Posts Tagged ‘invertible counterpoint’

Mahler, Symphony No. 6, 2nd movement

Tuesday, June 1st, 2010

I keep thinking of non-Mahler topics I would like to tackle here, but things have been busy.  I have some time over the next few weeks, so perhaps they will pop up, but for now, here are some observations on the Scherzo from the Sixth Symphony.

The  first time I ever heard this piece, in April 1995, as performed by the Cincinnati Symphony, I heard the Scherzo as a sort of reimagining of the first movement.  I feel less and less that this is true, but the opening bars of each bear a striking similarity with their pedal A and melodic figures that rise toward the meat of the piece–a Schenkerian inital ascent, as it were.

What is really interesting about the first section of the Scherzo is that it seems to be related to a device that Mozart and Hadyn used from time to time in their menuetto movements–the spot that later composers used for the Scherzo.  In a few of their minuets, Mozart and Hadyn employ a strict canonic construction, and if Mahler’s use of canon isn’t strict, it is at least suggested–very clearly in places like mm. 7-9, in which motives are repeated directly, and in Mahler’s use of invertible counterpoint.  It is, really, the same old trick that Zarlino teaches–using invertible counterpoint, write two sections of music at the same time.  Again, Mahler isn’t strict, but his motivic choices allow him to layer and relayer his material.

Orchestrationally, there is a great deal of sort of “standard” writing, with mixed scoring that is effective, but not particularly colorful.  Lutoslawski, with his single movement symphonic plans, criticized the Romantic composers for making two large statements in their symphonies–typically the first and last movements.  He had Brahms in mind, but surely Mahler is no less guilty, if not more so.  In the Sixth, the last movement is by far the most significant, with the first movement probably next so, if not least for beign the most memorable.  Where, then, does that leave this piece, the middle child?

In constructing a piece of this length, is it possible to fully engage the audience for the complete duration of the symphony?  It is difficult to imagine the audience not becoming slightly fidgety at some point.   In Shakespeare, there is frequently a pause in the dramatic arc at the beginning of the last act–some ceremony, or comic relief.  In the same way, Mahler has moments of intense drama that are contrasted with moments of thoughtfulness and repose–even, moments that are simply “vamp” that have us waiting patiently for a scene change or to let us relax.  Is it lazy to think of Mahler in this way?  He was a man, not a god.

This movement spends a great deal of time on the subdominant of its various keys, for example, in m. 44ff.  There is also a fair amount of sequential motion, although generally up or down by second.  This aids in getting to more remote keys, as at m. 62, which sees a modulation to C-minor.

The concept of key is beginning to feel a little stretched in some places, as in the long “D-major” section beginning in m. 273, which never arrives at a tonic chord (although, characteristically for this movement, it lands on the subdominant in m. 299).  At the same time, there are more meter changes in this movement than in any of Mahler’s work so far.  While the outer sections are somewhat canonic in structure, the frequent meter changes disrupt this by throwing a simple-meter wrench into a compound-meter machine.

The major-minor motto of this piece makes its appearance at some of the crucial formal junctures, but most importantly in the coda, beginning at m. 419.  The harmony moves down by step, with AM-am, GM-gm, FM-fm in the trumpets and flutes.  The motto returns again in A, and is repeated several times against motivic material from this movement. 

Berlioz and Tchaikovsky brought such motives into their symphonic writing; in a way, Mahler’s concept of the symphony owes a great deal to Symphonie Fantastique.  Mahler has been self-referential before, but this is the first instance of a “motto” in any of his symphonies, and so there can be little wonder about the attachment of such importance to it by musicologists.  As a composer, though, I am more interested in the musical effect–what does the listener with no knowledge of Mahler’s biography or any explicit or implicit “program” to the symphony make of this device?  It is a unifying element, certainly, but its application seems slightly ham-handed at times.  The motive itself, as I mentioned in my previous post, is clear and direct, and distinctly unconventional–a relatively rare occurence in tonal music.  Could Mahler have dealt with it in a way that is not so obvious?

Another month with this symphony, then, so another month to ponder such questions.

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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 111

Sunday, June 28th, 2009

Here it is… the last one. 

Two big, beefy substantial movements.  Lutoslawski justified writing one-movement symphonies by saying that Brahms’ and Beethovens’ symphonies tended toward two big-idea statements per piece, presumably the first and last movements, although it is often possible that Beethoven is trying for three or four (perhaps in the Eroica).  It would be impossible to accuse Beethoven of overreaching his grasp in this case.  The two movements are well-balanced–a muscular, decisive sonata-allegro paired with an expansive set of variations. 

First things first–the proportions of the first movement are not especially large or striking–in my (G.Schirmer) edition, the development section scarcely lasts a page.  Once again, Beethoven is not the composer of long, overwhelming development sections the way we were all taught.  A glance at the score suggests that the proportional model for sonata-allegro is largely intact.   Why do we teach undergraduates that Beethoven’s development sections are overgrown?  My experience with the piano sonatas suggests that they are not.  On the other hand, motivic development technique often appears in unexpected places–codas, transitional sections, and within themes–places that in Haydn or Mozart would be simple or sequential repetition in Beethoven are more fully ornamented.  An example is the second theme of this movement.

