue subject from the principal subject first presented in m. 17. But since in their new metrical guise the notes of the underlying sequence no longer receive equal durations, stretto is no longer feasible. Buxtehude instead engages in an impressive display of triple counterpoint, featuring inversions at both the octave and twelfth. The counterpoint is triple because, even though Buxtehude really employs only two subjects in this section, the chromatically inflected “tail” of the subject (C–C-sharp–D) overlaps with almost every subsequent entry of the two subjects, and indeed sometimes appears separately. Not surprisingly, by now, the “second subject” of this section (first encountered in the tenor at m. 63) may also be considered to be derived from the original sequential subject, insofar as it mostly parallels the principal subject (now elaborated in the bass voice) in thirds. Were it to adhere fully to the sequential paradigm, the tenor melody at m. 63 would need to start on a c′; this is not possible, however, because of the c′′-sharp in the soprano, so Buxtehude instead begins the tenor on an e′, a good choice in any case, since it gives the “second subject” a more distinctive profile.

5.3 BuxWV 158 illustrates Werckmeister’s improvisatory principles especially clearly, but sequential subjects appear in many other Buxtehude organ works as well, including the following:

Praeludium (Prelude, Fugue and Chaconne) in C Major, BuxWV 137, mm. 12–18 (Example 7): Buxtehude presents a rising-fourth, falling-third sequential subject in a three-voice texture, with the two uppermost parts proceeding in thirds beginning in m. 14. Praeludium in G Minor, BuxWV 150 (Example 8): having presented the subject in various forms and contrapuntal combinations throughout the work, Buxtehude in the penultimate system leads the subject through an extended, sequential stretto. Both the sequential subject and extensive doubling in thirds that characterize this piece are improvisational strategies addressed by Werckmeister. Chorale fantasia “Ich dank dir, lieber Herre,” BuxWV 194 (Example 9): after introducing the first phrase of the chorale in homorhythmic style in mm. 1–3, Buxtehude launches into a rapidly syncopated eight-measure sequential stretto based on a falling-third, rising-second figure with passing tones. When a third voice joins the first two half-way through, the outer voices move in parallel tenths. The same stretto appears again at m. 24, but expanded to thirteen measures and incorporating all four voices. When the bass enters at m. 32, the outer voices move in contrary motion. Many other instances of quasi-canonic writing permeate the remainder of this work. Magnificat primi toni, BuxWV 203: Buxtehude employs a gigue-like sequential subject in the fifth verse (Example 10). Contrary to first appearances, after the alto entrance in the second measure the sequence continues in the tenor (left hand), but with octave displacements and altered rhythm.

6. Parallel Thirds as a Key to Improvising Invertible Counterpoint

6.1 Because each unit of Werckmeister’s sequential subjects typically contains only two pitches, any texture of more than two voices inevitably requires parallel imperfect consonances, whether thirds, sixths, or tenths. According to Werckmeister, the easiest way to implement these subjects in dense imitative textures, and ensure subsequent invertibility, is to employ parallel thirds in each hand: “Also dass unten lauter Tertien, und oben lauter Tertien bleiben, das ist also der Schlüssel zu allerhand Arten von den Canonibus und gedoppelten 3. und 4. fachen Contrapuncto” (Parallel thirds below, and parallel thirds above, are the very key to all types of canons and double, triple and quadruple counterpoint).41 Manipulation of these two sets of parallel thirds is aided by knowledge of their possible vertical combinations: the three special Sätze (chords) or Griffe (hand positions or chord voicings) advertised in the title of Harmonologia. If each hand plays one of a triad’s two constituent thirds, then only three root-position or first-inversion voicings are possible for any given triad: (1) both hands can play the root and third, omitting the fifth; (2) the left hand can play the root and third while the right hand plays the third and fifth; and (3) the left hand can play the third and fifth while the right hand plays the root and third. (Playing the original third and fifth in both hands would of course amount to playing the root and third of a new chord.) Werckmeister illustrates the three Sätze as shown in Example 11. In the first three chords, the left hand remains stationary and the right hand moves; in the second set of three, the right hand remains stationary and the left hand moves.42 If both hands move, they must do so in contrary motion in order to avoid parallel perfect intervals. In Example 12, Werckmeister demonstrates how the subject C–D–E–F, presented in the lowest voice, may be harmonized in simplex counterpoint using two pairs of parallel thirds. The first three chords each represent one of the three Sätze or Griffe, moving stepwise in contrary motion in a simple voice exchange; by the fourth chord the right hand has run out of available pitches in its stepwise descent and must leap downward to obtain a new relation to the cantus firmus.

