canon

PACHELBEL, PURCELL, AND BIBER: THE VARIATIONS CANON Term Paper, MUSIC 9521Y (Introduction to Research in Music Theory)

Completed by MATTHEW NACE (250547326) for DR. K. MOONEY JANUARY 11, 2015 UNIVERSITY OF WESTERN ONTARIO London, ON, Canada

Pachelbel, Purcell and Biber: The Variations-Canon

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Contents I.

Abstract ............................................................................................................................................ 3

II.

Foreword .......................................................................................................................................... 3

III.

Definition of Canon .......................................................................................................................... 9

IV.

Variations Forms ............................................................................................................................ 12

V.

The Affinity between Canons and Chord Patterns......................................................................... 15

VI.

The chords of the Pachelbel canon ................................................................................................ 19

VII.

Characteristics of the Pachelbel Progression 1: Stretto ................................................................ 22

VIII.

Characteristics of the Pachelbel Progression 2: Plagal Sequence ................................................. 26

IX.

The Variations of the Pachelbel Canon .......................................................................................... 28

X.

Macro-structural Form of the Pachelbel canon ............................................................................. 34

XI.

Voice-leading: Canon in D as a Stacked Canon ............................................................................. 39

XII.

Notable characteristics of specific variations of the canon ........................................................... 51

XIII.

Biber’s Chaconne-canon in unisono............................................................................................... 61

XIV.

The Chords and Ostinato of the Biber Ciacona-Canon .................................................................. 64

XV.

Aside: Biber and Merula ................................................................................................................ 66

XVI.

Form of the Biber ciacona-canon ................................................................................................... 67

XVII.

Variations of the Biber ciacona-canon ........................................................................................... 68

XVIII.

Voice-leading: Catch-like elements of the Biber ciacona-canon................................................... 73

XIX.

Notable characteristics of specific variations of the ciacona ......................................................... 78

XX.

Purcell’s Chaconne (Two in one upon a Ground)........................................................................... 82

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XXI.

Purcell and the Ground – Relationship to Dido and Æneas, and the Chords of the Two-in-one .. 83

XXII.

Characteristics of the Passacaglia progression .............................................................................. 85

XXIII.

Variations and Form of the Purcell Two-in-one ............................................................................. 88

XXIV.

Notable characteristics specific variations of the two-in-one ....................................................... 89

XXV.

The Canons of the Goldberg Variations ......................................................................................... 91

XXVI.

Writing a Canon over a Cantus Firmus ........................................................................................... 97

XXVII. On the Importance of Sequences ................................................................................................ 109 XXVIII. Recapitulation .............................................................................................................................. 111 XXIX.

Appendix A ................................................................................................................................... 114

Schenkerian and Westergaardian Analyses of the Pachelbel Canon XXX.

Appendix B ................................................................................................................................... 116

The Gigue from “Kanon und Gigue in D-Dur fur Drei Violinen und Basso Continuo” by Johann Pachelbel XXXI.

Appendix C ................................................................................................................................... 117

The Clarke Chaconnes XXXII. Appendix D ................................................................................................................................... 125 The Merula Chiacona XXXIII. Appendix E ................................................................................................................................... 128 Dido’s Lament (Phrase vs. Ostinato) XXXIV. Bibliography ................................................................................................................................. 129

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Pachelbel, Purcell and Biber: The Variations-Canon I.

Abstract

T II.

his article will discuss the interaction of the formal elements of variations forms such as Chaconne, Passacaglia, Ground, etc. with the technique of strict canonic imitation to create a hybrid form, here referred to as a Variations-Canon. In addition, analyses will be presented (particularly with respect to, but not limited to, the expression of the characteristics of this hybrid from) of three examples of this hybrid form: Johann Pachelbel’s Canon in D for Three Violins and Basso Continuo, Heinrich Ignaz Franz von Biber’s Ciacona (Canon in unisono), Partita III in A Major, mvt. VI from Artificiosa-Ariosa, and Henry Purcell’s Chaconne (Two in one upon a ground) for two flutes and bassoon from the beginning of Act III of Dioclesian.

Foreword

I

n the late seventeenth century (dates around 16801 are popularly cited, but poorly documented, and the exact date is uncertain), the German composer Johann Pachelbel (c.1653 - 1706) composed his Kanon und Gigue in D-dur fur Drei Violinen und Basso Continuo

(Canon and Gigue in D-Major for three violins and basso continuo). Describes it thus: The three-part canon over a bass is one of Pachelbel’s most admired works. In it he combined two of the strictest contrapuntal techniques in a fine display of technical mastery: the bass, a two-bar ostinato, is the foundation of 28 variations, while above this the three violins proceed in two-bar sections in a relentless canon.2 Pachelbel was music teacher to Johann Christoph Bach (older brother and music teacher of Johann Sebastian Bach), and was invited to contribute music for the J. Cristoph Bach’s wedding in 1694;3 it has

1

(Canon and Gigue in D major (Pachelbel, Johann) 2014) (Butt and Nolte, 851) 3 (C. Wolff 2001) 2

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been suggested that he may have written the canon and gigue for this occasion,4 but this claim has gained only limited acceptance. Historically, Pachelbel is a composer known chiefly as a composer of organ music, and the canon and gigue seem to have been largely forgotten for two centuries, until it was published in 1919.5 In the years since, the canon portion has become one of the rare few classical compositions to become popular outside the classical world. It is a rare high school music student who has not played some arrangement of this piece before graduating, and elements of it have been borrowed frequently in the genre of popular music. Unfortunately, it has also (perhaps consequently) become relatively unpopular, or at least, not taken seriously, in the classical world. It is among the most frequently requested pieces at weddings and similar ceremonies, and working classical musicians are typically called upon to perform the piece far more often than they would like (particularly, more often than cellists would like, as they are called upon to play the same eight notes over and over again, nearly thirty times before the end of the piece). It is also a popular hobby among classical music amateurs to collect lists of modern popular songs that use the same chord pattern as the Pachelbel canon (there are a surprising number of these, given that the chord pattern is substantially more complex than is typical in most popular songs).6

4

(Pachelbel's Canon 2015); a citation for this claim is given, but unclear, and I have been unable to verify it. (Canon and Gigue in D major (Pachelbel, Johann) 2014) 6 These lists often include considerable misinformation. For instance, comedian Rob Parovian, on his 2001 album “American Cheese”, recorded a rant about the Pachelbel canon that listed many songs that are supposedly based on the same chords, demonstrating as he sings the songs and strums the chords on his guitar; however, as the rant progresses, he quietly starts altering his chords, without mentioning the fact to his audience, and the songs he quotes begin to have less and less in common with the Pachelbel canon. By the end of his rant, the songs often 5

have only the first three out of eight chords in common (i.e. The Beatles’ “Let It Be”:

I – V – vi – IV – I – V – IV – I;

the Pachelbel canon: I – V – vi – iii – IV – I – IV – V). Unfortunately, many enthusiasts have taken the rant as literally true, and his spurious list shows up all over the internet; at various points in time, it has even been found, listed in full, on the Wikipedia article (Pachelbel's Canon 2015). At the time of writing, this has fortunately been removed from the article; however, there is heavy debate on the article’s discussion page (http://en.wikipedia.org/wiki/Talk:Pachelbel%27s_Canon) regarding whether or not the lists should return to the article, and there is a link to an older version of the page where the list used to appear (http://en.wikipedia.org/w/index.php?title=Pachelbel%27s_Canon&diff=next&oldid=130852425).

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I, myself, first came to know the canon in a high school music class, through an arrangement for concert band with the appalling title “Famous Canon”. Although I very much enjoyed playing the arrangement in high school, that arrangement, and many others like it, have come to frustrate me to no end in the years since. The title itself is one source of irritation: from a pedagogical view point, such a title fails to teach the students the conventions for identifying a piece of music, or the importance of knowing who wrote it or how to find an authoritative copy of it; most of my high school friends still remember this canon, but are unable to identify it as anything other than “Famous Canon” – and this phenomenon is not limited to high school students. Indeed, during my research for this very paper, I came across a rather substantial essay, written by a university student, that includes an analysis of this canon, but analyses an arrangement for flute and piano, rather than the original form of the composition (this is only one of many shortcomings in that paper).7 The second problem I have with this arrangement, and many others like it, is that it was not presented as a true canon, but merely as a chaconne (indeed, many such “arrangements” do not even keep the same variations, suggesting that they ought more appropriately be entitled “Variations on a Theme by Pachelbel”, or something like it). As a student, I spent many years wondering why the piece was called “Canon”, when it did not appear to make use of imitation after the first dozen measures or so. The reason for this is that the piece is, in fact, organised according to two different organising principles: it is a true canon, but it is also a chaconne. I would hypothesise that Pachelbel chose to call it a canon because canons are more difficult to compose than chaconnes, and so the canonic principle would have occupied much more of his attention during composition than the chaconne elements. It is also worth noting that Pachelbel wrote many chaconnes,8 and naming the piece a canon may have helped to distinguish it from them. However, it is typically the chaconne elements of the compositions which have

7 8

(Paiz, 20-2) “Pachelbel’s fondness for variation form […] is demonstrated in his six ciaccone.” (Butt and Nolte, 849)

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principally occupied the attention of arrangers creating arrangements of the piece for mass consumption by amateur performers. As a music teacher myself, when I ask students what a canon is, I commonly get a response along the lines of: “Oh!,it’s like that piece, Canon, you know, the one that goes like this …”, but they have no idea what makes the piece a canon (probably because the arrangements they have played or heard did not retain the canonic elements of the piece). If this poverty of critical examination were limited only to high school music classes, it would be of little concern; however, even at more advanced levels, there seems to be a shortage of good, quality analyses of the piece that take into consideration all of its characteristics. Perhaps this is because of the previously mentioned tendency among classical musicians not to take the piece seriously, or perhaps it is the consequence of the deceptive appearance of simplicity pervading the foreground of the piece (the complexity of the piece is not evident to the performer; it is not technically demanding,9 and strongly resembles much less complex chaconnes, of which Pachelbel composed many). Most analyses have failed to notice that the much vaunted constantly repeating chord pattern is somewhat flexible; they cannot seem to agree on how many variations there are, and there seems to be only very poor analyses of the significance of the order in which those variations occur; certainly, I have found no analyses that take into consideration the effects that the canonic principle must have had on the compositional process. This, then, is one of the purposes of the present article: to provide some long overdue attention to this composition – including the application of the older treatises on composing canon over a cantus firmus10 and the more recent developments in canon theory made over the last two decades, especially in regards to the concept of stacked canon and compositional procedure.

9

“From a technical point of view, his music for strings makes no virtuoso demands and never exceeds the third position.” (Butt and Nolte, 851) 10 From Latin, “Fixed Melody”; in early polyphony, it was traditional to begin with a previously existing melody, and create harmonies by adding new melodies to it; most treatises after 1600 deal with composing canon independently, without concern for a cantus firmus, but it is a feature of some Renaissance treatises.

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Although the forms of chaconne and passacaglia have often been said to be related to canon in some ways, the Pachelbel canon is one of the rare few pieces in the literature that combines both of these forms together. Doing so has some very specific and interesting consequences, and results in a hybrid form I call a Variations-Canon. I am aware of only two other pieces that fit perfectly into this form. The first is a ciacona by Franz Biber; it is the sixth movement of his Partita III in A-Major from his ArtificiosaAriosa, and is identified as a Ciacona first, with the identification of the canonic principle added as a subtitle. This piece, known to have been written around the same time that the Pachelbel canon is thought to have been written (though without an exact date on the Pachelbel canon, we cannot know which was written first), is remarkably similar to it – so much so that we might begin to wonder if one might not have been based upon the other, or if they might perhaps both have been modelled upon another, earlier work that has not come down to us. The other extant variations-canon is by Henry Purcell, from the beginning of Act III of Dioclesian, labelled first as a Chaconne (I shall refer to it as a passacaglia, for reasons I shall discuss in section IV), and then secondarily as a “two in one upon a ground”. The second purpose of this article, then, is to describe the variations-canon form, and to provide analyses of these two examples. Through my research, I have also come to the conclusion that the canons displaying certain characteristics seem to have an affinity for repeating chord patterns or bass ostinati; many canons that were not originally written with bass ostinati seem to be capable of carrying such ostinati, a tendency that I shall discuss in this article. From the flip side, there are numerous other forms related to the chaconne and passacaglia that were traditionally used for composing variations (i.e. the passamezzo, the folia, etc.) that would seem to be possible candidates for canonic treatment, yet no such examples seem to exist in the literature; I shall explore the specific characteristics of the chaconne and the passacaglia that may explain why they have been chosen for this kind of treatment, while other forms have not.

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There are also, no doubt, numerous examples of variations forms that employ some degree of imitation briefly, before moving on to free composition; of these, I will concentrate some attention here on Merula’s Chiacona. For contrast, I will also give some attention to a pair of chaconnes attributed to Jeremiah Clarke11 that bare an uncanny resemblance to the Pachelbel canon, which will help to determine those aspects of Pachelbel’s canon that result from the canonic principle, versus those that result from the variations principle. (The score for the Clarke chaconnes can be found in Appendix C). Finally, I can hardly write a paper on the combination of canon and variations without touching upon the Goldberg Variations; I will show that the Goldbergs represent a very different way of combining these techniques, which does not fit into the form I will be describing; nevertheless, I will also show that Bach’s supplement to the Goldbergs has much in common with the other pieces we will be examining. A brief note about intended audience: I have written this paper in fulfilment of a course requirement toward a Masters of Music Theory degree at the University of Western Ontario. However, it is also my intention, once this paper has reached its final form, to publish this paper in some form, to help correct the generally poor treatment of this subject as noted above. Because of the generally dismissive attitude of most serious music scholars toward the subject, also mentioned above, it seems most appropriate to take the knowledgeable music enthusiast as my target audience, insofar as possible, while still maintaining the level of academic rigour expected for my course requirements. As a result, I have gone to some effort to explain some concepts that would be obvious to the average music scholar, yet

11

There have been two important composers named Jeremiah Clarke. The better known of the two (c. 1674-1707) was a younger contemporary of Purcell, and likewise a student of John Blow. His best known works are the Prince of Denmark’s March (better known to most according to its form, a Trumpet Voluntary, long misattributed to Prucell) and the Trumpet Tune from the Island Princess. This, however, is not the Jeremiah Clarke thought to have composed the chaconnes in question. They are commonly attributed to the other Jeremiah Clarke (c.1743-1809), a classical English composer and perform (violin and keyboards) who is generally less well known. However, the authorship of these two chaconnes is disputed (see (Gifford and Shaw 2001)), though the likely actual composer is unknown. For simplicity, I shall here refer to Clarke as the composer.

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not so to the average music enthusiast (a basic knowledge of harmony, including the syntax of Roman numeral analysis and figured bass is assumed). At such moments, I ask indulgence of the scholar. Additionally, the purpose of the assignment is to engage with the literature and source material used for research in music. However, this paper, by nature, is at least as heavily analytical as it is research based; it is my intention that the extended length, and the breadth of topics covered, should compensate for this by increasing the opportunities to incorporate research into this paper, in spite of the fact that there remain many pages of this work upon which no citations are present.

III.

Definition of Canon

T

he modern use of the term “canon” is rather different from the way it was originally used in music. In its original form, a canon was a “rule” (it is in this sense that the word is used in ecclesiastical canon law, and in “the literary canon”). In the earliest examples, a musical line

was written that was called the dux (from Latin, the “leader”), and the “canon” was the rule according to which an additional part, called the comes (the “follower”), or perhaps multiple additional parts, might be derived from it. Usually, each performer would read the same part, but would realise the part in different ways to create harmony (a canon written out this way is said to be “unresolved”; once it has been written out in full score, it is said to be “resolved”).12 The canonic rule might mean delaying the start of the comes, or transposing it,13 augmenting it, diminishing it, inverting it, retrograding it, removing some notes, or any combination of these elements. For many kinds of simple imitation at an interval with a delay, the rule might simply appear as multiple clefs at the beginning of the line (indicating the pitch level of the comes)

12

(Canon, 138); Albrechstberger uses the terms clausus (closed) and apertus (open), respectively. (Albrechtsberger, 232-4) 13 Some theorists are more particular about the kind of transposition permitted. For Zarlino, only perfect intervals (i.e. unison, octave, fourth, fifth) could support true canon, because they could maintain exact intervals, while transpositions at the second, third, sixth or seventh would necessarily involve switching freely between major and minor versions of the intervals, and hence could only be termed “imitation”. (Zarlino, 126, 135)

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separated by rests (indicating the length of the delay).14 For instance, for a canon written in the treble clef, starting on C, imitated a measure later at the fifth below, the composer might give first the mezzosoprano clef (indicating the pitch of the comes) and a whole rest, followed by the treble clef of the dux (Figure III-1). Sometimes, the composer might not even tell you how to change the part, or give you a riddle to help you determine how to derive the part15 (this is called an “enigma canon”). In contemporary usage, the word canon is generally taken to refer to the composition itself, not the rule according to which the additional parts are derived,16 and it will be in this sense that the term will be used in this article. In general, if a piece is referred to simply as a “canon”, the comes will be identical to the dux, except begun after a brief delay (canons at the unison). Any other form of canon will generally

Figure III-1 How the rule of a canon might be notated and realised (A) As the canon would be written; (B) as it would be understood, in the given clefs; (C) transcribed into modern clefs.

14

(Morley, 118) “In the Middle Ages and Renaissance, such rules were sometimes stated in cryptic fashion and might entail a variety of ways of interpreting a composition’s notation […] for example, in the Agnus Dei of Dufay’s Missa L’Homme Armé, which has the rule ‘Cancer eat plenus et redear medius’ [rom Latin, crab goes full and returns half – crab canons are retrograde canons, so the line is performed backwards] (tenor in full note-values backward, then forward in half note-values; […]).” (Canon, 137) 16 Some early musicians viewed this label with contempt. Thus, Zarlino: “Because the Greek word for rule was κανών, some not very intelligent musicians designate canon what should be called a fugue, …” (Zarlino, 130). Indeed, the terms fugue or fuga were once used to describe canons; we will not use them here because they are now more typically used to describe a compositional form or procedure of a different style, based on sections of temporary and non-strict imitation alternating with free composition in a heavily contrapuntal texture. 15

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be described explicitly, such as a “canon at the fifth”, or an “augmentation canon”, and so on. Most of the canons that will be discussed in this article will be simple canons at the unison. In addition to the previously mentioned canonic techniques, in the Renaissance, it was common to write canons over a cantus firmus, just as the cantus firmus was used so frequently for most composition in the Renaissance. This tradition is discussed by Thomas Morley in his Plaine and Easie Introduction to Practicall Musicke17, Gioseffo Zarlino in Le Istitutioni Harmoniche18, and William Bathe in A Brief Introduction to the Skill of Song.19 It is reasonable to assume that the existing tradition of composing canon over a cantus firmus would exert an influence on the process of composing a canon over a ground bass. A quick note is warranted here regarding notation: since the principle involved in canon is that it should be possible to play the canon from a single written line, it is therefore common to write the canon in this form, along with the rule describing it, as noted above. As to the rule itself, the time delay might not be directly specified in writing, but may be added as marks above the music. It is traditional to mark the current progress of the dux at the moment each comes enters with the symbol

(this is the symbol

musicians normally use for a dal Segno repeat, but in the context of canon, it is referred to as a presa)20, and if the comes voices are meant to end together with the dux (rather than trailing off one at a time), it is traditional to mark the progress of each comes at the moment the dux ends (and therefore, the moment

17

(Morley, 108-19) (Zarlino, 126-40, 215-20) 19 (Bathe, "The" - "flat", ~32-35) (n.b. page numbers are not printed in this edition; pages can be identified by a caption at the bottom that indicates the first word of the following page; the page numbers I have cited here are based on my count, and are approximate). 20 (Zarlino, 130) 18

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the comes should stop) with the symbol ! (which visually looks identical to a fermata, but in the context

of canons, is traditionally referred to as a coronata).21

IV.

