The 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*

Abstract

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

References

Examples

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 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.”]

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