6.2 Noting that Werckmeister “reduces the arts of writing both double counterpoint and canons to manipulations with two sets of parallel thirds,” Kerala Snyder observes that Harmonologia musica represents a culmination of this compositional strategy.43 Pervasive doubling in thirds did indeed have an extensive history before Werckmeister: Zarlino, for instance, had discussed it at length with respect to both contrapunto à mente and worked-out invertible counterpoint in Le istitutioni harmoniche (1558). Zarlino’s teachings were transmitted across the Alps by various students, including Seth Calvisius and Jan Peterszon Sweelinck. Sweelinck was in turn the teacher of Jakob Praetorius and Heinrich Scheidemann, who taught Matthias Weckmann and Johann Adam Reincken, respectively. Weckmann and Reincken, together with Christoph Bernhard, Johann Theile, and Dieterich Buxtehude, constituted a school of learned counterpoint centered in Hamburg and Lübeck during the early 1670s.44 Each of these men except for Buxtehude penned manuscript treatises addressing the composition of multiple counterpoint, including the role of parallel thirds.45 For his part, Buxtehude inscribed a four-voice duplex augmentation canon in the signature album of his friend Johann Valentin Meder (Example 13).46 Like many of the canons by his Hamburg colleagues, Buxtehude’s canon for Meder employs continuous doubling in thirds, a means of ensuring both invertibility and a full sound. As Alfred Mann and others have noted, this sort of canon sine pausis (canon without pauses—duplication of the original melodic line in thirds, tenths or sixths by simultaneous commencement of the voices) suggests the strengthening of vertical harmonic thinking that characterized the contrapuntal technique of the high Renaissance and its theory of double counterpoint.”47

6.3 Werckmeister does not restrict continuous parallel thirds to sequential subjects; he also applies the technique in non-sequential canons similar to those by Buxtehude and members of the Hamburg school, as well as in genres such as chorale preludes. Non-sequential melodies are more difficult to treat imitatively in improvisation:

Wann aber die Melodeyen in ungleichen Intervallen einhergehen, weiss ich nicht ob sie ex tempore können in allerhand Fugis ligatis und Imitationibus hingesungen werden: denn die Regeln so Zarlinus und seine Nachfolger davon gegeben haben, sind gar weitläufftig und obscur, dass man sich in einem Moment nicht leichte besinnen wird. Unsere aber bestehen aus den Sätzen, wann unten und oben allezeit eine Tertia gesetzet wird, so kan es nicht fehlen.48 [But when melodies proceed by dissimilar intervals, I do not know whether they can be sung extemporaneously in all sorts of canons and imitations: for the rules which Zarlino and his followers have given regarding this are so complicated and obscure that one cannot easily in a moment recall them. Our rules, however, derive from the chords, when below and above a third is always placed, and thus cannot fail.]

6.4 In Example 14, one of 12 examples elaborating “Wenn wir in höchsten Nöthen seyn,” Werckmeister demonstrates how a cantus firmus and its counterpoint, each doubled in thirds, may be combined in invertible counterpoint.49 The point of such displays is not the Haupt-Satz itself (as Werckmeister terms the original contrapunctus), but rather the manifold resolutiones in two, three, and four voices that can be derived from it through inversion. Working out the Haupt-Satz in paired thirds ensures subsequent invertibility in two-voice renderings at the octave, tenth, or twelfth, depending on which of the original four voices are included in a given resolutio (as Werckmeister calls the replica, revolutio, or evolutio).

6.5 For all its simple ingenuity, this contrapuntal strategy is not without its limitations. In order to avoid parallel perfect intervals, the two pairs of thirds must always move in contrary motion, but this cannot always be done gracefully. To gain a new axis of symmetry, the composer must sometimes resort to leaps of an octave or sixth, as Werckmeister himself advises. Such leaps are occasionally evident in Werckmeister’s cantus-firmus setting in Example 14, in Buxtehude’s augmentation canon in Example 13, and in similar works by Theile and Weckmann. The resultant awkward lines and disrupted voice-leading are not fully consistent with the strict style.

6.6 Parallel thirds can be useful in improvisation quite apart from invertible counterpoint, of course. In the main body of the treatise, Werckmeister discusses chorale settings at length, illustrating his observations with twenty-eight settings of the first phrase of “Vater unser in Himmelreich,” ranging in texture from simple homorhythmic continuo realizations to a permutation fugue. Addressing right-hand embellishment of the cantus firmus, Werckmeister recommends that

Wer nun pedaliter spielen wil, der nehme die zwo mittel Parteyen in die lincke Hand, den Discantum allein in die rechte, so kan man, wenn der Bass schlecht fort gehet, die Mordanten, Accente und andere Figuren, im Canto, als in der Formal-Stimme, desto besser anbringen; Es können dann auch die Passagien in der rechten Hand auff allerley Arth gemachet werden.50 [Now whoever wants to play with pedals should take the two middle voices in the left hand, and the discantus alone in the right hand; thus one can, if the bass proceeds in a simple manner, better introduce mordents, trills, and other figures in the melody, as the leading voice; and then too can all sorts of diminutions be made in the right hand.]

6.7 Because parallel thirds require little thought to execute, playing them with the left hand frees the mind to concentrate on more involved embellishments in the right hand. Inspection of Buxtehude’s chorale preludes reveals that in about half of them (primarily the less imitative ones), parallel thirds seem to be the normal or “default” voice-leading pattern in the left hand when other voice-leading considerations do not overrule. In the remainder, parallel thirds in the left hand occur only incidentally, not as a stylistic feature in their own right. Among Buxtehude’s chorale preludes with extended passages of left-hand parallel thirds are “Ein feste Burg,” BuxWV 184; “Es ist das Heil uns kommen her,” BuxWV 186; “Es spricht der unweisen Mund wohl,” BuxWV 187; both settings of “Nun bitten wir den heiligen Geist,” BuxWV 208 and 209; and both settings of “Komm heiliger Geist, Herre Gott,” BuxWV 199 and 200. Snyder writes that “we can assume that [Buxtehude’s chorale preludes] are written-down versions of the introductions that he routinely improvised before the congregational hymns; every one of them is an elaboration of a hymn included in the Lübeckisches Gesangbuch.”51 Werckmeister’s teachings thus illuminate the improvisational basis of Buxtehude’s chorale preludes.

