Contents

Introduction
Structural - conceptual
Treatment of dissonances
’Mistakes’ against voice leading
Chromaticism
Fugal writing: some technical aspects
Sequences - Linear Intervallic Patterns
Rhythmic motion and rhythm
Metre
Bach the Virtuoso Harpsichordist
Terminology
Footnotes

Introduction

Giovanni Pierluigi da Palestrina (1525 or 1526-1594) enjoyed the esteem of his composer colleagues in the seventeenth century, and his method of composition was considered to be the ideal way of writing polyphonic religious music. This kind of music was called stile antico, which was synonymous with stile da Palestrina. The admiration for Palestrina was not restricted to the seventeenth century, but also in the eighteenth century his music was considered as ideal for training young composers. In this context Johann Joseph Fux published, in 1725, Gradus ad Parnassum,¹ a systematically constructed treatise in Latin in, according to the author, the style of Palestrina.²

Bach admired Fux the composer,³ but it is much harder to find out how he liked Fux the theorist. Bach owned a copy of Gradus ad Parnassum,⁴ but – according to Carl Philipp Emanuel Bach – did not use it during his lessons: ‘… In composition he started his pupils right away with what was practical, and omitted all the dry species of counterpoint that are given in Fux and others…’⁵ From this testimony it appears that Bach was a highly practical composer, and that L’art-pour-l’art was strange to him.⁶

There is, nevertheless, some evidence that the point of view of Carl Philipp Emanuel with regard to Gradus would seem to be a little harsh. In recent years several manuscripts have been found which prove Bach’s (theoretical) interest in stile antico. In 1999 a section of a study of canon and counterpoint partly in stile antico, in the form of a written conversation between Johann Sebastian and Wilhelm Friedemann, has been found in Kiev. This manuscript, however, dating from 1736-1738, does not appear to be a dialogue between teacher and pupil (as in Gradus), but rather an exchange of ideas between colleagues. ⁷ In a publication of 2003 Walter Werbeck discussed two previously unknown manuscripts by Bach, probably written between 1739 and 1742. These manuscripts give a survey of the most important contrapuntal techniques in stile antico, and were probably used for teaching purposes.⁸

In his pioneer biography on Johann Sebastian Bach, Philipp Spitta writes that Sebastian esteemed Gradus very highly, and that he encouraged one of his students, Lorenz Christoph Mizler (1711-1778), to make an annotated German translation of it, ‘under Bach’s very eye, as it were’,⁹ which was published in 1742 in Leipzig.¹⁰ In spite of the fact that Christoph Wolff confirms this,¹¹ there is no historical evidence that Mizler was Bach’s pupil, or that Bach would have been involved in the annotated German translation. Consequently, a direct appreciation by Bach of Fux’s treatise does not exist. Of course, it would have been a remarkable coincidence that Mizler should have published his translation at the moment that Bach occupied himself very strongly with the study of the stile antico.¹²

From the surviving musical material in Leipzig we know that Bach performed Palestrina’s six-voiced Missa sine nomine¹³ and his four-voiced Missa sopra la Cantilena Ecce Sacerdos Magnus,¹⁴ which shows his appreciation for this composer.

This interest in stile antico in the realms of composition and pedagogy, however, does not mean that the Art of Fugue was written in this severe style – as Christoph Wolff elaborately proved¹⁵ – in spite of the fact that the work shows some influences of it (the general ‘serious’ tone, the used ‘older’ alla breve time signatures (4/2, 2/2)¹⁶ with their typical minims, crochets and quavers (only rarely semi-quavers) and an occasional use of modal elements (Contrapunctus 1-7, 12 and 13 all end with a plagal cadence).¹⁷ The style of composition Bach uses for the Art of Fugue is situated in between the stile antico and the newer baroque style, from which occasionally elements of both styles are taken.¹⁸ Christoph Wolff names this kind of compositions alla breve-pieces.¹⁹ Though Bach must have recognized the value of Fux and the value of codifying principles of classic polyphony (perhaps reflected in Mizler’s translation), his Art of Fugue exceeds the parameters set out by Fux. Others, of course, had been expanding these limits – that is, after all, one of the key stories of the “Baroque”. Yet it is in this uniquely Bachian expansion – conceptually, technically and rhetorically – of the Fuxian paradigm that many of the most interesting, telling – and as yet – under-explored moments of the Art of Fugue are to be seen (and heard!).

In this article I hope to demonstrate, from several angles, that Bach, in his Art of Fugue, pushed every textbook rule and concept to the limit.

Structural - conceptual

To have a clear view of the context in which Bach conceived the Art of Fugue, we must go back to the seventeenth century, where the instrumental opposite of the motet was the ricercare and fantasia. The young Bach made himself familiar with this imitative keyboard music, as Carl Philipp Emanuel hastily writes to Forkel: ‘Besides Froberger, Kerl, and Pachelbel [sic], he heard and studied the works of Frescobaldi, … Fischer, Strunck [Nicolaus Adam Strungk], some old and good Frenchmen, Buxtehude, Reincken, N. Bruhns, and … Böhm… Through his own study and reflection alone he became even in his youth a pure and strong fugue writer. The above-named favorites were all strong fugue writers.’²⁰ Ricercari and fantasie were still written in the 18^th century, but were influenced more and more by features of the baroque fugue.²¹ While thematic development in great density rules the stile antico,²² the stile alla breve (named after the (used) time signature used) differs from it on a structural level by the use of episodes (confutatio), which are used in contrast to the thematical Durchführungen (confirmatio).²³ Looking at the Art of Fugue, Bach clearly applies this concept of alternation between confirmatii and confutatii (the most concentrated Contrapunctus 7 has only one episode), but expands and personalizes this rhetorical contrast and – more generally – the baroque structural concept, which will be made clear from the following examples.

After the exposition of Contrapunctus 2 (example 1),

Bach writes a confutatio which is followed by a confirmatio, the counter-exposition (from bar 17 onwards). This confutatio, however, ends in a manner which we normally would not expect in a fugue. In bar 21, the contradevelopment with the final motif of the theme inversus and rectus is stopped, and from there the soprano is left all alone with that motif above a pedal note in the bass, while the harmony stays unchanged during those two bars (V7/V). Because of this, the following counter-exposition begins in an unusual manner, as the alto begins not with the first but with the second note of the theme. This is a consequence of the harmony; the V7/V first has to resolve into the dominant, in which the d¹ does not belong. Bach turns this problem to his advantage, and develops the inversion of the final motif of the theme through the alto, tenor and bass voices, while the alto, after its initial use of the final motif of the theme, goes on with the complete theme from the second note on. In that way, it appears that a new episode is starting in bar 23, while it is actually the start of the counter-exposition.

