Posts Tagged ‘fugue’

Mahler, Symphony No. 5, fifth movement

Saturday, May 1st, 2010

This movement has gone, over the last weeks, from being a piece that I’ve long admired to something of an analytic enigma.  Simply put, I am at odds to determine whether this “Rondo-Finale” is best considered as a rondo, a sonata-allegro or a fugue.

First, to the title, if that can be a clue for the analyst.  Does “Rondo-Finale” suggest “rondo-as-finale” or “rondo-then-finale?”  My hearing suggests that there is indeed a rondo here, and that it is followed by a lengthy coda, so that the second possibility seems stronger.  In this case, the coda could perhaps begin in m. 581, at a key change to A-flat major, far-removed from the home key of D major.  The melodic and motivic material is related to the rondo theme (mm. 24-55), but this late harmonic move away from the home key suggests a coda.

The music up to this point is highly suggestive of rondo technique, specifically of five-part rondo with its three statements of the rondo theme with interspersed refrains.  The second refrain (beginning in m. 167, the “C” of “ABACA”) is the longest, and is heavily reliant on developmental techniques, especially exploration of remote key areas and contrapuntal recombination of motivic material.

It is, however, the first refrain (beginning at m. 56, the “B” section) that is most striking.  It suggests a four-part fugal exposition, first with a running eighth-note subject, then with various countersubjects introduced over the eighty bars of this section before the return of the rondo theme.  The second refrain can then be cast as a continuation of the fugue.  At m. 273, the original fugue subject appears in counterpoint with one of the countersubjects.  This countersubject becomes the second subject of a double fugue that dominates much of the rest of the second refrain (development).  Contrapuntal technique abounds, with the inversion of the second subject appearing in the violins at m. 457.

The final entrance of the rondo theme appears in a highly modified form at m. 497, leading not back to the beginning, then, but toward the “finale” section of the movement.

There are also intimations here of sonata-allegro, or at least something along the lines of a hybrid sonata-rondo, as found in another wonderfully contrapuntal work, the finale of Robert Schumann’s Piano Quintet, Op. 44.  The first rondo and first refrain would correspond to a sonata exposition, the second rondo and second refrain to the development, and the last refrain and “finale” to the recapitulation and coda.

It is a fascinating feature of this movement that it not only is a highly compelling piece of music, but that it also embodies these three formal procedures .

A final issue with this symphony is its harmonic plan.  The home keys of the five movements are, in order, C-sharp minor, A minor, D major, F major and D major.  Mahler uses this narrative tonality in other places.  If Mahler had in mind a major-key finale to a minor-key symphony, in the manner of Beethoven’s Fifth and Ninth Symphonies, perhaps he felt that the technical challenges involved in C-sharp major might be too much for his orchestra, given the already stiff demands of the music.  At any rate, it is also simply possible that Mahler is moving away from the single-key concept of a symphony.  The five movements appear appear to be held together from a motivic standpoint, rather than from harmonic consistency, but in a traditional sense, they are no more related than a suite of pieces extracted from an opera or ballet.  It is a testament to Mahler’s compositional technique that the piece feels completely unified without sharing a common key center.

Now on to the Sixth–four large movements, so two weeks each.

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Bach, WTC I, Fugue in E minor

Friday, November 20th, 2009

This post is for a friend of mine, who is trying to perfect the art of wooing women using his music theory and analysis skills. In the manner of Cyrano de Bergerac, however, he needs a little assistance. But at least he doesn’t have a big nose.

My comments refer to the Kalmus edition of Bach’s The Well-Tempered Clavier, edited by Carl Czerny.  Not the best edition ever, but fairly readable, as they don’t try to cram too much onto a page like some older editions.  I trust it fairly implicitly.

This is the only two-voice fugue in either book of the WTC.  The structure is more reminiscent of a two-part invention than a fugue, but the line between these is somewhat blurry.  The subject has a strange chromatic feel to it following the rising arpeggio that forms the head motive.  It modulates–once the subject has been played, the music is not in the same key it was when the subject began–the subject always ends on v of the key in which it began.  The first entry of the subject takes the piece from E minor to B minor (note D natural, not D sharp on the downbeat of measure 3).  The answer begins immediately, as expected.  This is a real answer, with the exception of the last note–where the subject always ends with a descending whole-step, the answer always ends with a descending half-step, meaning that after the answer, we may remain in the same key.  The half-step allows the subject to end on V (of the answer) rather than some temporary i.  Measures 1-4 are the first exposition.  A final feature of note is the material beginning on beat 2 of m. 3 in the soprano voice, which represents recurring contrapuntal material and accompanies every subsequent entry of either subject or answer.

