[AUTHOR'S NOTE: This paper was originally submitted on 12 December 1994 for New England Conservatory of Music's Interpretive Analysis seminar, taught by Professor John Heiss. Please refer to a good score as you read this article.]
Beethoven's Sonata Op. 102, #2 for piano and cello is one of his most dramatic works. The last movement, a fugue, is a prime example of his revolutionary and powerful use of the form. Although the movement starts with a simple, even dance-like subject, the music quickly begins to travel through enormously different emotional, textural, harmonic, and oratorical domains. The work derives great power from Beethoven's transcendental sense of form, phrasing, harmony, and motivic treatment of the subjects and countersubject.
I. Beethoven's Form�Great Emotional Impact
The movement starts with no hint of the drama that follows. After a five-measure introduction there is a relatively conventional four-voice exposition (mm. 4-30). A short episode (m. 30 to the upbeat to m. 35) is followed by an entry in the cello in the tonic lasting until m. 42.
But from here, things start to heat up. The piano soprano entry in m. 47 is the highest-pitched entry yet. Also new here is the doubling of the countersubject in both the left hand of the piano and the cello (in thirds). What is special about this setting of the countersubject is the pitches on which they start, a G and E. In m. 10 -- another entry on the dominant (as here) -- the countersubject had begun on the pitch A. Although the listener might have initially felt that the subject and countersubject were harmonically simple, it is already apparent that Beethoven is quite capable of setting them up for some dramatic settings.
From mm. 72-84 the drama of the form is heightened to yet another level. Here Beethoven composes a stretto section with an entry in the piano's middle-voice (mm.72-79), interrupted by an entry in the piano-soprano (mm. 78-81). This in turn is interrupted by an entry in the cello (mm. 80-83).
Balancing out that activity is a long episode (mm. 84-100). Then as a foil to the growing tension Beethoven gives us a pianissimo entry in the cello (m.101-107), which is also the first entry to be inverted. The gentle falling motion of the subject�from d1-d�also helps to relieve the tension.
Next comes a very long episode (mm. 108-143) in which Beethoven reaches deep into the depths of his harmonic and motivic abilities, energetically escalating the music to a thunderous climax. Then Beethoven surprises us with a most unexpected move, a second subject. Simple, gentle, legato; this subject is the antithesis of the emotional tone of the movement thus far. In order to let this effect sink in, Beethoven composes two more successive entries in the piano-soprano (mm. 147-150) and in the piano-bass (mm. 150-153).
Of course the next logical step (if you are composer worth your salt!) is to combine both the subjects. This Beethoven does twice in mm. 154-164, allowing both instruments a go at each subject. A short episode is followed by the last appearance of the second subject in the piano-bass (mm. 169-172) where it is again coupled with the first subject in the right hand of the piano.
From this point in the movement, Beethoven provides a short episode (mm. 165-169), which leads directly into an AA pedal trill in the piano. This is accompanied by stretti in the cello (mm. 185-191), and piano-soprano (mm. 187-191), both instruments fortissimo. This is the last full entry until the end of the work. What follows is the longest episode of the movement which in turn builds to the final extended entry in the cello and piano-bass, marked fortissimo (upbeat to 233-240).
II. Phrasing�A Struggle Between Stability and Instability
As a complement to the larger architectural techniques described above, Beethoven adds to the drama by carefully manipulating the "big rhythm" (the phrase groupings). He breaks up the longer phrase-groups into small segments. Even within the larger segments, he interrupts an otherwise seamless flow by adding sforzandi or accents to second and third beats. Indeed the weak beat sforzando or accent is an integral part of the rhythmic identity of this movement. On the other hand, sometimes an even-measured phrase will balance out instabilities in the harmony.
The exposition is made up of regular six-measure sub-phrases (starting on upbeats and ending on second beats). Despite this stable phrasing, the second-beat accents and later sforzando-piani hint at the tumult to come. In the next entry after the exposition (upbeat of m. 35 to m. 41) there is already a feeling of unbalance in the big rhythm, sensed partly because of the seven-measure phrase. In mm. 62-69 the water starts to boil�along with the breakdown of the subject there are many alternating second and third-beat sforzandi, a total of seventeen in just seven measures (see part III of this paper). Then, in mm. 73-84 the phrasing becomes further obfuscated by the stretti entries. In m. 90 stability of phrasing is restored with a brief canonic passage between the cello and the right hand of the piano, but the chromatic bass line along with its second-beat accents creates both rhythmic and harmonic instability, sustaining the drama.