Variations Forms

T

here are a substantial number of forms that have been used for the composition of variations over the years. From the late Baroque onwards, the Theme and Variations was typical. This form is different from the kind of variations used in a variations-canon, in that the variations

are generally created by altering a pre-existing melody, called the theme. In the older forms, the theme is not so much a melody as it is a bass ostinato, or even a chord progression. The variations that are written upon it are drastically different, and often have very little to do with each other except for the underlying harmony (to separate the two techniques, as an oversimplification, we typically reserve the term “Theme and Variations” for the classical version, while the older style is called “continuous variations).22 The ostinato or chord pattern is often played once by itself, then any number of variations may be written upon it in no particular order, except that the first few are generally quite simple, and the variations gradually become more and more complex. The Harvard Dictionary of Music notes that: “The first written versions in Spanish guitar books of the early 17th century show a simple progression sometimes repeating often enough over the bass line to create a series of ostinato variations; this suggests an originally improvised practice.”23

21

(Zarlino, 130) “Variation forms are not necessarily always sectional and strophic. If the theme is a short ostinato or ground bass, its repetitions will generate a continuously unfolding piece over which figuration and textures change with each statement of the theme; this is known as continuous variation. […] Haydn’s variations in their turn span the range of possibilities of Classical variation. At first writing only constant-bass and constant harmony sets [continuous variations], he gradually added greater numbers of melodic-outline variations [Theme and Variations] until this type began to predominate around 1770 […]” (Variation, 939, 941) c.f. “The term ‘variation’ should be understood very loosely, however, as in chaconnes, there is generally no underlying melodic theme tying the variations together but at most a harmonic rhythm or bass formula, […]” (Silbiger, Chaonne, 411) 23 (Chaconne, 155) 22

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Many of these variations forms originate from Renaissance dances. The chaconne, for instance, is based on a primal dance in three-four time, in a major key, from Latin America, which was imported in a sanitised form into Spain, and quickly swept the continent.24 It is associated with a traditional chord pattern of I –V – vi – I.25 Similarly, the passacaglia began as a three-four time minor-key dance with a traditional chord pattern of i – v^ – VI – V.26 The two forms are very similar in most respects, and over time, the traditional chord patterns became less relevant; there are a few other differences occasionally noted, but exceptions are too frequent to be useful in distinguishing them.27 For instance, chaconnes tend to be faster, and passacaglias tend to have more lyrical and expressive melodies. Even the tendency for chaconnes to be in major keys and passacaglias to be in minor keys has many exceptions.28 Over time, these characteristics began to be less important than the overall concept of repeating a chord pattern and playing variations over it. By the nineteenth century, the chaconne and passacaglia were hardly distinct from each other, and where composers did still observe a distinction, they did so in different and inconsistent ways.29 For our purposes here, we will be interested primarily in harmony, and so we shall define the terms in relation to the previously identified chord patterns. One characteristic that is common, but not normative, for these forms is the repetition of each variations.30 As an example, of the two Grounds by Jeremiah Clarke that will be used as a point of comparison to the Pachelbel canon, the second, in three-four time in F-Major (the more prototypical of the two) repeats every variation, while the first, in common-time in C-Major (the one more like the Pachelbel canon) does not. However, the Pachelbel canon generates a rather interesting manifestation

24

(Chaconne, 155) (Silbiger, Chaonne, 412) 26 (Silbiger, Passacaglia, 192) 27 (Jenne and Little, 200) 28 (Jenne and Little, 200) 29 (Chaconne, 156) 30 (Jenne and Little, 200) 25

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of this property: while each violin plays each variation only once, the canonic principle ensures that each violin plays the variation in sequence, so that it is, in fact, repeated – not only twice, but three times (in fact, the nature of the variations themselves, in some sense, causes the variations to seem to be repeated four times, a detail that will be discussed in section IX below). Another characteristic of these variations forms is that they tend to be on the longer side frequently as much as one hundred measures, rather than the two dozen or so measures typical of most dances (of which most of these variations forms are examples) – and both the music and the dance choreography gradually increases in complexity from the exposition, reaching a climax near the end, followed by a brief period of decreasing complexity to a simpler ending.31 This pathway parallels traditional narrative structure, with its introduction, rising action, catharsis, and denouement. The Pachelbel canon partly follows this structure, but not perfectly, as will be discussed in section X below. Another relevant characteristic of chaconnes is a relationship to the Sarabande, another dance form with which it shares some characteristics. The sarabande, a Baroque dance form, is not based on continuous variations, as the passacaglia and chaconne are; however, it is a descendent of the older Zarabanda, a form frequently associated with La folia, another traditional chord pattern used for variations – which, however, does not seem to be a suitable candidate for canonic treatment. A particular characteristic which the chaconne frequently shares with the sarabande is the rhythmic syncopation pattern typical to the sarabande, characterised by the emphasis on the second beat of a three-four measure.32 Although this characteristic does not appear in the Pachelbel canon (which is not in threefour time), it is strongly characteristic of the theme and many of the variations of the Biber ciaconnacanon.

31 32

(Jenne and Little, 200) (Jenne and Little, 97-8)

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There are a variety of other traditional dances that were the basis of variations throughout the renaissance (for instance, the previously mentioned la folia, the romanesca, and the passamezzo, etc.). However, these patterns do not seem to have retained their popularity for variations forms in later periods of music history, and more particularly do not seem to be the bases for any canons. This suggests that there may be certain characteristics of the chaconne and passacaglia patterns that lend themselves particularly well to canonic treatment.

V.

The Affinity between Canons and Chord Patterns

O

ne of the challenges involved in writing imitative music in general, and canons in particular, is that each note that you write has harmonic implications at multiple points in time. Assume, for instance, that you are writing a three voice canon at the unison, at a delay of

one measure. The first note you write will belong to the chord on the downbeat of the first measure, but it will also belong to the chord on the downbeat of the second measure and the chord on the downbeat of the third measure (assuming that it is not a non-harmonic tone – accented non-harmonic tones are less common in canon than in free composition, because they must be prepared and resolved appropriately in every canonic imitation). As a result, it can be difficult to create harmonic variety. Already, by the third measure, there are two chord tones carried over (in the dux and comes 1 in the previous measure, now in comes 1 and comes 2, respectively). If they happen to be a perfect fifth apart – and assuming that we want to have complete chords as often as possible, and so avoid duplicating chord tones – the dux will almost certainly play the dividing third. If there are a third or sixth apart, they could belong to only two possible triads: the first is the one of which they were a part in the previous measure, resulting in a repeated harmony; the other is a chord whose root is a third away from the chord in the previous measure (see Figure V-1) – chords which often express the same function (think iii and I^,

IV and ii^, or V and viio^ – for simplicity, I shall refer to all of these as mediants, even where that term is

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A

B

C

Figure V-1 Harmonic consequences of the canonic principle (A) When the remaining notes form a fifth, the dux is almost obligated to pick the third. (B) If the remaining notes form a third, there are two possible options. (C) However, one of those two options is the same harmony as the previous chord, so the other is more likely.

not strictly accurate).

This limitation of which harmonies are available can become even more

pronounced when voice-leading considerations come into effect, such as the avoidance of parallel fifths, etc. Take, for example, the round. The round is one of the simplest forms of canon. It is generally short, and often repeats at the performer’s discretion. The delay between entrances is generally quite short, and the interval of imitation is almost always the unison. In such cases, the harmony, almost by definition, typically becomes a strict alternation between the tonic and either the dominant or subdominant (or sometimes both). This is even true, if to a somewhat lesser degree, when the rounds are written by history’s greatest composers (see Example V-1). Only by making the time delay between entrances longer does it become significantly easier to create harmonically diverse surface progressions. If, as in the case of the Pachelbel canon, the delay is two measures, at a slow tempo with a crochet33 harmonic rhythm, there is time for eight chords before

33

For efficiency, I have chosen to use the traditional names for rhythms throughout this paper: minim = half note; crochet = quarter note; quaver = eighth note; semiquaver = sixteenth note; demi-semiquaver = thirty-second note.

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Example V-1 A selection of traditional rounds, notated harmonically into a single cycle Note how the rounds tend to be harmonically simple, often mere alternation between one primary chord and the next. Even those written by great composers are much less harmonically sophisticated than the music those composers typically write. All these rounds are transcribed from (Dyk and Taylor).

the canonic principle adds substantial pressure to repeat the initial chord. In such a case, the pressure to repeat a harmony is actually somewhat greater again, because all of the chords must interact with each other appropriately, and the possible progressions formed by using some of the alternate chords (the “mediants”) may not be as effective as the original progression.

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The consequence of this is that the degree of harmony becomes more complex, but at the same time, the chord progression becomes even more rigid – which leads us to something very much like a chaconne or a passacaglia. If the degree to which this is true is dependent upon the length of the delay between entrances, it is also dependent upon the number of voices. If there are only two, there is lots of flexibility. The more voice that are included, the less flexibility there is, because each voice contributes to the carried-over harmony. However, this need not be restricted to actual voices. If there is a degree of polyphonic melody in use, a single real voice may incorporate two or more conceptual voices, and hence multiple notes contributing to the carried over harmony. In the case of the Pachelbel canon, several variations contain melodic polyphony, which contributes to the canon’s tendency to stick to a consistent harmony. One of the consequences of this effect is that many rounds that do not already contain a ground bass are often well suited to supporting one. Consider, for instance, the well-known round Dona nobis pacem, which has been popularly (if uncertainly) attributed to Palestrina. It is in three-four time, at a delay of eight measures between entrances, and is harmonically quite similar to a chaconne (hence to the Pachelbel canon). It works quite well to superimpose upon it a ground bass consisting of a descent, one note per measure, from the tonic to the mediant, followed by a cadential

creates an overall harmonic progression of

2 – 5 – 1 (Example V-2).

This

I - V% - vi& - I$ -ii^ – I^ - [ii - V$=!] – I, supporting voice-leading

similar to that of the Pachelbel canon, or more generally, a chaconne. It is certainly not the only such example of a canon that can carry a chaconne-esque ground bass.

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Example V-2 Dona Nobis Pacem (Atrtibuted to Palestrina) with Implied Ground The implied ground bass pattern is quite similar to the bass and chord patterns we shall see in our examinations of variations-canons, including the Pachelbel canon and the Biber ciacona-canon. Canonic voices transcribed from (Dyk and Taylor).

VI.

The chords of the Pachelbel canon

P

achelbel’s Canon in D (

Example VI-1) is a version of the chaconne pattern. According to the Grove Dictionary,

“the most common progression for the chaconne was I – V – iv – V; in later variants, the

final dominant was often extended by a standard cadential formula.”34 In Italy, “The characteristic chaconne formula, […], commence with I – V – VI, and then return to V, either directly or by way of intermediary harmonies such as

IV – V or I^ - IV – V.”35 If we combine the intermediate

extension (substituting iii for I^, about which, more later) with an expanded cadence (i.e. replace V with I

– IV – V), we end up with the traditional description of the chords of the Pachelbel canon: I – V – vi – 34 35

(Silbiger, Chaonne, 411) (Silbiger, Chaonne, 412)

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Figure VI-1 The first, unclear, use of the ii6 chord in the Pachelbel canon This first use of the ii6 chord is delayed by a suspension, and may even be nothing more than an anticipation toward the following chord; however, it does a good job foreshadowing the later use of this chord.

iii – IV – I – IV – V.

This, however, is an oversimplification.

In truth, the Pachelbel canon is built upon a bass ostinato of 8 – 5 – 6 – 3 – 4 – 1 – 4 – 5 , with a certain flexibility regarding the exact chords that are creating in the variations composed over it. Craig Stuart Sapp, in his brief analysis based on computerised analytical methods, notes that “There is some variation in the harmonic structure, […], representing a chord of FS minor […]. In [some] cases the chord with a root on FS is changed into a chord with a root on D.”36 It is interesting to note that this substitution is the same substitution already remarked upon in the discussion of the expanded chaconne formula, but it reverse. It happens quite frequently throughout the canon, and will be discussed again later, in the context of voice-leading procedures. In addition to this frequent change, the final

IV of the progression is thrice

changed to ii^; the first time, in the 24th variation, it is discreetly disguised through a 7-6 suspension that resolves in such a way that it could equally be interpreted as an anticipation of the 2 over V , causing us to question the analysis as a ii^ at all, but it also foreshadows the less ambiguous change of the final two variations, allowing a very strong cadential closing of ii^ - V – I.37

36

(Sapp, 84-5) Interestingly, this strong cadence is not actually a perfect authentic cadence, as it resolves to 3 in the highest voice, leaving the canon feeling slightly incomplete, providing a slight push onwards toward the accompanying gigue.

37

Pachelbel, Purcell and Biber: The Variations-Canon

Example VI-1 Kanon in D-dur fur drei violinen und basso continuo (Canon in D) by Johann Pachelbel Transcribed by this author from manuscript Mus.MS 16481/8.

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These changes, however, are entirely in the upper voices. The bass ostinato is kept unaltered throughout, and is realised in straight crochets. This line is repeated in the cello twenty-eight times. The first statement of the ostinato is treated as a theme, and is played alone, as is typical for a chaconne. For the remaining twenty-seven repetitions, melodic variations of varying degrees of complexity are added to the violins (note that although there are twenty-seven repetitions after the initial statement, it may not be appropriate to say that there are twenty-seven variations – more on this in section IX below).

VII. Characteristics of the Pachelbel Progression 1: Stretto

A

s has already been sketched out in brief, the chord pattern used in the Pachelbel canon has a number of remarkable characteristics that make it very well suited to both canonic treatment and variations treatment.

We now proceed to an examination of these

characteristics, from a variety of perspectives. The chord pattern of the Pachelbel canon is based on a sequence of descending thirds. For this discussion, we shall adopt the form I – V – vi – iii – IV – I – ii^ – V – I, with one half-measure per chord as the model for the chord pattern.38 To begin with, the sequence unit is one measure, so the chord pattern can be reduced to an essential chord at the beginning of each measure – except the final measure, in which the final chord is the essential chord, as it resolves back to the tonic to begin the next variation. The remaining chords are simply passing chords, leaving the overall pattern I – vi – IV – V – I. This pattern is, of course, a highly normative chord pattern for tonal music. If we include the ii chord, it also has the

38

In its original form, the Pachelbel canon was written with four chords per measure. However, it is occasionally convenient to rewrite the canon in longer note values, doubling the length of each note. Throughout this paper, I will uses which ever form best displays the relevant characteristic being discussed. In this case, setting two chords per measure sets the sequence pattern in clearer relief.

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Figure VII-1 Evolution of the descending thirds sequence The first chord is transformed by the 5-6 technique; the sixth of the resulting first inversion triad is then transferred down to the lowest position, creating a root position triad a third below the original chord. The resulting two chords share two notes in common.

useful property that the motion between the beginnings of measures is always the same (down a third) – except for the cadential gesture back to the beginning of the pattern. As a result, it offers a convenient method of creating variations, almost mechanically, by sequencing a one-measure motive over three and a half measures, followed by a cadence. This sequence has an important characteristic. It can be created by a simple pair of transformations: first, apply the 5-6 technique to the existing chord (I ) p ^ yielding I – vi^), then revoice the newly created chord, so that it is in root position, giving I – vi (Figure VII-1). This pattern is then highly conducive to the principle of stretto canon. Stretto canon is canon in which the time delay between entrances is rather short.39 As already discussed, there are relatively few harmonic possibilities when delays are short. The canonic principle strongly predisposes the harmony to follow a continuous cycle of a single type of intervallic motion related to the interval of imitation (for instance, a canon at the fourth above will spiral a cycle of ascending fourths) – meaning that a canon at the unison will repeat the same harmony ad nauseum – with the next best choice being a chord related to it as a mediant (in our informal sense of “mediant”). This mediant relationship is precisely the nature of the first chords of each measure in the Pachelbel canon. It would, then, be possible to create a stretto canon at the unison in two voices40 upon

39

(Gauldin, 29). This distinction is somewhat fuzzy - it is difficult to decide how short is short enough, or how long is too long, to call a canon a “stretto canon”. Since we are dealing with canons with repeating chord patterns, for our purposes, it is convenient to define stretto canon as a canon for which the delay is shorter than the chord progression. 40 Adding a third voice would almost certainly cause parallel fifths.

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this chord progression, at a delay of two chords, in which each variation would be the result of sequencing a motive over the chords, followed by a cadence (as is true of many of Clarke’s variations). One would be forced to use a figuration that could be interpreted alternately as a passing tone and a suspension in order to deal with the cadence. Pachelbel does not make use of this particular characteristic of the progression, but Biber does: his earliest variation (and one that appears a few times as a kind of recurring theme) is itself a brief kind of stretto canon not only upon the ground, but also of the ground. Meanwhile, the Pachelbel canon does

Figure VII-2 Stretto-canonic techniques on the Pachelbel progression (A) Shows a section of the Pachelbel canon as written. (B) Shows a similar effect created using a stretto-canon approach. Note the close similarity of the material between the double bar lines to part (A). Also notice that the stretto canon is constructed by beginning the dux one measure before the beginning of the variation. (C) Shows that this was the process used to construct the first variation of the Biber ciacona-canon.

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Figure VII-3 A hypothetical realisation of the continuo using the stretto-canonic principle Taken from the moment Variation A enters, notice that the hypothetical continuo part imitates the dux in stretto fashion, at a delay of one measure, and harmonises well with the existing parts.

make strong use of imitation at the third throughout its variations, often resulting in equivalent effects to what would occur if he used this stretto canon technique (Figure VII-2). It should also be noted that the Pachelbel canon is composed for three violins and basso continuo,41 and a skilled continuo player might choose to take advantage of this property to add a stretto-canonic part to the piece in some variations (Figure VII-3). From a voice-leading perspective, however, these chords as stated provide few acceptable voiceleading options. Chords moving down by third provide only three typical voice-leading patterns. All three involve the fifth rising to become the new root: the 5-6 technique (keeping the common tones and moving the remaining tone by step), lowering the root and third (creating a triple-root), and voice exchange (of the two tones that would otherwise be kept common) (Figure VII-4). Of these, the first provides minimal

41

Basso continuo was the practice of accompanying the music by an instrument capable of playing chords (i.e. harpsichord, organ, lute, etc.), improvising a part based on the notated bass line (often with annotations that represent the intervals of the various notes in the harmony relative to the bass line).

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Figure VII-4 The possible voice-leading patterns for descending by a third

motion, systematic use of the second would create too many incomplete chords, and the third results in constant alternation between close and open structure. None is particularly effective for sequences.

VIII. Characteristics of the Pachelbel Progression 2: Plagal Sequence In order to increase the voice-leading options, we apply the 5-6 technique again, but descending this time, creating passing chords, each leading to the dominant of the preceding chord (Figure VIII-1) – what is sometimes called a back-related dominant. This creates a sequence that is alternatively called the descending thirds sequence, the descending 5-6 sequence, the plagal sequence, or – most importantly – if the bass line is revoiced into root position, it is often called the Pachelbel sequence (though it is, of course, far older, predating tonality itself; it was described by Guilelmus Monachus in his treatise De perceptis artistic musice et practice compendiosus libellus in the late 15th century42). The sequence gives

I - V^ - vi - iii^ - IV - I^ - ii - vi^.

Of course, ii is a typical “mediant” substitution for IV, though we have to

break the sequence at the final chord to create a V, leaving us with the chord progression of the Pachelbel

Figure VIII-1 The use of the descending 5-6 technique to generate the Plagal Sequence The outer fifth of the first chord is expanded to a sixth, this time by lowering the root, and it reharmonised as a V6 chord.

42

(The Parallelism (the Pachelbelsequenz) n.d.)

Pachelbel, Purcell and Biber: The Variations-Canon canon.

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Derivations of this progression according to the principles of Schenkerian Theory and

Westergaardian Theory can be found in Appendix A. This chord pattern has several more options for voice-leading. Even using only a single voiceleading pattern, there are four completely mobile voices perfectly conducive to sequencing. Two of these voices descend by step every chord; the third voice remains oblique within each measure, and skips down by third between measures; the fourth voice alternates skipping down by fourth with rising by second. This voice-leading pattern is fully invertible quadruple counterpoint; even when chords are inverted to second inversion, they resolve (somewhat imperfectly) as either double suspensions or passing six-four chords (Figure VIII-2). This makes all four voices useful for creating variations (except perhaps the descending fourth/ascending second voice, only because it remains constantly in the bass of the Pachelbel canon, and so would cause parallel octaves). And, of course, the new motion within measure (motion by fourths) is one of the most flexible voice leading motions, allowing lots of other (non-sequential) variations to be constructed.