7. Conclusions

7.1 Our appreciation of Werckmeister’s place in history should not be limited to his design of circulating temperament systems, for his teachings in Harmonologia musica yield important insights into the contrapuntal and improvisational strategies of his era. Moreover, Werckmeister’s ideas merit inclusion in the modern recovery of Baroque performance practices, for as his friend Buxtehude noted, he “has considered well … and excerpted what is useful to art” (see par. 2.1). Werckmeister’s ideas on contrapuntal improvisation surpass mere speculation: they belong to the realm of real musical creation, as Buxtehude’s own use of these techniques confirms. Werckmeister’s most novel contribution to north German contrapuntal theory, the use of stepwise sequential subjects as a way of ensuring combinatoriality in fugal improvisation, is not, strictly speaking, “his”; he did not so much invent it as discover, like Columbus did the New World, something that had been present, indeed inhabited, for a long time. Even so, Werckmeister is due considerable credit for the ingenuity to work out the integration of horizontal and vertical contrapuntal elements in so sophisticated a manner. In the words of none other than Buxtehude himself in a verse honoring Werckmeister,

So sagen ins gesamt von Ihm die klugen Geister Mit höchsten Ruhm und Preis, lobt dieses Werck den Meister.52 [So say of him the brightest minds together: “With highest fame and praise this work does laud its master.”]


* Michael R. Dodds (doddsm@ncarts.edu) is Head of Music History at the North Carolina School of the Arts. Thanks to recent Fulbright and NEH fellowships, he is currently completing a book addressing relationships among modal theory, liturgical performance practices, and keyboard music in the Baroque era. An earlier version of this article was presented in paper form at the 1998 GOArt Organ Academy, Göteborg, Sweden.

1 Andreas Werckmeister, Harmonologia musica (Frankfurt and Leipzig: Calvisius, 1702; reprint, with Hypomnemata musica and other writings, Hildesheim: Georg Olms Verlag, 1970; reprint, with Musicalische Paradoxal-Discourse, Laaber: Laaber-Verlag, 2003).

2 Invertible counterpoint involves the writing of two or more contrapuntal lines so that any given voice may serve as lowest in the texture without compromising the rules of correct part-writing. Among its most vexing challenges are the proper treatment of dissonances and parallel intervals. For example, in writing invertible counterpoint at the octave one must avoid fifths, because in inversion they will become fourths, which are considered dissonant in the strict style. In inversion at the tenth, normally acceptable parallel imperfect consonances generate unacceptable parallel perfect intervals in the inversion. The most common types of invertible counterpoint are at the octave, twelfth, and tenth. Werckmeister usually refers to invertible counterpoint as “Gedoppelten Contrapunct or drey- oder vier-facher Contrapunct,” depending on the number of voices involved in the inversion; the original contrapunctus he calls the “Haupt-Satz,” while he refers to the inversion as the “resolutio”or “replica,” depending on the context. (The approximate synonyms “revolutio”and “evolutio”are also found in contemporaneous sources.)

3 For a notable example of historically informed recovery of improvisational practice, see William Porter, “Reconstructing 17th–Century North German Improvisational Practice: Notes on the Praeambulum with a Report on Pedagogy Used in December 1995,” GOArt Research Reports, vol. 2, ed. Sverker Jullander (Güteborg: Güteborg Organ Art Center, 2000), 25–40.

4 Harmonologia musica, unnumbered prefatory page.

5 English translation from Kerala Snyder, Dieterich Buxtehude: Organist in Lübeck (New York: Schirmer Books, 1987),127.

6 “Werckmeister (welchem zu Gefallen an. 1704 nach Halberstadt gereiset, von welchem auch nach der Zeit mit etlichen Briefen und Buxtehudischen Clavier-Stücken beehret worden bin) und nachgehends guter Componisten Arbeit in Partitura, nechst Gott, zu dancken habe.” Letter of 3 October 1729 from Johann Gottfried Walther to Heinrich Bokemeyer; German text from Johann Gottfried Walther, Briefe, ed. Klaus Beckmann and Hans-Joachim Schulze (Leipzig: VEB Deutscher Verlag für Musik, 1987), 70; English translation from Snyder, Dieterich Buxtehude,128. For more on the relationship between Werckmeister and Buxtehude, see Snyder, Dieterich Buxtehude,85, 126–8, 135, and 321.

7 Snyder, Dieterich Buxtehude, 84–5.

8 See George J. Buelow, Grove Music Online, s.v. “Andreas Werckmeister” (accessed 15 June 2006); and Rolf Dammann, “Zur Musiklehre des Andreas Werckmeister,” Archiv für Musikwissenschaft 11 (1954): 206–37.

9 Werckmeister, Harmonologia musica, §184. Unless otherwise indicated, all translations are my own.

10 Werckmeister, Harmonologia musica, “Vorrede.”

11 Werckmeister, Harmonologia musica, §95.

12 The reaction of Columbus’s naysayers, “O! Diese Kunst können wir auch?” seems better translated as an exclamation than a question. Such a rendering better conveys the critics’ apparent belittlement of a solution difficult to invent but obvious once proposed.

