On an invention of Johann Sebastian Bach (Herschkowitz)

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On an invention of Johann Sebastian Bach
(To the problem of the genesis of Viennese classical sonata form)

автор Philip Herschkowitz
Источник: Invention • In Russian: Об одной инвенции Иоганна Себастьяна Баха. Translated from Russian by D. Smirnov-Sadovsky.


Philipp Herschkowitz:

On an invention of Johann Sebastian Bach:

to the problem of the genesis of Viennese classical sonata form[1]

Translated from Russian by D. Smirnov-Sadovsky, text edited by Guy Stockton

When speaking of “polyphonic forms”, i.e. about forms of polyphonic music in the epoch of Bach or before, one touches on something that is not really the form – or the structure – of a piece, but something different.For example, is it possible to regard a fugue as a musical form, when almost every fugue by Bach is built differently? An analysis of a fugue or an invention, or any other work by J. S. Bach whose genre they regard as a form, is really only an analysis of the writing[2] and the harmony of the piece.To search for the form one has to go deeper.Musical form in its essence does not depend on the epoch that the piece of music belongs to. But on the other hand, the level of development of musical form does depend on the epoch. If one looks from this point of view, it becomes clear that form could be hiding behind the texture of a polyphonic composition – form that would reach its full development later, in the epoch where the polyphonic style gave up its place to the homophonic. An especially striking example of a work that combines polyphonic writing with an embryonic homophonic form is the three-part Invention in F minor by J. S. Bach. In the structure of this invention, which is written in a triple counterpoint that is distinguished by its extreme richness, there are ten appearances of the theme, and three interludes: one of which appears only once, and two others – twice each.The ten appearances of the theme occur in the following order of keys:

1. F minor
2. C minor
3. F minor
4. A-flat major
5. E-flat major
6. C minor
7. D-flat major
8. A-flat major
9. F minor
10. F minor


– and it is possible to say that the solution of the reason to this order is the main problem in analysing this work.Indeed, why does the theme appear four times in F minor, twice in C minor and A-flat major, and only once in E-flat major and D-flat major? Why are the appearances of the theme in F minor 1, 3, 9 and 10, the appearances in C minor – 2 and 6, in A-flat major – 4 and 8, in E-flat major – 5, and in D-flat major – 7?

The solution of this problem becomes possible if one takes into consideration the very distinctive relations of these ten appearances of the theme with the interludes placed between them. The five appearances of the three interludes (1 + 2 + 2) and ten appearances of the theme make together a very strict symmetry. At first sight, this symmetry is contained only in the fact that every interlude is placed between two appearances of the theme, one of which is a double or dual, and the other is a simple or single appearance.

Scheme 1

Bach-Invention-Image002.jpg

However, if we turn our attention to the order of interludes, if we establish the fact that after the appearance of all three interludes the second and the third appear again,

I¹    I²    I³    I²΄    I³΄


we can see a more precise picture of the symmetry within the work. As it turns out, the two appearances of the second interlude divide the Invention into three similar segments:

Scheme 2


the last two of which are identical, and the first differs from them by its interlude only.

Now, when the structure of the Invention begins to be visible, we can come nearer to the understanding of the transpositions of its theme:

1st Segment
T F minor
{
T C minor
T F minor
2nd Segment
T A-flat major
{
T E-flat major
T C minor
3rd Segment
T D-flat major
{
T A-flat major
T F minor


T F minor

The above table, showing the distribution of these transpositions into segments, clearly reveals that each of the three pairs of appearances of the theme at the beginning of each segment relate to each other as tonic and dominant:

1) F minor – C minor
2) A-flat major – E-flat major
3) D-flat major – A-flat major

At this stage of the research it is worth taking into consideration that: 1) In the first segment, unlike the other two, the third appearance of the theme harmonically coincides (F minor) with its first appearance; 2) The similarity of both of the last segments concerns not only their formal structure, but also their harmony – they represent a relation that is determined by an exact transposition.However, there is one difference: under the conditions of triple counterpoint, the voices in the third segment are placed in a different order compared to the second segment.


It is necessary to turn special attention to the harmonic similarity of the first and third appearances of the theme in the first segment. But we also need to turn special attention to the fact that at the beginning of the segment there is a double appearance of the theme, and at the end – a single appearance. This layout clearly resembles a correlation of the first and third parts in a three-part song[3], where the first part is a period (consisting of antecedent and consequent)[4], and the third part is the repetition of the antecedent or the consequent.In these conditions, an interlude placed between the double and single appearances of the theme, – between the “period” and the repetition of the “antecedent” or “consequent”,– finds itself in the position of the second part of a three-part song.

A three-part song is one of the types of principal theme.Taking into account the fact that the first segment of the Invention begins and ends in the same key (the main key of the piece), we have every reason to consider this whole segment, with all its previously mentioned structural qualities, as the principal theme of this work, which is not only and not simply an invention, but also something else.


[Example 1]

Bach-Invention-Image006.jpg


Of course, there is a big difference between this kind of “three-part song” and Beethoven’s three-part songs that constitute the principal themes in the slow movements of his sonata cycles.There we can find the highest stage of structural development of this type of principal theme, and here we are faced with its embryonic state. This state is characterised as such by the fact that a fragment of this three-part song theme (the antecedent of the period) has to be considered– in relation to the polyphonic writing of the piece – also as a theme. From one point of view the theme of the piece consists of eight bars, but from another, not less grounded, – it consists only of two bars. Behind polyphonic writing there is always a homophonic form, no matter how embryonic it is; and these are inseparably linked, like the soul of the piece with its body. This form – a sonata form as will later be shown – enlightens the invention. The essence of the succession of the epochs of Bach and the Viennese classics is hidden in the alloy of polyphonic and homophonic writing. Anton Webern considered polyphony as a manifestation of a musical idea in the sum of four voices and homophony as, occurring in the process of development, a concentration of a musical idea in one voice accompanied by the rest of the voices. On the basis of such a conception it is possible to imagine that this concentration was preceded and favoured by the origin of homophonic form in the entrails of polyphonic writing. This form was that new musical space, in which “the sum of four voices” as the main bearer of a musical idea soon had to be transformed into an accompanied principal voice.