Figure VIII-2 Voice leading of Pachelbel Sequence as quadruple counterpoint Examples of the voice leading with all four voices in the bass, and with all pairs of voices inverted relative to each other. Notice that in (C), the six-four chords resolve as passing chords, while in (D), they resolve as double suspensions.

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IX.

The Variations of the Pachelbel Canon

T

he Pachelbel canon begins with a statement of the ground bass, and then repeats this ground an additional twenty-seven times with variations composed above it in the first violin (with twenty-six and twenty-five of these being imitated by the second and third violins,

respectively). It is tempting, then, to suppose that there are twenty seven variations. This, however, is the result of too shallow of an engagement with the variations themselves, and I will refer to these as subvariations. Kathryn Welter identifies only twelve variations, mostly consisting of two repetitions of the bass ostinato.43 This analysis comes from the recognition that nearly all of the twenty-seven subvariations come in pairs of similar figuration, generally in parallel thirds or sixths to each other, and consequently, these subvariations should be grouped together as a single variation. This produces the list

Table IX-1 Description of the groups of related subvariations

43

Group 1 2 3 4 5

Measures 3-6 7-10 11-14 15-18 19-22

6 7

23-26 27-30

8 9 10 11

31-34 35-38 39-42 43-46

12

47-50

12.5*

51-52

13

53-56

(Welter, 207-8)

Description Straight crochets, mostly descending (Theme) Straight quavers Straight semi-quavers Crochets and quavers, with some de-emphasized strong beats, and large skips Running scales in demi-semi-quavers, with some semiquavers is strong positions Alternating quavers and rests in roughly static voice-leading “Polyphonic” semi-quavers in coupled octaves, based on theme, figured primarily with neighbour tones Repeated semi-quavers, largely static within chords Semi-quavers, with second semi-quaver of each beat divided in half Slow rhythms, mostly one note per chord Dotted-quaver semi-quaver motive with some semi-quaver neighbours and passing tones Syncopated quavers and crochets creating suspensions, mostly following theme Mostly crochets, with dotted-quaver semi-quaver passing tones added to some beats, along oblique/third skips voice “Polyphonic” quavers coupled in octaves, mostly following theme

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of variations in Table IX-1.44 This is a superior analysis, but it, too, falls rather short of the mark. One of the consequences of the strict canonic principle is that each variation must be repeated in full by each voice (in the case of the Pachelbel canon, three violins). If all the variations were to be complex, they would constantly contest each other for the listener’s attention. If we accept the premise that pairs of subvariations sharing the same type of figuration are really a single variation, as in the groupings above, then we can also recognise a pattern between groupings: there is a rough alternation between complex and interesting variations with quick rhythms and melodic contours (generally, the odd-numbered variations) and simpler variations with slow or repeated rhythms and generally static contours that follow the simplest voice-leading that the chords will allow (generally, the even-numbered variations).45 This pattern is unusual among chaconnes. As a point of comparison, consider again the Clarke chaconnes. These chaconnes are particularly useful because they are remarkably similar to the Pachelbel canon, but are arguably less complex; this is a very important observation, because in spite of the generally lower complexity, none of the variations in the Clarke chaconnes are as simplistic as the simpler variations of the Pachelbel canon. Generally, Clarke plays a variation in the right hand, with a simple accompaniment – really just the bass line – in the left hand, then repeats the variation in the left hand, with a simple accompaniment – just simple block chords – in the right hand, and occasionally repeats it again in both hands together, then moves on directly to the next variation. There is no sense of moving

44

Note that my list is slightly different from Welter’s, but built according to the same principles. Also, note that for this list, I have described rhythms and measure numbers according to the original, four chord per measure notation. 45 I first became aware of these patterns during my undergrad; while performing in a guitar ensemble for a course requirement, we decided to do an arrangement of the Pachelbel canon, and being the only theorist in the group, I was asked to prepare the arrangement. Since some members of the group were either inexperience had too little time to practice the whole canon, my approach was to divide the canon into its variations and try to spread the most complicated variations evenly amongst the group, reducing the demands on everyone, so that each performer would need to learn fewer complex variations; obviously, the first task was to analyse the variations and sort them by complexity. When the arrangement came together much more quickly and easily than anticipated, it was apparent to me that the canon must have been composed already organised in this fashion.

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back to a simpler variation before returning to complexity; this characteristic is distinctly Pachelbel. On the other hand, the idea of the variation passing from one hand to the other is quite similar to Pachelbel passing the variation from one violin to the next, which supports the interpretation that these more complex variations are the true variations, while the simpler variations correspond better to the simple bass line and block chord accompaniments found in Clarke’s de-emphasised hand. This alternation of principle and accompanimental groups of subvariations is rather significant. It is a consequence of the canonic principle and the need for variations not to vie with each other for the listener’s attention. Since there are three violin parts, and each of the violins are separated by one statement of the ostinato, it takes four full statements of the ostinato for each of the complex subvariations to run its course (Figure IX-1). This is, in fact, an interesting play upon the usual idea that many chaconnes repeat each variation; in this case, however, each variation is repeated three times, in an overlapping fashion, which still results in the variation lasting for double its discrete length. If you were to remove the second violin, the repeat structure would appear perfectly conventional, except that each theme would be passed back and forth between the first and third violins, and the first and third violins would correspond remarkably well to the right and left hands of the Clarke chaconnes – however, the parallel thirds or sixths that actually exist between halves of the same variation of the Pachelbel canon would be lost. Violin 1

Var. A

Var. A’ Var. A

Violin 2 Violin 3 Ostinato

1

2

Var. B Var. A’

Var. B’ Var. B

Var. A

Var. A’

3

4

5

6

Var. B’ Var. B

Var. B’

7

8

Figure IX-1 Effects of the canonic principle on the length of variations. The notation Var. A and Var. A’ represent the first and second subvariations of the first variation. Note that if Violin 2 were to be removed, the resulting combination would resemble a simple, standard repeat structure.

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As you can see from Figure IX-1, the practical consequence of this effect is that in each canonic voice, there are two empty cycles between important variations. To prevent the conflict for attention with the principle variations, the material that is written into these measures must be simple and unassuming – and that is, in fact, precisely what we have observed. In truth, then, the material that fits between these variations are not true variations – in the same way that the bass ostinato is not a variation – but rather accompanimental material. As such, it could be defined as actually being and extension of the variation before it; however, it could equally be defined as being part of the following variation. Since there is no reason to prefer one interpretation or the other, if we are to label these subvariations at all, we need a label that describes both possibilities. As it happens, musicians already have such a labelling scheme inherent in the concept of enharmonic equivalence; we can apply the rather whimsical but effective label of Variation AS/Bs. Figure IX-2 shows the order of variations, with the accompanimental material added. Violin 1

Var. A

Violin 2

Var. A Var. B

Var. A

Var. A’

Violin 3 Ostinato

1

S s

Var. A’

2

S s Var. AS Var. Bs

Var. A ’ Var. B ’

Var. A

Var. A’

3

4

S s

S s Var. BS Var. Cs

Var. B

Var. B’

Var. B Var. C

S s Var. AS Var. Bs

Var. B

Var. B’

Var. A ’ Var. B ’

S s

Var. B

Var. B’

5

6

7

8

Var. A ’ Var. B ’

Var. B ’ Var. C ’

Figure IX-2 Variation scheme, with accompaniment added.

S

s

The notation Var. A and Var. B represents the accompanimental material between Variations A and B.

This concept, philosophically, also implies that the first group of subvariations also ought not be thought of as a variation. It is important to remember that while the theme of a variations form like a chaconne may be a bass ostinato, it may equally be a chord pattern, and it is traditional to introduce the theme before beginning the variations. Since the piece is being spun out as a canon, the first three voices needed to fully outline the chord progression would take three cycles of the ostinato to complete. Hence,

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Figure IX-3 Variations Analysis of the Pachelbel Canon The sharp and flat notations mark variations that are strictly accompanimental. Prime markings (‘) indicate the second half of a variation, generally in parallel thirds or sixths. Where they are separated by more than an octave, grace notes are added at the octave to make the parallels more apparent. Page 1

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Figure IX-3 Variations Analysis of the Pachelbel Canon Page 2

the first group of subvariations ought to be considered part of the theme. According to that viewpoint, the piece begins by stating the theme, followed by only six true variations – each repeated according to the odd, overlapping repetition scheme described above – counterpointed by accompanimental material. T here is only one tiny interruption to this scheme, in that the accomapnimental material between the

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last two true variations contains one extra cycle. The reason for this extra cycle appears to be the provision of an appropriate third upper voice to counterpoint the final group of subvariations for the final cycle and cadence. The material that would be present in that voice if not for this additional cycle contains some quirks that would make it less suitable for the final cycle and cadence (these quirks will be discussed in section XII). A complete breakdown of all the variations according to their functions is presented as Figure IX-3.

X.

Macro-structural Form of the Pachelbel canon

W

e have already observed the combination of pairs of subvariations into related groups, and the alternation of principle variations with accompanimental material, so that there are, in total, only six variations beyond the theme. There is, in fact, a further

level of organisation. As already described in section IV, it is typical for variations forms such as the chaconne to steadily increase in complexity to a climax near the end, before slowing down to a simpler ending. However, the Pachelbel canon does not follow this pattern exactly. Considering only the six primary variations, there is also a kind of rough alternation between slower and faster variations. Variation A is semiquavers, then Variation B is demi-semiquavers; Variation C is back to semiquavers, while Variation D again includes demi-semiquavers; Variation E returns to semiquavers, and Variation F slows to quavers. As a general description, then, the variations increase in complexity to measure 22, then fall back and increase again to measure 38, then slow back down to the end. This overall pathway can be understood in multiple ways. In terms of a comparison to narrative structure, it is a somewhat superior comparison than the simple model of increasing complexity normally associated with a chaconne. In complex narratives following the structure or Aristotle’s Incline, there is an introduction, and then rising action to a local maximum at a crisis – a kind of failed climax in which the

Pachelbel, Purcell and Biber: The Variations-Canon

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hero is unsuccessful, suffers a loss of faith, and has to drag himself back up and try again, initiating another stretch of rising action to the true climax, followed by a dénouement. In the Pachelbel canon, the introduction is obviously represented by the statement of the theme, followed by the first wave of increasing complexity through Variation B to a failed climax, a fall back to Variation C and the second wave of increasing complexity to variation D. Note that while the “wave of increasing complexity” seems to only last two variations, the canonic texture and the fact that the two halves of each variation are composed in parallel thirds and sixths means that each variation in and of itself contains two levels of increasing complexity, giving the overall rise in complexity four levels. Following the climax, the denouement is represented by Variations E and F. The final group of accompanimental subvariations between Variations E and F is unique amongst all the other accompanimental subvariations, in that it encompasses three subvariations. Since all of the other accompanimental groups are two subvariations long, and the gap between principle variations caused by the canonic structure is also two subvariations long, therefore all of the other accompanimental groups are “hidden” behind the more complex variations they accompany; since this last one, however, contains an additional strain, it is long enough to be in the foreground for one brief strain; in a sense, it is a kind of failed principle variation. It is also in some sense actually slightly more complex than the final variations, in that it while it is slightly slower, it also contains syncopation and dotted rhythms; it therefore

Figure X-1 Comparison of Narrative Structure with the Pattern of Complexity in Pachelbel’s Variations

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contributes to the overall decrease in complexity to the end. Figure X-1 compares the level of complexity of the variations to traditional narrative structure. Another way to describe the overall form is according to traditional musical forms. There are two obvious possibilities. The first is to see the whole chaconne (after the introduction) as being structured in three parts consisting of two principle variations each. The first variation of each section (Variations A, C, and E) is semiquavers; the second variation of the first two sections (Variations B and D) is then demisemiquavers, while the second variation of the final section (Variation F) breaks this pattern to go to the traditional simpler ending. The other approach is to see the piece as a kind of strange rondo. Although the melodic material is not repeated in an alternating fashion, the degree of complexity does alternate in a predictable way. If the semiquaver variations are collectively labelled A, and the demi-semiquaver variations are collectively labelled B, then the structure does indeed show the expected form A B A B A, followed by the final principle variation, which would serve as a codetta. Even the extra accompanimental subvariation might then be described as a bridge to the codetta.

Example X-1 Exception in the otherwise strict canon in the manuscript From manuscript Mus.MS 16481/8. Highlighting added by author. (The other difference between parts, the hatch marks across the stems of the crochets, is an articulation sign for a measured, single-pitch tremolo, a shorthand for semiquavers, and so is not actually different).

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There is one more interesting formal element – if we allow ourselves to indulge in a certain amount of speculation. Up to this point, we have assumed that the Pachelbel canon is a strict canon from start to finish; however, this may not be true. In the surviving manuscript, there is one very small exception: in

S

the final chord of subvariation C ’, which runs in repeating semiquavers (tremolo style), both comes voices

S

S

begin with one semiquaver C 6, then drop down an octave for the remaining three semiquavers at C 5;

S

however, in the dux, all four semiquavers remain at C 6 (see Example X-1). My first instinct for this was to identify this as a copyist’s error. However, on closer analysis, there are reasons to question this assumption. To begin with, the exception occurs in the dux, not the two comes voices. A typical copyist, copying a canon, would normally begin by copying the dux, then copying the comes voices, probably from the dux he has just copied, rather than from the original source (it is easier and more efficient to copy a part from two measures away than to keep looking back and forth between the original and the copy). If this procedure had been followed, it is unlikely that the changes in the two comes voices could have been introduced by mistake. And it is immensely more likely for a figuration like this to be accidentally missed than to be accidentally introduced. In addition, the note that has been transposed is a leading tone on the dominant chord, which is then resolved in its own register in the two comes voices. It is also less likely that a musically literate copyist would miss the fact that the leading tone is not resolved. Under such circumstances, while it is clearly not conclusive, it is certainly plausible that the single difference between the dux and the comes voices may be deliberate. This hypothesis is supported by the fact that the single exception occurs at a particularly suggestive

S

moment. It occurs right at the end of variation C, both in the sense that it is the end of subvariation C ’ in the dux, and the end of subvariation C’ in the second comes, so that it is the end of the last time the principle portion of variation C is heard, and immediately before the beginning of variation D. If we ignore

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the presentation of the theme and the accompanimental variations associated with it, as we have done in our analysis of the canon as a three part form, then the exception occurs at the exact middle of the piece (after the third of six variations). The exception is a particularly high leading tone, leading to the expectation of a strong cadence. The conclusion to draw from this is that Pachelbel may have intended for this moment to be formally significant, suggesting a kind of two-part form. The idea that the canon may be a kind of two-part form would seem to conflict with the idea that it may be a kind of three-part form. However, there is precedent for this concept. We have already acknowledged the indirect relationship between Pachelbel and J.S. Bach (whose older brother was a student of Pachelbel). One of Bach’s crowning achievements, the Goldberg Variations, while not a variations canon as such, has much in common with the variations canons that we have been discussing (more about this in section XXV). It also has a characteristic particularly relevant to the present discussion: it is well established to be divided formally in a variety of complex ways, including ways that overlap with each other. The most obvious division of the Goldberg Variations is in half. Discounting the opening aria (the theme) and its da capo, there are thirty variations, and the sixteenth – the one beginning the second half – is an overture. However, there are other ways in which the variations can be grouped. One of these46 is to divide them into three overall groups. Both variations ten and twenty – the dividing points for thirds – display notable characteristics. The tenth variation, in particular, is a fughetta, and fugues are frequently A

A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Figure X-2 Groupings of Variations Noted in the Goldberg Variations Note that the variations, while grouped principally into threes and in half, are also grouped in three sections of ten, which overlaps the other two groupings, allowing the Goldberg Variations to display elements of both a large-scale binary organisation and a large-scale ternary organisation at the same time.

46

We shall examine other patterns in section XXV.

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placed as the last movement in a suite of movements, which implies that the tenth variation is the end of the formal section. The twentieth variation is a virtuosic toccata with rapid hand crossing, which also gives the sense of being the climax of a section of variations, just as the more complicated variations of the Pachelbel canon might mark the end of a formal section. In summary, the Goldberg Variations display characteristics of both two-part and three part-forms (see Figure X-2). In Bach’s case, we know that it was prompted by his fascination with numerology; however, if we can say that Bach intended his Goldberg Variations to display elements of both two- and three-part forms, we should equally be able to suppose that Pachelbel, working in a similar geographical and historical context, and even with an indirect connection between the composers, may similarly have chosen to incorporate elements of both two- and three-part forms into his variations in the canon.

XI.

I

Voice-leading: Canon in D as a Stacked Canon

n the last couple decades, there has been a resurgence of interest in canon. Of particular relevance to the Pachelbel canon is the concept of stacked canon. Alan Gosman coined the term47 to describe a canon of at least three voices in which the canonic rule describing how the first comes imitates the

Figure XI-1 Example of a stacked canon at the fifth with a one measure delay Note that the relationship of comes 2 to comes 1 is precisely the same as the relationship of comes 1 to the dux – one measure delay, imitation at a perfect fifth.

47

(Gosman, 290)

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dux is then reapplied identically to describe how each additional comes voice imitates the previous comes voice. For instance, the in a stacked canon at the fifth at a delay of one measure, the dux begins, then one measure later, the first comes enters a fifth higher, and then another measure later, the second comes enters an additional fifth higher (overall two measures higher and a ninth above the original dux), and so on (Figure XI-1). Gosman’s article was released almost simultaneously with articles on canonic compositional technique by Robert Morris48 and Robert Gauldin.49 Both of these articles discuss similar issues to Gosman, but they do not directly address the concept of stacked canon. The concept of the stacked canon was expanded upon by almost immediately by David Burn,50 and has since been elaborated upon by other authors, and recently in particular by Jurjen van Geenen.51 It is important to note at this point that Gosman, the earliest author to have worked directly with the concept of stacked canon, explicitly excluded canon at the unison from the definition of a stacked canon, even when multiple voices exist that follow the same canonic rule. He noted that “Renaissance composers judged this [stacked canon] as a particularly interesting effect when the interval of the canon was not a unison, and I include this condition in my definition.”52 Burn, the next author to deal with stacked canon directly, followed suit.53 However, in spite of the deference normally owed to those who coin a term, it is my opinion that it was inappropriate to exclude imitation at the unison from the definition of stacked canon with no better reason than that it is “less interesting”. The conclusions that Gosman and Burn reach, and the characteristics that they describe, apply equally well to three or more voices following the same imitation scheme at the unison as they do to those employing imitation at other intervals.

48

(Morris) (Gauldin) 50 (Burn) 51 (van Geenen) 52 (Gosman, 289) 53 (Burn, 74) 49

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Questions of form and technique are not generally restricted to those cases that are “interesting”. There are many uninteresting compositions in the repertoire, but we do not exclude them from their formal categories on such grounds: a fugue that is not “interesting” is still a fugue; a concerto that is not “interesting” is still a concerto; and a stacked canon that is not “interesting” is still a stacked canon, etc. And in this sense, the Pachelbel canon is a stacked canon: it has three canonic voices, each related to the previous according to the same rule. Gosman and Burn may disagree with my assertion, but the techniques that they, and those that followed them, have described will nevertheless be useful in our examination of this piece. It may appear on the surface that composing a canon should not be any great challenge. If one has mastered the basic rules of counterpoint, it would seem reasonable that to write a canon delayed by one measure, one could compose a measure in the dux, then copy that measure into the next measure in the comes, and write the following measure of the dux as counterpoint against the newly copied comes. In principle, this method should work, but it has some flaws. There is no guarantee that the new dux measure, having been written to work well against the previous measure, will lend itself to good counterpoint in the next measure – that is, when you write the end of the dux measure, then copy the measure into the comes, will the melodic notes implied for the beginning of the new measure by the last note of the dux agree with the notes that are harmonically appropriate to sound against the first note of the comes? (Figure XI-2). Additionally, a canon written one measure at a time can feel aimless and

Figure XI-2 The problem with writing canons one measure at a time Note that in the third measure of the dux, the melody from the previous measure – particularly the leading tone – strongly recommends the C; however, the D in the comes is forced by the previous measure of the dux, and the C forms a second against it. The second measure harmonises well according to the rules of counterpoint, but it makes the continuation difficult.