13 Examples 1–4 appear in tablature in Harmonologia musica, §168–72.

14 Vincenzo Lusitano, Introduttione facilissima et novissima di canto fermo, figurato, contraponto semplice, et in concerto (Rome, 1553; Venice, 1558, 1561; Lisbon, 1603; reprint of 1558 edition, Bologna: Libreria musicale italiana editrice, 1989). In the 1558 edition, Lusitano addresses sequential counterpoints against a cantus firmus on fols. 12r–15r and sequential canons on fols. 17r–20v. For a discussion of sixteenth-century approaches to contraponto fugato and double counterpoint, including Lusitano’s, see Peter Schubert, “Counterpoint Pedagogy in the Renaissance,” in Cambridge History of Western Music Theory, ed. Thomas Christensen (Cambridge: Cambridge University Press, 2002), 503–33, especially 510–6.

15 Tomás de Santa María, Libro llamado Arte de tañer fantasia (Valladolid, 1565; reprint, Geneva: Minkoff, 1973), fols. 48r–51v.

16 Orazio Tigrini, Il compendio della musica nel quale brevemente si tratta dell’arte del contrapunto (Venice, 1588; reprint, New York: Broude Brothers, 1966), 116–23.

17 Pedro Cerone, El melopeo y maestro (Naples, 1613; reprint, Bologna: Forni, 1969), 1:592.

18 Lodovico Zacconi, Prattica di musica (Venice, 1622; reprint, Bologna: Forni, 1983), 2:180–96.

19 Alessandro Poglietti, “Compendium oder Kurtzer Begriff, und Einführung zur Musica, Sonderlich einem Organisten dienlich,” A-KR L. 146. The manuscript is dated 1676 on its title page. See especially the last, unlabeled example on p. 32; the passegiatti syncopati and scherzi on p. 33; and the risposti on p. 34. For a more detailed discussion of Poglietti’s treatise, see Helmut Federhofer, “Zur handschriftlichen  berlieferung der Musiktheorie in Ésterreich in der zweiten Hälfte des 17. Jahrhunderts,” Die Musikforschung 11 (1958): 271; and Dodds, “The Baroque Church Tones in Theory and Practice”(Ph.D. diss., University of Rochester, 1999), 178–80.

20 Bamberg and Würzburg, 1670–7, 4 vols. It is discussed in Bruce A. Lamott, “Keyboard Improvisation According to Nova instructio pro pulsandis organis (1670–ca. 1675) by Spiridion à Monte Carmelo” (Ph.D. diss., Stanford University, 1980). The only complete exemplar of Nova instructio is a manuscript copy preserved in D-Dl; the surviving printed volumes of parts 1 and 2 were used by seventeenth-century Franciscan organists in Prague and Vienna. No printed volumes survive for parts 3 and 4, the latter of which addresses imitative improvisation. From the evidence of these sources and manuscript concordances for the musical examples, Lamott concludes that the Nova instructio “circulated among German Catholic organists as far from Bamberg as Vienna and Prague, and was in use at least as late as 1717” (30–3).

21 Spiridion himself “played the organ in Rome” in 1643, the year of Frescobaldi’s death, though no evidence exists that he studied with the great organist of Saint Peter’s basilica (Lamott, 19).

22 Lamott, 173–82.

23 Lamott, 120–1.

24 See, for example, Gioseffo Zarlino, Istitutioni Harmoniche (Venice, 1558; reprint, New York: Broude, 1965), part 3, chapter 55, p. 227: “‘Quando lecito di usare in una parte della Cantilena due, o più volte un passaggio, & quando non.’ Si come la varietà delle cose apporta piacere & dilettatione; così la cosa istessa troppo usata, alle volte genera noia & fastidio.” (“When a Passage in One Voice of a Composition May Be Repeated and When Not.” As variety brings pleasure and delight, so excessive repetition generates boredom and annoyance); trans. Guy A. Marco and Claude V. Palisca, Gioseffo Zarlino, The Art of Counterpoint: Part Three of Le Istitutioni Harmoniche, 1558 (New Haven: Yale University Press, 1968), 153. Solmization primers from the sixteenth through eighteenth centuries, on the other hand, are replete with such sequential melodies because of their usefulness in teaching sight-singing.

25 For a facsimile of the portrait, which is in private possession in Cremona, see Maurizio Padoan, “Tarquinio Merula nelle fonti documentarie,” in Contributi e studi di liturgia e musica nella regione padana, ed. Giampaolo Ropa and Vittorio Gibelli (Bologna: Antiquæ Musicæ Italicæ Studiosi, 1971), 281–2.

26 Carl Philipp Emanuel Bach, “Biographische Mitteilungen über Johann Sebastian Bach,” as transcribed in Dokumente zum Nachwirken Johann Sebastian Bachs, 1750–1800, Bach-Dokumente 3, ed. Hans-Joachim Schulze (Kassel: Bärenreiter, 1972), 285. The letter, which is undated but most likely dates from late 1774, is presented in both facsimile and transcription in Bach-Urkunden: Ursprung der musikalisch-Bachischen Familie, Nachrichten über Johann Sebastian Bach von Carl Philipp Emanuel Bach, Veröffentlichungen der Neuen Bachgesellschaft 17, no. 3, ed. Max Schneider (Leipzig: Breitkopf und Härtel, [1916?]).