In Bach’s archaic three-part song the antecedent and consequent of its first part – the first 2+2 bars – form a sequence and are therefore in contradiction with the essence of the period that they make. However, this sequence constitutes a tonic-dominant relation: i.e. the exact relation of functions that provides the harmonic opposition of antecedent and consequent, which is the main characteristic of a period.It is not out of place to note that it is possible to find periods in Beethoven’s music where antecedent and consequent form a sequence (the theme of the Scherzo from the Twelfth Sonata, Op.26-II, the theme of the Allegretto from the Fourteenth Sonata, Op.27/2-II)[5]. But if these periods belonging to Beethoven coincide with sequences, in this case the sequence plays the role of a period. There the period is primary and the sequence is secondary, but here is the opposite.

In the simplest examples of Beethoven’s periods – the first parts of three-part songs – the antecedent ends with a half-cadence (on the dominant) and the consequent with a perfect cadence (on the dominant or tonic). The third part, that ends in these examples with a perfect cadence on the tonic exclusively, is the repetition of the consequent of the first part even in those cases where the harmonic content of the perfect cadence of this consequent is not the tonic, but the dominant. The type of cadence, but not its harmonic content, determines what is repeated in the third part – antecedent or consequent. In this way, it seems that in the examined three-part song, the third part also repeats the consequent, because it is placed on the tonic but has the same perfect cadence as the consequent that is placed on the dominant. However, the antecedent that is placed on the tonic also ends with a perfect cadence, and the third part is principally identical to it.Therefore, the third part is in the first instance the repetition of the antecedent but not the consequent. This undoubtedly constitutes one more primitive feature of the given three-part song, together with the primitive features of its first part (the sequence and, in connection to this, – the identical perfect cadences of both antecedent and consequent)[6].

The second part (the first interlude) holds very little resemblance to the structure of Beethoven’s second parts.However, the main feature of a second part is present here: it dwells on the dominant.Here the dominant pedal is not as obvious as in the second part of the principal theme from Beethoven’s First Sonata, Op.2/1-II; it is not even as clear as the abstract (hidden) dominant pedal in the Largo appassionato of his Second Sonata, Op.2/2-II.But nevertheless, in the conditions of the embryonic state of this three-part song, the fact that the interlude (the second part) begins with the dominant triad (though minor) and ends with a real dominant means that we can’t treat this any different to a dwelling that is typical for a second part.

Whilst continuing to examine the structure of the piece, we could take into consideration the fact that the other two segments also consist of three parts, like the first.The first and third parts of all three segments are relatively identical to each other; however, together with this, the second parts of the second and third segments differ from the second part of the first segment. This difference between the first segment on the one hand, and the second and third segments on the other, is linked to one more difference: while the first segment(principal theme) begins and ends with the same principal key (F minor), each of the following two segments end in a different key to their beginning.

[Example 2]

Bach-Invention-Image008.jpg

The second segment begins with a period in A flat major with the antecedent in the tonic and the consequent in the dominant, as in the initial period of the principal theme.The structure of the third part, like the principal theme, relates to either the antecedent or consequent (which are identical by their structure). However, and this is the main point, it differs from both of them by its harmony: it is in C minor — not in A flat major.[7] (See example 2)

Speaking on the second part of this segment we have to state that it is not the embryonic dwelling on the dominant, like in the three-part song (principal theme), but a complex sequence that provides the connection by modulating between the keys of the first and third parts. The third segment is the precise repetition of the second segment transposed a perfect fifth down.So, the keys A flat major and C minor are replaced with the keys D flat major and F minor.


[Example 3]

Bach-Invention-Image010.jpg

It is necessary to give an account on the importance of the harmonic difference between the first and second as well as between the second and third segments.If the first segment represents the principal theme, what do the two following segments represent?And there is another question: why do we not have a fourth segment that would be harmonically identical to the first, and serve as a recapitulation of the principal theme?

It would be more appropriate to answer the second question first: the last bars of the piece that follow the third segment, formally coincide with the third part of the principal theme and represent its shorter recapitulation.

[Example 4]

Bach-Invention-Image012.jpg

Returning to the first question it is necessary to mention that the second segment is characterised by its harmony (A flat major is the relative major key; C minor is the minor dominant) like the subordinate theme of the piece written monothematically.[8] However, this “monothematicism” has another primitive feature here: it is not only the thematic elements of the principal and subordinate themes that are almost identical, but their structures also.

The third segment, which is a precise copy of the subordinate theme transposed a fifth down (and with a different order of voices relative to each other) represents its recapitulation. It is characterised as such by its third part that is placed in the main key after the dominant on which the third part of the subordinate theme was placed in the exposition.

So it follows that this work has a mirror recapitulation: the principal and subordinate themes appear here not as in the exposition, but in reverse order.

Finally it becomes clear that the examined monothematic piece with a shortened mirror recapitulation represents a sonata form: the first appearance of the second interlude functions as a transition,

[Example 5]

Bach-Invention-Image014.jpg

and its second appearance (between the second and third segment) – functions as a development section.

[Example 6]

Bach-Invention-Image016.jpg


* * *


Anton Webern, who regarded Beethoven’s creative output as the highest point of the development of musical form, promulgated the necessity of the analysis of any musical composition written before or after Beethoven on the basis of Beethoven’s formal principles.This kind of research gives possibility, on the one hand, to retrace a primitive state of these principles in the works of the masters of the pre-Beethoven time, and on the other hand – to learn their modifications that have been undertaken in the following processes of their development in the works of the masters of the second half of the 19th century and the first half of the 20th century.