With Contrapunctus 11 we go a step further in Bach’s complex working out of form. This piece is by far the most daring composition of the Art of Fugue in the realms of harmony, counterpoint and structure, where harmony and counterpoint control the structure, in opposition to, say, Fux’s treatise, where form is determined beforehand. Before embarking on a discussion of the piece, it needs to be said that not every scholar agrees that this piece is a triple fugue - for me the only correct designation possible - because of the systematic combination of the three themes (which in fact are the three inverted themes of Contrapunctus 8) in the last section of the piece.²⁴ Kees van Houten and Marinus Kasbergen are convinced that Contrapuncti 8 and 11 are double fugues (that is fugues constructed on only two themes): ‘In the next group of four Contrapuncti [Contrapuncti 8 to 11 inclusive] the Art of Fugue theme is also combined with other themes. This group of four double fugues is less homogeneous than that of the first four Contrapuncti’²⁵ Marcel Bitsch shares this opinion.²⁶ This view is explained by considering the second theme of Contrapunctus 8 (bar 89) and its inversion, the third theme of Contrapunctus 11 (bar 89), not as independent themes but as counter-subjects to the first theme of Contrapunctus 8 and the third theme, its inversion, of Contrapunctus 11. Others, like Wolfgang Graeser, think of Contrapunctus 11 as being a quadruple fugue, considering the chromatic countersubject to the second theme (bar 28) as an independent theme.²⁷ This, in my opinion, is wrong, because this subject does not appear any more in the combination of the different themes in the last section of this fugue. This view is based on the fact that most of Bach’s fugues on several themes do not use regular counter-subjects (in order to have the combination of the different themes as a rhetorical climax), contrary to what is mentioned in one of the classics among textbooks on fugue, André Gédalge’s Traité de la fugue: ‘A melodic or rhythmic figure heard in these conditions in a fugue takes the name of NEW SUBJECT. This NEW SUBJECT must be written in counterpoint – reversible with the PRINCIPAL SUBJECT at least, and if possible with the COUNTER-SUBJECT of the principal subject. The new subject may also be accompanied be a countersubject, reversible or not with the principal subject and its counter-subject.’²⁸ The fact that the second theme is accompanied by a countersubject can be considered an exception in the oeuvre of Bach. The confusion is even bigger because the second subject in Contrapunctus 8 (from bar 39 on) is introduced together with the first; in Contrapunctus 11 it seems that Bach applies the same technique from bar 27 on, while it is ‘only’ a counter-subject to the second subject. This shows that Bach prefers to exhaust all the possibilities of a certain type of fugue which goes way beyond the usual baroque concept of fugues with several subjects.

In the realms of dispositio, this piece is the piéce de résistance of the Art of Fugue. Instead of constructing this triple fugue ‘in a regular way’ Bach makes the following ‘deviations’: (1) in contrast to Contrapunctus 8, Bach does not give the combination of the first and second theme, a consequence of Bach’s counter-subject to the second theme, and (2) both themes, when combined in the final section of the fugue, are always accompanied – without the counter-subject to the second theme – by the third theme.

Equally remarkable is that Bach writes a whole section with only the inversus of the first theme (bars 71-89). But normally the inversion of a theme – if used – will mostly appear in the same section as its original theme (as in the first section of Contrapunctus 1).²⁹ This separate presentation (and thus a) structural peculiarity) can be seen as a suspensio (the listener can expect something ‘special’ with both forms of the theme). This expectation is satisfied in the last section of the fugue, in which Bach, after a first combination of the three themes (bars 145-150), gives simultaneously the first theme and its inversion (bars 158-162), which, moreover, offers the possibility of double counterpoint at the tenth (bars 164-168). From this it is marvellous to see how Bach is also a master rhetorician on a large scale!

Treatment of dissonances

When speaking of Bach’s bold musical language, one cannot omit a profound discussion of dissonance and possible deviations from what was generally accepted.

The traditional suspension (ligatura or syncopatio) of the strict counterpoint (minim dissonant on the downbeat (sometimes ornamented with an anticipation, an anticipation and a lower adjacent note or with a consonant chord note), prepared with a consonant minim on the upbeat)³⁰ is very rare in the Art of Fugue, because the other voices would have to adopt this immobility - there is, most of the time, a rhythmic motion by quaver. This will be thoroughly discussed in the chapter on Rhythmic motion and rhythm. There are, however, some examples of the stile antico ligatura in the most ‘strict’ passages of the work, namely in the first section of Contrapunctus 14, for example in bars 35-36 in the soprano and in only one more case, in Contrapunctus 10 bar 19-20 also in the soprano (example 2).

Bach gives, as opposed to the technically very limited possibilities for the use of dissonances in strict counterpoint, a great variety of realizations:

1. Dissonant crotchet (or quaver followed by a consonant or even a dissonant quaver before resolving on the next crotchet (Contrapunctus 1 bar 17 in the tenor and Contrapunctus 1 bars 26-27 in the tenor)) prepared by a consonant minim (example 3).

2. Dissonant quaver prepared by a consonant minim (Contrapunctus 1 bars 23-24 in the soprano) (example 4).

3. Dissonant crotchet (or quaver followed by a consonant or even a dissonant quaver before resolving on the next crotchet) prepared by a consonant crotchet (Contrapunctus 1 bar 27 in the alto, in Contrapunctus 5 bar 39 in the soprano and bars 50-51 in the alto). The first suspension Bach writes in the Art of Fugue is one of this type, by which he shows from the first moment of polyphony (the transition from bar 6-7 in Contrapunctus 1 in the counter-subject against the comes) that this work will be an exploration of the contrapuntal limits (example 5).

4. Dissonant quaver prepared by a consonant crotchet (Contrapunctus 1 bars 47-48 in the bass)
(example 6).

5. In certain instances the dissonant is even longer than its preparation; we find examples of this in Contrapunctus 10 bars 118-119 in the soprano and in Contrapunctus 14 bars 42-43 also in the soprano (example 7).

6. A note has two dissonant functions before resolving (diminished fifth and perfect fourth in Contrapunctus 1 bars 72-73 between soprano and bass; seventh and fourth in Contrapunctus 1 bar 69 between tenor and bass) (example 8).

In spite of Bach’s highly baroque ligature, he is very consistent in using them, and writes quasi systematically a suspension when the possibility occurs. As a result I find it quite striking to see that in certain cases he avoids the evident suspension. I want to explain two instances which clearly prove that Bach does not follow any other logic but his own.

In Contrapunctus 5, the first of the three stretto fugues, bar 27 is very interesting (example 9).

Bach writes in the bass a minim followed by a quaver rest and three quavers a-g-f. The first thing we notice is that Bach does not imitate the soprano from bars 26-27 exactly - no tie. Why? When we look at analogous places, it becomes evident that it is only in bar 27 that Bach does not tie the consonant minim to the dissonant quaver (resulting in a suspension), but, looking at the voice leading, omitting the tie was Bach’s only option, in order to avoid octaves between soprano and bass resulting from the suspension resolving into g. This would mean that the generally accepted rule, given among others by Gédalge, must be reconsidered: ‘Every time that a note, heard at the end of a bar, is repeated in the bar immediately following after a rest and descends a scale step, one must consider it to be a retardation, and treat it harmonically as if the syncopation had been sustained; ...’³¹ Gédalge gives only an example in a 3/4-bar but it is evident that this view also counts for the two minims in a 2/2-bar.

Omitting evident ligature is not restricted to Contrapunctus 5. Also in Contrapunctus 8 Bach does not write the obvious suspension in the counterpoint against the answer in bar 7 and 8, but consciously opts for a quaver rest (example 10).

We have to wait until the beginning of the first episode (bar 15) to see (and hear) this suspension for the first time, by which this confutatio clearly distinguishes itself from the exposition. We find the answer for the omission of the suspension when looking at the second theme and its combination with the first. In every place where Bach had first written a rest, we do find in the combination the typical dissonance previously avoided - true, not as a real suspension (the note is not tied), but as an appoggiatura (repeated note), which characterizes the second theme (bars 40-41). From this – again – it becomes clear how Bach’s counterpoint helps to control structure.

As with the suspension Fux also heavily restricts the use of other non-chord notes and melodic diminutions, only admitting the passing tone (transitus) and (complete) neighbour tone (quasitransitus), both on weak beats.³² Bach, of course, expands this limited amount of non-chord notes and also uses, next to these figurae fundamentales, accented (dissonant) passing notes and appoggiaturas (Contrapunctus 4 bar 131 in the alto). Incomplete (upper) neighbour notes (échappées), forbidden by Fux, are also a great curiosity in the Art of Fugue (the only examples can be found in the Canon alla Decima among others in bars 3-4 in the bass).³³ But the most striking example of Bach’s bold approach to non-harmonic notes from the Art of Fugue can be found in Contrapunctus 6 in Stylo Francese. In this piece there is one particularly ‘baroque’ note, namely the last semiquaver of the soprano in bar 26 (d²) (example 11).