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meandering, since it leaves the harmony to become whatever it will, rather than preplanning it – in particular, this methodology does not lend itself to writing good cadences at appropriate moments. 54 Clearly, more sophisticated techniques are desirable. Each of the authors mentioned above has examined the techniques that may be used to compose a melody that will form an acceptable canon according to whatever rule is selected, and several different ways of approaching the problem are proposed. Most are concerned only with the formation of acceptable counterpoint in the Renaissance style, which is appropriate, since most of the articles are specifically interested in Renaissance canonic procedure. This method involves composition first of a simpler sketch of the canon in what is called “first species canon”. This is analogous to the concept of species counterpoint, in which the first species is to write one note in the contrapuntal voice for each note written in the cantus firmus; in first species canon, we write one note in the dux for each cycle of the canon (i.e. if the delay of the canon is one measure, then first species canon writes one note in the dux per measure, etc.). Interestingly, the first approach to applying this methodology to stacked canon (Gosman) achieved acceptable harmony through instructions on how to handle not the harmony itself, but rather, the melody. The method is a kind of chart that shows which melody notes are possible in the next measure, based on which note is present in the current measure. For instance, if we are dealing with a stacked canon at the fifth, and the current note in the dux is G, we know that in the next measure, we will have a D in the comes. As a result of the D in the comes, the dux can only support same note or notes related by a third, a perfect fifth, or a sixth to the D (that is, D, F/F#, A above/G below, or B). Rather than thinking ahead about which note will be in the comes (D), and deciding which notes will form acceptable harmony, we instead simply associate these possible pitches melodically as the pitches that can follow G. In this way, we created good harmony, while ignoring

54

(Gauldin, 49)

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harmony, but following melodic rules. Although, from our modern perspective, this seems like an unusual way to handle harmony, it has a very long tradition, going back at least to the thirteenth century; this is, for instance, the method employed in ars organi (a.k.a. the Vatican Organum Treatise) to describe how organum (a very early form of mediæval polyphonic music) was to be created. Creating organum has some similarity to creating canon: just as, in a canon, the comes is pre-written to match the previous measure of the dux (and so cannot be changed to accommodate the harmony, as it would change the previous harmony in which that part has already participated), in organum, one part is also already prewritten – the pre-existing traditional melody we call the cantus firmus, which likewise cannot be changed (by tradition) – and the upper voice, called the organum, is written above it.55 The treatise imparts thirty-one rules for the creation of the organum voice. The first rule says: “Si cantus ascenderit duas uoces et organum incipiat in dupla, descendat organum 3 uoces et erit in quinta”56 – more or less: if the cantus firmus moves up by the melodic interval of a second, and the organum voice begins at a harmonic interval of an octave compared to the cantus firmus, then the organum voice should descend by a melodic interval of a third, so that they will together form the harmonic interval of a fifth. All thirty-one rules follow this basic model. Here, we again see an example of creating the desired harmony by giving instructions on how to move the melody. The only real difference is that for organum, we must know both the original harmony and the melodic motion of the cantus firmus before we can

55 They even use similar terminology to describe parts: the cantus firmus, in older treatises, is frequently called the “preceding part”, and the organum voice is often called the “following part”, just as dux and comes mean “leader” and “follower”, respectively. For instance, in the early twelfth century organum treatise Ad organum faciendum (“On how organum is to be made”), we find: “Sciendum est enim organales voces affinitatem habere cum præcedentibus” (Huff, 43) – approximately, “Indeed, the notes of the organal voice are to be understood to have an affinity with the preceding notes [præcedentibus]”. (Huff notes in his translation, which differs from mine only in phrasing, that the word præcedentibus should be understood as meaning the notes of the cantus firmus) (Huff, 42). 56 (Anonymous)

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decide how to move the organum, while in canon, we need only know the original pitch, because the canonic rule gives us the rest of the information. However, this method still results in relatively poor control of harmony (even though the harmony is guaranteed to be acceptable, it is still created incidentally) – and this is perfectly acceptable in the Renaissance, since it is quite typical of Renaissance music for the harmony to be generated incidentally by the voice leading. On the other hand, Pachelbel was a mature Baroque composer, and questions of strong harmonic progression, voice-leading, and complete chords and doubling would have been of greater significance to him than to Renaissance composers. In particular, the aforementioned approach does not work well when the harmony is predetermined, with many chords between the beginnings of subsequent cycles, as is true of the Pachelbel canon. (It will become useful when we discuss the Purcell canon in section XX). Instead, we need a method that accommodates those chords between cycles – between notes in the first species canon. This can be achieved by writing the sketch in second species canon (two notes per canonic cycle), which gives two chords per cycle; this method is proposed by van Geenen, based on relative chord tones. Van Geenen’s approach is based on the assumption that the inclination of a canon to maintain the

Figure XI-3 A stacked canon at the second below, showing the chord-tone cycle. The harmony of this canon follows the circle of fifths. In each voice, not only are the notes imitated, but they also maintain the same function (root, third, fifth) within the chord; this only works because the cycle is maintained perfectly; any chords substituted by its “mediant” would also disrupt the cycle of chord tones.

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same kind of chord motion in a cycle (i.e. canon at the fifth following a cycle of chord motion by fifth, canon at the unison maintaining consistently the same chord at the delay length of the canon) is not only a strong inclination, but is, in fact, desirable. When you follow this cycle, there is imitation not only of notes, but also of chord tones. If the note in the dux is the root of the chord, it will also be the root of the chord when it reappears in the comes (Figure XI-3). The consequence of this fact is that a composer can choose to completely ignore the actual notes themselves, and concern himself exclusively with the chord tones. Let us simplify for a moment, and imagine that there are only two chords. For the sake of argument, suppose that we are writing a stacked canon, again at the lower second, this time for four canonic voices, and that the chord cycle is therefore likewise a continuous cycle of descending seconds. In addition, suppose that there are two chords per cycle. Combining these two ideas allows us to create the most familiar continuous chord pattern in Western music: the descending fifths sequence – we’ll even make the second chord per measure a secondary dominant seventh chord. We can actually ignore the notes that are needed, and just create a path – one complete path that covers each chord tone once, in a single circuit – between the chord tones of the chords. It is even a fairly simple task to control the voice-leading. The voice-leading concerns are straightforward: the only way to create parallel fifths (other than illusory ones caused by non-chord tones in the process of embellishment that will come later) is to connect fifths to each other directly (i.e. in the same voice); we will avoid this – that is, we will not allow a line between the fifths of the two chords. We also need to resolve the seventh downward, which in the cycle of fifths should resolve to the third of the following chord, so we will obviously draw the line between the seventh and the next third first. Also, ideally, we would like to resolve the third of the secondary dominant upward to the following root, so we will draw this line next. I also would like to have complete chords, which cannot occur together with all of the aforementioned priorities as long as both chords are in root position; fortunately, we also probably don’t want all the chords to be

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in root position anyway, because it would result in a never-ending sequence of perfect authentic cadences. Instead, we will put the secondary dominant chords in first inversion (we have already decided how the third should resolve), so the fifth will resolve down to the root, and the root will hold over to the fifth – we will add these lines. The resolution of the first chord to the second is traditionally less rigid; we will still not allow parallel fifths. Next, we will take care of the bass line, with the root moving to the bass (third) of the secondary dominant. The remaining tones will be connected in whatever way is necessary to ensure that all the chord tones are connected in one single path. This is important, and it has an important effect: when we resolve the third of the secondary dominant – the bass of the first inversion seventh chord – to the root of the next chord, we need to resolve it to the octave of the root, not the root bass. If we connect the two roots together in both directions, which we would ordinarily do in a non-canonic composition, we would be separating the bass notes from the rest of the chord tone cycle, and we need all the tones connected in a single cycle. The correct root position of the chords on the downbeat will be taken care of anyway, as each new voice is added; later in the canon, when no new voices are added, we can correct the position of the chord by moving the voices mid-chord. The seventh also needs to be prepared, usually

Figure XI-4 Voice-leading diagram Shows the voice-leading of a simple stacked canon at the second below, with a repeating chord cycle of .

ii - V&

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from the third of the previous chord. Unfortunately, we could end up with the same isolation problem, with the third preparing the seventh, then the seventh resolving back to the third. As with the root/fifth problem, the solution is to prepare from and resolve to different thirds. Since we obviously don’t want to begin the chord with a doubled third – and since we don’t need to keep the doubled root, we can shift from the root to the third (I have labelled them as octave and tenth) before switching chords. We will probably also want to do this to avoid parallel fifths, by resolving the first fifth to an octave, before switching to the fifth. Figure XI-4 shows this diagram of our hypothetical canon using the methods described by van Geenen. This very small and simple diagram is a snapshot of the entire voice-leading pattern of the canon. A possible realisation of this diagram is shown in Figure XI-5.

Figure XI-5 Hypothetical stacked canon at the lower second based on relative chord tone diagram in Figure XI-4 Upper system is directly from the relative chord tone diagram (more or less “First Species”). Second system is the same canon, decorated with nonharmonic tones.

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Since the Pachelbel canon, under our slightly modified definition, a stacked canon, it needs to follow this procedure as well. If a canon cannot be described according to a similar diagram, then some of the chords will be incomplete – not necessarily a problem, but not ideal – and there may also be inappropriate chord doublings. Of course, the Pachelbel canon has many more chords in between the beginnings of subvariations. This, in theory, allows much greater flexibility; however, in practice, Pachelbel does not choose to take advantage of this flexibility to anywhere near the degree that he could. In none of his chords does he chose to follow the cycle exactly, but each of the chords is dominanted by following the cycles more often than not. For instance, the chord tones for the first chord of each subvariations are: 3

1

1

3

1

1

3

1

5

3

1

5

3

1

5

3

1

3

5

1

3

5

1

3

5

1

3

5

As you can see, after some initial variation brief non-conformance, the rest of the canon contains only one breach of the cycle on the first chord of each subvariation; even this brief non-conformance almost fits: in both the third and sixth occurrences of the chord, where a root appears when a fifth was expected, the diminutions57 of the melody are a filled-in ascent from root to fifth (the third as an arpeggio of quavers, and the sixth as a scale of semiquavers) that arrives on the next beat, not an unusual way to create a

Figure XI-6 Voice leading correction after Variation C' While the A (root of the V chord, fifth of the I chord) is common, and could be kept, this is a cadential gesture leading into the next variation, and the oblique resolution would sound odd; Pachelbel first resolves to the third, then returns to the fifth.

57

n.b. In this context, the term diminution does not refer to either a diminished interval (diminished fifth, etc.) or rhythmic diminution (repeating a given rhythmic figure again, in shorter rhythmic values, but maintaining the proportions of durations between notes); instead, the term diminution here is used, as it often was in old counterpoint treatises, to mean the breaking of a long note in a slower species of counterpoint into multiple notes, creating a higher species of counterpoint – essentially the process of turning a simple counterpoint exercise into a true composition OR the process of creating a more complex variation on a previously stated melody.

Pachelbel, Purcell and Biber: The Variations-Canon

Subvariation A

s

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Subvariation A’

s

Figure XI-8 Beginnings of subvariations A and A’ In both of these subvariations, the melody rises from an opening tonic to a fifth, a typical opening gesture, and it is easy to understand these figurations to suggest that the A that ends this initial ascent is the intended tone.

variation on a predetermined simple melody (Figure XI-8); in that sense, the entire cycle of first beats of variations could be understood as being made up of only two cycle groups right from the beginning. It is similarly important to note other cases when the critical note of the variation is not the first

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note. Consider Variation C , for instance, in which the first semiquaver is not the expected note, but the entire rest of the beat consists of repeated semiquavers on the expected note. At times, this is done to allow a proper resolution of the preceding note before moving onto the preferred note. For instance, although we might expect the root of a passing

V chord to resolve to the fifth of the I chord, we would

not expect that to occur in a cadential gesture between the ending of one variation and the beginning of the next. This is what happens in the end of variation C’ moving to variation D. It would sound quite odd to play the neighbour tone figure on the previous beat, then proceed obliquely to the downbeat; hence, the resolution down to the third is used to smooth out the resolution, before immediately moving back to the fifth that the cycle requires (Figure XI-6). In addition, in some cases, even the moment when the cycle breaks can be smoothed over either by reversing the sequence so that one version overlaps, as in

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the cycle of the first chord (shown above), or by providing both tones, as in subvariation E ’ of the cycle

Figure XI-7 Cycle-pivoting in Variation E#’

1

5

In this measure, the D ( ) is expected, and it is given, but the beat is split, and the A ( ) is added, which agrees with the notes needed for the new cycle group.

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Table XI-1 Chord tone cycles in the Pachelbel canon

Note that for the purpose of these cycles, 6 and 5 will be treated as the same chord tone, since they represent the single fluid chord tone not shared between two mediant-related chords. Lighter circles indicate an approximate, but non-strict, cycle (individual discrepancies noted with smaller elipses).

of second chords of second measures (Figure XI-7). The chord tone cycles of each of the chords in the chord progression are shown in Table XI-1. Of course, with so many chords per cycle, there is no reason why all of the chords need to maintain the cycle exactly (it is normal for some chords to occasionally be incomplete), and indeed, there is only a single four cycle stretch of the canon that maintains a strict cycle for all eight chords – hardly long enough to be significant. However, we should expect to see large sections in the canon in which the voice-leading of some significant chords should hold to the cycle rather well. To verify this, we must determine which chords are significant. There are four likely candidates. The first is the chord exactly at the midpoint of the progression ( each measure (

IV), significant because of its prominent metrical position.

Similarly, the midpoint of

vi and the other IV, respectively) might also be significant, those to a correspondingly

lesser degree; these are also the chords previously indicated as the basis of the sequence. The fourth

Pachelbel, Purcell and Biber: The Variations-Canon candidate is the final chord, a

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V chord that participates in the cadential gestures between variations,

which needs resolution. Reviewing the chart above, we can see that there are many cases of multiple chords agreeing with each other for quite long periods of the canon, especially the first two chords. However, two cases are particularly significant, involving chords we have identified as significant, and collectively lasting nearly eighty percent of the length of the canon. Beginning at the seventh subvariation, the middle chord maintains a strict cycle for forty percent of the canon (coincidentally, the middle chord of the first measure also fits for this whole stretch, but not the middle chord of the second measure), and the final chord maintains a strict cycle for the entire last forty percent of the canon. Notably, the closer we get to the end of the canon, the more chords participate in the strict canon, until five cycles from the end, when only the second last chord does not participate – this shouldn’t surprise us, because it is related to the following chord by a second, and therefore must be carefully controlled to avoid parallel fifths. This gradual standardisation of the voice-leading towards the end aligns nicely to the usual characteristic of chaconnes winding down in complexity as they near the end.

XII.

Notable characteristics of specific variations of the canon

T

he above table (Table XI-1) is also useful for another reason: it allows us to determine when the substitute chords are used. The substitute chords are all identifiable by the presence of the sixth chord tone. The sixths appear in only two rows: the

iii chord and the second IV chord.

These are the two chords that are substituted with their “mediant” first inversions, already discussed the use of the

I^ and ii^.

We have

ii^ chord near the end of the canon, first ambiguously as a decoration of

IV that anticipates its use as a true ii^ during the final cadence. However, the list of chord tones provides us even more detail, because each chord tone remains in effect for three subvariations. Interestingly, the last five chord tones are 6 3 5 6 1, and the two sixths are only three cycles apart (and less than three cycles

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from the end) – note that the sixth showing up every three subvariations is yet another expression of the principle of the chord tone cycles. That means that the sixth remains in effect from its first appearance

S

(in subvariation E ) right through until the end of the piece. However, the fifth chord tone also appears among the last three subvariations, together with the sixth. That means that the last three subvariations actually use not the

ii^, but rather the ii%.

This is one of the only uses of a seventh chord in the entire

canon, which only intensifies the sense of expectation of the end of the piece. The

I^ chord, meanwhile, clearly makes its appearance at the eighth subvariation (Subvariation AS’),

and reappears three subvariations later, so remaining in effect for the next six subvariations (until subvariation C). It then reappears at after an additional eight subvariations (at Subvariation E), and again repeats another three subvariations later, so it once again remains in effect for six subvariations, lasting until one subvariation before the end – it is essentially present for most of the first and last thirds of the large-scale three-part structure we have noted, and is generally absent in the middle section, reinforcing (admittedly weakly) our analysis of the form in three parts. Notice that it returns to the more prototypical IV chord for the final subvariation. In the case of

iii and I^ chords, none of the sixths overlap with a fifth,

so it never forms a seventh chord in this way. There is another notable characteristic to some of the variations. We have already noted the degree of harmonic stability in the canon. Indeed, in a joint article in Congressus Numerantium, Phyllis Chinn, Wesley Chinn, and Sami Shumays use the Pachelbel canon as an illustration of the harmonic patterns of simple canons, and remark that: The harmony in this particular canon “works” because each phrase is constructed over the same progression of harmonies, which are repeated throughout the canon in pairs of measures, any of which could be played simultaneously without sounding dissonant. The trade-off for this easy way to assure that the chords created by the translation in the melody sound consonant is that the piece never “goes anywhere”

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harmonically. The only change in sound as the piece continues is in rhythmic and melodic variation.58 [emphasis added] This, however, is not strictly true. We have already remarked upon the chord substitutions – for which, if certain variations were to be combined, intervals of a second or a seventh might result (for instance, if one variation used a 1 as the root of a I^ chord, while another used a 7 as the fifth of a iii chord). This, however, is not the most drastic such example. It is true that there are precious few accidentals of any kind in the Pachelbel canon. Virtually all the chords are diatonic. However, there are three C-naturals in this canon in the key of D-major, all in close proximity to subvariation E (Figure XII-1). These variations clearly cannot be combined with any variation containing a C-sharp in the same position, as this would create not only a second/seventh, but also a “split tone” (the use of two versions of the same scale degree or chord tone), which sounds substantially more dissonant than the same interval taking place between different tones. To a modern listener, steeped in the tonal tradition, these C-naturals appearing from nowhere in the midst of such a heavily diatonic composition sound somewhat jarring, and are difficult to explain, particularly the first two, which are in the context of G-Major chords. We are always on shaky ground when we attempt to guess the reason why a composer made the decisions he made; however, since one of the themes of this paper is the consequences of applying the canonic principle to a variations form, it is worth a look to see whether or not these unexpected C-naturals might have been predisposed by the

Figure XII-1 The unusual C-naturals of the Pachelbel canon

58

(Chinn, Chinn and Shumays, 180)

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constraints of the canonic form. Since the first two are more difficult, and since they come chronologically sooner, we shall consider these first. There are two possible conventional traditions according to which we might try to explain why Pachelbel chose to write in these accidentals: the modal tradition, no longer dominant at the time when Pachelbel was composing, but still maintaining an influence, particularly in education (indeed, the study of counterpoint based on modal models, as exemplified by such writers as Fux, remained typical long after the time period in question, even – to a lesser degree – up to the present) and the tonal tradition, newly the dominant, but by no means the exclusive musical system in use. Since a typical reason under the tonal system does not immediately suggest itself, and the use of the accidentals is unexpected to those familiar to tonal music, it is logical to look first at the modal system. The modal system would suggest two typical reasons why an accidental (in modal parlance, musica ficta) might be used. The first is the avoidance of the tritone, according to the traditional phrase mi contra fa est diabolus in musica59 (“mi against fa is the devil in music”: the simultaneous use of two notes respectively representing mi, the third note, and fa, the fourth note, of two adjacent hexachords60 – A

B

Figure XII-2 Mi contra Fa in Pachelbel (A) Mi from the natural hexachord against Fa from the hard hexachord forms a tritone. (B) Transposed into the key of D-Major, this occurs between C-sharp and G, which would come into effect during the G chords of the canon.