The basis of Webern’s teaching about form consists of the recognition of two different states of musical structure (“fixed” and “floating”) that oppose each other; the presence of both of them in a musical work, i.e. the presence of their opposition, is its indisputable condition. The highest level of musical form, which reveals itself within Beethoven’s creative output, is apparent when “fixed” and “floating” are clearly differentiated from one another.

The principal theme in Beethoven’s music, as a rule, is built “fixed”; the subordinate theme, transition and development section are built “floating”; furthermore they are built “floating” in varying ways. Later we will see in particular that there are principal differences between the “floating” of a subordinate theme and the “floating” of a development section. Concerning the co-relation of “fixed” and “floating” in a musical work, it is necessary to state that it is a balancing co-relation: a less floating subordinate theme can correspond to a less fixed principal theme. A musical problem within a work can even stand the relationship of the two themes on its head: the principal theme can have a floating structure and, correspondingly, the subordinate theme can have a fixed structure.

Musical composition represents a system of repetitions of different categories and scales. The repetition could reproduce the pattern that is repeated in whole or partly, literally or modified. It could be applied only to the pitch (ignoring note duration,) or vice versa (only the duration of the notes without taking their pitch into account.) This is repetition on the smallest scale, or better to say – on the lowest level.At a higher level the principal theme, that is a complex of repetitions, in turn, appears in the role of a repeated pattern – only now, the repetition is very far from being precise.

There are particular types of principal theme.One of them is a three-part song, which is the type of principal theme that we are examining in this particular piece. In the basic examples of the types of principal theme the essence of their fixed structure is revealed, in general, in a simple or complex combination of formal similarity with a harmonic opposition of their compound parts.

In musical composition a floating element is conditioned by a fixed element. While the fixed principal themes have their own types, there are no types of floating subordinate themes in nature: the principal theme is fixed by itself; but the subordinate theme is floating in relation to the principal theme.

The subordinate theme begins as if it belongs to one of the types of principal theme. But it only begins like this; the continuation of this beginning proceeds, as Webern said, by way of a "free fantasy". This "freedom" and this "fantasy" are "a free fantasy" only with respect to the fact that the fixed elements that appear in its initial part are ignored in the development of the subordinate theme.But despite this there is no real "freedom" here at all, because everything here is placed in a certain definite, though mysterious, relation to the principal theme in this or that way. The subordinate theme, avoiding the logic of the types of structure of principal theme, uses an idea association as an engine of its realization.In other words,the motives and the thematic elements of the principal theme (and sometimes of the transition), appear in new co-relations that force them to fundamentally change their appearance, sometimes making them unrecognizable. These elements become the part of the structure that differs from the structure of the principal theme, let us say, like the shadow differs from the subject to which it belongs.The floating subordinate theme reflects the fixed principal theme like a shadow, which does not exist by itself, but is only a vague reflection of the real subject that cast it.However, the subordinate theme is only perceived as an especially new and contrasting phenomenon because of the opposition of the floating structure to the fixed structure.In connection with this, everything that has appeared in the principal theme is changed, when it reappears in the subordinate theme, as if it has obtained a different physical state.

Whilst the essence and meaning of the subordinate theme is to be a bearer of the subordinate key (as opposed to the principal theme, that bears the main key), the transition and the development section, by their floating qualities – that are different from the "floating" qualities of the subordinate theme – correspondingly represent the way from the main key to the subordinate key, and vice versa: from the subordinate key to the main key.The latter deserves special attention.

It is necessary to understand the structure of the development section not only from the point of view of its function, but also of its scale. If the subordinate theme represents the repetition of the principal theme, the development section is the repetition of something that has a much bigger scale: in sonata form (of which we are speaking here,) the recapitulation is not the only repetition of the whole exposition, the development section also fulfills this role.If the exposition, that is characterized by an exchange of the main key with the subordinate key, receives in the recapitulation its repetition in which the main key is indivisibly predominant, this could only happen because of the presence of another repetition of the exposition, where as the result of a struggle with the returning main key, the subordinate key is suspended from the position that it gained.In the development section the fight of the two keys is reflected in the formal plan by the highest level of ‘floating’, the scale of it defines its essence, which fundamentally differs from the essence of the floating subordinate theme.

In the development section all the motives and thematic elements are "developing".By combination, they could be subjected to very far-reaching variation, and as a result it is often a wrong impression created about the presence in the development section of new elements that have nothing in common with the exposition, or, in contrary, about the absence of the important elements of the exposition. The development section is formulated with sequences; moreover the sequenced segments themselves are very often composed of sequences. The sequences organize the processes of annihilating the gravitating forces of the tonic of the subordinate key and return this tonic to the position of degree subjected to the main tonic, which obtains its hegemony again.

The sequence is the essence of the "floating" within the development section. The difference between the "floating" of the subordinate theme and the "floating" of the development section is defined by the fact that the subordinate theme resting on the subordinate key is static, while the dynamism is the main feature of the development section that makes the return journey from the subordinate key.The tool of this "dynamic floating" within the development section is the sequence with its labile structure, which is constantly in motion and resembles quicksand, and for which, the repetition is the everlasting occasion for the multiple and multi-faced modifications.

Very often in the development section it is possible to distinguish two sections, and each of them has, as a rule, its own "model" – that is sequenced – and its sequential repetitions.The border between these sections latently represents the moment of the final breakaway with the subordinate key together with the appearance of the first signs of the restoration of the hegemony of the main tonic.