We have yet to discover a name for such a non-chord note; it is an unprepared dissonant (anticipated by a rest; the chord of the last crotchet of the bar is F major), is on a weak part of the beat and ‘resolves’ with a leap. This description is reminiscent of an échappée, but without being approached from a harmonic note one scale step below or above. Moreover, it forms a cross-relation with the first beat of the tenor in the following bar (db¹). It would appear that Bach preferred to use thematic material in the soprano at this point (the descending fifth, which emerges through this, imitates the tenor from the beginning of that bar) instead of writing a harmonic-contrapuntal ‘correct’ realization (he could have written c² instead of d² with a descending fourth as a result).

’Mistakes’ against voice leading

When discussing the Art of Fugue, which is an elaborate demonstration of contrapuntal and fugal writing, one is obliged to investigate to what degree the technical ‘correctness’ of writing is within tradition. From what will follow it will become quickly evident that Bach was not too concerned with existing rules (certainly not when breaking them!).

Fux’s rules concerning hidden successions of fifths and octaves are very severe.³⁴ For Bach, however, these successions are in certain circumstances (as appears from the examples) perfectly acceptable, even between soprano and bass with a leap in the soprano. It is striking that Bach uses only the descending hidden succession of fifths with a leap in the soprano while an inner voice has a dissonant suspension (Contrapunctus 11 bars 62-63, Contrapunctus 14 bars 66-67), while a hidden succession of octaves with a leap in the soprano is only used in an ascending way (Contrapunctus 8 bar 121) (example 12).

Bach’s approach with regard to successions of fifths or octaves is also freer than in Gradus where we read the following: ‘It should be said that the skip of a third cannot prevent a succession of either two fifths or two octaves. The intervening note on the upbeat is regarded as hardly existing, since owing to its short duration, and the small distance between the tones, it cannot compensate to such an extent that the ear will not notice the two succeeding fifths or octaves … It is different if the skip is of a greater interval; e.g., a fourth, fifth, or sixth. In such a case the distance between the two tones causes the ear to forget, as it were, the first note on the downbeat until the next note on the downbeat.’³⁵ Again for Bach this is acceptable, which becomes clear from the following examples in Contrapunctus 2 (bar 14 between tenor and bass: fifths on both beats) and in Contrapunctus 5 (bar 14 between alto and bass; octaves on first and second beat while the bass has only a melodic diminution of a third) (example 13).

Also in Contrapunctus 5 we seem to have similar successions of fifths in the bars 17-18 between soprano and alto, but, after a closer analysis, it becomes evident that Bach writes here a linear intervallic pattern³⁶ 10-6 in which the inner voice imitates the soprano (which has the theme) in stretto, also a typical feature of this voice-leading design (example 14).³⁷

This shows us that the fifths on the second half of bar 17 and first and second half of bar 18 are actually not an issue.

Similar striking examples of the free way Bach approached his counterpoint are to be found in Contrapunctus 8 measure 16 (successions of octaves between bass and soprano) and in the Canon alla Duodecima bar 24 (successions of octaves between bass and soprano which become fifths in the evolutio, see bar 57).

Another ‘mistake’ against voice leading is the ottava battuta which occurs when ‘both voices [move] from a tenth into an octave, leading them stepwise, the upper part down and the lower up. This octave … is prohibited.’³⁸ Bach does not write it very often, but one can still find several instances of it, of which the clearest example can be seen in Contrapunctus 4 bar 29 (example 15).

Bach’s relaxed attitude towards rules does not limit itself to (hidden) successions of fifths and octaves (as well as to the ottava battuta), but also shows itself in, at times, highly irregular resolutions of dissonants (seventh and fourth) and his chord choice. The most striking liberty that Bach allows himself is a dissonant ligatura (seventh) in the soprano, which does not resolve stepwise and downwards, but which makes an ‘incorrect’ ascending leap of a fourth. I have found two examples ofthis in the Art of Fugue: Contrapunctus 10, bar 21 and Contrapunctus 14 bar 102 (example 16).

In this second example (the irregular resolution is a consequence of the presence of the subject), one could say that the seventh does resolve, but only in bar 105. It remains anyway a bold voice leading procedure, and, even for Bach, an exception. Other instances can be found in the last bar of the Canone alla Terza from the Goldberg Variations BWV 988 (in the alto) and in the third bar of the Fantasia in A minor BWV 904/1, where the seventh does not resolve ‘correctly’ either, but makes a similar ascending leap of a fourth (example 17).

The passage in BWV 904/1 is even more interesting because the second highest voice doubles this seventh but resolves correctly.

Another interval which does not resolve ‘correctly’ is the fourth a¹-e¹ in the second half of bar 7 of Contrapunctus 1 (the a¹ in the soprano should have been resolved to g#¹). Bach uses the same counterpoint fragment once again at the moment that the four voices enter from bar 13 on. In the second half of bar 15 Bach uses the e in the bass as the fifth of a chord of A minor, with the result – according to the strict rules of counterpoint – of a forbidden six-four chord.

The use of 6/4-chords and seventh chords in strict counterpoint has always been very restricted.³⁹ Bach however, as was usual in baroque music, uses both very often. One instance is so special and ingenious that it deserves to be mentioned separately. If we look at the two lowest voices in the penultimate bar of Contrapunctus 8, the first half of that bar seems to be a cadential 6/4-chord, which resolves in the second half of the bar (example 18).

But if Bach meant to have this harmonization he should have written in the soprano the following semiquavers in the first half of bar 187: d¹-c#¹-d¹-e¹-f¹-g¹-a¹-f¹ (or d¹) or d¹-a-b-c#¹-d¹-e¹-f ¹-d¹ (or e¹). As he does not do this, we should interpret the second crotchet as ii° (on a short dominant pedal point).

Chromaticism

The use of chromaticism was, since the seventeenth century, generally accepted even in the strict style.⁴⁰ When Bach uses chromaticism in the Art of Fugue it is not only much more pushed and daring, but helps to structure the piece. In Contrapunctus 8 chromaticism is an essential part of the structure; the first subject is the first new subject since the original Art of Fugue subject and makes a modest use of chromaticism. With the introduction of the second subject (bar 39), which is immediately combined with the first, the degree of chromatic intensity increases considerably (also by applying it to several voices at the same moment). But Bach expands chromaticism even to the limits of enharmonic modulation. In bars 152-153 Bach does not start the harmonization of the first subject in the obvious key of C major but in E minor, in which the a#¹ (functioning as the leading tone of V/V) is used enharmonically as bb¹ (the seventh of the secondary dominant chord to ii6) to make the modulation to the key of C major, which was initially avoided (example 19).

The first subject is not well suited to major tonalities, and this instance is the only one in the piece where Bach more or less uses it in a major key.⁴¹

Contrapunctus 11, which uses the inverted themes of Contrapunctus 8, has a similar construction, but still goes a step further; it contains not only the most daring harmonic passages from the Art of Fugue but, in addition, it shows a strictly worked out and advanced use of chromaticism which controls the whole piece. As an illustration of ‘pushing the limit’, Bach does not restrict it to one or two voices (notice in bars 68-70 the six spectacular parallel chromatic fourths), but applies it even in several voices at the same moment in contrary motion (bars 44-47), which results in – apart from a bold harmony – unusual successions of vertical intervals occurring (bar 46, middle voices: minor seventh, augmented fifth and diminished fifth) (example 20).