27 Translation from Hans T. David and Arthur Mendel, The Bach Reader, rev. ed. (New York: Norton, 1966), 277. Forkel incorporated the Bach son’s account in his seminal biography, Johann Sebastian Bach: Leben, Kunst und Kunstwerke (Leipzig: Hoffmeister & Kühnel, 1802), 46–7.

28 After heaping derision upon singers who presumptuously double the bass in thirds or tenths without regard for other voice-leading considerations, Zarlino concedes that “É ben vero, che’l porre alle volte la parte, che si aggiunge distante per una Decima, overo per una Terza dall’una delle due, torna molto commodo: Ma bisogna avertire, che quando le proposte fussero per una Terza lontane l’una dall’altra, & quella che si aggiunge cantasse per una Decima; tra la aggiunta & l’una delle due; che sarebbe la grave, quando la parte si aggiungesse nell’acuto; overo sarebbe l’acuta, quando l’aggiunta fusse più grave; sempre si udirebbe la Ottava; & così dico, quando fussero distanti per una Decima, & la aggiunta cantasse per una Terza; Onde se le due proposte havessero molte Terze, o Decime l’una dopo l’altra; come si fogliono porre alle volte, si udirebbono con l’aggiunto tante Ottave senza mezo alcuno: quante erano le Terze, o le Decime contenute tra le parti; & per tal maniera si verrebbe à fare errore. Però cosa buona, anzi necessaria il vedere il Contrapunto delle due proposte, per potere schivare gli errori, che potessero occorrere: percioche quando si facesse altramente, sarebbe impossibile di far cosa buona; se almeno non si havesse alla memoria ciascuna delle parti.” (It is true that to add the voice a third or tenth from one of the given parts is often convenient. But one must watch that the two given parts are not already a third apart when another is added at the tenth, because if that is the case, consecutive octaves will occur with one of the given parts wherever thirds or tenths occurred in the original. The same will happen if the original voices are a tenth apart and the added part is sung at the third. To avoid such errors, it is essential to see the two given parts, unless they have been memorized in advance. Otherwise nothing good will be produced.) (Zarlino, Istitutioni harmoniche, part 3, chapter 64, p. 260; trans. Marco and Palisca, 222.)

29 For further discussion of the Hamburg school, see Kerala Snyder, “Dieterich Buxtehude’s Studies in Learned Counterpoint,” Journal of the American Musicological Society 33 (1980): 544–64; and Dieterich Buxtehude, 107–20, 212–6; see also Christoph Wolff, “Das Hamburger Buxtehude-Bild,” Musik und Kirche 53 (1983): 8–19.

30 “Die General-Regeln aber des Zarlini, Pontii, Tigrini und Artusi sind folgende: (1.) Man muss im Principal keine Sextam gebrauchen, denn diese giebt in der Replica keine Consonanz. (2.) Müssen die Stimmen nicht über eine Decimam von einander stehen. (3.) Soll die oberste Stimme nicht unter die unterste, noch die unterste über die oberste gehen. (4.) Es soll auch die Septima Syncopata verhütet werden. (5.) Soll man auch keine Decimam minorem vor der Octava oder Duodecima her, oder die Tertiam minorem vor dem Unisono oder Quinta hersetzen, wenn die Stimmen motu contrario procediren; Denn auff solche Weise hat man in replica, tritonos und harte relationes. (6.) Bringet eine jede Duodecima im principal, in der replica einen Unisonum und eine jede Quinte eine Octavam. (7.) Man soll in Principal nicht zwo Tertien, noch zwo Sexten nach einander setzen, denn aus denselben werden in replica, lauter Unisoni, und aus den Sexten, Quinten. (8.) Soll man keine Syncopen, darinnen eitel Consonantien sind, gebrauchen. (9.) Sollen die Stimmen so viel möglich gradatim procediren, denn die Saltus per Quintas und Quartas verursachen offte relationes non-harmonicas [recte non-harmonici] intolerabiles. In Summa es werden fast alle Intervalla verbothen, die doch in den Special-Regeln wieder statt haben; Die Special-Regeln aber benebst den Exempeln aus besagten Italiänischen Autoribus hieher zu setzen, wolte eine grosse Weitläufftigkeit verursachen, weil ich derselben Regeln in die [§]222. befinde. Denn alle desjenige, was dort durch so viel dunckele Regeln abgehandelt wird, das haben wir in unsern dreyen Sätzen oder Griffen, schon besser, klährer und vollkommener. Doch möchten auch die General-Regeln einem oder dem andern noch dienlich und angenehm seyn, weswegen ich dieselben mit beyfügen wollen.” (But the general rules of Zarlino, Pontio, Tigrini, and Artusi are as follows: [1] One must use no sixths in the principal counterpoint, for these do not produce consonance in the replica. [2] The voices must not be more than a tenth away from one another. [3] The highest voice must not go beneath the lowest, nor the lowest above the highest. [4] The syncopated seventh should also be avoided. [5] One should also use no minor tenth before the octave or twelfth, or minor third before the unison, if the voices are proceeding in contrary motion; for that can produce tritones and hard relations in the replica. [6] Each twelfth in the principal counterpoint produces a unison in the replica [at the twelfth], and each fifth in the principal produces an octave in the replica [at the twelfth]. [7] In the principal counterpoint one should place neither two thirds after one another, nor two sixths, for in the replica [at the tenth] these will produce parallel unisons and fifths, respectively. [8] One should use no syncopes containing unnecessary consonances. [9] The voices should proceed by step as much as possible, for leaps by fifth and fourth often produce intolerable cross-relations. In sum, almost all intervals come to be forbidden, except for those found in the special rules [i.e., for individual intervals of inversion]; but were the special rules placed here together with the examples from the above-mentioned Italian authors, that would occasion great complexity, while I relegate the same rules to [the musical example in §]222. For all that is dealt with there [in the Italian authors] through so many obscure rules, we have formulated better, more clearly, and more completely in our three chords or hand-positions. Yet some will also find the general rules still useful and agreeable, on account of which I want to include them) (Werckmeister, Harmonologia musica, §187). I thank Jürgen Thym and Frederick Gable for advice on the translation of certain phrases of this passage; any infelicities of translation are my own.