Returning after this short summary of Webern’s concept of musical form[9] to the research of Bach’s invention that possesses a sonata structure, it is now possible to understand that the primitiveness of this structure is stipulated by the low level of differentiation between the two elements of musical construction – “fixed” and “floating” – that characterize the given composition.While the principal theme is not sufficiently “fixed”, the “floating” of the subordinate theme is reached only indirectly – not by formal but by harmonic means, which could not be the contrary because of the so far reached likeness between the subordinate theme and principal theme. And, of course, the reason for this peculiar structural co-relation of the two themes, and for the rudimentary character of the transition and development section, is because all of the parts are conditioned by each other.

The defect of the “fixed” element of the principal theme (three-part song) is displaced, to a certain extent, in each of its parts.

The illusory nature of the fixed quality of the first part is defined by the fact that both antecedent and consequent of the period constitute a sequence.It is possible to name this sequence as a period only in a metaphorical way; however, at the same time, in the conditions of the birth of the new form, it possesses the ability to perform the function of a period.

The “fixed” element is undoubtedly lost also because of the fact that its third part, being harmonically identical to the antecedent of its first part is, as we have already stated,arather definite repetition of the antecedent, but not of the consequent that concludes the period.However, this gap in the “fixed” element is bridged here to a certain extent by borrowing the important characteristic features of the consequent – despite the fact that it doesn’t represent the repetition of it (or, it does represent a repetition of the consequent, but not in the first instance). This creates a synthesis of the antecedent and consequent of the initial period (as in the third parts of Beethoven’s most exemplary three-part songs): copying the harmonyof the double-voiced antecedent (with the similar structure of both antecedent and consequent), the third part takes the appearance of the consequent that is characterized by the triple-voiced texture and triple counterpoint.

And the second part, like the other two parts, favors by its own structure the striving of this three-part song (principal theme) for rather pretending to be a fixed construction, than for actually being a fixed construction. Normally, the size of the second part is much more often equal to that of the antecedent or consequent of the preceding period and, as a rule, it is divided into two halves, the second of which relates to the first as the repetition to the repeated. In this particular case the second part, consisting of two bars (i.e. corresponding to the indicated norm), represents not a bar and its repetition, but a completely different structure: while the first bar is divided into two different halves, the second bar is composed of two repetitions of the second half of the first bar. Nevertheless, the first half of the first bar has also not been deprived of a repetition – peculiar however, – because the second part as a whole is set out as a two-voiced canon, and therefore everything that appears in one voice is reproduced, i.e. repeated, by the second voice. It is worth taking into consideration the fact that all of the interludes (and only them) are composed here as canons. Another fact worth noting is that this first interlude, the only interlude without a repetition and the only one that is double-voiced and structurally asymmetrical, does not performthe function of a modulation – in opposition to the rest of the interludes. Thus, as previously stated, one could and should interpret its harmony as dwelling on the dominant of the principal key. Because the other two interludes (both of which appear twice, are composed of three voices and are clearly symmetrical in their inner repetitions) have modulatory functions, the opposite characteristic features of the first interlude, including its structural asymmetry, stress the fact of the absence of modulation contained within it.This shows the presence of the certain "fixed" qualities of the principal theme: the fixed principal theme – i.e. the principal theme that is not an exception as regards to the structure, does not modulate.

Nevertheless, the weakened “fixed” qualities of the principal theme of this work are conditioned in particular by this peculiar division of the second part together with the examined deviations of the other two parts from something that had to become their structural archetype in later years.Completing the study of the principal theme it is necessary to mention the fact that also in these later years, under the conditions of predominating homophony, the second parts of three-part songs often, in contrast to the other two parts, preserved an appearance of imitative polyphony (see for example the principal themes of the slow and final movements of Beethoven’s Second Sonata, Op. 2/2-II and IV).

The weakened “fixed” qualities of the principal theme correspond with the weakened “floating” qualities of the subordinate theme.Moreover, because of the fact that the structures of both the principal and subordinate themes represent in principle the same three-part song, it is possible to state that the weakened “fixed” element of one theme and the weakened “floating” element of the other are mutually determined.

The weakened “floating” (i.e. being on the edge of “fixed”) qualities of this subordinate theme are contained in the fact that it appears in the shape of a three-part song.However, here is only the framework of a three-part song in which the essence of it is absent. A “three-part song” where the first part is placed in one key and the third part in another is not a real three-part song.Its harmonic instability is the essence of its “floating” qualities, which conceal themselves behind the appearance of a fixed construction.

The harmonic dualism that characterises the subordinate theme of this invention-sonata occurs very often within Bach’s works written in minor keys.If in later times, only one of the two keys (the dominant minor or the relative major) could be the bearer of the subordinate idea of a work, in Bach’s music the subordinate idea, which had not yet emancipated from its embryonic state, inevitably demanded, in its seeming awkwardness, that both of the mentioned keys had to jointly serve as its bearer.

Of course, in the conditions of the examined mono-thematic[10] work, stipulated by the archaic aesthetic, the presence of two subordinate keys that are opposed to one main key represents the means of the principle opposition of the subordinate and principal themes: i.e. the means of indirectly creating a floating structure from one theme in relation to a fixed structure of another.

However, the presence of two keys in one theme also presupposes the presence of a modulation that joins them together.And inasmuch as one key occupies the first part and another – the third part of the subordinate theme, the modulation, of course, is placed in the second part.Therefore, the principal and subordinate themes have different second parts because their functions are different.