I do not agree with Peter Schleuning when he claims that the harmony in bar 46 clearly suggests C major, and that Bach arrives in A minor in bar 47 via enharmonic modulation.⁴² Schleuning says that the last crotchet of bar 46 forms a dominant ninth chord without its fundamental note g, when the soprano descends chromatically from a¹ to ab¹, which is notated as g#¹ by Bach. This harmonic analysis must be incorrect; the chord on the last quarter of bar 46 is iiø6/5 in A minor, which becomes a diminished seventh chord (vii°4/3) on the second quaver of that beat. This harmonic progression is also used by Bach on the second beat of bar 44 and 45, a passage not mentioned by Schleuning.

Even in the most ‘strict’ passage of the Art of Fugue, the first section of the quadruple fugue Contrapunctus 14, which is written in a serene and much less virtuosic style, Bach refuses to omit his daring chromaticism - and daring it is. Through chromaticism augmented sixth chords occur (bars 24 and 56), and in this piece chromaticism also helps to structure it. Because of the nature of the third subject of this contrapunctus, based on Bach’s own name - B, A, C, H in the German system stands for Bb, A, C, B - this section is the most chromatic one of the fugue. In spite of this, the underlying harmonic progressions remain quite clear (see bars 223-225).

Fugal writing: some technical aspects

Conceiving a contrapuntal or fugal piece starts with an inventio, that is, the choice of which initial material will be used. Fux gives the following rule regarding the choice for the first note of a fugue subject: ‘… the voices of a fugue cannot start at intervals other than those that constitute a mode, that is, intervals other than the unison, octave, and fifth …’⁴³ Though it was common in Bach’s time to let a subject also start on the mediant (as the second subject of Contrapunctus 14 does (bar 114)), other choices than tonic, third or dominant are rare.⁴⁴ In the Art of Fugue, however, we find four such subjects: (1) the second subject of Contrapunctus 8 (triple fugue) starts on the submediant bb¹ (bar 38), (2) the first subject of Contrapunctus 10 (double fugue)⁴⁵ starts on the leading tone c#¹, (3) the third subject of Contrapunctus 11 (triple fugue) starts on the flatted supertonic eb (bar 89, this subject is the inversus of the second subject of Contrapunctus 8; see above), and (4) the third subject of the Contrapunctus 14 starts on the submediant bb (bar 193). Knowing that there are nine other themes, in addition to the Art of Fugue subject and its variations (I count the secondary subjects of Contrapunctus 8 and 11, which are each others inversions, as separate subjects), it is striking to see that four start on a ‘wrong’ note, again clear proof of Bach’s intention to push all the rules to the limit and, in this case, beyond.

The dux of Contrapunctus 14 ‘correctly’ starts on the tonic (and its inversus on the dominant), but there is a striking inconsistency when the inversus is quoted in the soprano in bar 98. In bar 101 there is a melodic/rhythmical variation in bar 101 within the theme (example 21).

It is remarkable that Bach makes this adaptation only at this one place in the whole piece. When we look at Bach’s manuscript it is clear that the first note of bar 101 was originally a minim, which afterwards was corrected to the version we know, and which also appears in the first edition. The NBA and the critical reports of all the editions I consulted do not mention this. The question, of course, is did Bach make this correction himself? As far as I know, this observation is new. This ornament, strictly speaking, is not necessary for the rhythmic motion (it seems to imitate the bass from the bar before). Is it too bold to assume that Bach, perhaps in his haste to finish the piece, overlooked thhe had put a correction in the inverted first theme?

When answering a dux Bach is usually very consistent and applies the basic rule concerning tonal answering: the fifth at the beginning of the dux becomes a fourth in the comes.⁴⁶ But in Contrapunctus 11, however, Bach gives a real answer (example 22).

This shows how important it was for Bach to keep the melodic character of his subject, even when a hard rule has to be broken (the repeated d², the logical consequence of the theoretically correct tonal answer, would deform the melodic line of the answer too much). Tovey takes a clear stand on this issue: ‘Note that Bach does not spoil his answer by trying to make it tonal. That could be done either by making it go into the subdominant [a¹-d²-c², E.D.], or by stumbling over two D's in the first bar. Common sense forbids such pedantries.’⁴⁷

Anything but pedantic is Bach’s great craftsmanship in applying stretto (congeries). Fux also uses stretto, but in a more elementary, academic way, especially to close off fugues.⁴⁸ The subjects in Gradus, which are treated in stretto, are fully written out (not cut off). With Bach (among many others) this is not the case. In many instances the first entrance in stretto is cut off when it becomes clear that it will be impossible to keep the stretto until all voices have completely finished their subject. But Tovey says that we cannot interpret bar 48 and the following bars in Contrapunctus 1 as a stretto between alto and soprano, because Bach reserves that contrapuntal technique for Contrapuncti 5 to 7 inclusive.⁴⁹ For me, this passage perfectly illustrates the peculiarities of a stretto; close thematic entrances, where the attention goes principally to the beginning of the theme. In that context, it does not really matter if the theme is complete or not. In bars 29-36 a real stretto appears, which Bach will elaborate fully in Contrapunctus 5, and where, during the second entrance of the bass (bar 32; the first entrance of the bass starts on the second half of bar 29), he adapts the theme melodically (bar 33-34), and gives it a form on which Contrapunctus 4, which uses the inverted theme, will be based (example 23).

It does not happen often that, in a stretto, the first entrance of the theme is given completely, while the second one is cut off. This happens in bars 21b-25 of Contrapunctus 8, where the theme is given by the middle voice (example 24).

Half a bar later there follows a clear imitation at the fifth in the bass, clearly suggesting stretto. Tovey again does not find it appropriate to speak about stretto here, because ‘stretto is not the business in this fugue’.⁵⁰

Sequences - Linear Intervallic Patterns

Specialists as Knud Jeppesen⁵¹, Alfred Mann⁵² and Christoph Wolff [See chapter on Rhythmic motion and rhythm] all agree on the fact that sequences (and linear intervallic patterns) are rather exceptional in Palestrina’s music, while they are important components of many baroque (and other) compositions. In the Art of Fugue the most subtle and striking ones can be found in the Canon alla Ottava. They deserve some closer attention because they are easily missed and show Bach’s supreme craftsmanship of voice leading (example 25).

In bars 67, 71 and 75 we recognize three linear intervallic patterns associated with the descending-fifth sequence.⁵⁴ What is striking about these examples is that the harmonic rhythm is not regular. It does not evolve, as Peter Schleuning claims, by a dotted quaver but by a semi-quaver followed by a quaver.⁵⁵

The 6-5 linear intervallic pattern of bar 67 is particularly interesting because of what precedes it in the bass. On the second and third beat of bar 66 Bach establishes a dominant harmony, which resolves into i6 on the first beat of bar 67; on that second beat of bar 66 he writes an a, on the third beat a g. Both these beats in the bass are embellished with a complete neighbour note. This neighbour note pattern seems to continue into bar 67, but instead it changes through the linear intervallic pattern.