31 Werckmeister, Harmonologia musica, §214.

32 Werckmeister, Harmonologia musica, §219.

33 Per arsin et thesin can refer either to an answer in contrary motion, or more frequently, to stretto in which the subject enters on successive (i.e., “up” and “down”) beats. The latter sense is intended here.

34 Werckmeister’s fascination with canons intersects another of his “practical” interests, circulating temperaments, in the context of modulating or spiral canons. In Harmonologia musica, §167, he includes a perpetual canon on “Vater unser in Himmelreich”that modulates alternately upward by fifth and downward by fourth, so that after twelve iterations it will have circumnavigated the circle of fifths—just the sort of feat that Werckmeister’s circulating temperaments allowed. Werckmeister does not indicate the canon’s interval of imitation, but a comes presented at the octave above proves to be invertible at the twelfth. In either inversion, doubling of either or both parts in thirds or tenths is possible.

35 For a discussion of Werckmeister’s teachings on fuguein other treatises, including Musicae mathematicae hodegus curiosus (Frankfurt and Leipzig, 1686) and Cribrum musicum, oder musicalisches Sieb (Quedlinburg and Leipzig, 1700), see Paul Mark Walker, Theories of Fugue from the Age of Josquin to the Age of Bach (Rochester: University of Rochester Press, 2000), 234–43.

36 See also Werckmeister, Harmonologia musica, §95–8, 183, 197, and 209, as well as 191.

37 Werckmeister, Harmonologia musica, §210.

38 “Darum wann man einen rechtschaffenen Organisten probiren wil, so muss man denselben nicht lassen spielen was er wil, man gebe erstlich einen, so sich vor einen perfecten Organisten ausgiebet, zur Probe für, ein Thema zu einer Fuga, dass dasselbe auf unterschiedliche Arth tractiret werde” (Werckmeister, Harmonologia musica, §129). This is but one of many such statements in primary sources indicating that skill in fugal improvisation was expected of competent Baroque organists.

39 For a similar approach in recent improvisational pedagogy, see Porter, “Reconstructing.”

40 BuxWV 158 may be securely dated before 1688, at least fourteen years before the publication of Harmonologia musica, although Buxtehude may in fact even have composed it in Helsingor prior to his appointment to the Marienkirche in Lübeck in 1668 (Snyder, Dieterich Buxtehude, 324–6). Snyder (p. 352) sees the paired canonic entries and real answers of the fugue’s exposition as further evidence for the earliness of this work.

41 Werckmeister, Harmonologia musica, §176.

42 Wann man nun aus einer gantzen Modulation oder Melodie, da unterschiedliche Intervalla darinnen vorfallen, oder da der Comes nach etlichen Intervallis dem Duci folgen soll, so hat man wider drey Griffe, wo das gantze Wesen innen beruhet, als wann ich wieder den Clavem c. pro radice, oder Bass gebrauche, so nehme ich die Tertiam e. in der lincken Hand darzu; In der rechten kan ich g′ und e′ nehmen, ist ein Satz, oder zu diesen c. e. noch einmahl c.e.′ ist der ander Satz, der dritten wird verrückt, wann e. und g. unten gesetzet wird, so bleibet aber c.′ und e.′ darzu; Unterweilen lauffen sie auch wohl gar also, dass Unisoni daraus werden, weil der Motus Contrarius muss gebrauchet werden; Als:

g           f e           d e           f c           d

In dieses Exempel steht der Alt und der Bass im d. und der Tenor und Discant im f. und also zwey und zwey im Unisono.

Wir wollen zu mehrer Erklärung dieser dreyen Sätze eine Tabell setzen, da der Clavis C. das Fundament seyn soll; Als:

ece c

gee c cae c oder  gee c geg e geh g
1. 2. 3.  