The modulation is created here by means of a complex sequence: one-and-a-half bars and their repetition (a fourth lower = a fifth higher) represent the sequence themselves, where the half-bar is repeated twice (a second higher and a second higher again). The second part of the subordinate theme, like the second part of the principal theme, is exposed as a two-voiced canon (however with the addition of the third voice that seems to be “free” and “separate”).The “follower” (answer) of the repetition (the second one-and-a-half bar) is identical to the “leader” (subject)[11] of the repeated (the first one-and-a-half bar):


[Example 7]

The second part of the principal theme and the second part of the subordinate theme, despite their difference, are related to each other:

1) In both of them the answer is placed a fifth lower than the subject

2) In both cases the answer follows the subject with the same time-interval: a crotchet later

3) The unrepeated motive of three quavers, which has appeared in both voices of the canon in the second part of the principal theme, represents, in contrary, the only motive, i.e. the only repeating element, in each of the two voices of the second part of the subordinate theme, where it appears in both of the one-and-a-half bar segments, and in them – three times in each of the two voices:

[Example 8]

Bach-Invention-Image020.jpg

i.e. exactly the same number of appearances of the motive that is repeated in [the “subject” of][12] the second part of the principal theme:

[Example 9]

Bach-Invention-Image022.jpg

and in one case it is a gradual descending repetition, and in the other – gradually ascending.

Under more careful examination it appears that the third voice[13] of the second part of the subordinate theme is not “free”, however, it is “separated”.By the duration of its sounds (there are only crotchets throughout) it represents the reproduction of the voice that descends through a chromatic scale in the first and third parts of both principal and subordinate themes.However, it is also of the same “bone and flesh” as the canon: by the abstraction of the rests, we can state that the third voice is composed as a chain of simple repetitions and of repetitions of the retrograde inversion form of the augmented three quaver motive:

[Example 10]

Bach-Invention-Image024.jpg

Incidentally, the retrograde inversion of the motive is also present in a chain of repetitions (ignoring the rests) when it appears in its normal non-augmented form:

[Example 11]

Bach-Invention-Image026.jpg

In other words, there are good reasons to consider the second part of the subordinate theme as a three-voiced canon.

In the conclusion of the examination of the subordinate theme, it is necessary to stress once more that, as in the principal theme, where the consequent of the initial period represents the dominant of the main key, but not C minor as an independent key, the consequent of the first part of the subordinate theme also does not represent E-flat major as an independent key, but is subjected harmonically to the key of the antecedent (A-flat major) as the bearer of its dominant.Only the third part of the subordinate theme could and should claim harmonic independence – it really is placed in C minor.Schoenberg’s concept of harmony, and his teaching on artificial dominants in particular, allow one to distinguish something that is a modulation from something that is not: to distinguish extended tonality from something that lies beyond it.On the basis of this teaching it is easy to realise that here the subordinate theme is resting on two different keys, one of which is represented by the tonic and dominant (the first “part”) and the other – by the tonic only (the third “part”).A third key is not present here.

As previously mentioned, in the mirror recapitulation of this piece, the subordinate theme that precedes the principal theme is transposed a fifth lower in relation to the exposition, without any modification.Therefore its third part appears in the tonic F minor, and the first part in D-flat major – the subdominant (or to be more precise, the subdominant of the relative key).So, this piece represents the prototype of those works by Mozart (and Schubert), where the recapitulations find themselves in the main key (in the tonic of the principal key) after beginning in the subdominant (in the key of the subdominant).

It is now important to clarify why the recapitulation of the principal theme is shortened. Consisting of two bars[14] that represent the tonic of the main key, the principal theme in the recapitulation is identical to two of the elements from the whole principal theme – its third part and the antecedent of the initial period. However, the essence of the matter lies in the fact that these two bars appear immediately after exactly the same bars (also placed on the tonic) that represent the third part of the recapitulation of the subordinate theme. Three times – twice in the exposition (in the principal and subordinate themes) and once in the recapitulation (in the subordinate theme) – the antecedent and consequent, composing a period and appearing in different keys (F minor, A-flat major and D-flat major) had the same co-relation: tonic and dominant.Only here, when after the enclosing two bars of the recapitulation of the subordinate theme the only two bars of the recapitulation of the principal theme appear, the two sentences [the antecedent and consequent] both placed on the tonic, though belonging to the different themes, follow each other. The main tonic at the end of the recapitulation of the subordinate theme, preceding the main tonic of the beginning of the recapitulation of the principal theme, renders the rest of the principal theme as needless and impossible.It is possible to say that the tonic-consequent following the tonic-antecedent is the dominant sublimated into the tonic.This is the same dominant that always – three times – immediately followed the tonic.The two appearances of the tonic, one after the other, represent, in the given circumstances, the legitimate and natural end of the piece.

Of course, it is very interesting that the “missing” – as if evaporated – six (of the eight) bars of the recapitulation of the principal theme are nevertheless, by their quantity, present within the piece. However, it is not six bars that are missing, but only four: the similarity of the remaining two bars with two elements from the whole principal theme not only permits, but demands that they should be counted twice, and considered not as two, but four bars (the more so because they appear and are perceived as the echo of the third part of the recapitulation of the subordinate theme which precedes them). The other four bars, really missing in the recapitulation, could be found (it is necessary to repeat that here we are speaking only about the quantity, but not about their formal and harmonic essence) in the second part of the subordinate theme and its recapitulation (one bar in each of them) and in the development section (two bars). In other words, as the second part of the principal theme consists of two bars, the other second parts consists of three bars (of one-and-a-half bars and their repetition); as the transition also consists of two bars, the development section – the repetition of the interlude that serves as a transition – consists of four bars. The bars concealed in the third interlude and its repetition, and also in the repetition of the second interlude, have shortened the recapitulation of the principal theme. This is not mechanical bookkeeping however – it is organic. Such quantitative structural compensations occur very often in the music by all Great Masters (and by Beethoven first of all).