Rhythmic motion and rhythm

The concept and use of rhythmic motion and rhythm in the Baroque era differs strongly from that in the Renaissance. Knud Jeppesen makes the following statement: ‘The Palestrina music moves in free, prose-like rhythms in contrast to the poetic, strict rhythmic pattern of the eighteenth and nineteenth centuries …’⁵⁶ In the Renaissance there is much less the ‘compulsion’ of the rhythm, but rhythmical variety is highly esteemed, while in the late-baroque era, and also in the Art of Fugue, the rhythmic drive/motion is very important. The biggest difference in this domain between the Renaissance polyphony and the late-baroque music is that, in the Renaissance, rhythm and melody merge together, while, in the late-baroque, rhythm and melody become rather independent values. Rhythm, which of course is as important in stile moderno as in stile antico, has, by this, a different meaning and realization in both styles. In the stile antico rhythm is mainly used for the horizontal organisation of the melody, and has a certain weightlessness without accents. There are no metrical groups, and rhythmical symmetry is avoided. Instead a free mixture of note values dominates, as in the prose-like rhythm and melody of the stile da Palestrina. Immediate successions of small and big note values are avoided; transitions between both are often realized thanks to ligature. The quiet flowing, without uniform rhythm motion, visibly very apparent from the domination of the ‘white’ notes, is the most important feature. The melody in the stile moderno (and stile alla breve) has metrical repetitions, sequences, motives and symmetry.⁵⁷

No further explanation is required to assert that, in the Art of Fugue, the forceful and propulsive force of the rhythmic motion is almost constant. Yet Bach respects his own rhythmic logic, and sometimes even abandons this typical baroque drive. This feature is discussed by Kent Kennan as a general guidance for the writing of baroque counterpoint: ‘Abrupt halts in the rhythmic motion are to be avoided. Where there is a choice between motion on a strong beat and motion on a weak beat, the latter arrangement is generally preferable, so that there will be a sense of propulsion into the strong beat, and so that the length of the note and the strength of the beat will be mutually supportive… But this principle must not be taken too literally, for exceptions are frequently brought about by special considerations in the musical pattern.’⁵⁸ Bach indeed wittily abandons this principle sometimes, guided by ‘special considerations’. In the transition from bars 23 to 24 of Contrapunctus 1 (example 26),

he stops the continuous movement of quavers, in order to use the motif that he introduced in the canon between tenor and soprano in the episode from bar 17 on. Bach could also have put in the bass – second half of bar 23 - an imitation of the soprano from the beginning of that bar (d-B-c#-a), but clearly chose to use the preceding motif. In the next bar the continuous movement of quavers is also missing to make room for the expressive suspension a¹ that resolves on g¹.

Another example of a deliberate halt in the rhythmic motion, in favour of a ‘special consideration’, is to be found in Contrapunctus 10 (example 27).

The definitive version of this piece is conceptually quite different from Bach’s first version from 1742. In the manuscript this fugue seems to be – again – a stretto fugue instead of a double fugue. This first version only starts on bar 23 of the definitive version, from which the incomplete alto entry in bar 24, as well as the bass in quavers, are lacking. In the definitive versionit seems that, at this point, Bach, by stopping the continuous flow of quavers in the first half of bar 26, wants to make it clear to us that the entry in the tenor is the actual second, ‘correct’, entry, while that of the alto was only ‘fake’.

Within the contrapuntal texture of the stile antico, it is out of the question that the music stops completely in a brusque manner and afterwards continues in a purely chordal way (noèma). But this is exactly what happens in bars 70-73 of Contrapunctus 1. In bar 70 there is a sudden silence (apokope - suspensio) after an unresolved diminished seventh chord in last inversion (which is a very strong Parrhesia); as a result the forceful and propulsive force of the rhythmic motion is gone. After a hesitating 6/4-chord (dubitatio), the rhythmic motion is being reinstalled in bar 73 (confirmatio). We find a similar example on the third beat of bar 58 of Contrapunctus 7 (example 28),

where the music also suddenly stops after a diminished seventh chord, with the difference that at that point the soprano starts a little solo, cadence-like passage. Apart from the fact that these sudden halts in the rhythmic motion are to be avoided, and that such cadences are foreign to the stile antico (see the chapter Bach the Virtuoso Harpsichordist), there is also something peculiar about the length of the lower voices on this third beat of bar 58. The length for these voices, a quaver, is equal for all the voices except for the bass, which has a crotchet. As a result, it would appear that Bach slightly favours the outer voices to the inner voices (compare bars 21-22 of Contrapunctus 2, where Bach really writes a pedal point in the bass while the inner voices rest). Bach could have written a quaver in the bass, as in the inner voices, instead of a crotchet. We find this difference in the manuscript as well as in the first edition, from which we can prudently assume that this was what Bach meant, and which also shows the importance of the function of the bass (cfr. basso continuo). The NBA and the critical reports of all the editions I consulted do not mention this. Both examples show Bach as a pure baroque composer and master of rhetoric!

Rhythmic motion also explains apparently strange or inconsistent choices in Bach’s horizontal writing. Contrapunctus 8 is based on a clear rhythmic motion from a weak to a strong beat (as a consequence of the use of an upbeat, chromaticism and a trill which all appear on the weak beats of the first subject), which gives the piece a very propulsive character. As a consequence, the simultaneous use of minims on a weak beat between different voices never occurs (for Fux, of course, this is perfectly acceptable; the examples in Gradus are legion), because there would be too little direction to the next strong beat. This also clarifies why Bach avoids the expressive chromatic note in bars 63 and 109 (he does not write a minim c# on the weak beat of these bars but a crotchet rest followed by a crotchet note c#). This decision even causes an open fifth in the second half of bar 109 (example 29).

When we compare bar 42 (or 46) with bar 53, we see that Bach writes a crotchet rest in the middle voice of the second half of bar 53, instead of the dotted minim of bar 42 (example 30).

This cannot be an inconsistency in horizontal writing, but is yet another instance of how precisely the realization of the counterpoint controls the general structure. In bar 42 a new motif is introduced in the bass (which will be used for the construction of extensive episodes) at a moment when the two top voices each have their own theme (it seems as if the second theme ends with the second half of bar 43, but later it will be clear that it only continues until bar 42). In bar 53 the situation is different, because the bass starts this confutatio with the new motif introduced by a suspension, which is imitated by the middle voice in the same bar. To make this imitation clear Bach writes a rest (bar 52 now also clearly shows that the second theme ends on the tonic). Notice that Bach had to make a typical baroque octave displacement ⁵⁹ to make the passage playable with two hands; the logical continuation of the line in bar 52 would have been a G. If we compare this way of thinking to that in earlier works by Bach we see that the latter seem to be less consistent. A fine example is the fugue in A minor BWV 904/2, in which the use of rests instead of tied notes seems quite arbitrary (compare bars 10-12 to bars 22-24), and which illustrates this clearly (example 31).

Why did Bach make a clear distinction in the soprano in the second half of bar 11 (crochet a² followed by semiquaver note rest) and the sequence in the next bar (tied g²)? A possible explanation lies in the fact that Bach only writes the tied note, followed by a repetition of the same note, when the tied note is dissonant and resolves stepwise downwards (as systematically occurs in the soprano in bars 23-24; in bar 11, however, this tied a² would have been consonant).

In the Canon alla Ottava (which uses the subject rectus and inversus) Bach uses an equal precision in the rhythmical notation now with regard to the closings of phrases; end-notes are nearly always written as quavers followed by a semiquaver rest (for instance in the right hand in bar 23, example 32).

But at one crucial place Bach deviates from this; in bar 40 he finishes the phrase in the right hand with two dotted quavers (descending fifth e²-a¹). This is the only place in the piece where he does this, and this is structurally very significant. At this point the section using only the inversus of the main theme is being finished, after which, in the next section, the rectus is introduced. The effect of this closing makes the entrance of the rectus even more striking. Yet another example of Bach’s subtle contrapuntal art! So this feature is not a consequence of a linear, syntagmatic approach, as is usual in Fux, but shows how Bach’s – as they seem to be – contrapuntal details generate structure.