59

c.f. (Fux, 51); (Schulter, 2.4) In Mediæval and Renaissance theory and vocal pedagogy, a hexachord is a six-note scale with only a single semitone, placed between mi and fa, so that, by extension, any note directly below a semitone can be understood as the note mi, while any note directly above a semitone can be understood as the note fa. The three traditional hexachords are the hard (starting on G), the natural (starting on C), and the soft (starting on F), and the adjacent hexachords are spaced a perfect fourth apart from each other. (Schulter, 1.3) 60

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Figure XII-3 A conspicuous case of the mi contra fa rule not being applied in the Pachelbel canon Note that the figuration here is quite similar to the first two C-naturals of the canon, yet the C is not lowered in this case.

alternatively phrased as the mi and fa, coming from two hexachords, which form a perfect interval harmony – tends to create dissonance, particularly the tritone, and should be avoided). In principle, this rule would dictate that any C-sharp played during the context of a G major chord might reasonably be expected to be altered to a C-natural (Figure XII-2). This might possibly explain these first two C-naturals; however, they are not the only examples of C-sharps over a G-Major chord, and the others are not altered. There are examples in measures 14, 20, 28, and 38, incorporating a variety of different motions, including a similar context to those in which the C-sharp has been lowered, yet they have not been likewise altered (Figure XII-3). The second typical reason for an alteration in the modal system is the lowering of some upper neighbours to form semitones, according to the maxim una nota super la semper est canendum fa61 (“one note above la always is to be sung fa” : when an ascending melody ascends to one note above la - la is the sixth note, so ascending to the seventh – of a given hexachord, then turns back around, the note A

B

Example XII-1 Una nota super la rule, with a conspicuous non-conformance in the Pachelbel canon (A) The note above la doesn’t exist in a traditional hexachord; we solve the problem by mutating into a different hexachord. While we could mutate into a hexachord on G, and sing the extra note as mi, the rule says that since we have only one note above la, we must sing it as fa in the hexachord on F. (B) However, the Pachelbel canon does not follow this rule. In this case, the note is not fa (it would have to be C-natural, but it hasn’t been lowered in this way).

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(Schulter, 3.4)

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should be treated as fa, the fourth note, of the adjacent hexachord, and hence should be only a semitone higher than the sixth); this reason does not apply to the C-naturals in question, which are in the context of descending motion, and might potentially apply to one case, measure 28, which is not altered. We have established that it is possible but not necessarily convincing to attribute these C-naturals to modal practices. Unfortunately, as we have already noted, they also do not seem to be easily explicable according to tonal practices. The most typical uses of a C-natural in the context of tonal music would be as borrowed chords (especially the applied dominant chord

V/IV), which does not apply to these cases,

or 4-3 suspensions, which these also are not. A third possibility, which might work, but is atypical, would be the choice to make the lower neighbour tone a semitone (this is somewhat analogous to the Mediæval una nota super la rule – by comparison, in Latin, it could be phrased as una nota sub ut sæpe est canendum mi: a note below do often is to be sung as mi62) – in this case, we would want the lower neighbour (B) to be a semitone below the C-sharp (the note as it exists in the key signature). This is atypical because the usual way to make this happen is to raise the B to B-sharp; however, since the C-sharp is also a semitone below the D, it would result in two consecutive semitones, which would not suit the style; instead, the Csharp is lowered to a C-natural, which does create the same effect, but it is not a typical way to create it. One possibility does suggest itself, in that they are both followed by a skip. As a C-sharp, they would be quite dissonant against the G in the bass, and traditionally it is considered poor voice leading to skip away from a dissonance; indeed, in the case of Variation E, the only option other than a skip would be oblique, which – much like the V chord already discussed at the end of Variation C’ – would sound quite odd after the preceding lower neighbour figure, so the skip would seem to be appropriate. On the positive side, this is a characteristic unique to these two instances (it does not appear in the melodic trajectory of the other candidate C-sharps over G-Major chords), so the rest of the canon does not contradict the

62

In the original form of solmisation (do-re-mi, etc), the first note was not called do, but ut. Anachronistic translation provided by author.

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hypothesis. Yet, lowering the tritone to a perfect fourth still produces a fourth against the bass, traditionally considered a dissonance in this context, and so still produces a leap from a dissonance, though admittedly a less extreme instance. So again, we have a possible but unconvincing explanation. In fact, unsurprisingly, the most convincing explanation is drawn from the canonic principle itself.

I^

S

The one case we have not examined closely so far is the case over the chord in variation E ’. This instance is much easier to explain, as it creates a secondary dominant chord, to the expected

V%/IV, which does, in fact, resolve

IV chord. The significance of this is that, in a canon, cause need not precede effect; that

is, the use of the C-natural in this subvariation can have an impact on other subvariations that, due to the canonic principle, will sound simultaneously with it – not only those that will follow it, but also those that preceded it and still remain in effect. In this case, this includes Variation E’, one of the two C-naturals we have been endeavouring to explain. If the C had been left as C-sharp, we would have a C-sharp and a C-natural separated by only a single semiquaver – a false relation (a.k.a. cross relation, two differently inflected versions of the same scale degree being used in to close proximity to each other), which would produce a poor effect and should be avoided. By a similar argument, the C-natural in Variation E is within the influence of Variation E’, and so a false relation must also be avoided here; we should note that the C-sharp in the context of the A-Major chord that appears at the end of variation E’ should not be flatted to avoid a false relation, since it participates (as a

V chord) in the cadential gesture leading to the next variation, and is needed to keep

the dominant chord major. Of course, the weak point of this particular interpretation, however much it may appear to be the best explanation of the first two C-naturals, is that we first need to explain why, in only a single variation in a piece that otherwise uses exclusively diatonic chords, Pachelbel chose to use an applied dominant seventh chord. The variation in question does not display any particular characteristics that would imply

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Figure XII-5 Examples of apparent seventh chords that are merely results of decorations

a need for or benefit from such an intensification; it is neither a particularly important variation, nor a particularly important moment within a variation. However, a brief examination of the theory of the seventh chord, not as it is currently conceived, but as it was understood at that time, as represented by Rameau’s Treatise on Harmony, might suggest a reason for this choice. The concept of the seventh chord, while simple in theory, is thorny in practice. It is not difficult to identify combinations of notes that appear to be spelled as a seventh chord, but it can be misleading, as the apparent seventh, or sometimes the apparent root, may result from decoration and non-harmonic tones (Figure XII-5). Indeed, the concept of the seventh seems to have originated from non-harmonic tones, especially as a suspension from the previous chord, or as a descending accented passing tone, though it is also commonly introduced as an unaccented passing tone within the context of a single chord (Figure XII-4); however, in this last case, it is debatable whether the chord ought to be identified as a seventh chord, or merely a chord with a passing tone. To determine whether or not it represents a

Figure XII-4 Typical preparation of seventh chords A: The seventh is suspended from the previous chord. B: The seventh is an accented passing tones. C: The seventh is an unaccented passing tone.

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A

B

Figure XII-6 The possible seventh chord of Variation E#’, compared to a trivial seventh chord (A) This seventh can also be described as a passing tone, and so only context can decide whether it represents a true seventh chord – a judgment influenced by the use of sequencing in this variation. (B) This seventh, on the other hand, is trivial; it behaves as a passing tone only.

seventh chord, we must make a subjective judgment about the prominence of the seventh chord tone. If it is very prominent (perhaps half of the total length of a reasonably long chord), it can easily be seen as a seventh. If it is fleeting (a small fraction of the length of a passing chord), it may merely be a passing tone.

S

I^

This is, in fact, the judgment we must make in variation E ’ (Figure XII-6). The chord is , and the melody is a descending passing tone from the root, in which the seventh spans half the duration of the chord. The status of this note as a seventh or a passing chord is ambiguous. However, we must remember that we are working within the context of a sequence, and although the chord

I^ is a substitute for the

iii

expected , the repeated use of this sequence primes us to hear it as a sequence nonetheless; the melody continues as a true sequence. In the first statement of the melodic sequence, the syncopated note is consonant with the first chord, then is held as a 6-5 suspension over the second chord, such that the resolution note is the actual chord tone, the target of the sequence motive. By analogy, we expect in the second statement that the syncopated note will again be consonant with the first chord – it is – and to be suspended over to the next chord and resolved to the target chord tone; however, since the chord is instead of

I^

iii, the “suspension” is, in fact, consonant, and the “resolution tone” dissonant, yet the

sequence still predisposes the listener to hear the seventh as the target note. In summary, then, this passing tone, unlike other similar passing tones (e.g. measure 37), must be understood as a true seventh, and not a passing tone.

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The definitive identification of this note as a passing tone is important. Unlike modern theory, in which it is generally understood that any diatonic triad may also carry a diatonic seventh, seventh chords were still a relatively new harmonic resource during Pachelbel’s era, even more so on non-dominant chords, and it was not typical to build a true diatonic seventh chord on the tonic of the key. Rameau, in his Traité de l’harmonie (Treatise on Harmony), published in 1722 (roughly twenty to forty years after the estimated date for the composition of the Pachelbel canon), under the chapter heading “How to use the Seventh on every Note of a Key in a diatonic progression”, writes: Only the tonic note should always bear the perfect chord [i.e. the major or minor triad, without a seventh], while the seventh chord is appropriate for all other notes. Notice, though, that in the progression of an ascending fourth to a perfect chord or to a seventh chord, all notes [and hence, the chords built upon them] should be regarded as [secondary, or applied] dominants and may thus bear the seventh chord.63 In other word, in Rameau’s historical context – and likely Pachelbel’s as well – one could build seventh chords freely on most degrees of the scale, but if I is created upon the tonic of the key, it must be created as an applied dominant towards the subdominant chord – hence, the C-natural. The influence of this C-natural, and the avoidance of false relations, then, may ultimately explain the other two, as previously described, as yet another example of the consequences of applying the canonic principle to the composition of a variations form such as a chaconne. If we had no other information to apply to the problem, it would likely rest here. However, the same curious characteristic appears, even more prominently, in the Biber ciacona-canon, and this explanation is not sufficient, leading us to propose another possible explanation, which however shall wait until our examination of the same characteristic in the Biber work in Section XIX below, at which point the discussion is more relevant.

63

(Rameau, 284)

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XIII. Biber’s Chaconne-canon in unisono

H

einrich Ignaz Franz von Biber (c.1644 - 1704), later von Bibern (after he was elevated to the status of Knight), was a Bohemian born composer and violin virtuoso contemporary of Pachelbel, active primarily in Salzburg, and known best for his violin sonatas,64 many of

which employ alternate tunings notated in scordatura (retuning the instrument, then writing down the notes that the fingering would create if the instrument were tuned in standard tuning). In 1696 – near the estimated date of composition of the Pachelbel canon – Biber composed a series of seven partitas (instrumental suites) for string duet with basso continuo, mostly with altered tunings and scordatura, called Harmonia Artificiosa-Ariosa. The sixth movement of the third partita, in A-major, is a ciacona, subtitled canon in unison. So, unlike the Pachelbel canon, there can be no question that the composition is both a chaconne and a strict canon. The fact that the composition dates for the two pieces are so close to each other is rather telling, given just how similar they are. However, there are several differences. In general, the Biber ciaconacanon is closer to a traditional chaconne than the Pachelbel canon in several ways. First, it is in three-four time, as most chaconnes are, rather than the common-time of the Pachelbel canon. Second, it uses the syncopated rhythm common to the chaconne and sarabande in which the second beat is the principle stressed beat. And thirdly, like a typical chaconne, but unlike Pachelbel, it does seems to increases in complexity steadily from start to finish, after a fashion, which will be discussed in more detail in section 0 below.

64

GROVE

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Example XIII-1 Ciacona-canon in unisono from Harmonia Artificiosa-Ariosa by Heinrich Ignaz Franz von Biber. Page 1. Transcribed by the author from (Nettl and Reidinger).

Pachelbel, Purcell and Biber: The Variations-Canon

Example XIII-1 Ciacona-canon in unisono from Harmonia Artificiosa-Ariosa by Heinrich Ignaz Franz von Biber. Page 2.

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XIV. The Chords and Ostinato of the Biber Ciacona-Canon

T

he bass ostinato of the Biber ciacona-canon is similar in contour to the Pachelbel canon, and the chord progression (as presented in the theme) is quite similar, with only one chord

I

different: the sixth chord, which was a chord in the Pachelbel canon, has been substituted for

a

iii chord (Figure XIV-1).

This is an interesting change, because it is qualitatively the same kind of

substitution we saw in the Pachelbel canon, except backward, and moved to a different position in the progression. However, in spite of being the same kind of transformation (what Neo-Riemannian theorists call a Leittonwechsel, or a “leading-tone exchange” transformation), in context there are three substantial differences between the way Pachelbel uses the transformation and the way Biber uses it. First, Pachelbel uses it less often, generally maintaining the true chords of the Plagal Sequence; Biber, on the other hand, uses the substitute chord more frequently, generally decreasing the degree to which his chord pattern reflects the plagal sequence upon which it is based.

iii

Second, Pachelbel replaces a less common chord ( , which is perhaps the least frequently used

I^ chord (one of the more frequent diatonic triads); Biber, instead, replaces the common I chord with the less common iii chord.

diatonic triad) with the

Finally, Biber makes the substitution just prior to the succession

IV - V cadential gesture, leaving the

iii - IV - V, a succession of three stepwise triads, all in root position.

This kind of chord

succession must be treated carefully even under normal circumstances, because of the heightened risk of

Figure XIV-1 The ostinato and chords of the Biber Ciacona-Canon Note that the iii chord is found where a I chord is expected.

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parallel fifths. In a canon, the situation is even more difficult, because the comes voice is predetermined by the previous cycle, so the avoidance of parallel fifths may dictate which note is written in the dux, and hence dictate the completed harmony. If this substitution is apparently so undesirable, why would Biber chose it? As always, we should be careful when attempting to analyse a composer’s decision-making process; however, it is true in this case that the theme of the ciaconca-canon was carefully constructed to display certain characteristics (to be discussed in section XIX below) that dictate that it will use the

iii chord rather than I chord, and it may be

that Pachelbel choses to continue that choice frequently throughout the canon. Aside from this substitution, the chords are largely the same, other than inversions – half of the chords have been recast as first inversion chords. The chord pattern of the theme, then, is:

iii^ - IV^ - iii - IV - V.

I - V^ - vi^ -

However, the Biber ciacona-canon is much more flexible harmonically than the

Pachelbel canon, including lots of suspensions, secondary dominants, and root position chords built where the first inversion chords were in the theme (built upon the same notes from the bass ostinato). Unlike the Pachelbel canon, the Biber ciacona-canon is indisputably not a stacked canon, because it is only for two canonic voices; in principle, it would seem that the previously discussed difficulties in changing the harmonies would seem not to apply to this canon in the way that it did to the Pachelbel canon – and, indeed, as we have already seen, there is more harmonic flexibility here than in Pachelbel. However, Biber was a violin virtuoso, and much of the music he composed for violin was likewise virtuosic, including the ciacona-canon. Among the virtuosic elements displayed by the ciacona-canon is lots of multiple stops,65 from double stops right up to quadruple stops in a single canonic voice. While not precisely the same a polyphonic melodies, it has precisely the same consequences: once a multi-stop has

65

More than one note at a time, played on a single instrument.

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been written into one strain of the canon, it becomes difficult to create a different harmony in the next strain.

XV. Aside: Biber and Merula

J

ust as we noted a very close relationship between the Pachelbel canon and the Clarke grounds, we can also note a counterpart for Biber’s ciacona-canon in a Chiacona by Tarquinio Merula (c.1594 – 1665). This chiacona predates any of the variations-canons we shall examine, and it

doesn’t quite fit into the category of variations-canon, since it is not a canon; however, it does begin in canon, and maintains strict canon for the first thirteen measures, then returns to strict canon at measure fifty-five for an additional seven measures, out of a total of sixty-nine measures, making it a strict canon for approximately thirty percent of the chiacona. This is actually not terribly surprising. Throughout the Renaissance and into the early Baroque, imitation up to a certain point was a highly desirable characteristic. Thomas Morley, in his Plaine and Easie Introduction, refers to this practice as Fuga,66 and he at one point criticises his student for not having continued the imitation for long enough before having broken it off.67 Even the Gigue that accompanied the Pachelbel canon, while not a canon itself, enters in imitation (see Appendix B). What makes Merula’s piece interesting for our purposes is that it is a pure chaconne, and its ostinato is nearly identical to Biber’s (other than time signature – six-four instead of three-four – and key), with only one note (the first lower neighbour) missing (see Figure XV-1); Merula’s and Biber’s variations technique also

66

Fuga is a word Zarlino used to refer to strict canon – at a perfect interval – as opposed to imitation – at any other interval (Zarlino, 126, 135); in modern usage, the term fugue is used almost exclusively for imitation at the fifth, or rarely at the fourth, with a number of other standard characteristics included. 67 “We call that a Fuge, when one part beginneth and the other ſingeth the ſame, for ſome number of notes (which the firſt did ſing) […].” (Morley, 84); “Take the deſcant of your owne way, which was in the eleuenth, or fourth above, and ſing it as you did begin (but in the fift below under the plaine ſong) and it will in a manner goe through to the end, whereas yours did keep report but for fiue notes.” (Morley, 86). These excerpts have been previously cited in (Grimshaw, 662).

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Figure XV-1 Comparison of the ostinato of the Biber Ciacona-canon and the Merula Chiacona Caratted numbers represent the pitch’s scale degree relative to its key centre. Note that while the Biber ciacona-canon is notated with the key signature we now associate with D-Major, its key centre is A, and the G-sharp in the first measure is diatonic to A-Major (it is common in Baroque music to notate with one fewer sharps).

share some commonalities. The complete score of Merula’s chiacona can be found in Appendix .

XVI. Form of the Biber ciacona-canon

A

s already mentioned, the form of the Biber ciacona-canon is more conventional of chaconnes than is the Pachelbel canon. In general, the complexity of the piece increases steadily until measure ninety-seven, whence it winds down to a restatement of the theme until the time

change. Then, something happens that is particularly interesting when compared to the Pachelbel canon. Up to this point, we have all but ignored the fact that the Pachelbel canon is, in fact, the first movement of a two movement suite, followed by a gigue. We have not concerned ourselves with the gigue largely because it is not a canon, and therefore not particularly relevant to our topic (the score for this gigue can be found in ). However, after the Biber ciacona-canon winds down to recap the theme, it transitions from three-four to nine-eight time – essentially a gigue, just like the Pachelbel canon, except that this time, the “gigue” is still a canon. The rest of the form is also slightly inflected, similar to the way the Pachelbel canon is inflected, based on the characteristics of the variations, into two main parts; this will be described after a brief examination of the variations themselves.

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XVII. Variations of the Biber ciacona-canon

A

nother way that the Biber ciacona-canon is different from the Pachelbel canon is that it is more difficult to clearly separate the variations from each other. While it is still a chaconne, characterised by thirty-two repetitions of the bass ostinato, the variations written over it are

of varying lengths. Many of the variations do follow the familiar pattern of playing a principle variation, followed by a complimentary variation, and then one or two throw-away accompanimental variations. It should be noted that since this canon is for only two canonic voices, a single accompanimental variation is completely sufficient to prevent the voices from competing for the listener’s attention. However, Biber does sometime use more accompanimental variations, which has an interesting effect: since the Pachelbel canon always uses just enough accompanimental variations to keep the player busy while the other two violins play the principle variations, it doesn’t really ever come to a resting point, the momentum continues almost perpetually (as is true of most baroque fugues, another genre that Pachelbel and the composers associated with him are well known for); meanwhile, the extra accompanimental variations used by Biber sometimes cause the principle variations to become separated from each other, which creates a sense of repose and separation, a kind of bridge, giving some relief from the perpetual motion that might otherwise result. At other points in the ciacona-canon, Biber doesn’t seem to make use of accompanimental variations at all, particularly towards the end, when the whole piece appears to be through-composed. This is particularly true of the nine-eight section, when the variations begin to become somewhat nondescript and not easily distinguished from each other, with no clear sense of separation between them, further strengthening the notion that this portion represents a distinct section, separate from the rest of the chaconne.