In mm. 94-101 there is a sequence which helps to return stability in the phrasing. Then starting from the upbeat to m. 102 to m. 107 one begins to sense more steadiness in the big rhythm created by a regular six-measure entry. (Note, however, that since the entry is inverted, there isn't a feeling of absolute tranquility). In mm. 108-112, a five-measure phrase, the piano left hand has a chromatic descent moving on the weak second beats at first. Then in mm. 113-130, Beethoven composes a sequence which, along with the cello playing rhythmically with the left hand of the piano, presents a clear, regular big rhythm. Still again there are many sforzandi which have the affect of unsettling the overall regularity. From mm. 131-142 the big rhythm is regular again in a twelve-measure phrase (made up of groupings of four + four + four measures). This is needed here because practically everything else about the music feels as though it is breaking down.
The unpretentious second subject is clearly a phrase of four measures (mm. 143-146); but hot on its heels one senses instability in the phrasing created by the stretti of the two subjects. In mm. 176-185 more instability arises with the line fragmenting into two-beat segments. The tension rises in the third beat of m. 185 with a bone-crushing fortissimo entry accompanied by a trilled pedal-point in the bass. To add to the drama, the section up to m. 194 contains stretti and subject fragments which further obscure the big rhythm.
From mm. 207-226 there is a mix of stability and instability. In. mm. 207-221 there is stability due to the regularity of the phrasing (eight + four + four); mm. 222-226 is an unstable five-measure phrase; and mm. 227 to the upbeat to 232 is a stable six-measure phrase.
From m. 233 to the end there is a stable twelve-measure phrase (two + six + four), but the slurred groups of two quarter notes create a hemiola and obfuscate the large rhythm. Also of note, these two-beat groups comment of the previous�they are syncopations which correspond to the off-beat stresses in the first subject and episodes. Even the phrasing of last four measures is unclear in that the dominant chords in mm. 241, 241, and 243 sound as if they should be on an upbeat. The icing on the cake is the final tonic chord which occurs on the fourth measure of the last four-measure sub-phrase, normally the weakest measure of a four-measure phrase. This may leave the listener wondering whether or not they really heard a four-measure phrase at all!
III. Subjects and Countersubject�Treated Motivically
There are two subjects and one countersubject in this movement. In general the drama of Beethoven's fugal technique comes from the way he treats subjects as motivic�breaking them down and developing and juxtaposing certain parts of them. Equally important is how and where he does this.
Subject #1 can be divided into three parts. The first whole measure of subject #1 I call "x". It consists of a rising, stepwise scale covering the range of a sixth. The next two measures of the subject I call "y". Both measures are really the same idea�a quarter note (passing or neighbor note) followed by a half (or quarter) note on the second beat which is given an accent or sforzando. The fourth (and sometimes fifth and sixth) measure of the subject is what I call "z". It consists of two eighth-notes falling stepwise to a quarter note on the second beat (outlining a third), then rising up a fourth or fifth to another quarter note (this last quarter sometimes being omitted in later transformations.
The countersubject mainly consists of a tied upbeat quarter-note connected to a descending line. The first full measure descends at the rate of quarter notes. The second measure consists of a descending group of four eighth-notes followed by a quarter which is tied over to the next measure, a kind of melodic turn of four eighths and a quarter. The next part includes a quarter rest followed by an ascending line of four eighth-notes leading to a quarter note in the last measure.
Subject #2 is quite simple with a general outline of four dotted half-notes. The second pitch is a third lower than the first, while the third pitch is the highest, and the fourth pitch usually the lowest.
As the movement progresses, Beethoven dramatically alters and combines these motives. Again, although the exposition is comparatively stable, the accents and sforzandi on second beats of "y" hint at the drama to come. In mm. 20-22 there is the first appearance of "x" extended out to three measures. This extended form is used to keep or build momentum with its ascending and descending, rolling eighth-note lines. In mm. 26-29 "x" is in the middle voice of the piano and ties the counterpoint together for four measures now. In mm.38-40 "x" is in the piano-soprano.
The next transformation occurs in the episode from mm. 41-45 where "x" appears in the piano-bass along with fragments of "z" in the piano right hand and cello. Also new here is a variation of "z" which appears on the upbeat to m. 40 through the second beat of m. 41. It is a kind of extended form of the motive created by adding an extra quarter note tied over to the two eighths (see "Variations" section of "Subjects and Countersubject").