Diese beyden Tabellen sind einander in der Sätzen gleich, doch sind diese Sätze den Ordinar Sätzen, davon wir oben gemeldet haben, gantz zu wider und gantz unvollkommen; Dem in der ersten [p. 101] Tabella hat der erste Satz allhier zwo Tertien, und mangelt die Quinta g. Im andern Satze ist die Quinta und hat der natürliche Ordnung zu wider, zwo Tertien, und die Octava mangelt, nemlich c. Der dritte Satz verliehret gar die Natur der Triadis harmonicae, und da sonst die Quarta oben in die Octavam kommen solte, nimmt dieser die Tertiam oben, und setzet die Quartam in die mitte, im Alt, und Tenor. Die andere Tabell hat eben diese Eigenschafften, nur dass im Discant der obere Clavis der Triadis stehen bleibet. Denn der erste Satz der ersten Tabell kömmt mit dem andern Satze der andern überein: Als von untersten Clave zu rechnen sind zwo Tertien, und eine Octava darinnen, und im andern Satze der ersten Tabella sind wieder zwo Tertien und eine Quinta wie im ersten Satze der andern Tabelle; Der dritte Satz in beyden Tabellen sind einander wieder gleich: Denn sie setzen die Quartam in die mitte. Das sind also drey unvollkommene Sätze, und wird doch in der Replica oder Versetzung, eine völlige und natürliche harmonia, die da so wunderlich sich durch einander herschränket und verändert, dass man fast nicht weiss wie es zu gehet, und entstehet aus derselben Unvollkommenheit eine Vollkommenheit, welches ein Spiegel der Natur und Ordnung Gottes ist. Also ist bald die Quinta, bald die Tertia, bald die Octava oben; Man kan auch mit zweyen von diesen Griffen etwas ausrichten, denn wann die Quinta oben kömmt, hat man alle drey haupt Contrapuncta per Oct. Dec. und Duodec. doch hat der Satz in der Quinta wieder seinen Nutz.

[Now when one must derive a comes from a complete line or melody made up of diverse intervals or in which the comes follows the dux at a certain interval, then one has for that purpose three chord voicings [hand-positions], which contain the whole essence of the matter. For example, if I use the note [lit. key] c as a root or bass and then take the third, e, along with it in the left hand, and in the right hand I take g′ and e′, then that is one chord. If with this c and e [in the left hand] I play c′ and e′ [in the right hand], that is a second chord. The third chord is inverted, if e and g are placed below while the c′ and e′ remain. Meanwhile they work very well to produce unisons, as long as contrary motion is used. For example:

g           f e           d e           f c           d

In this example the alto and bass are on d and the tenor and soprano on f, and thus doubled in unison.

For greater clarification of these three chords we wish to present a table, in which the note [lit. key] C is the foundation. For example:

ece c

gee c cae c or  gee c geg e geh g
1. 2. 3.  




These two tables are the same as one another with respect to their chords, yet compared to the ordinary chords that we have addressed above [in the section on thoroughbass], these chords are altogether different and incomplete. For in the first table [p. 101] the first chord has two thirds, and lacks the fifth, g. In the second chord the fifth is present, but contrary to the natural order it has two thirds and no octave, namely c′. The third chord quite abandons the nature of the triadis harmonicae, and rather than taking the fourth in the octave above [i.e., in the right hand], it takes the third above and sets the fourth in the middle, between the alto and tenor. The second table has these same properties, except that in the soprano the top note [lit. key] of the triad remains the same. For the first chord of the first table corresponds to the second chord of the second table, in that from the lowest note [lit. key] are reckoned two thirds and an octave, and in the second chord of the first table are two thirds and a fifth, which are comparable to the first chord of the second table. The third chords in both tables are once again the same, for both set the fourth in the middle. They are thus three incomplete chords, and yet in the replica or inversion they become a full and natural harmony, in which they are so wondrously interlaced and transformed that one can scarcely understand how it comes to pass that out of the selfsame incompleteness arises completeness—which is a mirror of nature and God’s order. In sum, sometimes the fifth is on top, sometimes the third, and sometimes the octave. One can also work out something with two of these hand positions, for when the fifth is in the top voice, one has available all three of the main inversions, at the octave, tenth, and twelfth; and so the chord with the fifth in the top voice remains very useful (Werckmeister, Harmonologia musica, §175)]. I thank Jürgen Thym and Frederick Gable for advice on the translation of certain phrases of this passage; any infelicities of translation are my own.

43 Snyder, “Studies in Learned Counterpoint,” 552.

44 For a discussion of Sweelinck’s transmission of the Zarlinian legacy to the Hamburg contrapuntists, see Walker, ch. 7, “Invertible Counterpoint and the Hamburg Circle of Theorists,” 204–17.