It is also necessary to note that the last bar of the piece stands apart from the recapitulation of the principal theme. The bar is wholly occupied with the final chord of the last cadence, which strongly differs from all other multiple cadences by the fact that it is the only concluding chord – the only end – while each of the other similar chords also act simultaneously as the beginning of the next segment of the composition. Final chords of the cadences within each segment – we are speaking in particular and especially about the antecedents and consequents of the first parts and also about the third parts – are placed outside their borders and therefore it occurs that the harmonic formation and formal arrangement of each segment do not coincide by their size: the first of them is longer than the second. This is why every “harmonic end” is a “formal beginning”. But there is only one exception: the last bar.By its form it means nothing.It is a formal vacuum.(However, because of its fermata [or pause], the last bar becomes equal to two bars, i.e. the same duration as the main formal element of the work.) But because of a lack of the necessity to be a beginning for any other repetition, it can perfectly perform its harmonic function: it embodies not only the end of the piece, but also the sum of all those points of rest that the previous cadences could not actually perform because their closing chords had to tear themselves between the functions of beginning and ending – to unite in themselves the essence of the beginning and the essence of the end.The presence of this sum within the last chord is concretised by its duration that is doubled by the fermata (the pause).

In the conclusion of the examination of the themes and their recapitulations we can not leave without attention to the fact that the coinciding of the end of the antecedent with the beginning of the consequent within the initial period is also one of the features of a weakened “fixed” quality within the principal theme.

The only things left to investigate are the transition

[Example 12]

Bach-Invention-Image014.jpg

and the development section.

[Example 13]

Bach-Invention-Image016.jpg

They are remarkable in two respects: firstly, because they are short; secondly, because, despite the difference of their functions, they are built in the same way and moreover – from the same motives.Both of these qualities make the transposition and the development section primitive when compared to the transition and development section of Viennese classical sonata form.The transition is composed here of one bar and its sequential repetition, and the repeated bar itself, in turn, can be considered as a bar that can be broken down in two structurally identical and sequential half-bars.[15] The development section represents the doubling of the transition, and this is not just a quantitative doubling: the two bars that represent a sequence in the transition, here in the development section are sequenced as a whole. The transition and development section are sequenced at different intervals: the second bar of the transition is placed a perfect fourth higher than its first bar; the third and fourth bars of the development section are placed a major second lower than its first and second bars.Because of the difference between the intervals of these two sequences, a very close relation is created between the transition and development section: if the first two bars of the development section are placed a perfect fourth lower than the transition, the last two bars appear a perfect fourth higher than the transition. Therefore the second bar of the first couple of bars of the development section is identical to the first bar of the transition, and the first bar of the second couple of bars of the development section is identical to the second bar of the transition. It is easier to say, the transition as a whole is identical by its pitch to the middle – the second and third – bars of the development section; one of which belongs to its repeated, so called “model” (or “pattern”) and another – to the sequential repetition of the latter.

In other words, the development section is the same as the transition to which two bars have been attached – one before it and one after.By the addition of these two bars the function of the transition has been transformed – taking into account the harmonic qualities of the subordinate theme and its recapitulation – into its opposing function.This defines the meaning of the symmetrical presence of the whole transition in the development section.And at this point it would be appropriate to state that everything that we perceive as the musical power of expression, as musical character, is in the end the emanation of structural relations, and the co-relation of the transition and development section in this work can serve as an example of this.

The mechanisms of the transition and development section could be better observed if one pays attention to the co-relation of their half-bars: not only is thesecond bar of the transition placed a fourth higher (fifth lower) in relation to the first bar,

[Example 14]

Bach-Invention-Image030.jpg

but the second half of the first bar is placed at the same interval in relation to its first half.

[Example 15]

Bach-Invention-Image032.jpg

The noted distance between two halves of the first bar is kept the same in the second bar, but the first half of the second bar and the second half of the first bar are identical by their pitch.


[Example 16]

Bach-Invention-Image034.jpg

This relationship is kept between any two adjoining bars of the development section. And exactly in this way, if proceeding from F minor (by the transition), two steps by fourths (during the course of two bars), one will reach the dominant of A-flat major.Likewise, if one proceeds from C minor (by the development section), four steps by fourths (during the course of four bars) one will reach the dominant of D-flat major.

The common character of the motives and structures of the transition and the development section is conditioned by the “monothematicism” of the piece: if the transition is a reflection of the principal theme and the development section is the reflection of the exposition as a whole, then it is quite logical for them to be homogeneous. This is because the principal theme represents the complete structural prototype of the subordinate theme (the second of the two main components of the exposition) and contains all of its motives and thematic elements. This common character of the transition and thedevelopment section is really a primitive feature of this sonata form, however it is conditioned as such by the more fundamental primitive features of it.[16]

Every half-bar of the transition and the development section consists of three motives placed vertically:

[Example 17]

Bach-Invention-Image036.jpg

1) Two crochets ascending chromatically, borrowed from the voice in the first and third parts of the principal and subordinate themes, which consists of crochets descending chromatically.(In both half-bars this motive is placed in its pure unmodified form only at the beginnings of the transition and the development section; in the first half of the second bar of the transition and in the last three bars of the development section it is modified).

[Example 18]

Bach-Invention-Image038.jpg

2) Figuration borrowed from the conclusion of the row of three appearances of the three–quaver–motive within the first and third parts of the principal and subordinate themes:


[Example 19]

Bach-Invention-Image040.jpg

3) The motive of three quavers.


The texture of the transition and development section, like the texture of the second part of the subordinate theme and its recapitulation, represents a canon of two voices with the addition of a third voice that is separated from the canon. But if in the noted second parts the canon is created from the three–quaver motive, here, in the transition and development section, this motive, in contrary, makes up the separated voice. On the other hand, the crotchets that make up the separated voice in the second parts here serve as one of the two motive elements of the canon.