The first episode lasts from m. 5 to m. 10, and consists of a descending-fifths sequence, with Bach’s very typical alternation of material between hands, until m. 10, when the sequence breaks down in order to tonicize the next key area, G major.  Between measures 10 and 11, there is almost a cadence–we are denied the stereotypical Baroque cadential 6-4 pattern, but harmonically, most of the pieces are there.  In particular, the last beat of measure 10 is very typical of the approach to a cadence, but the next passage begins on a first inversion chord (because the subject begins on the root of the chord and enters in the soprano). 

The second exposition lasts from m. 11 to m. 14 and presents subject and answer in G major, with both accompanied by the countersubject introduced in mm. 3-4.  This is an excellent example of Bachian use of invertible counterpoint, as the countersubject “works” either above or below the subject or answer.

Measures 15-19 constitute the second episode, again, a falling-fifths sequence until m. 19, which introduces the dominant of the next key area.  Measure 19 is also notable for the use of both hands in octaves, a practice not often seen in Bach.  Bach again uses inversion in this episode–note the alternating first-inversion and root position chords.

The third exposition is in measures 20-23.  This is the first time that the subject and answer have switch places, so that the subject appears in the bass and the answer appears in the soprano.  This exposition is in the key of a minor.  Because the subject always moves to v, the answer is in e minor, ending on V (B major).

The third episode lasts from measure 24 to measure 29.  The first four of these bars are measures 5-9 in inversion, another example of invertible counterpoint.  Like the first episode, they constitute a descending-fifths sequence.  The parallel construction would seem to suggest that C major be the next key area, but this is prevented by a minor change in m. 29.  Between the second and third sixteenth-note of the run in thirds here (inversion of the run in sixths from m. 10), Bach leaps up by a fourth instead of a third, then continues a step higher to tonicize D minor instead of C major.  Like its counterpart, m. 29 is almost a cadence.

D minor is not a closely-related key to E minor, and our arrival here requires explanation; D major or C major would be more harmonically typical.  I would suggest that, given the tight construction of this piece, Bach wished to continue in this manner.  If Bach had continued to C major, as the parallel construction would suggest, he would have found himself nearing the end of the fugue in a poor situation.  To continue the parallelism from C major, would require a move to D minor (following pattern of the second and third expositions in G major and A minor, respectively).  He finds himself moving further and further from home, when it is now necessary to head back toward the tonic.  Thus, C major was not an acceptable choice for the next exposition, as it leads away from rather than back to the home key.  D major, on the other hand, would require an answer in A major, similarly leading away from the home key.  However, since the subject modulates to its own v, a subject in D minor leads to an answer in A minor, the iv chord of the piece.  Thus, by stepping out of the key, Bach brings us back to the key.  This odd entrance of the subject (m. 30) is necessitated by the modulating properties of the subject itself, and the exposition leaves us in a very good position to end the piece.

The final episode, mm. 34-38, is an inverted version of mm. 15-19, complete with the hands in octaves (m. 38).

In m. 39, Bach uses the head motive and the characteristic “bariolage” section of the subject to suggest stretto.  Since neither the subject nor the medium of two-voice fugue are really well-suited to stretto, he reverts to the format of the two-part invention and merely reminds us of the subject.  The only true cadence in the piece occurs in the last measure.

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Opus 110

Saturday, May 30th, 2009

Here’s the May 2009 installment of my series of posts on the Beethoven Piano Sonatas.  This month is Sonata No. 31 in A-flat, Op. 110–next month will be the last month in the cycle, which means I will need a new analysis project–let me know if you want to start one with me and dialog on the compositional aspects of pieces from the standard repertoire.  I could, of course, spend another few years going back over the Beethoven Piano Sonatas, but there is so much great music out there that I’ve never even touched, that I feel like it would be better for me to move on.  So… I haven’t decided on my next project yet, but I do have some ideas… if one or more people were interested in working through some pieces with me, I would let them have some input in the decision.  I’ve considered the Mahler symphonies, Bach’s St. Matthew Passion, Beethoven’s string quartets, Chopin’s Preludes… let me know what you think!