The entry starting in m. 46 heightens the drama further by putting the countersubject in both the cello and the left hand of the piano. In addition (as previously mentioned) this is the highest-pitched entry yet (in the piano-soprano) ranging from d2 to f# 3.
By now one will notice that "x" is generally substituted for the end of the countersubject ("b") as in mm. 50-52. After m. 52 it also has the function of eliding into the next episode built on "x", descending "z" in the cello, and "z" variant in the piano right hand. These two forms of "z" are dramatically placed rhythmically right "on top" of each other, thus creating further tension.
The next level change is in mm. 60-62 where Beethoven puts "x" in the cello, mostly on the bright A-string, thus emphasizing the motive. In the episode of mm. 63-72 Beethoven adds to the feeling of unrest in a few ways: First of all, the piano, right hand, now plays the inversion of "b" while the left hand is working with the "z" variant. The cello plays inversions of the first two beats of "z" (mm. 63-66) and then joins the piano left hand in playing the "z" variant. Additionally, all the "z" variants and the inverted "b" motives are marked with those seventeen sforzandi on their first notes. This creates second and third beat accents adding to the unsettled feeling.
By now it may have become clear that there is a tight relationship between these motives (see "Motive transformations" in "Subjects and Countersubject"). By merely adding two more eighth-notes, "z" becomes the inversion of "b". This motive, in turn, can also be extended two eighth-notes to become "x". Thus part of the drama of Beethoven's motivic technique is the way he transforms and relates the motives.
In mm. 84-93 the episode is built on the melodic turn (first occurring in the third measure of the countersubject) in the piano right hand and cello, separated by one beat (mm. 84-88), and then by a whole measure (mm. 90-93). Along with the tension-building crescendo, Beethoven also brings in the "y" motive (the syncopated, rising bass line).
Again in mm. 94-101 the drama is created by the cello and piano echoing each other at the distance of one measure. Also, the piano left hand is playing an augmentation of "x". The entry in m. 101 presents something new�a simultaneous stretto in the cello and piano left hand, both in inversion.
By mm. 112 to 139 the highest level of tension has been attained. Everything is breaking down. Starting on the last beat of m. 112 to 120 both the cello and piano have pieces of subject #1 which are alternated every third measure (i.e. piano left hand, cello, piano right hand, in a repeated pattern). From the upbeat to m. 120 the piano and cello seem to join in their fragments of the subject, but the catch is that it doesn't lead anywhere except to dead-end sforzandi. From m. 131 the fragments are disintegrating into ever smaller segments until mm. 136-138 where there is nothing left but clangorous one and two-beat fragments. This is the highest level of tension in the whole movement.
Perhaps, then, it only makes sense that after a "resolution" in F# major in mm. 139-142, Beethoven introduces a new subject. Although this subject is simple and marked piano, its appearance here is quite dramatic because one was unprepared for it at the beginning of the movement. The drama of mm. 143-175 is derived from the juxtaposition of subject #2, subject #1 or the motives derived from subject #1, and the countersubject. For instance, in m. 147 subject #2 is in the piano right hand while the left hand and the cello have an inversion of the "z" variant. Perhaps the most obvious example of this juxtaposition is the stretto section (mm. 154-164). Although the motives are still used in this section to the end, the countersubject has now been completely displaced by subject #2.
In a further transformation, from m. 172 there are no more appearances of subject #2 (hence this is not a real double fugue). In m. 176 there are fragments of "z" in the piano right hand, and then the cello in m. 179. On the third beat of m. 185 Beethoven begins a stretto section (entry in the cello) with a massive fortissimo, accompanied by a pedal-point AA trill. All this signals the listener that the end of the work is coming.
From mm. 195-206 most of the motivic material is based on "x" and is used both to outline harmonies and keep up the forward motion. With the return to the tonic tonality in m. 207 we have a return to some familiar motives--while the cello plays "x" the piano right hand plays both "z" (soprano) and "b". To strengthen this sense of stability, in m. 211 the instruments switch off with the piano playing "x" and the cello playing "z". From mm. 214-230 the motivic material is all "x" (except for the pedal). This motive coupled with the subdominant in mm. 214-220 further alerts the listener that the work will soon draw to a close.