Another typical feature of baroque rhythm we find in Contrapunctus 2, which uses the main subject rectus but with a rhythmical variation at its conclusion (dotted rhythm). Bach makes such a consistent use of this variation in the whole piece (the rhythmic motion is built only with this dotted rhythm), so that this contrapunctus, like the 6th, could have been called in Stylo Franscese. The dominance of this dotted rhythm is not compatible with the free flowing rhythmical concept of the stile antico.

Metre

The baroque concept of metre is well documented.⁶⁰ The most striking feature, which is discussed in all treatises, is the difference between ‘good’ and ‘bad’ notes, depending on their metric position in the bar, and by some writers even related to their (harmonic) function. Georg Muffat (1653-1704) writes in the preface to his Florilegium secundum (1698): ‘Of all the notes found in any composition to be played, there are those that are good …, and others that are bad …. Good notes are those that seem naturally to give the ear a little repose. Such notes are longer, those that come on the beat or essential subdivisions of measures, those that have a dot after them, and (among equal small notes) those that are odd-numbered and are ordinarily played down-bow. The bad notes are all the others, which like passing notes, do not satisfy the ear so well, and leave after them a desire to go on’. From this can be concluded that barlines precede good notes and mark recurring patterns of organization, most of which reflect dance steps. Good and bad notes are identified by their position in the measure’.⁶¹ Johann Gottfried Walther (1684-1748) gives very important supplementary information concerning the possible functions of a ‘good’ or ‘bad’ note: ‘Tempo di buona (ital.) is the good part of the beat. Under the equal tactus, the first of two minims, or the first half of the beat is g; also the first and third of four crotchets, the first, third, fifth and seventh of quavers and so forth, because these tempi, or odd-numbered parts of the beat are suitable for the placement of a caesura, a cadence, a long syllable, a syncopated dissonance, and above all a consonance (from which comes its name – di buona).⁶² Temp di cattiva, or di mala (ital.) is the bad part of the beat. In the Tactu aequali or beat with two equal strokes, the second of two minims or the second half of the beat is bad; also the second and fourth of four crotchets, sixth and eight of quavers, because these tempi or even-numbered parts of the beat are all different from the above-mentioned parts, and are their opposites’⁶³ It is fascinating how Bach varies this general baroque concept at certain places and interchanges the metric place of the ‘good’ and ‘bad’ notes, a consequence of the changed function of these notes; syncopated dissonances occur on the metrically written ‘bad’ beats, while their resolutions are on the ‘good’ beats. As a result new imaginary barlines emerge, which precede the new ‘good’ notes. The end of Contrapunctus 3 gives a clear demonstration of this (example 33).

In bar 67b the soprano and tenor have a syncopated dissonance (‘good’ note), which resolves in bar 68a (‘bad’ note), while the harmony (bass) remains the same (iv6). The same occurs between bar 69b and 70a, where the syncopated dissonances are in the soprano and alto. In both these examples ‘good’ and ‘bad’ notes last a minim. But in bar 68b-69a the situation is different. Through the use of two syncopated dissonances, which last a quaver and a crotchet respectively (in the tenor in bar 68b, in the soprano in bar 69a) instead of a minim, ‘good’ and ‘bad’ notes last only a crochet.

A similar construction can be seen in bars 51-53 of Contrapunctus 5; bar 51b and 52b form the ‘good’ beats, 52a and 53a the ‘bad’ beats (syncopated dissonance and resolution both last a minim). But bar 51a has a ‘good’ and ‘bad’ note which last only a crochet.

What makes bars 37-41 from Contrapunctus 10 so interesting, apart from the fact that again the metric place of the ‘good’ and ‘bad’ notes has been interchanged, is that Bach, in Contrapunctus 3, writes quasi the same music but notates it ‘correctly’ (Compare bars 39-41 from Contrapunctus 10 with bars 27-28 from Contrapunctus 3).

Through this knowledge it can be seen that Bach even changes a binary bar 2/2 into the ternary bar 3/2. In Contrapunctus 5 bars 28b-32 – four and a half bars written in 2/2 – actually sound as three 3/2-bars; this passage is a linear intervallic pattern 3-6 (example 34).

The three voices imitate each other using the inversus of the closing motive of the subject of Contrapunctus 5, a descending fourth in quavers. From the second 3-6-occurrence the 9-8-suspension in the tenor is characteristic (‘good’ note); in the first 3-6-occurence there is a 4-3-suspension and an incomplete neighbour note in the soprano (bar 28b forms the ‘good’ beat). But this first 3-6-occurrence stays metrically ambiguous through the presence of the subject (bar 28b-29); on the one hand there is the harmonic resolution on the first beat of bar 29 (dissonances of bar 28b resolve on the first beat of bar 29 where the soprano also stops), but on the other hand the subject forces us to hear the f1 as the ‘good’ beat. A similar construction can be seen in bars 81-83.

The most complex complex passage of the Art of Fugue in this domain can be seen in bars 21-23 of Contrapunctus 7 (example 35).

The fourth beat of bar 21 and the second and fourth beats of bar 22 function as ‘good’ notes (systematic harmonic resolution on a stationary bass from written weak to strong beat), a consequence of the fact that the sixth note of the subject (which starts in bar 20b), the f, lasts twice ‘too long’ in relation to the proportion of the theme. And the entrance of the theme in the alto makes this passage even more confusing, because it does not follow the new metric place of the ‘good’ notes (this entrance starts on the written first beat of bar 23), while the entrance of the theme in the tenor a crochet later does. This is the most brilliant metric deception of the Art of Fugue!

Bach’s choice of a 9/16 time signature for his first canon alla ottava (three beats in a bar, and every beat has a ternary division) implies almost certainly metric shifts. Bars 19-21 are in this respect the most interesting passage, and almost form what is commonly called a hemiola (example 36).

On the third beat of bar 19, Bach writes a iv with an accented passing note (‘good’ note) which resolves via an embellishment on the first beat of bar 20 (‘bad’ note), while the bass remains immobile over the bar line on the fundamental note of the chord, g. The first beat of 20 is thus once again a resolution of the written ‘weaker’ beat of bar 19. As a result, two imaginary 6/16-bars emerge, which also become visibly apparent because of the broken chords in the right hand on both ‘bad’ beats of these new imaginary 6/16-bars. After this, the situation is even more complex, because the written second beat of bar 20 is not a dissonance (as could be expected from what preceded it), but the third beat is, forming a double appoggiatura of i and so representing the ‘good’ beat, which resolves on the written first beat of bar 21 (‘bad’ note).⁶⁴ Bars 65-66 of the same canon present a clearer hemiola (example 37).

The first and second beat of bar 65 form the first degree (i), the third beat of bar 65 and the first beat of bar 66 the second degree (ii°; in the left hand the bass remains immobile over the bar-line) and the second and third beat of bar 66 the fifth degree (V). Bach stresses the hemiola through the presence of the trills in the right hand, one on the third beat of bar 65 and one on the second beat of bar 66.

Bach the Virtuoso Harpsichordist

Within the linear concept of Palestrina’s music, virtuoso, quasi not-contrapuntal passages do not belong in polyphonic pieces at all. But in Contrapunctus 8 there are three such passages which show Bach more as the great keyboard virtuoso than as the contrapuntal genius (bars 90-92, 119-123 and 178-179). In spite of the limited use of counterpoint during these three passages, they still form musical and rhetorical climaxes at important structural points. The first closes the section with the first two themes, after which the third theme, the inverted Art of Fugue theme, is introduced. The second passage, closing with two bars of outrageous parallel chords (bars 122-123), and thus obtaining a powerful climax, introduces a new section, in which the Art of Fugue theme does not appear. The last virtuoso passage leads to the final cadence of bars 180-181, which is anything but academic (example 38).