On to the piece:  A study in contrast this one, and highly indicative of the “official” traits of Beethoven’s late style as it has been taught to me.  I’ll dive right in.

The first movement, if not in textbook sonata form, at least seems to reference it.  I’m not Donald Tovey, who looked for sonata form in every piece he ever analyzed (the last movement of Schumann’s Piano Quintet is a Rondo with sonata aspects, not a straight sonata-allegro), but it seems reasonable to assert that Beethoven is working with thematic groups and a strong sense of motivic unity.  His use of core technique is somewhat fascinating, as it is built on a descending thirds sequence instead of the usual stepwise sequence.  I’m puzzled by the modulation to E major in what corresponds to the recapitulation.  This isn’t Beethoven opening up a window to another tonal world but rather knocking out a wall–a very unexpected place, although it makes sense that something different needs to happen where the exposition modulated to E-flat (the modulatory technique to E is an enharmonic respelling of a borrowed chord… IV becomes iv, which is vi in the new key; Beethoven gets out of that key by a fascinating use of common-tone technique and sequence).

To understand Beethoven’s use of sequence is often to gain understanding of his medium-scale structure (and in some cases, large scale, as in the “Spring” Sonata).  In Las Cruces last week, I spoke with Fred Bugbee about NMSU’s music theory track, and eventually the conversation came around to sequences.  One reason I’ve decided to part company with my current theory textbook, Kotska & Payne’s Tonal Harmony is that their treatment of sequences simply lacks body.  The new generation of theory textbooks is much more realistic about the use of sequence in tonal music, and, truthfully, it was teaching from Clendinning & Marvin’s The Musician’s Guide to Theory and Analysis that really got across how important sequence is.  My study of Beethoven has only reinforced that.

The second movement, Allegro molto is diminutive in proportion, but as will all Beethoven’s scherzi, I am amazed at the sheer craft involved.  Every time I make the mistake of listening to a Classical or Romantic scherzo as merely a light, intermezzo sort of movement, I tend to realize that I’m not doing it justice.  With some composers, it’s an easier mistake to make than with others.  Much of Mendelssohn’s genius seems to lie in his scherzi, for example, while Dvorak has a tendency to revert to folk dances.  Nearly every time I look closely at a scherzo, however, I see a level of compositional craft that equals the outer movements.  It is as though composers were freed from the strictures of sonata-allegro or rondo (although most rondos have wonderfully original moments) and could pull out the tricks they worked on as students–canon, invertible counterpoint, rhythmic surprises, and the works.  What fun!  Beethoven doesn’t use contrapuntal tricks, but in this tiny scherzo, he gives us the most rhythmically ingenious and formally cogent plan of the piece.  Why should this tiny movement have a coda when the first movement has none?  I suspect it is more necessary here because we have heard the A-section twice, and the listener needs to have a fuller sense of closure than a simple cadence.

I could puzzle over the last movement for quite some time.  Here is Beethoven’s late-style interest in counterpoint (the fugue, complete with a second exposition in inversion), side-by-side with harmonic innovation (a common-tone diminished-seventh chord with a modulating function), and a confusion about rhythm and key signatures that simply doesn’t make sense at this point.  To wit:  for much of the piece, the key is A-flat minor, at least until the start of the fugue, but the expected seven-flat key signature never appears.  Instead, the movement begins in B-flat minor, shifts to E major and then is written in E-flat minor.  Are these key signatures simply flags of convenience?  At the same time, Beethoven indicates “Recitative,” and breaks out of the signified meter (common time).  How free is this meter?  And how, precisely, is the performer to understand the subsequent barlines?  The “Klagender Gesang” in 12/16 meter is another puzzling aspect–it is almost as though Beethoven is writing a fantasia, a written-out improvisation, at the end of which he launches into the fantastic three-voice fugue. 

Then this full-bodied G minor and G major review of earlier material–the “Klagender Gesang” in G-minor paralleling the A-flat minor section and the fugue (in inversion) in G-major (although we get only an exposition and a long episode).  At last, the retuirn (recapitulation?) of the fugue subject in the original key–part recapitulation, part coda, really. 

One more Beethoven sonata–I look forward to Ludwig’s valedictory effort in the genre.

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