Finally after a false entry in mm. 230-232 Beethoven composes the final entry (m. 232) stated with a firm fortissimo in both the cello and piano left hand. From mm. 235-240 the entry is extended by use of "y". The hemiolas add further excitement and drive to the final cadence. The V-I cadences in mm. 241 and 242 are really parts of "y", especially as it appeared in the third measure of subject #1 (m. 7).
IV. Pitch range
The pitch range is not to be overlooked in its dramatic contributions to this movement. The lowest pitch in the movement is an FF# occurring in m. 23, and the highest pitch of the piece is an a3 occurring in m. 218 and m. 240. It is also interesting to note that the golden section (.618) of this range turns out to be a1, or A440, the so-called "concert a". This is important in that the piece is in D major and the pitch A is the fifth scale degree in a D major scale thus invoking a fifth relationship.
As we have seen, in m. 137 the subject is breaking down and Beethoven is pounding out sforzandi on his way to a great turn-around in the piece (in F# major). Here we find that Beethoven uses one of the widest ranges (FF#-f#3) of the work to complement the drama of the harmony, motivic treatment of the first subject, etc. By way of contrast, in the section immediately following where Beethoven introduces the comparatively simple (harmonically, rhythmically) second subject the range stays much narrower for quite some time. This augments the restful feeling, at least up until the reintroduction of the first subject (mm. 143-153) the range is G-f#2.
V. Harmony�Excursions Into the Exotic
In general, the drama of Beethoven's harmonic technique comes from his rapid-fire exploration of remote areas and dramatic changes of harmony. The overall picture is that of rapidly changing areas of harmony (mm. 1-142) followed by a section of searching for the tonic harmony with the aid of two pedal points on an A and D, respectively (mm. 143-end).
In m. 63 Beethoven gives us the first real break from D major, the tonic key of this movement. One has traveled through areas of minor keys, but without cadences. For example, in m. 66 the music is in E minor, but by m. 75 the tonality is B minor. From mm. 84-88 it is difficult to know what key the music is in because of the numerous diminished seventh chords. A "taste" of G major is felt in m. 89, but by m. 94 the music is in B major, and again we are in G major by m. 102. Beethoven is able to do this by "slipping" through these areas without having to cadence in them. This gives an unsettled feeling, heightening the drama.
In mm. 108-112 there is a highly chromatic descent in the cello line and left-hand piano line. The quarter-note displacement of the lines in the first three measures add even more to the unsettled feeling.
From mm. 113-139 the harmony is unstable. In mm. 113-121 the cello and piano line rises sequentially by step, and in m. 122 there is a hint of A major. By m. 132 the bass goes down to a GG# with the right hand and left hand in contrary motion. This again is the moment of highest drama. There is a thick texture, great range, increasing breakdown in the phrasing, and more accenting within a fortissimo. Finally in m. 139 the harmony "resolves" to F# major with the piano cascading down the keyboard in arpeggi.
In m. 143 there is a moment of rest with the introduction of subject #2, marked piano. It turns out the music is in B major so the previous F# major chord can be thought of as a dominant chord. Even though this is a moment of rest, it is still highly dramatic within the framework of the whole piece because of the clear establishment of this new key (not a transitory suggestion as before).
In m. 158 there is already a hint of the home key of D major with the entry of subject #2 in the cello and subject #1 in the piano-soprano. The goal of D major is strengthened in m. 169 with the grand entry of the subject #1 in the piano-soprano along with subject #2 in octaves in the piano-bass.
As was mentioned above, in m. 185 the entry of the AA trill in the piano-bass signifies the oncoming of the end of the movement. Still, in m. 195 Beethoven extends the wait with a temporary area of B minor. Then in m. 199 there is an entry of a D pedal in the piano left hand. Although the harmony isn't D major (from m. 202 it is probably best thought of as being in G minor, giving a feeling of a minor plagal inflection), all the music centers around this anchoring D pedal.
In m. 207, the music clearly is in D major. After a moment of subdominant (mm. 214-220) the cello adds an f# to the pedal D, strengthening and stabilizing the tonic tonality. From here to the end the tonic tonality is secure.
It is widely accepted that Beethoven is a master of form. Perhaps more important, however, is how skillfully Beethoven intermixes other compositional elements within his forms. The last movement of the Sonata Op. 102, Nr. 2 bears this out with a transcendentally masterful combination of harmonic, textural, motivic, rhythmic, and fugal techniques. These elements, elaborately integrated, work together to create great drama and variety. The genius of Beethoven in this movement, at least, turns out to be his attention to every detail of his composition.
Edited, 7 October 2002
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