In this connection it is necessary to make two more precise definitions:

1) The two motive elements that compose the two voices of the canon of the transition and development section are as follows:


[Example 20]

Bach-Invention-Image042.jpg

In one voice they appear in one order, and in the second voice they appear in the reverse order:


[Example 21]

Bach-Invention-Image044.jpg

Hence, it follows that each of these two voices plays the role of the “leader” and “follower” (or the subject and answer) simultaneously:

[Example 22]

Bach-Invention-Image046.jpg

2) In the exposition as well as in the recapitulation the first and third parts of the principal and subordinate themes are written with real triple counterpoint. However, the triple counterpoint of the second part of the subordinate theme (in the exposition and recapitulation), transition and development section is only potential: two voices here have the relations of active double counterpoint, i. e. each of the repeated patterns and the repetitions appear not only in one and the same voice; but the third [reclining] voice concentrates in itself the repeated elements together with the repetitions despite the fact that it is quite able to enter into the relations of triple counterpoint with the other two voices. It is the “reclining” voice.However, it is a “reclining” voice that is not “separated”, but is the one of the two voices of the canon:

[Example 23]

Bach-Invention-Image048.jpg

the other “voice” of which is really distributed between the other two voices

[Example 24]

Bach-Invention-Image050.jpg

that become partners in the double counterpoint of the “separated voice”, which is also split up into two voices:

[Example 25]

Bach-Invention-Image052.jpg

One can see that it has become necessary to distinguish between the voice (without inverted commas) and the “voice”.In this, one more duality is reflected (together with the main duality of the piece: an invention that is also a sonata) – the duality that originated by these “reclining” voices: if they were absent, there would be no reason to speak about the canons!But in speaking about the canons it is necessary to speak about their “voices”! And it is also necessary to speak about the voices (in the different meaning of the word) – because these canons are conditioned by the potentially triple, but actively double counterpoint.

These “reclining” voices appear as follows: the lower voice in the transition, the middle voice in the development section, and the bottom voice again in the second part of the subordinate theme and its recapitulation.They appear in every part of the piece where a “floating” structure is present.Therefore, these voices are a sort of “trade mark” for the “floating” structure here.

The common textural features of the transition and development section, along with the second parts of the subordinate theme and its recapitulation, have a profound meaning. In this composition it has become possible to distinguish between the rudiments of “fixed” and “floating” musical constructions.However, they are only the rudiments.In their mature state in the music of Beethoven, these structural dissimilarities are more differentiated. In Beethoven there is not only “fixed” that is opposed to “floating”, but there are also two types of “floating” that oppose each other: the “floating” that reveals itself in subordinate themes, and the “floating” that becomes apparent in development sections. It may seem that such a kind of opposition is completely absent in the examined invention-sonata: here the subordinate theme is constructed by way of combining the elements borrowed from the principal theme (the first and third parts), and additionally, the second part is built on the basis of the principles that form the construction of the development section. At this stage in the development of musical form, the sequence (the tool of the development section) is the only means of constructing the “floating” structure. As such, it even intrudes on the principal theme, preventing it from obtaining a really fixed structure.Hence, the antecedent and consequent of the first part of the principal theme represent some likeness to a sequence. It is only the third part, consisting of only one sentence[17] and participating decisively in the creation of the contour of a three-part song, that indirectly defines the meaning and function of the sentences of the first part as components (the antecedent and consequent) of a period – the “fixed” formal phenomena.

And nevertheless, despite the fact that the sequence is present here throughout the whole piece and represents the main structural element of the transition and development section as well as of the second part of the subordinate theme and its recapitulation, there is a very essential distinction between the former and the latter. This distinction is contained in the fact that, in contrast to the normal canons of the second parts, the canons of the transition and development section are ambivalent: as previously noted, each of their voices plays the role of the “leader” and “follower” (subject and answer) simultaneously. This distinction reflects the essence of the difference between the two harmonic categories. Firstly, in the transition and development section, which are characterised by ambivalent canons, the fundamental modulations of the piece take place (one that connects the principal theme with the subordinate theme and the other that joins the exposition as a whole with the recapitulation as a whole).[18]] And secondly, in the second parts of the narrow frames of the subordinate theme and its recapitulation, that are characterised with normal canons, the inner modulationsoccur, which link the two keys that relate to each other like two sides of the same coin.The two “harmonic categories”, embodied in a formal context by the canons of different kinds, already represent at this early stage the phenomena that at a later time had to create an opposition, on a far greater scale of harmonic and formal differentiation, between the static “floating” of a subordinate theme and the dynamic “floating” of a development section.However, the rudiments of this dynamism are accented here by the fact that the development section is similar to the only interlude where – in just one of the voices (the “reclining” voice) – one of the motives is transformed during its repetitions (the importance of change due to variation within a development section has already been discussed).

Perhaps, it is necessary to stress something that needs no explanation: the perfection of a piece of art does not depend on the time that it was created – i.e. at an earlier or later moment within the development of art. However, the character of this perfection is defined by this moment. Under the conditions of the invention its organics are only provided by the primitiveness of the sonata. This is the organics of the intermediate phenomena of musical fauna, the phenomena that is as transitional as, in the world of animals, a species like the duck-bill (or platypus) that belongs to the mammals, but propagates by eggs like a bird. Only by the primitiveness of its structure can a sonata be identified as an invention, giving to the latter that distinctive ambiguity, that vast increase in richness and depth.

Bach’s Three-part Invention in F Minor is not his only composition in “Viennese” sonata form. There are more. However, in this connection it is really amazing that this isnot the only work where Bach manages to combine sonata form with triple counterpoint. A thought inevitably comes to mind: Bach, being aware of the importance of the innovation that he was creating, decided to unite this innovation with something ancient, but having no less importance, and this was able to serve as a catalyst for the introduction of the new musical phenomena into the creative practice, which was still unable to seriously change its own essence.The ancient law of triple counterpoint stands as an exclamation mark to the new law of sonata form. Perhaps, it is permissible to say, that here the sonata has appeared from triple counterpoint, like Aphrodite appeared from the sea. This maybe the truth or some likeness of the truth, that permits one – to a certain extent – to express admiration for the Greatest Master, whose work(if to speak about the assimilation of his music by the following generations), represent a map, on which there are still more blank spots than on the map of Africa in the 19th century.We must recognise, however, that Bach’s blank spots will never be completely understood. And that is only because the greatness of a Great Master is endless. But one of the main targets, or in general, the foremost aim of musical science, is still attainable: to find every possible way of enlarging our perspective on the correlation of the closest past of musical art with a past that is more remote.