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As to the variations themselves, we can distinguish, discounting the theme and its da capo, some eight variations with varying lengths and with or without accompanimental variations to separate them. Interestingly, there does seem to be a pattern regarding which variations include accompanimental variations. Figure XVII-1, below, shows the variations of the ciacona-canon. Discounting the opening theme and the closing gigue-like sections, we have an overall two-part structure. The Theme (Variation Group A) is tied to Variation Group B through accompanimental variations – though it is slightly separated from it by using two accompanimental variations. Group B and Group C, both relatively simple, are connected directly through a single accompanimental variation. Group D, on the other hand, is more complex, with more subvariations, and is separated from those around it by rests on both sides. Group E starts with an accompanimental variation (after rests, so it does not overlap with the previous group, and so behaves like a bridge, just like after the theme, and helps to articulate the two-part form. Groups E and F are both relatively simpler again, and are again connected directly via a single accompanimental variation, while Group G is again more complex, with more subvariations, and is once more set apart from the other subvariations by rests on boths sides. The pattern, then, after the theme and before the “gigue” is two simpler, shorter variations connected by an accompanimental variation, followed by a longer, more complex variation bookended by rests; this pattern repeats twice. Table XVII-1 Characteristics of Variation Groups in the Biber Ciacona-canon

Variation Group A (Theme) B C D

Preceded By Basso Accomp. Accomp. Rests

Number of Subvariations 2 1 1 6

Followed By 2 Accomp. (1 shared) Accomp. (shared) Rests (shared) Rests

E F G

Accomp. Accomp. Rests

1 1 4

Accomp. (shared) Rests (shared) Rests

H (“Gigue”)

Accomp.

6

(end)

Notice that, if we ignore the theme and the closing gigue-like section, the remaining six groups form two groups of almost identical properties.

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Figure XVII-1 The Variations of the Biber Ciacona-Canon Blue rectangles show principle variations of the groups, while accompanimental variations are outside the rectangles. Red lines show accompanimental variations that are shared between adjacent groups of variations. Page 1

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Figure XVII-1 The Variations of the Biber Ciacona-Canon The accompanimental variation highlighted in light blue in group E is not shared, and serves as an introduction to the second half.

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Figure XVII-1 The Variations of the Biber Ciacona-Canon Page 3.

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Table XVII-1 summarises the properties of the variations with respect to the groupings and the presence or absence of accompanimental variations.

XVIII. Voice-leading: Catch-like elements of the Biber ciacona-canon

T

he Biber ciacona-canon has a particularly interesting characteristic: it borrows elements of the catch. The catch is a specialised form of the round, in which the interdependence of the various cycles of the canon is increased by careful design of each cycle so that the true lead melody is

not apparent from a glance at the score, and will not be perceived until all parts are present, at which point the true (hidden) lead part pops out of the mix as a composite of the various strains.68 Think of it like a handbell choir. In a handbell choir, each member has only a small selection (often two) of the bells needed to make up the scale. The music will be distributed among the parts so that, whichever note is called for, it will be written into the part of the performer who has that bell. Looking at any one part, it is generally not possible to find the melody, but when all parts are played together, the melody pops out. In the same way, the true lead part of a catch is distributed among the different cycles of the canon, such that it doesn’t become apparent until all the parts are present.

Figure XVIII-1 The distribution of notes in a handbell choir The melody is the first phrase of the familiar Christmas carol “Joy to the World”. Each chorister has only two bells, and the notes are written into the correct part at the moment in which it is called form. Only when all are heard together is the true melody apparent.

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(Dyk and Taylor, viii)

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In a true catch, the lyrics typically play a substantial role in this effect. The words will be carefully selected so that the sounds needed to create the hidden message are distributed among various parts. Often, the hidden message conveyed by the composite is ribald, or otherwise risqué,69 while the composer, if confronted about the impropriety of it, can claim ignorance and provide a copy of the otherwise mundane lyrics that each singer is actually signing, to “prove” that the hidden message came about by accident. Composing a catch often relies on the technique of rhythmic counterpoint, where the beginnings of notes in one voice (or canonic cycle) are offset from those in the next canonic cycle. A simple example would be to have (at a delay of one measure) two minims in the first measure, and then a syncopated figure of a quarter rest, then a minim, then a crochet in the next measure, so that, when heard together, the notes in one voice will begin on beats one and three, while the notes in the other voice will begin on beats two and four. The result will be that, in the composite, it will seem that there is one note per beat,

Figure XVIII-2 A simple catch The regular lyrics say “This ache is catching.” When overlaid upon itself, it seems to say “This is ache catch,” which would sound like “This is a catch”. Compare this to the handbell choir (Figure XVIII-1).

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(Dyk and Taylor, viii). See also (Catch, 152): “The 17th-century catch became a sophisticated and often intricate genre, developing the manner of treating the words that was to remain characteristic. This involved calculating the words so that the interplay among the parts produced new combinations, usually comic or (especially during the Restoration period) bawdy in effect.” c.f. (Mann, The Study of Fugue 1958/1987, 9, footnote 3): “Hawkins defines [a catch] as that species of round ‘wherein, to humor some conceit in the words, the melody is broken, and the sense interrupted in one part, and caught again or supplied by another.’ (Grove’s Dictionary, 3 rd ed., 1946.)” [emphasis added]

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but in truth, these notes are distributed among both voices (Figure XVIII-2). A more spectacular example can be found in Figure XVIII-3. Obviously, the canons we are considering cannot be true catches, because they are not rounds (they do not repeat the same few measures over and over again), and they do not have lyrics. Nevertheless, they can display catch-like characteristics – within certain provisions. A typical catch often is able to make its message pop out based on the prominence of various vocal articulations (i.e. hard consonants tend to stand out). A non-vocal canon must use alternative methods, such as voice-crossing, or especially strategically placed rests to achieve this effect. While many composers use rests to solve harmonic and

Figure XVIII-3 A more complex catch While many short ribald ideas pop out of the mix, the highlighted section is particularly clear, and much more risqué than the linear lyrics. Transcribed by author from (Dyk and Taylor, 217).

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voice-leading problems while writing canons,70 the way that rests are used in a catch and catch-like passages in canons is an altogether more sophisticated phenomenon. Strictly speaking, there is nothing particularly special in the idea of using offset rests to cause an alternation of notes between voices – this idea has a long tradition, going back to the hocket of the mediæval era.71 The catch is that, in a catch (or similar canon), the rests are not alternating between two distinct parts, but between two cycles of the same part, which requires a more careful compositional technique, because – as always with a canon – the decisions you make in one cycle to complement the previous cycle will also impact the next cycle.

S

S

For example, the Pachelbel canon contains two subvariations (B and B ’) that display a catch-like effect between them (Figure XVIII-4). Both are characterised by alternating quavers and eight rests. However, the former places the notes on the beats, and the later places the notes on the off-beats, so that they together create a single part. It was not commented upon during our examination of the Pachelbel canon because it is a rather unspectacular example, and because it is an accompanimental

S

Figure XVIII-4 The catch-like texture of variations BS and B ’ of the Pachelbel canon These two parts, back-to-back in the dux, are arranged to alternate notes, together forming a single part. In this case, as an accompanimental variation, spreading the melody out over two parts thins the texture, resulting in even less music competing with the true variation for the listener’s attention.

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Consider for example: “A way of easing the prevention of chord-tone doublings in the resulting canon, is the insertion of rests between two short duces…” (van Geenen, 203). 71 “In polyphony of the 13th and 14th centuries, a stylistic device or a self-contained composition characterized by the distribution of a melodic line between two voices in such a way that as one sounds, the other is silent.” (Hocket, 392)

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Figure XVIII-6 Catch-like texture of the Merula Chiacona

variation that is generally not heard in the contest for the listener’s attention. A similar technique can be seen in the Merula chiacona in measures 17-20 (Figure XVIII-6). The Biber ciacona-canon, on the other hand, contains a much more spectacular implementation of this technique. Beginning in measure thirty-seven, Biber begins to include entire measures of rests, alternating with entire measures of scalar passages. Like our previous examples, for the entire first cycle, the scale comes first as the first half of the cycle, followed by the rest for the entire remainder of the cycle; in the next cycle, the rest comes first, followed by the scale. The composite makes it sound as though the various scales are all played by a single violin, while in fact, they are played by alternating first and second

A

B

Figure XVIII-5 Similar catch-like texture in the Biber ciacona-canon and the Merula Chiacona (A) Composite scales in the Biber Ciacona-Canon; (B) the theme of the Merula Chiacona.

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violins. A similar technique can be seen in the opening of the Merula Chiacona where the theme is followed by a rest to allow it to be presented again in the comes (Figure XVIII-5). It is important to note that in the Biber ciacona-canon, the extended rests between these scalar passages are not truly silent; there is still the basso continuo, with the cello continuing to repeat the bass ostinato, and the harpsichord likely taking a larger role during this rest to smooth out the awkward sound created by ending a scale on the last subdivision of a measure, without any notes on the next downbeat. A skilled continuo player might even decide to anticipate the call-and-response-like texture that will exist between the dux and comes in the next cycle, and chose to answer the scale the same way (in a sense, behaving like a kind of ante-dux, as it would be like playing the part one cycle ahead of the dux itself).72

XIX. Notable characteristics of specific variations of the ciacona

O

ne of the most remarkable variations in the Biber ciacona-canon is the theme itself. We have already remarked how the Biber ciacona-canon is not a stacked canon (as the Pachelbel canon was); neither is it technically a stretto canon. Yet Biber cleverly crafts his

theme to display characteristics of a stretto canon. Although the chords of the Biber canon are nearly identical to those of the Pachelbel canon, Biber’s selection of inversions creates a bass ostinato that is less symmetrical than Pachelbel’s. This allows him to use the bass ostinato itself, transposed up two octaves, as the melody of his theme. In order to make it work, the melody begins one measure earlier than the beginning of the cycle, so that each measure of the melody in the dux appears one measure before the same melody in the ostinato, almost as though the ostinato were actually a comes, just for a moment, and as though the canon were only delayed by one measure, instead of four.

72

A similar texture is written into the suggested continuo realisation in (Nettl and Reidinger 1956)

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Figure XIX-1 The theme of the Biber ciacona-canon as imitation of both stretto and stacked canon

As if this were not enough, Biber uses a clever repetition scheme to intensify the effect: it will, of course, be repeated beginning four measures later in the true comes; Biber then waits an additional two measures and repeats the same melody again (admittedly with adjustments at the cadence) in the middle octave in the dux, so that it begins one measure after the cycle begins, so that it appears one measure behind the ostinato, and two measures behind the true comes (Figure XIX-1). In this way, we actually have the basic melody overlapping itself at a distance of both one measure and two measures at the same time. For this brief moment, the Biber ciacona-canon imitates both a stretto-canon and (disregarding octave) a stacked canon at the unison, while truly being neither. As mentioned in section 0 above, this careful construction of the theme has consequences for the chord progression. Because of the desire to imitate stretto canon here, all three voices are predetermined for most of the variation, and the sixth chord (already discussed at some length) has only two distinct

I

pitches: C-sharp and E. Technically, these are both part of both the chord and the

I

iii chord; however, it

would be a rather odd chord, missing its root. This may be part of the reason why Biber prefers the

iii

I

chord over the . There are a few minor observations to make about some of the other variations. The variation beginning measure seventy seven contains the most interesting harmony of the piece: the beginning of

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Pachelbel, Purcell and Biber: The Variations-Canon

iii chord, is in this case is a borrowed III chord (n.b. it does not resolve to

vi chord as a secondary dominant would normally do). There is little to say of the next variation, beginning measure eighty-five, other than that it is the

variation of greatest speed, where the violinist’s technical proficiency is most audible (the use of quadruple stops is arguably a greater display of skill, but in the context of the full orchestration, it is less audible). The second last variation, immediately prior to the recap of the theme, while not strictly a repeat of the first variation after the theme, is strongly reminiscent of it, which helps to increase the sense of coming back to rest after the point of greatest complexity (as chaconnes generally do), and gives a sense of partial symmetry to the movement. The transition to the final, gigue-like section is quite interesting, particularly as a canon, in that the time signature changes at different times in the two violin parts – it is, of course, delayed by one canon cycle in the comes voice. However, the final variation of the three-four section is simple, with no notes beginning off the beat; this means that when the dux switches to nine-eight, the comes part blends with it just as though it were already in nine-eight (it should also be noted that the ostinato switches from

Figure XIX-2 The time-change of the Biber ciacon-canon Note that the time-change is delayed in the comes (Violin I).

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Figure XIX-3 Catch-like texture of the closing “gigue” section of the Biber Ciacona-Canon This excerpt from the ciacona-canon shows how the texture of the “gigue” section consists of almost perpetual triple quavers, though the triples are passed back and forth from dux to comes (that is, between one cycle and the next). Nearly the entire “gigue” section is constructed this way.

crochets and minims to dotted crochets and dotted minims, to match the new time signature) (Figure XIX-2). As for the final section itself, the variations themselves are generally not particularly remarkable, or easily distinguishable from each other. One characteristic worthy of note, however, is another instance of the same catch-like technique already discussed. The composite texture of the final variations is one of perpetually moving eighth quavers, something that is not visually apparent from reading only the dux, because the quaver groups are distributed among consecutive cycles (and so, among both violins) (Figure XIX-3). Returning briefly to the Merula chiacona, it is important not only to remark upon the ways that they are similar, but also upon the ways that they are different. Near the middle of the piece (beginning measure 37 – see score in Appendix ), Merula takes the opportunity to break the monotony of the ostinato by writing some very simple, accompanimental variations in the violins (not unlike the accompanimental variations of the Pachelbel canon), while writing the true variations in the cello (normally playing the ostinato). This is a common technique in chaconnes,73 but it is a technique that none of the three true variations-canons we are examining display. While it is not strictly ruled out from a variations-canon, the

73

c.f. “[…] in chaconnes there is generally […] at most a harmonic-rhythmic or bass formula, which tends to be treated rather freely or may even be abandoned altogether.” (Silbiger, Chaonne, 411)

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fact that the canonic principle occupies so much of the composer’s attention likely discourages the idea of writing interesting variations into the bass – in a sense, a canon of this sort is a kind of solo piece (albeit for multiple soloists), so writing solos for the continuo part would be akin to writing a random solo for a non-featured instrument into a concerto: not strictly unheard of, but not typical.

XX.

Purcell’s Chaconne (Two in one upon a Ground)

H

enry Purcell (1659 – 1695), a contemporary of Pachelbel and Biber, was the last great A-list homegrown English composer prior to the twentieth century, and a well-known composer of rounds and catches.74 The third act of his opera Dioclesian begins with a variations canon

– a chaconne subtitled “Two in one upon a ground” (the construction “two in one”, “three in one”, etc., was a common designation used by English composers to describe the number of parts in a canon that are read from the same dux).75

A. Dido’s Lament

B. Two-in-one upon a ground

Example XX-1 Comparison of Ground Bass patterns for Dido’s Lament and the Two-in-One Upon a Ground Note that both the overall contour and the principal chord functions are the same.

74

(Catch, 152) “Definition of two parts in one[:] It is when two parts are ſo made, as one ſingeth euery note and reſt, in the ſame length and order which the leading part did ſing before […].” (Morley, 108) 75

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A ground is similar to the other variations forms we have discussed, but it is usually a distinctly English variant. Richard Hudson notes, “Many later Baroque grounds are essentially extensions of the passacaglia bass, to which a cadence has been added”,76 and Purcell’s two-in-one upon a ground is no exception; in spite of being entitled “chaconne”, it actually is essentially a passacaglia (at least according to the definitions in use in this paper), and in that way is different from the other variations canons we have examined so far.

XXI. Purcell and the Ground – Relationship to Dido and Æneas, and the Chords of the Two-in-one

A

ccording to Richard Hudson, “The English ground reached a highpoint in the vocal and instrumental music of Purcell (numerous examples, including the famous lament for Dido and Aeneas) […]”.77 This is a very interesting comment, since the passacaglia two-in-one is quite

similar to the lament from Dido and Æneas. Both are, of course, passacaglias, with a chord progression based on a descending bass ostinato in the minor key, plus a cadence. The only significant difference between them is the use of chromatic passing tones in Dido and Æneas. Example XX-1 shows the comparison between the ground bass of the passacaglia two-in-one and Dido’s Lament. One of the stand-out characteristics of the English ground that the two-in-one shares with Dido’s Lament is that the phrasing of the melodies written above the ground bass often does not align with the cycles of the ground bass itself. The two-in-one and Dido’s Lament are based on a six-measure and fivemeasure repeating ground bass, respectively, but the melody is not built in six-measure or five-measure

76 77

(Hudson, 448) (Hudson, 449)

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phrases – or even the same number of measures per phrase throughout the work; the phrase length is

Example XXI-1 Chaconne: Two in One upon a Ground, from Dioclesian (Act III) by Henry Purcell Original scoring is bassoon for the ground bass, with the melody on two flutes – which, in historical context, should be understood as recorders. Also included is phrasing (which is irregular), shown in blue boxes, vs. the repeating ground bass cycle, shown in green underscores; note that these are not aligned with each other. Transcribed by author from (Bridge and Pointer).

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quite fluid in both pieces (this is actually a characteristic more common of vocal grounds than instrumental78 - hence its use in Dido’s Lament – but it is no less present in the two-in-one). Appendix shows Dido’s Lament, with the ground bass cycles and the phrasing labelled; the phrasing and ground bass cycles of the two-in-one was included in Example XXI-1.

XXII. Characteristics of the Passacaglia progression

A

s noted, in spite of Purcell’s designation for it, the chord pattern of the Purcell two-in-one is a passacaglia, instead of a chaconne, which gives it a rather different chord progression, based on the descending tetrachord ground bass plus a cadence, implying the chord pattern

i – VII – VI – V – i – V, though it is more typically realised with ”mediant” type substitutions (as we have defined them),79 just like the substitutions in Pachelbel and Biber, for reasons that will be discussed momentarily. In this case, Purcell favours the realisation

i – v^ – VI – V – i – V.

One of the major characteristics of the Pachelbel progression was its genesis through sequencing, and so its predisposition to sequencing in the variations. This is much less true of the passacaglia chord progression. Up to a point, this chord pattern can support a sequence, in that the motion is consistent from one chord to the next (descent by one scale step), but it is a fundamentally different kind of sequence; it is an essentially melodic sequence, rather than a harmonic sequence, because each

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“While the nature of the bass itself does not necessarily differ between instrumental and vocal pieces […], the treatment of the upper parts diverges considerably, and different structures result. In instrumental pieces, the phrases of the upper lines usually coincide with the structure of the ground bass, so that continuous variations result […]. In vocal pieces, however, the ground bass tends not to delimit individual variations, but rather acts as the underlying support for a freely and often irregularly phrased setting of the text in the upper line. The vocal melody, expressly designed not to match the phrase structure of the ground thus creates an affective continuity whose subtle conflict with the bass may build to a moving climax, as in Dido’s Lament …” (emphasis added). (Ground, 366-7). Perhaps Purcell chooses to use the vocal style in this instrumental composition because of its place within his opera Dioclesian, essentially a vocal work with periodic instrumental interludes. 79 For a discussion of the chord pattern of the passacaglia based on the descending tetrachord, see (Silbiger, Passacaglia, 192).

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Figure XXII-1 Sequencing in the Purcell Two-in-one that do not align with the chords The expected sequences, based on the chord pattern, would be one or two measures long; this one is three measures long, and crosses the threshold between one statement of the chords and the next.

statement of the sequence motive is fully enclosed by a single chord.80 One could also create a sequence based on two pairs of chords:

[ i – VII ] – [ VI – V ]; however, it is typical to run sequences for at least

part of a third statement, and these chords will only permit this sequence to run twice, so this particular option is trivial (though Purcell does occasionally take advantage of it in this piece). Purcell also finds ways to incorporate melodic sequences of other lengths and displacements - consider a three-measure motive, sequenced in measures twenty-three through twenty-eight (Figure XXII-1); however, since the canon runs at a delay of two measures, this will also create uneven sequencing. In one substantial way, however, the passacaglia progression is strongly reminiscent of the Pachelbel progression: both are dominated by descending motion. Recall that the Pachelbel progression has four distinct voices: two that descend stepwise, one that alternates between oblique and descent of a third, and one that alternated descent of a fourth with ascent of a second; this progression is an example of fully invertible quadruple counterpoint, in which half of all possible arrangements of voices would result in a steadily descending bass line – just like the descending bass line of the passacaglia progression.