Instead of constructing this material to place it in the perfectly authentic cadence (with result that the piece should have been finished there), the middle voice does not take this harmonic phenomenon into account, and stubbornly goes on with its own motif (derived from the inversus of the second theme). As a result, the effect of the cadence is weakened, and this also explains the function of the little coda (with the last combination of the three themes).

Other similar passages are to be found in Contrapunctus 2 bars 21-22 and in Contrapunctus 7 bars 58-59.

Terminology

The term Contrapunctus is used by Fux on the one hand for the theory of counterpoint, on the other hand for a counterpoint against the cantus firmus. Bach, however, gives each separate piece in The Art of Fugue this title (except for the canons). Joseph Kerman reflects on this: ‘As for Bach’s use of the solemn term “contrapunctus,” that accords with his evident intention in The Art of Fugue to control counterpoint as a universal principle, rather than simply the genre of fugue’.⁶⁵ Otherwise put, ‘part of the function of the word Contrapunctus in The Art of Fugis to describe the technique of setting different genera of counterpoint against a principal subject’.⁶⁶

Yet Bach was not the first one to use this term as a title for a piece. Buxtehude already used the name Contrapunctus for his setting of Mit Fried und Freud which ‘is made up of four movements grouped into two pairs, each made up of a contrapunctus and its harmonic inversion, called an evolutio’.⁶⁷ Just as in the Art of Fugue these four movements are in D minor, and are written in the stile alla breve. The technique of harmonic inversion is used in The Art of Fugue in exactly the same manner in Contrapunctus 12 en 13, which both are written as rectus and inversus (the evolutio). From this appears, as David Yearsley suspects, that Bach must have known Buxtehude’s composition.⁶⁸

Apart from the highly original title Contrapunctus, Bach also uses some technical terms to describe some contrapuntal mechanisms. Davitt Moroney is convinced that the title in Hypodiapason is the correct one, and that alla Ottava, the one used in the first print, would not be by Bach. ‘Bach's title in the autograph score is more likely to be correct than that in the printed edition; not only is it in his own hand, but it is in a classical style, comparable to most of his other Latin titles in Die Kunst der Fuge. The Italian title was probably added by one of his sons.⁶⁹ I am not fully convinced by this reasoning, because the terminology of the two following canons, alla decima and alla duodecima (also Italian indications, from which Moroney concludes that these terms are also not by Bach), already appears earlier in the Art of Fugue, namely with Contrapuncti 9 and 10, which have the subtitles alla duodecima and alla decima respectively (there is no reason to assume that these terms are not by Bach). It seems plausible to me that Bach abandoned the original title of the first canon in favour of more uniformity with the two following canons.

Bach also changed the name of the fourth and last canon from Canon in Hypodiatessaron to Canon per Augmentationem in Contrario Motu. This change of name could have originated because of the lack of a clear description of the mechanism of this canon. With the title in Hypodiatessaron it seems that this piece would ‘only’ be a canon at the fourth, while Bach uses here the most complex canonic technique, namely a canon in augmentation and contrary motion.

This desire for accurate titles for his canons, seems in contradiction with the title of the Canon alla Decima Contrapunto alla Terza. Not only is it a bit deceptive, but also it is wrong, so that it is probably not by Bach. The term alla Decima clearly has two meanings in this canon; not only is the distance between the two canonic voices in the first part a tenth, but the whole canon is moreover written in double counterpoint at the tenth (and not at the third like the title suggests, because with this last technique the voices are not exchanged). As a result, after the exchange of voices (bar 44), there originates a canon alla ottava. (The same remark holds good for the Canon alla Duodecima Contrapunto alla Quinta.)

It also has to be said that Fux does not deal with canon in his treatise. A subtle hint from Bach against Gradus?