(1967-70s)



Notes

  1. [1]Филипп Гершкович: «Об одной инвенции Иоганна Себастьяна Баха (К вопросу о происхождении классической венской сонатной формы)» в сборнике «Учёные записки Тартусского Государственного Университета: 467, Семиотика текста; труды по знаковым системам XI» Тарту, 1979, cc. 44-70 Ob odnoi inventsii Ioganna Sebastiana Bacha: k voprosu o proiskhozhdenii klassicheskoi venskoi sonatnoy formy’, in the collection ‘Uchenye zapiski tartuskogo gosudarstvennogo universiteta: trudy po znakovym sistemam’ (Tartu State University Scholarly Notes: 467, Semiotic of a text; Studies in Semiotics XI) Tartu, 1979, pp. 44–70 in Russian.(DS)
  2. Meaning the way of writing, which includes texture, style etc. (DS)
  3. The author uses Webern’s terminology in order of principal necessity.For those who are unfamiliar with this classification of formal phenomenon it should not be difficult to understand what is hidden under the terms “ three-part song” or “ model”. (PH)
  4. In Herschkowitz’s terminology it is literally “consisting of two sentences”. The same is also accepted in Russia: for “antecedent and consequent” they usually use “the first and second sentences”. (DS)
  5. From here onwards we mean Beethoven’s Piano Sonatas when no instrument is indicated. (PH)
  6. Only after the publication of this article did I notice that I had made a severe mistake: the first segment ends (at the beginning of b.9 – DS) not with a perfect cadence, but with a cadence that is usually called “imperfect”.I committed this blunder as a result of too much familiarity with the score of the Invention: I was working by heart!And became a victim of triple counterpoint. However, it is also untrue that I– by heart – was not aware of the absence of a perfect cadence at the end of the first segment.The sixth chord that concludes the first segment never, even for an instant, escaped my mind.But because the other (following) two segments really end with perfect cadences I, influenced by the observation of the beauty of the functionally differentiated similarity of all three segments, let myself be carried away with the kaleidoscopic qualities reached by the triple counterpoint in this work.If the triple counterpoint should be absent here, it should be undoubtedly a perfect cadence at this place. Moreover, it even exists here.But it is a perfect cadence, transfigured by triple counterpoint.So, in the end I did not make a mistake, but only stated a thought that needed to be explained. (PH) <This note was made by the author after the publication of the article.However, it was “abolished” later.(LH – Leni Herschkowitz)>
  7. It is important to stress that the understanding of the structure of the piece demands consideration of the consequent of the period – contained within the first part of any of these three segments – not as the representative of an independent key, but only as the representative of the dominant of the key of the antecedent.
  8. Monothematic usually means having only one theme – a composition based on one subject.However here we have to speak of two themes (the principal and subordinate) that are similar in shape but that function differently. (DS)
  9. The defect of this summary lies in its conciseness. However, the author could not consider the possibility of extending it to the necessary scale, for it would greatly exceed the size of the research itself.Neither could he just put aside a summary of Webern’s formal principles, because the essence of this research is contained within them. (PH)
  10. In this instance it is necessary to widen the meaning of the term mono-thematic: here it concerns not only the motives of the principal and subordinate themes, but their structures as a whole. (PH) [This fragment was originally part of the main text – GS].
  11. Instead of the “subject” and “answer” Herschkowitz here uses older Russian terms: «вождь» (the “leader”) and «спутник» (the “follower”). (DS)
  12. Added by the translator. (DS)
  13. Here it is important to understand that the “first”, “second” and “third” voices of these three bars do not simply correspond with “soprano”, “alto” and “bass” voices, but have a more entangled and complex relation.Thus, the “first” voice of the canon (“leader”) begins in the “soprano” but in the middle of the next bar it has moved to the “alto” or “middle” voice. The “second” voice of the canon (“follower”) is placed into the “bass” voice throughout these three bars, and therefore it is a “reclining” voice. The “third” voice of this three-voice combination is “separated” from the canon, because it is not imitated immediately in any other voice.It begins in the “middle” voice and then in the middle of the next bar it moves to the “soprano” voice.So, these three bars are divided exactly in the middle and the second one-and-a-half bar segment is the sequential repetition of the first, but with a different order of voices: if the “first” (“leader”) and “third” (“separated”) voices exchange positions (between soprano and alto) and enter into the relations of double counterpoint, the second voice (follower) remains “reclining” in the bass and has only“potential” abilities to enter into the relation of triple counterpoint (if it also should be moved to any other voice). However, this is not the case, and instead of triple counterpoint we really have to speak only about active double counterpoint here.Herschkowitz wanted to stress that Bach possibly did this intentionally to indicate the difference between “fixed” and “floating” types of construction. (DS)
  14. The last bar will be discussed later. (PH)
  15. From here onwards, when speaking about the “repetitions”, “identical structures”, and “precise sequences”, one must ignore the fact that the order of the voices in the repetitions is always different to the repeated pattern. (PH)
  16. Speaking about the fundamental features Herschkowitz was referring to the “fixed and floating” elements. (DS)
  17. Here the term “sentence” is used for indicating one of the two elements of a period(the antecedent or consequent) as distinct from a “sentence” as an independent form of principal theme. (DS)
  18. The modulations of the transition and development section are the opposite of each other, and just this opposition is reflected in the ambivalence of their (identical) canons. To be more precise, it is not the canons that are ambivalent, but their voices: each of them, being the “leader” and “follower” simultaneously, unite in themselves two opposing qualities, like the one and the same twice repeated second interlude unites in itself the opposite qualities of the transition and development section. (PH)


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