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“[La marche d’harmonie tonal] sera considérée comme telle toute reproduction […] systématique immédiate, […], d’un modèle […] correspondant à un ensemble harmonique d’au moins 2 termes (fonctions), dote le plus souvent d’une configuration mélodico-rhythmique qui lui est propre. […]. D’abord, on exclut la marche dans laquelle le modèle renvoie à un seul accord, même si les configurations mélodiques et rhythmiques sont reproduits symétriquement.” (The harmonic tonal sequence will be considered as one of complete, immediate, systematic reproduction, of a model corresponding to a harmonic group of at least two chord functions, endowed most frequently which a melodic-harmonic configuration that is particular to it. First, we exclude the sequence in which the model reflects a single chord, even if the melodic and rhythmic configurations are symmetrically reproduced.) (Beaudet and Ménard, 1.3 (B) 1)

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Likewise, the Purcell two-in-one is also potentially dominated by descending motion. I say potentially because of the glaring problem of parallel fifths and octaves that would present themselves if all voices continually descend; this is most certainly the reason why Purcell typically replaces the with a

v^ chord.

VII chord

Certainly, there are other voice-leading options available for these kinds of chord

progressions that avoid the forbidden parallels; however, these patters tend to be less smooth, or contain more incomplete chords. At any rate, Purcell’s two-in-one is nevertheless dominated by descending motion, which implies that this smooth, descending characteristic, found in both the Pachelbel progression and the passacaglia progression, is a particularly valued characteristic in the composition of variations canons (this will be discussed more fully in section XXVI below). Another useful characteristic of the passacaglia progression is that, like the Pachelbel progression, every second chord of the main (precadential) portion of the progression is built on a bass note a third lower. This is a characteristic that lends the progression quite well to stretto canon. We saw how this could potentially have been applied to the Pachelbel progression in Figure VII-2, and how it was actually applied to the Biber ciacona-canon in Figure XIX-1, in both compositions at the level of one measure (i.e. two chords per measure, so the lower third relationship appears every measure, and the stretto canon is at a distance of one measure), but in neither case was this the dominant canonic rule for the composition. In the Purcell two-in-one, the harmonic rhythm is slower (only one chord per measure), so this lower third relationship appears only every second measure, and indeed, this two-in-one is actually realised as a stretto canon at a distance of two measures. One additional noteworthy characteristics of this progression: although it is described as a descending tetrachord with a cadence tacked on the end, the descending tetrachord itself ends with a cadential pattern (

VI – V), known as a Phrygian half cadence, which could then be said to resolve i^

(imperfectly) to the following chord. Even the substitution of the

v^ chord for the VII chord contributes,

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albeit weakly, in the sense that every second chord is a dominant chord of some sort; all this provides extra opportunities for cadences within the progression, which helps to produce the irregular phrasing already commented upon and the lack of agreement between phrasing/variations and the ground bass cycle. (It also is reminiscent of the rounds described in section V, in that it is a constant alternation between tonic and dominant chords, with the only exception being the

VI chord which is a leading-tone

exchange substitution for the tonic chord).

XXIII. Variations and Form of the Purcell Two-in-one

A

s already stated, the phrases of the Purcell two-in-one are not of a uniform length, and do not align with the ground bass cycles; hence, it is difficult to predict from the score where one variation ends and the next begins. Additionally, while there are some rests in dux, none

are long enough (more than two measures) to cause a true pause in the composite composition; there are also precious few points in the dux that could be labelled as clearly accompanimental – certainly none long enough to create a bridge like those seen in the Biber ciacona-canon. The result of all this is that the two-in-one, like the Pachelbel canon, has a sense of perpetual motion to it. There are, nevertheless, moments in the piece when a point of transition can be heard – usually with the beginning of the new variation elided with the end of the previous variation, so we are not aware of the cadence as we hear it, but rather, reinterpret it as a cadence a moment later, when it becomes clear that the texture has changed. These transitions occur in measures 19, 25, 37, 44, 49, 59, 67, 75, and we can mark these as the beginnings of variations. Although the groupings are substantially different from the phrasing marked in Example XXI-1, they likewise do not form a regular pattern and do not match the cycles of the ground bass.

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XXIV. Notable characteristics specific variations of the two-in-one

T

here are relatively few notable characteristics within specific variations of the Purcell two-inone. One interesting point, relative to the Pachelbel and the Biber, is that Purcell is perfectly willing to allow a certain amount of dissonance to occur within the canon as a consequence of

greater compositional freedom. Gauldin, near the conclusion of his article, touches briefly on the process of converting first species canon to florid canon, and notes that the use of non-harmonic tones can pose special challenges in canon compared to their use in non-imitative composition, because they must be carried over into the next canonic cycle – and must still be prepared and resolved appropriately,81 and we have, in fact, seen that both Pachelbel and Biber treat these tones quite carefully. Purcell, however, doesn’t seem terribly concerned with this issue. Specifically, while the two canonic voices are generally either consonant with each other, or make use of traditional non-harmonic tones, properly prepared and resolved, Purcell nevertheless allows the canonic voices to be dissonant against the ground bass. Consider measure twenty-nine (Figure XXIV-1).

Figure XXIV-1 The different treatment of the suspension in the comes compared to the dux in the Purcell Two-in-one Note that in the dux, the suspension resolves in the traditional fashion, but in the comes, the “resolution” occurs simultaneously with a movement in the bass ostinato, so it continues to be dissonant, and true resolution is delayed an additional measure.

81

(Gauldin, 49)

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The comes voice is an imitation of a 4-3 suspension from the dux in measure twenty-seven, but of course the ground bass has changed. Instead, we have a properly prepared 7-6 suspension – but it “resolves” downward concurrently with a descending step in the ground bass, so that it remains unresolved at the interval of a seventh. This is a rather unusual pattern, using parallel dissonances. There is one traditional case in which parallel dissonances are common: the so-called “Corelli Cadence”, a cadence typical of the era, but usually associated with Italian composers. This is an authentic cadence that combines the use of two similar but non-complimentary non-harmonic tones. The first is a 4-3 suspension of the dominant chord, which delays the leading tone and creates the interval of a second against the tonic. The other is an anticipation of the tonic, descending from the fifth of the dominant chord (the second scale degree) – the tonic against the leading tone likewise creates the interval of a second. Either of these non-harmonic tones is a typical decoration of a cadence, but the use of both together can create a chain of parallel seconds, so that the first does not resolve in the expected way. In fact, Purcell himself barely avoids the Corelli cadence at the end of the two-in-one through a slight adjustment of the rhythm. Figure XXIV-2 shows the ending of the Purcell two-in-one as it would appear as a true Corelli cadence and as it is written; the only difference is the transformation of the double quaver to a double semiquaver – essentially rushing the last two notes.

Figure XXIV-2 The end of the Purcell Two-in-one as a Corelli Cadence (A) shows the cadence as it would appear as a typical Corelli cadence, while (B) shows it as it is actually written.

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In this way, Purcell hints at the Corelli cadence (and so another instance of the parallel dissonances already seen in measure twenty-nine) without actually creating one.

XXV. The Canons of the Goldberg Variations

J

ohann Sebastian Bach, whose semi-close connection, musically, to Johann Pachelbel we have already noted, has long been noted as one of the greatest masters of counterpoint in history – and canon is a distinctly contrapuntal form, and indeed, Bach wrote many

important canons. He is also widely acknowledged for having brought the Baroque chaconne to its pinnacle.82 Hence, it is quite reasonable to suppose that Bach may have combined canon with variations forms – and in a way, he did, but not in the sense of the variations canons we have been examining. Bach composed the Goldberg Variations, which we have already discussed in section X above, in 1741, approximate half a century after the Pachelbel, the Biber, and the Purcell canons were written, and its close proximity to the Classical Era shows: although each of the thirty variations (plus the theme and its da capo) is written upon the same ground bass pattern, as is true of a chaconne, it is a much longer ground bass pattern, which is entirely self-contained – it even contains its own microform and repetition structure typical of a binary dance form of the Baroque, and indeed the theme itself is a Baroque sarabande. As a result, each of the thirty variations is a self-contained miniature composition that just happens to be written upon the same ground bass. This is more typical of Classical variations forms, which are likewise longer, often self-contained variations – though, unlike classical variations forms, Bach does not stick closely to the melody of the theme in his subsequent variations; we might say that the Goldberg Variations represent a transitional stage of the variations concept, midway between the baroque chaconne style and the classical “theme and variations” form.

82

(Silbiger, Chaonne, 414)

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Because each variation is self-contained, it is of course not a continuous canon, as the Pachelbel, Biber, and Purcell examples were. Many of the variations themselves are canons – specifically, every third variation is a canon, and the interval of imitation increases each time a new canon appears. Unfortunately, these canons themselves have little bearing on the Variations-Canon hybrid form we have been discussing. However, in 1974, Bach’s personal copy of the Goldberg Variations was discovered, and it included an additional page, containing fourteen canons written upon the first eight notes of the Goldberg ground.83 Much more so than the canons of the Goldberg Variations proper, these canons are reminiscent of the music we have examined in this paper. For one thing, the ground itself is shorter (only eight notes), and resembles the chord pattern and ground bass of the Pachelbel canon and the Biber ciacona-canon (see Example XXV-1). The same overall descending pattern is present, certainly for the entire first measure, and the second measure could be understood as having had the first note deleted and each subsequent note shifted forward (including the resolution of the dominant chord at the end of the measure). The first four of these canons, like the theme of the Biber ciacona-canon, are not so much canons upon the Goldberg ground as they are canons of the Goldberg ground – including inversions which we have thus far not seen in the variations canons we have considered. The remaining ten variations do substantially resemble the variations of the Pachelbel canon and the Biber ciacona-canon. These canons

Example XXV-1 The first eight notes of the Goldberg ground. The figured bass indications reflect the harmonies used in the theme (Aria – sarabande) of the Goldberg Variations. The exact harmony changes from canon to canon, but the bass ostinato remains the same.

83

F.

(14 Canons, BWV 1087 (Bach, Johann Sebastian)). A scan of the autograph of this page is included as Appendix

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Figure XXV-1 Catch-like elements in the 7th Canon on the Goldberg Ground

have a variety of characteristics we have already discussed; for instance, Canon #7 displays the same catch-like characteristics we noted throughout the Biber ciacona-canon (Figure XXV-1). Characteristic of Bach, the master contrapuntalist, they also include characteristics that none of the other three composers included in their variations-canons. Bach explores all the traditional canonic rules, not just straight imitation: some of his variations are based on canon at intervals other than the unison; some are built on retrogrades (a part played backwards); the last one is based on augmentation/diminution (playing the same parts at different speeds); nearly all of them make use of inversion (turning the melody upside down); and, typical of Bach’s chaconnes, Bach is much more free in the harmony that he applies over the ground bass pattern. This provides us with an opportunity to apply the characteristics we have noted, to see if we can construct a variations canon. There is no reason to believe that these canons of the Goldberg ground were ever meant to be used as a series of continuous variations in the way that the Pachelbel, Biber, and Purcell variations-canons are; however, it is entirely possible to do so, using the characteristics we have already noted in the existing variations canons – with a few provisos: like the Pachelbel, it would need to be a three-part (hence, stacked) canon at the unison, for polyphonic instruments (so that the larger canons in six voices can be performed by fewer than six instruments); it would make use of the stretto-like imitation pattern of the Biber theme; like the Purcell, the variations would not line up perfectly with the

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ground bass cycles; it would likely contain accompanimental variations that Bach hasn’t written; and we would likely need to re-order the canons to give coherence to the order of variations. I have provided a hypothetical arrangement of these canons as a variations canon, which includes every combination shown in the Fourteen Canons Upon the Goldberg Ground; rather than composing accompanimental variations (my own compositional skills are no match for Bach’s), I have simply left rests, or filled the gaps with a transposition of the subject (which results in parallel octaves that I will consider to simply be texture). The resulting arrangement will certainly not win any awards – after all, these canons were never conceived of as a single composition to be played in this way; nevertheless, it must be acknowledge that, at least from a theoretical perspective, it works – that is, the characteristics and techniques we have noted are sufficient to create a variations canon – and simultaneously presents us with a possible methodology according to which they might be created: while a true composer would have composed these knowing that they would be joined in this way, it is entirely possible that each variation might have been first composed as a self-contained canon, before being joined together.

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Pachelbel, Purcell and Biber: The Variations-Canon

Figure XXV-2 A Hypothetical variations-canon on Bach’s 14 Canons on the Goldberg Ground Page 1

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Figure XXV-2 A Hypothetical variations-canon on Bach’s 14 Canons on the Goldberg Ground Page 2.

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Figure XXV-2 A Hypothetical variations-canon on Bach’s 14 Canons on the Goldberg Ground Page 3.

XXVI. Writing a Canon over a Cantus Firmus

W

e have now come to the point of examining the reasons why the passacaglia and especially the chaconne have been used to create variations-canons, while the other typical variations forms, such as the Romanesca, the Favorita, the Passamezzo, the Folia, the Spagnoletta,

the Villano, the Canario, and others, have not. Classical guitarists will be particularly familiar with these kinds of pieces, as they represent a very large proportion of the early guitar (lute, etc.) repertoire,

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especially the Spagnoletta – yet the use of canon does not seem to be used in this portion of the repertoire. We have already made some observations about the characteristics common to all the variationscanons we have thus far examined. One characteristic we have noted is the general proclivity for steady downward voice-leading as often as possible in as many voices as possible. It is not immediately obvious why this should be so, but it turns out to be a consequence of another important characteristic. So far, our discussion of compositional technique for canon has focussed on the process of composing canon, and even stacked canon, in isolation – that is, in which the dux and comes (taken from the same part) must only harmonise well with each other. However, in our variations-canons, we must also harmonise well with the bass ostinato. It is at this point that we must now focus our attention on the process of writing a canon over a cantus firmus, when the dux and comes must also harmonise well against the cantus – an analogous

Figure XXVI-1 The effects of a cantus firmus on the writing of canon (A) In the upper version, the fifth created in measure two is consonant, but in the lower version (with a cantus firmus), this becomes a dissonant added ninth. (B) In the upper version, the fourth created in measure two is dissonant and normally unacceptable, yet with the addition of a cantus firmus in the lower version, it becomes a typical first-inversion triad.

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process to writing a canon over a bass ostinato.84 This normally will have the consequence of making otherwise consonant intervals into dissonant chords (such as a perfect fifth between the dux and the comes, a fifth above the cantus, produces a “chord” of 1-5-9, in which the ninth, previously consonant as a fifth, is now dissonant), although it can also make a previously dissonant interval into a consonant one (for instance, a perfect fourth between the dux and comes, normally dissonant, when placed a third above the cantus, suddenly becomes the upper fourth of a 6/3 chord, and as such, is perfectly consonant – see Figure XXVI-1). Modern musicians have long since given up on the tradition of keeping a cantus firmus in each new piece of music, in all but the most extremely traditional sacred situations. The tradition held sway throughout the Mediæval era and the Renaissance, but even by the Baroque era (the time of Pachelbel, Biber, and Purcell), the use of the cantus firmus was no longer the dominant method of composing, though it was still in wide use for educational purposes.85 In order to deal with the composition of canon over a cantus firmus, then, we need to look backward to the masters of the Renaissance. Three, in particular, included the cantus firmus in their discussion of canon: Zarlino, Morley, and Bathe.86 Morley’s Plaine and Easie Introduction to Practicall Musicke contains minimal textual instruction on the topic, but several musical examples, and as it has often been remarked, a picture is worth a thousand words. Zarlino’s Le Istitutioni Harmoniche III: The Art of Counterpoint contains somewhat more information on the subject, though the real jewel here is the least familiar: William Bathe’s Brief Introduction to the Skill of Song. Bath’s discussion of the concept is rather poorly

84

A canon of this kind is traditionally called a “chase” (caccia): “The term caccia […] designates, as a rule, works relating scenes of actual chasing or hunting, which were performed by two canonic voices moving above a free tenor part.” [emphasis added] (Mann, The Study of Fugue, 9, footnote 3) 85 Consider, for instance, the Gradus as Parnassum of J.J. Fux, long the standard method for studying counterpoint, published contemporaneously with the career of J.S. Bach, which makes very heavy use of the cantus firmus. 86 A survey of the contributions of these authors can be found in (Collins)

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written and difficult to interpret, but he provides an extremely useful chart, describing the types of motion allowable in canon at various intervals based on the motion of the cantus firmus. Recall that in section XI, we discussed how old treatises, going back at least to the twelfth century, described rules for achieving correct harmony according to how the melody should move.

The

instructions from the Ars Organi an approximately thus: If the cantus moves by melodic interval X, and the parts begin at harmonic interval A, then move the organum voice by melodic interval Y, so that the resulting harmonic interval is B. We noted that it is a simpler situation for canon, because the final harmonic interval is controlled by the initial pitch of the dux, not the initial interval, so theorists like Gosman can simply say: if the initial pitch of the dux is X, and the interval of imitation is M, then move the dux by the intervals P or Q to get an acceptable harmony. Now, however, with a cantus firmus involved, we must return to the more complex form of the rule: If the cantus moves by melodic interval X, and the dux begins at the harmonic interval of A relative to the cantus, and the interval of imitation is M, then move the dux by melodic intervals P or Q to get an acceptable harmony. This is approximately the logic that is presented in Bathe’s chart. Bathe’s chart is recreated in Table XXVI-1. Bathe’s explanation of this chart is difficult to understand, but it is worth the effort. First, he states that the chart is equally useful for those wishing to construct assorted canons without a cantus firmus, including in three parts (“the third part being under”),87 or simultaneously upon two “plainsongs” (simple melody in the style of a cantus firmus, whether or not it is a pre-existing, traditional melody), etc.88

87

(Bathe 1982, "C. you",~34) Bathe makes the general assumption that the cantus firmus will be in the lowest voice, and Collins follows this assumption (Collins, 118). This is one point in which Bathe and Zarlino differ, as Zarlino presents observations about how the dux will differ depending on whether it is above or below the cantus: “In a series of conjuct steps, it is proper in the first method (the [dux] above [the cantus firmus]) to have a descending run in the [dux] and in the second method ([dux] below) an ascending run, […]”. (Zarlino, 217). Meanwhile, we will also follow Bathe, because in the variations canons that we are examining, the melodic variations are typically well above the bass ostinato.

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Table XXVI-1 A Recreation of William Bathe’s Chart for the Creation of Two-In-One at Any Interval over a Cantus Firmus

The observations of the places up are six89

6 5 4 3 2 1

Places up.

Courses

down

Courses

up 1 2 3 4 5 6 7

7 6 5 4 3 2

8 vt ſu:1 Places down The observations of the places down are six.

1 2 3 4 5 6

85 1 7 2 5 1 1

11 7 6 1 6 2 7

10 6 5 7 7 3 6

9 5 4 6 1 4 5

8 4 3 5 2 5 4

7 3 2 4 3 6 3

6 2 1 3 4 7 2

1 3 5 6

6

1 3 5

1 6

3 5

1 3 6

5

6 1 3 5 1 6 3 5 1 3 6 5 1 3 5 6 1 1 5 2 7 1 5

1 3 5 1 6 3 5 1 3 6 5 1 3 5 6 6 2 2 6 3 1 2 6

1 6 3 5 1 3 6 5 1 3 5 6 6 1 3 5 3 3 7 4 2 3 7

3 5 1 3 6 5 1 3 5 6 6 1 3 5 1 6 4 4 1 5 3 4 8

1 3 6 5 1 3 5 6 6 1 3 5 1 6 3 5 5 5 2 6 4 5 9

5 1 3 5 6 6 1 3 5 1 6 3 5 1 3 6 6 6 3 7 5 6 10

1 3 5 6 6 1 3 5 1 690 3 5 1 3 6 5 7 7 4 1 6 7 11

The basic challenge in writing a canon over a cantus firmus is that we have to extend our error checking methods by an extra cycle. Remember that the basic process in writing a canon is to copy the current dux note forward into the next cycle (and transpose it by the interval of imitation, if it isn’t at the unison), then decide what kind of motion in the dux will create a suitable harmony against the new comes note; in this process, we need only concern ourselves with two cycles at a time: the previous one and the current one. It would be nice if the process of writing over a cantus firmus only required us to also make sure that the new dux note would only need to be consonant against the current cantus note; alas, we

89

Original: “The obſeruations of the places vp are ſixe”; “Courſes vp.”; “Courſes downe”; “Places dovvn”; “The obſeruations of the places down are ſixe.” It is not clear what is meant by the caption “8 vt ſu:”, so I have therefore left it un-modernised. 90 This cell contains a typo in Bathes treatise (it reads “3 6”); I have corrected it here.