Footnotes

[1] In Vienna by J.P. van Ghelen.
[2] Joel Lester: ‘It is clear from modern scholarship, especially that of Jeppesen, that the usages in Gradus do not agree in numerous ways with either the details or the spirit of Palestrina’s music’ (Joel Lester, ‘Compositional Theory in the Eighteenth Century’ (Harvard University Press, 1994), 35; for a critical study, see Knud Jeppesen, ‘Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century’ (New York: Dover 1992, reprint after New York: Prentice-Hall, 1939).
[3] Christoph Wolff, ‘The New Bach Reader’ (New York: Norton, 1998), 400, after BD III no.803.
[4] Christoph Wolff, ‘Der stile antico in der Musik Johann Sebastian Bachs’ (Wiesbaden: Steiner, 1968), 25.
[5] Christoph Wolff, ‘The New Bach Reader’ (New York: Norton, 1998), 399, after BD III no.803.
[6] Wolff, ‘Der stile antico’, 34.
[7] Peter Wollny, ‘Ein Quellenfund in Kiew – Unbekannte Kontrapunktstudien von Johann Sebastian und Wilhelm Friedemann Bach’, Bach in Leipzig – Bach und Leipzig – Konferenzbericht Leipzig 2000 (Hildesheim: Georg Olms Verlag 2002), 275-287.
[8] Walter Werbeck, ‘Bach und der Kontrapunkt – Neue Manuskript-Funde’, Bach-Jahrbuch (2003), 67-95.
[9] Philipp Spitta, ‘Johann Sebastian Bach’ (New York: Dover, 1992, reprint after Leipzig, 1873-1880 and the English translation by Clara Bell and J.A. Fuller Maitland (London: Novello, 1884-1885), volume 3, 125.
[10] Lorenz Christoph Mizler, ‘Gradus ad Parnassum oder Anfhrung zur regelmässigen Composition, aus dem Lateinischen ins Deutsche bersetzt, und mit Anmerkungen versehen’ (Leipzig, 1742).
[11] Wolff, ‘Der stile antico’, 28.
[12] Personal communication from Peter Wollny.
[13] Wolff, ‘Der stile antico’, 166-172.
[14] Barbara Wiermann, ‘Bach und Palestrina – Neue Quellen aus Johann Sebastian Bachs Notenbibliothek’, Bach-Jahrbuch (2002), 9-25.
[15] Wolff, ‘Der stile antico’.
[16] In 11 Contrapuncti one of these bars is used; for more information regarding this difference, see Wolff, ‘Der stile antico’, 38-53.
[17] Wolff, ‘Der stile antico’, 87 (footnote 98).
[18] I feel that this nuance cannot be clear enough. Certainly, as Peter Schleuning says, stile antico has ‘great significance in the Art of Fugue’ (Peter Schleuning, ‘Johann Sebastian Bach’s Kunst der Fuge : Ideologien · Entstehung · Analyse’ (Kassel: Deutscher Taschenbuch Verlag/Bärenreiter Verlag, 1993), 26.). When he says, however, that ‘Bach ... by no means applies the strict rules slavishly, but makes use of those freedoms, which the advanced style and continuo practice of his time allow, without problems or the apparent deviation from strict rules’, (Schleuning, ‘Kunst der Fuge’, 50.) it sounds a bit too much as if Bach was using strict style with here and there an exception.
[19] Wolff, ‘Der stile antico’, 121.
[20] Christoph Wolff, ‘The New Bach Reader’ (New York: Norton, 1998), 398-399, after BD III no.803.
[21] Wolff, ‘Der stile antico’, 94.
[22] ‘The most striking feature [regarding the construction of a piece in stile antico] is that the thematic and motivic material constantly appears, without adding contrasting elements, which results in a very rich but very concentrated texture. The Fortspinnung and development of theme and thematic material is characteristic.’ (Wolff, ‘Der stile antico’, 65.)
[23] The structure (dispositio) of a baroque fugue is usually based on the contrast between confutatio versus confirmatio; for more information, see the monumental study by Gerardus De Swerts, ‘Musurgia Rhetorica: Studien zur Affektenlehre des Barock’ (Antwerp-Köln, 1984).
[24] Because of the impracticality of the exact inversion of the second theme of Contrapunctus 8, Bach made an adjustment when using it as the third theme in Contrapunctus 11. The problem was that suspensions and appoggiaturas would resolve in an ascending way, which is against the nature of these non-chord notes.
[25] Kees van Houten and Marinus Kasbergen, ‘Bach: Die Kunst der Fuge en het getal’ (Zutphen: De Walburg Pers, 1989), 15.
[26] Marcel Bitsch, ‘J.S. Bach. L'Art de la Fugue. introduction, analyse et commentaires’ (Paris: Durand), 37 (footnote 1).
[27] Wolfgang Wiemer, ‘Die wiederhergestellte Ordnung in Johann Sebastian Bachs Kunst der Fuge: Untersuchungen am Originaldruck’ (Wiesbaden: Breitkopf & Härtel, 1977), 75 (Anhang IV: Die Anordnung Wolfgang Graeser).
[28] André Gedalge, ‘Traité de la Fugue’ (Paris: Enoch & Cie, 1901), 253.
[29]For the discussion of the order of the pieces in the Art of Fugue, see Gregory Butler, ‘Ordering problems in J.S. Bach’s “Art of Fugue” resolved’, The Musical Quarterly, Winter 1983, vol. LXIX, 44.
[30] Alfred Mann, ‘The Study of Counterpoint from Johann Jospeh Fux’s Gradus ad Parnassum’ (New York: Norton, 1971), 55; Peter Schubert, ‘Modal Counterpoint – Renaissance Style’ (Oxford University Press, 1999), 88.
[31] André Gedalge, ‘Traité de la Fugue’ (Paris: Enoch & Cie, 1901), 61.
[32] Fux only discusses some ‘special’ cases as the nota cambiata and the dissonant passing note. (Mann-Fux, ‘The Study of Counterpoint’, 50-52.)
[33] In Contrapunctus 4 there seem to be a lot of incomplete neighbour notes but they are a consonant embellishment of the suspension (bars 19–22 in the alto and soprano).
[34] Mann-Fux, ‘The Study of Counterpoint’, 32.
[35] Mann-Fux, ‘The Study of Counterpoint’, 43-44.
[36] This term was introduced by Allen Forte and Steven E. Gilbert (‘Introduction to Schenkerian Analysis’ (New York: Norton, 1982), 83-102.) and is a voice-leading design made up of successive recurrent pairs of intervals formed between the outer voices. The term sequence is sometimes used, incorrectly, to designate the linear intervallic pattern.
[37] What is interesting here is that the apparently unimportant passing notes in this variation of the Art of Fugue subject actually form the second vertical interval of the pair of intervals forming the linear intervallic pattern.
[38] Mann-Fux, ‘The Study of Counterpoint’, 37.
[39] For more information, see Joel Lester, ‘Compositional Theory in the Eighteenth Century’, 45.
[40] The seventeenth-century ricercar ‘frequently introduced into themes a mannered element of chromaticism … Several of the ricercars in Frescobaldi’s Fiori musicali (1635) do this, of which Bach acquired a manuscript copy in 1714.’ (David Ledbetter, ‘Bach’s Well-tempered Clavier – The 48 Preludes and Fugues’ (Yale University Press, 2002), 86.)
[41] To respect the original interval structure of this subject Bach should have written in bar 153-154 a¹-ab¹-g instead of a#¹- a-g.
[42] Schleuning, ‘Kunst der Fuge’, 114.
[43] Alfred Mann, ‘The Study of Fugue’ (New York: Dover, 1987), 80-81.
[44] Kent Kennan, ‘Third Edition Counterpoint’ (New Jersey: Prentice-Hall), 204.
[45] For more (analytic) information, see Joseph Kerman, ‘The Art of Fugue – Bach Fugues for Keyboard, 1715-1750’ (University of California Press: 2005), 39-49.
[46] ‘If the first part uses the skip of a fifth, the following part must use the skip of a fourth, in order not to exceed the limits of the mode or octave, and vice versa.’ (Mann, ‘The Study of Fugue’, 81.)
[47] Donald Francis Tovey, ‘A Companion to ‘The Art of Fugue’ J.S. Bach’ (London: Oxford University Press Third Impression, 1960), 28.
[48] Mann, ‘The Study of Fugue’, 83.
[49] Tovey, ‘A Companion to ‘The Art of Fugue’’, 3-4.
[50] Tovey, ‘A Companion to ‘The Art of Fugue’’, 18.
[51] ‘Genuine sequences do not occur. In general, Palestrina uses them only rarely (mostly in early works), since the balance of the linear treatment can easily be displaced by the overemphasis which they place upon a particular motive.’ (Jeppesen, ‘Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century’, 84.)
[52] ‘The forming of sequences (the so-called monotonia) ought to be avoided as far as possible.’ (Mann-Fux, ‘The Study of Counterpoint’, 54 (footnote).)
[53] See chapter on Rhythmic motion and rhythm.
[54] In bar 67 we see a 6-5 linear intervallic pattern, in bar 71 a 10-5 (or more correctly a 3-5) linear intervallic pattern and in bar 75 a 6-6 linear intervallic pattern.
[55] Peter Schleuning missed this; he only speaks about bar 75 in which Bach, according to him, writes three parallel seventh chords in last inversion. (Schleuning, ‘Kunst der Fuge’, 147.)
[56] Jeppesen, ‘Counterpoint: The Polyphonic Vocal Style of the Sixteenth Century’, 83.
[57] Wolff, ‘Der stile antico’, 54 & 64.
[58] Kennan, ‘Third Edition Counterpoint’, 16-17.
[59] For more information, see Malcolm Boyd, ‘Bach’s Instrumental Counterpoint’ (London, Barrie & Jenkins, 1977), 16.
[60] For more information, see George Houle, ‘Meter in Music, 1600-1800’ (Indiana University Press, 1987).
[61] English translation, Houle, ‘Meter in Music, 1600-1800’, 82.
[62] The fact that Walther says that a syncopated dissonance or a consonance can occur on a ‘good’ beat, means that there are two main distributions of notes in relation to the ‘good’ and ‘bad’ beats: (1) syncopated dissonance on the ‘good’ beat – consonance on the ‘bad’ beat, and (2) consonance on the ‘good’ beat – dissonance (passing note, neighbour note, anticipation) on the ‘bad’ beat.
[63] English translation Houle, ‘Meter in Music, 1600-1800’, 83.
[64] D minor harmony completely occupies bar 21. As a result the written last beat of bar 20 and the complete bar 21 form an imaginary 12/16-bar.
[65] Kerman, ‘The Art of Fugue – Bach Fugues for Keyboard’, 37.
[66] Ledbetter, ‘Bach’s Well-tempered Clavier’, 88.
[67] David Yearsley, ‘Bach and the Meanings of Counterpoint’ (Cambridge University Press, 2002), 13-15.
[68] Yearsley, ‘Bach and the Meanings of Counterpoint’, 13-14.
[69] Critical commentary of the edition realized by Davitt Moroney for Henle Verlag, 121.

Copyright © 2006 This article was written by Ewald Demeyere. You may freely distribute this work provided that it is unaltered and that no charge is made and this copyright notice is retained.
Contributed by Ewald Demeyere (April 2006)
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