45 For detailed discussions of the counterpoint treatises of the Hamburg school, see Walker, passim. In his “Musicalisches Kunstbuch” (1691), Johann Theile includes an augmentation canon with 8 two-part, 24 three-part, and 24 four-part resolutions in both motus rectus and motus contrarius that rely for their extra voices upon doubling in thirds, sixths, or tenths. The treatise survives in four manuscripts, all in D-Bsb: Mus. ms. theor. 913, Am. B. 451, Am. B. 452, and Am. B. 511. It has been edited by Carl Dahlhaus as Johann Theile, Musicalisches Kunstbuch, Denkmäler Norddeutscher Musik 1(Kassel: Bärenreiter, 1965). Similarly, the author of “Kurtze doch deutliche Regulen von den gedoppelten Contrapunctem,” a manuscript treatise formerly attributed to Sweelinck, recommends expanding two voices to four by adding parallel thirds above the top voice and below the bottom voice. The treatise is available in a modern edition as “Compositions Regeln Herrn M. Johan Peterssen Sweeling” in Compositions-Regeln, Werken van Jan Pieterszoon Sweelinck 10, ed. Hermann Gerhmann (The Hague: Nijhoff, 1901): 86–114. The original manuscript, D-Hs ND VI 5383, had been missing since World War II, but was returned by Russia in 1995/96; see Ulf Grapenthin, “‘Sweelincks Kompositionsregeln’ aus dem Nachlass Johann Adam Reinckens,” Hamburger Jahrbuch für Musikwissenschaft 18 (2001): 71–110. Walker (204, 206) surmised that Weckmann was the most likely author and composer of the Sweelinck treatise material on invertible counterpoint; see also Walker’s “From Renaissance ‘Fuga’ to Baroque Fugue: The Role of the Sweelinck Theory Manuscripts,” Schütz-Jahrbuch 7/8 (1985/86): 93–104. Grapenthin, however, concluded on manuscript evidence that the author of the “Kurtze doch deutliche Regulen” was Johann Theile. The duplex augmentation canon on p. 87 of the Sweelinck edition closely resembles Buxtehude’s canon for Meder (Example 13) in its use of rhythmic augmentation, simultaneous doubling in thirds, and occasional wide leaps. Christoph Bernhard also demonstrates how to increase the number of voices in a canon from two to four by adding parallel thirds; see his widely circulated manuscript treatise from ca.1657, “Tractatus compositionis augmentatus,” translated by Walter Hilse in “The Treatises of Christoph Bernhard,” Music Forum 3 (New York: Columbia University Press, 1973), 175–6.

46 Snyder’s solution of this canon is published in “Studies in Learned Counterpoint,” 548. Regarding Buxtehude’s chorale setting Mit Fried und Freud, BuxWV 76, a tour de force of invertible counterpoint modeled after Bernhard’s setting of Prudentia Prudentiana, Snyder observes that Buxtehude avoids continuous parallel thirds in favor of alternation between parallel thirds, sixths and tenths, particularly in the second Contrapunctus and Evolutio. Werckmeister’s treatise is of limited applicability in discussing these works of Buxtehude and Bernhard, since he does not venture into some of the more sophisticated and esoteric aspects of worked-out invertible counterpoint, such as simultaneous registral and melodic inversion, as do Zarlino and Bernhard himself in the Tractatus.

47 Alfred Mann, J. Kenneth Wilson, and Peter Urquhart, Grove Music Online, s.v. “Canon” (accessed 15 June 2006).

48 Werckmeister, Harmonologia musica, §194.

49 Harmonologia musica, §207; Werckmeister presents similar examples of non-canonic, four-voice invertible counterpoint in §§176, 198, and 205. Example 14 presents both typographical errors (emended per the listing of Druck-Fehler on the final unnumbered pages of Harmonologia musica) and a contrapuntal error. The contrapuntal error, on the downbeat of the final measure, affects both the Haupt-Satz and the resolutio at the twelfth.

50 Werckmeister, Harmonologia musica, §152.

51 Snyder, Dieterich Buxtehude, 98.

52 Buxtehude, congratulatory verse in Werckmeister, Harmonologia musica, unnumbered prefatory page.


Examples from Buxtehude’s organ works are based on Dieterich Buxtehude: Werke für Orgel, ed. Philipp Spitta and Max Seiffert (Leipzig: Breitkopf & Härtel, 1903–4; 2nd ed. 1950; reprint as Dieterich Buxtehude: Organ Works, Mineola, New York: Dover, 1988). The author thanks Anne Strickland for her collaboration in setting musical examples in Finale.

Example 1: Ascending Sequential Formulas for Contrapuntal Subjects

Example 2: Subject Ascending per secundas in Interlocking Imitation

Example 3: Subject Ascending per secundas in Stretto with 5–6 Syncopation

Example 4: Subject Ascending per tertias in Stretto

Example 5a: Subject Ascending per quartas, Voice-leading Paradigm

Example 5b: Subject Ascending per quartas in Stretto

Example 5c: Subject Ascending per quartas in Stretto with Diminutions

Example 6a: Buxtehude, Praeambulum in A Minor, BuxWV 158, mm. 17–20

Example 6b: Buxtehude, Praeambulum in A Minor, BuxWV 158, mm. 1–7

Example 6c: Buxtehude, Praeambulum in A Minor, BuxWV 158, mm. 32–44

Example 6d: Buxtehude, Praeambulum in A Minor, BuxWV 158, mm. 60–5

Example 7: Buxtehude, Praeludium (Prelude, Fugue, and Chaconne) in C Major, BuxWV 137, mm. 12–8

Example 8: Buxtehude, Praeludium in G Minor, BuxWV 150, m. 135 to the end

Example 9: Buxtehude, Chorale Fantasia on “Ich dank dir, lieber Herre,” BuxWV 194, mm. 1–16

Example 10: Buxtehude, Magnificat primi toni,BuxWV 203, mm. 76–91 (verse 5)

Example 11: Werckmeister’s Three Sätze or Chord Voicings for Invertible Counterpoint

Example 12: The Subject C–D–E–F Harmonized with Two Pairs of Parallel Thirds in Contrary Motion

Example 13: Buxtehude, duplex Augmentation Canon for Johann Valentin Meder,BuxWV 123

Example 14: Parallel Thirds in a Setting of “Wenn wir in höchsten Nöthen seyn”

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