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must also ensure that when it reappears (appropriately transposed) in the comes in the next cycle, that it will still be consonant with the next cantus note. In other words, we need now to consider not only the previous and current cycles, but also the cantus note of the next cycle. Bathe explains that the rows marked “Places Up” and “Places Down” refer to the interval of imitation,91 and that the number represent intervals, not the number of steps (hence, one place up or down is actually just the unison). He next explains, in a very confusing passage, that the columns labelled “Courses Up” and “Courses Down” refer to the melodic interval traversed by the cantus firmus during the time equal to the delay of the canonic rule (i.e. if the canonic rule is to delay imitation at a given interval by one measure, then these columns describe the melodic interval the cantus firmus will cover between the time that the note appears in the dux and the time that it reappears in the comes), over the following measure.92 The intersection of the chosen row and column will give you the harmonies (the interval between the dux and the cantus firmus) that will be acceptable in that measure – those that will make an acceptable harmony possible in the next measure.93 As Denis Collins notes, it is not quite enough information yet: we must then determine, from the desired harmony, which melodic interval in the dux will serve to create that harmony, and determine if that motion would be stylistically appropriate, in the way that Gosman describes. This encompasses

91

“Firſt it is to be vnderſtanded by this word place, is ment the diſtance [presumably melodic] of the following part, to the former part…” ibid. 92 “Next heere is to be vnderſtanded that by this word, Courſe, is meant the diſtaunce [melodic, as above] of that which followeth ſust ſo long after, as the following part [comes] reſteth to that which goeth before [dux], in the plaine Song or ground [i.e. the melodic distance, or interval, in the plainsong over a time period corresponding to the delay of imitation], …” ibid. 93 “… firſt looke in what place vp or downe, you would haue the following part to bee, […]. Then looke in what courſe vp or downe is the note of the ground, for which you would make, then looke what ſquare of the table meeteth with the place and courſe, and there you ſhall find noted by figures, what concord ſerueth for that courſe.” ibid.

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concerns such as parallel fifths, inappropriate leaps, etc. Collins provides a second chart for the purpose;94 however, we can calculate the melodic interval with simple math: The harmonic interval between the dux and the cantus firmus in measure 2 (we’ll call it η) results from the effect of the motion of the cantus firmus between measures 1 and 2 (call it Δγ) and the effect of the motion of the dux between measures 1 and 2 (call it Δδ) on the harmonic interval between the dux and the cantus firmus in measure 1 (call it α): η = α + Δδ – Δγ. Since we know the original harmony, and we know the target harmony and how the cantus moves, we simply rearrange to Δδ = η + Δγ – α. Calculating mathematically has two basic problems: it doesn’t account for voice crossing of the dux or comes below the cantus firmus; and it doesn’t account for stylistically appropriate melodies. However, for our purposes here, these are non-issues: in variations-canons, the ostinato and variations tend to be far enough apart in register that voice crossing is unlikely; and the delay of imitation in a variations canon is generally long enough, and the degree of melodic diminution generally complex enough, to rectify any unstylistic melodies. At any rate, the math is only helpful if we intend to write the dux independently, without having to write out the comes or the cantus with it. As an example of how to use this chart, consider Figure XXVI-2. This is a canon at the fifth, over a “cantus firmus” from the first few notes of the familiar children’s round “Three Blind Mice”. Normally, we can pick the first note of the dux at random, but against a cantus firmus, we must pick a harmony that works, not only against the current cantus firmus, but also against the next cantus pitch. Checking Bathe’s chart for the interval of imitation of a fifth and a descent of the cantus firmus by a second, we find that the intervals of unison, third, and fifth, will allow the comes to harmonise with the cantus in the following measure. We choose a third, and then copy it into the comes, transposed by a fifth.

94

(Collins, 118, 116)

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Now, we have another descent of a second in the cantus firmus, so we once again have the choice of unison, third, or fifth, between the dux and the comes. Although a casual glance at the score would suggest that a sixth would create an acceptable harmony, it would show up in the comes as a fourth against the cantus, which isn’t acceptable. This time, we’ll choose a fifth, and again transpose it into the comes. Next, we have an ascending third in the cantus firmus. We check Bathe’s table, and find that, in imitation at the fifth, with cantus motion of a third, dux should harmonise with the cantus at a unison, third, or sixth – a fifth (which would complete the C-Major triad) won’t work, because in the comes, it A

B

C

D

E

F

Figure XXVI-2 Creating a First Species Canon over a Cantus Firmus Note that we are not concerned about the parallel octaves created in E, because of the eventual diminutions.

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would create a seventh against the cantus. We choose the sixth, and transpose it into the dux. We come back to a pair of descending seconds in the cantus, and we continue the process to complete the excerpt. Note that we are not concerned about the parallel octaves between the cantus and the comes in measures 4 and 5, because we are dealing with the outline of a variations canon, and these parallel octaves will be broken up by diminutions. So, why is Bathe’s chart relevant to our consideration of why all the variations-canons seem to be chaconnes and passacaglias, but not spagnolettas, etc.? The critical detail on the chart is how the cantus firmus moves. What a holistic survey of the chart tells us is the number of options available in the dux based on the type of motion in the cantus. Looking at the column for canon at the unison, we can see that there are four options available if the cantus firmus stays at the unison from one cycle to the next – just as the Pachelbel canon and the Biber ciacona-canon do. There are also three options available if the cantus firmus moves down by a third from one cycle to the next – just as the Purcell two-in-one does. However, there are only one or two options available for cantus motion by second or fifth. In other words, Pachelbel, Biber, and Purcell each chose progressions that provided lots of opportunities. If a chord progression is not rich in these kinds of cantus motions that provide lots of opportunities, the chord progression will not be suitable for a variations canon. Of course, any progression that repeats itself on a regular basis will move by unison, and so could support a canon like Pachelbel’s. However, if you want to write it as a stretto canon, like Purcell, or at least incorporate imitated stretto, like Biber, you need a chord pattern in which the bass motion at the delay length of the canon consistently uses high-opportunity intervals. This rules out most of the traditional variations forms. Interestingly, if you are willing to write at other intervals, you could easily have just as much freedom writing a variations-canon over a chord progression of a cycle of fifths, at a delay of two chords and an interval of imitation of a second. Canons at intervals other than the unison tend, however, to be more difficult to compose, especially when written over a cantus firmus; Bathe is generally a less well-

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Figure XXVI-3 Differing consequences of ascending and descending voice-leading on canons at the unison (A) The voice-leading is ascending, and the resulting triads cycle through different inversions. (B) The voice-leading is descending, and the chords maintain consistent root-position triads.

known theorist than Morley or Zarlino, and his chart would likely not have been well known to many other composers. It is, perhaps, no surprise that no examples of a variations-canon have survived that use imitation at other intervals. We have so far glossed over one other point: there is a marked preference for steadily descending voice-leading. This makes perfect sense for a simple canon at the unison, because it ensures that the chords created remain in the same inversions from cycle to cycle (Figure XXVI-3). However, this effect doesn’t apply to variations-canons or canons over a cantus firmus, because as long as the cantus remains below the canonic voices, the inversions are controlled by the cantus, not the voice-leading of the canon. In the variations-canon, the bass ostinato is generally quite a bit lower than the canonic voices, and could ascend freely without crossing the canonic voices and losing the bass function. So it is unnecessary to stick to descending voice-leading, though it is not impossible that the composers chose to stick with the methodology that they found familiar. In general, though, we can simply acknowledge that descending voice-leading has always been a preferred kind of motion in the Western art music tradition, as far back as we have treatises discussing compositional procedure, though it is not entirely clear why. Even Schenkerian analysis, predicated as it is on the paramount importance of the

V - I progression, downplays the significance of the ascending

leading tone (traditionally the most important feature of this progression), relegating it to an inner voice,

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compared to the true, “structural” melody that can only be a descent from a tonic chord tone to the tonic. Schenker generates his theory from the harmonic series, and sees the overall progression of music as beginning higher-up in the harmonic series and progressing downwards to the fundamental pitch of that series, but much ink has been spilled debating the contrivances to which Schenker had to go to make his system coherent. Historically, some such preferences have been explained in ways that show why a pattern was common in the past, even if it is no longer so strongly followed today; for instance, the preference for rhythmic division into groups of three in the Mediæval era was explained as a reference to the holy trinity, and the different types of harmony predominating in different eras can be understood according to the differing cultural acceptance of complex harmonies and dissonance; various composers are well known to have incorporated various philosophies into their composing (Bach was well known for his taste for numerology). However, with respect to the predilection for descending voice-leading, the older theorist are not terribly helpful. Some, lacking any better explanation, simply attribute it (ridiculously) to the force of gravity,95 as though notes have a mass to be pulled upon. Another possible explanation is the association of the technique of canon with the techniques of double-counterpoint.96 Double counterpoint (as well as triple and quadruple counterpoint) is a subgenre of counterpoint that is fully invertable as some interval – that is, you can transpose the any of the voices by the specified interval such that a different arrangement of voices results, and yet the contrapuntal relationship between voices is still acceptable. We have already noted in section VIII above that the Pachelbel sequence is fully-invertible quadruple counterpoint. One of the challenges behind invertible

95

“The best and simplest explanation of this is the natural law of gravity; se Roth, Elemente der Stimmfuehrung, p. 89.” (J. J. Fux 1965/1971, 60, footnote 2). On the same page, Fux escribes the problem as “… harder to untangle than the Gordian Knot.” 96 Albrechtsberger provides examples of canons generated from double counterpoint in his treatise; see, for instance, (Albrechtsberger, 231).

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counterpoint is that when you rearrange the voices, the intervals between notes change, and often, intervals that were consonant become dissonant. For instance, invertible counterpoint at the octave (in two voices, this would mean that the upper voice would be transposed down – or the lower voice transposed up - an octave to be below the lower voice) swaps unisons for octaves, thirds for sixths, and seconds for sevenths, none of which typically cause any difficulties, but fifths are substituted for fourths, which has two effects: first, parallel first inversion chords (like the old technique of fauxbourdon) cannot be employed, because the parallel fourths that are typically found therein would become parallel fifths; and second, all fifths must resolve downwards, because when inverted, they would become fourths, which would need to be resolved like suspensions. Similar types of challenges result from invertible counterpoint at other intervals – in general, intervals that swap from consonant to dissonant must resolve downwards, just as the dissonant intervals must resolve downwards – in general, descending motion is even more dominant in invertible counterpoint that in simple counterpoint. The Pachelbel theme, for instance, could actually be generated from double counterpoint at the tenth, in the sense that the second half of the theme is approximately a transposition of the first half at the interval of a third (i.e. transposed down a tenth, as in double counterpoint at the tenth, then up an octave; technically, it is double counterpoint at the third, but it is traditional to deal with double counterpoint at compound intervals, and the transposition back up an octave will not change the characteristics of the voice-leading, since it will not cross back above the other part). The rules of double counterpoint at the tenth create the following interval exchanges: Table XXVI-2 Interval excahnges of Double Counterpoint at the Tenth

1

2

3

4

5

6

7

8

9

10

10

9

8

7

6

5

4

3

2

1

This means that thirds and octaves (here, we are talking about the relationship between the cantus and the first part of the variation) are perfectly good choices for maintaining the same chord progression,

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since they swap with each other essentially keeping the same chord tones; however, as always, parallel octaves are no good, and in this case, parallel thirds and sixths are not possible, because they would result in parallel octaves and fifths, respectively.

Unfortunately, the remaining choice for chord tones

(fifths/sixths) is limited, because it results in a “mediant” type substitution in the chords (the exchange of the fifth for the sixth, or vice-versa). As a result, the only real voicing possibilities that would create this theme in parallels as it does would be the exclusive use of octaves and thirds (hence the observation in section XI above that in the opening theme, there is no fifth in the chord tone cycle). This is by no means a perfect explanation for why descending voice leading must be used in these variations canons (ascending voice-leading is a perfectly viable option), but rather a desirable characteristic that is available with descending voice-leading that is not available in ascending voiceleading, which may help to explain why the composers seem to have preferred descending voice-leading.

XXVII.

A

On the Importance of Sequences lthough we have successfully shown that the chaconne and passacaglia bass patterns are highly conducive to stretto canon, and so have explained their prevalence in that context, we have also shown that, in theory, any repeating chord progression should be useful in creating

a canon delayed at the full length of the progression, like the Pachelbel canon. Can we suggest any plausible reason why other progressions have not been selected for this kind of treatment? This is quite a bit more difficult, as there is certainly no reason why it couldn’t be done. In the simplest sense, it could simply be a numbers game: since there are so few examples of variations-canons in the repertoire, we can hardly expect a complete cross-section of variations forms to be represented. We are instead trying to discover why composers have preferred the passacaglia and chaconne to other feasible patterns, and as already noted, we are always on shaky ground when trying to explain a composer’s æsthetic choices. That being said, the answer may lay in sequencing.

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We noted in section XXII above that all the canons examined herein are conducive to sequencing at some degree. This is most true of the Pachelbel canon, for which the chord pattern is very closely patterned on the traditional Plagal Sequence; however, even for the Purcell, the least obviously sequential of them all, if careful voice-leading is applied, sequencing is nevertheless possible. Is this a coincidence? Not likely; we have also observed in section XI above that canons often move in consistent sequences of chord progression – indeed, van Geenen’s approach to writing stacked canons based on relative chord tones is premised on this fact. The process of writing variations of this kind also frequently employs the use of sequencing. The various chord patterns we are considering all have their origins in the Renaissance, and we do have a Renaissance treatise that discusses variations of the kind that would have been created at the time. In 1553, Diego Ortiz wrote his Trattado de Glosas, which includes within it a discussion of variations. Unfortunately, Diego’s prose says very little other than, “To illustrate this way of playing I set forth here

1.

2.

Figure XXVII-1 Examples of sequencing in Ortiz’s treatise Although the sequencing does not look quite like the sequencing we typically expect to see, these incipits from the first two of Ortiz’s examples show, even at a casual glance, that sequencing is an important part of how variations were created over canti firmi in the Renaissance.

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six studies on the plain song which follows.”97 We are supposed to learn the technique through his examples. An examination of these examples does show a strong predilection for sequencing (see Figure XXVII-1). So far, all we have shown is that variations technique uses a lot of sequencing, and indeed, the forms we are trying to rule out (such as the spagnoletta) likewise use a lot of sequencing. There is an interesting difference, however. During the Renaissance, these patterns were all used as practical dance music, and complex polyphony such as canon would have been highly uncharacteristic of these types of pieces, just as these kinds of bass ostinati would not have been used as the basis of a more serious composition. By the Baroque era, it had become typical for these kinds of dances to become the basis for more serious compositions; however, by the Baroque era, harmony, and by extension, sequences, changed. The sequences in the Ortiz move all over the place, by any convenient interval. By the Baroque era, sequences were typically two chords per statement, and generally moved by second, or by descending third. Of all the chord patterns of the traditional variations forms, only the chaconne and the passacaglia consistently allow this kind of sequencing. Remember, the point here is not that the other forms wouldn’t work, only that a composer wishing to write a variations-canon would likely find the chaconne and the passacaglia more attractive than other traditional patterns because the traditional sequencing patterns are available.

XXVIII. Recapitulation

W

e have now considered three extant variations-canons, as well as several related pieces and a number of old theory treatises in our examination of the combination of variations forms, such as the chaconne and the passacaglia, with the technique of canonic imitation. We

97

(Farrell, 8)

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noted that canons tend to have limited choices harmonically, dictated by their intervals of imitation: canon at the unison tends to create static harmony, while canon at other intervals tends to progress by perpetual sequence according to that interval. We noted that canons that did not begin as variations forms can often support a bass ostinato to become a variations form, and that the strength of this association increases as the number of voices or the degree of polyphonic melody increases, and as the delay of imitation becomes longer. We concluded that the use of this technique in an existing variations form tends to succeed best when the chord pattern of the variations form provides opportunity for sequencing, and when the chords progress by intervals favourable to that interval of imitation especially if they favour consistent downward voice-leading and support invertible counterpoint. We have also noted that the use of the strict canonic technique within a variations form causes challenges to the composer that force him to find creative solutions. We saw that Pachelbel made use of parallel harmonies in consecutive variations, then filled in space with accompanimental variations, so individual variations would not need to compete for the listener’s attention. Pachelbel also showed that, even when using the strict canonic principle, it is still possible to create complex formal structures – and that the apparently simple surface of his canon is, in fact, much more sophisticated than it seems. Biber, on the other hand, made use of catch-like textures of alternating activity between two different parts of the same overall melodic line. We also saw the technique of imitating stretto canon in a non-stretto canon, a technique that is largely only feasible in chord patterns characterised by much sequencing, as a desirable attribute and a possible explanation for the prevalence of the chaconne form as a basis for a variations-canon. We also noted from a comparison of the Biber ciacona-canon to Merula’s Chiacona that the use of the canonic principle tends to discourage manipulation of the bass line itself, as is often done for the sake of variety in non-canonic chaconnes.

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In Purcell’s two-in-one, we saw that the technique can be applied just as effectively at a shorter delay than the entire length of the bass ostinato, and that the lengths of variations need not be closely aligned with the length of the ostinato. From the treatises by Morley, Zarlino, and especially Bathe, we have seen that it ought to be possible to construct variations-canons on other chord patterns than the traditional chaconne and passacaglia patterns, such as a cycle of fifths, by using intervals of imitation other than the unison. Given this possibility, and the fact that writing such a canon, if approached with some care and preplanning, is apparently easier to compose than it first seemed, we can only wonder whether such canons may once have existed, and have been lost to us.

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XXIX. Appendix A Schenkerian and Westergaardian Analyses of the Pachelbel Canon I have reserved the discussion of Schenkerian analysis of the Pachelbel theme to the appendix because it sheds relatively little light on the canonic aspects of the pieces presented here. However, as a general discussion of the reasons why the progression seems so ubiquitous, it is worth a quick look at how a Schenkerian would understand the chord pattern to arise. I have modelled my analysis after the generative style of Matthew Brown98.

Note that in the final stage, I have employed the “unfolding” notation uncharacteristically, for register transfers, rather than true chordal unfolding, only because the symbol helps to draw attention to the close relationships between these notes.

98

(Brown 2005)

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Westergaardian theory is an offshoot of Schenkerian theory. The following analysis according to the theories of Peter Westergaard comes from a blog99 from a Westergaardian theorist, who was responding to a question about the nature of a certain category of chord patterns. The blogger completely bypassed the point of the question, but did provide an analysis of the Pachelbel theme according to Westergaardian principles. I have adapted his analysis slightly here. First, the two chords I and V are presented. Next, the chords are voiced in such a way that the span of all the melodic motion prior to the V chord is outlined in each part.

The span is then separated into two notes, creating two smaller spans. These are then filled linearly (with an escape tone at the end of the second line), and finally the bass is altered to harmonise with the upper voices in sequence.

99

(Cook 2008)

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XXX. Appendix B The Gigue from “Kanon und Gigue in D-Dur fur Drei Violinen und Basso Continuo” by Johann Pachelbel

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XXXI. Appendix C The Clarke Chaconnes

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XXXII.Appendix D The Merula Chiacona

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XXXIII. Appendix E Dido’s Lament (Phrase vs. Ostinato)

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XXXIV. Appendix F Fourteen Canons on the Goldberg Ground

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