I have to admire Beethoven’s approach to the start of the Allegro con brio.  It is almost as though it takes three (or more) attempts to get the theme going, and the full theme doesn’t appear until after a fairly extended attempt.   There is wonderful invertible counterpoint in the transitional thematic area, and the ubiquitous fugato in the development.  Beethoven struggled in his counterpoint lessons with Albrechtsberger, but they seem to have paid off in the end, as his command of these devices is perfect.  I taught 16th-century counterpoint last semester, and we didn’t make it to invertible counterpoint.  I think that the next time around, I will take the option in our textbook (Peter Schubert’s Modal Counterpoint, Renaissance Style) to introduce it from the beginning, because of its power as a developmental tool in any style.

Stylistically, I’m a bit at odds with this movement–it doesn’t reek of Beethoven’s “late” style in the way that other pieces do.  Admittedly, I haven’t read up on current musicological ideas about this piece, but it seems as though it would fit fairly well with the Waldstein, and lacks the scope of Hammerklavier.  Note–this in no way detracts from my astonishment with this piece and my awe at its compositional greatness.

The theme and variations is masterful as well, despite some very interesting notational choices.  The tone called for by the first few notes is wonderfully dark and rich.  Finally, Beethoven has stopped writing full triads in the bass staff, an activity I am constantly telling my students to avoid.  The more open chord positions he chooses in the theme are dark but not muddy.  Has this composer finally come to terms with the more resonant instruments that were starting to become available to him?  What does it mean that, despite his deafness, he was able to figure this out?  More importantly, what does it tell the contemporary composer who must assimilate much greater and more frequent changes in technology that Beethoven could have imagined?

There is a wonderful sort of rhythmic accelerando amongst these variations.  The theme gives a basic compound-triple approach with homophonic chords.   Variation 1 now has an event on every division of the beat, and events are happening (roughly) two to three times as often.  Variation 2 is simply not in the correct meter.  6/16 implies two beats to the measure, and there are clearly three.  3/8 would make sense, if it weren’t for the marked metric modulation (eighth=dotted eighth) and/or the alternating 16th-32nd-note pattern that makes up the highest rhythmic level (highest in the Schenkerian sense of “most-complex”).  What appear as accompanying 16ths or eighths should be dotted notes… or the alternating 16th-32nd patterns should be under sextuplets… or the patterns should be dotted-32nd-64th!  What a mess!  I can only assume that in later editions to which I don’t have access, some wise editor has made a decision that clears this up.  On my reference recording, Ashkenazy plays the first and second options, at least to my ear.  The editors of my edition, Hans von Bulow and Sigmund Lebert chose to only comment on the situation rather than rectify it.

In variation 3 is another meter signature that would make my students cringe–12/32, again, not reflective of the triple-meter feel of the music.  What a mess, but the musical intent is clear enough.  The final four measures of this variation are wonderful.

In my own work, I need to accomplish what Beethoven does in the fourth and fifth variations–that is, build larger sections of single textures.  I feel like I accomplished this in several recent pieces, notably in South Africa.  It is, again, the old adage I’ve often told myself of letting the music breathe.  I have great admiration for my friend David Morneau and his cultivation of the miniature, especially in his project 60×365, but I feel that I need to cultivate a different approach.  Yes, brevity is the soul of wit, but our world is deprived of the long view, the long term and patience to understand them.  Film may be our best hope–I know so few people who really listen to music, but nearly all Americans shell out for multi-hour long movies.  All the same, music that is longer than three minutes and that doesn’t make its meaning purely through language is, I am discovering now more than ever, my big project for the time being.  As a composer, I need to be able to write a single movement that lasts 20 minutes while still saying something.  I don’t know where the commission, or even the performers will come from for this, because for the time being I’m not in the class of composers who get that type of work.  When I entered graduate school in 2004, I was writing movements of one-to-two minutes’ length on a regular basis, and a five-minute one-movement instrumental piece was a stretch.  I discovered the tactic of creating larger pieces by writing transitions–my Martian Dances is a fantastic example of this, and my Homo sapiens trombonensis has a fantastia-like form that is exciting, but lacks rigor and cohesiveness.  Nothing ever comes back.  I learned how to let a piece breathe and expand to its true length rather than simply become a rush of ideas.  Beethoven’s sonatas–indeed, the sonata principle–require that I build on this even more.  I need, simply, the right commission now, because a twenty-minute unaccompanied trombone piece just doesn’t seem like a good idea.  A string quartet, or a piano sonata.  My latest completed piece, my Piano Trio that I just shipped off to its commissioner, runs almost ten minutes in a single movement.  I’m getting there… I’m getting there.

I began my journey through Beethoven’s 32 piano sonatas in November 2006 as a way to start a project that looked beyong the end of my graduate work, and I feel that I have done myself a great service–so much so that July 2009 marks the beginning of a new project on the Mahler symphonies.  I kicked around some different possibilities–Bach, Chopin, a single large work like the St. Matthew Passion or a Mozart opera, but it seems that Mahler is calling to me the most, so it will be half of a Mahler symphony each month until the end of 2010 (yes, I may decide to include other Mahler such as the 10th symphony or Das Lied von der Erde, but I’ll think about that later).   Please feel free to join me on that trip.

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Opus 109

Thursday, April 30th, 2009

Back to the schedule at last–it’s the last day of the month, and I’ve actually been around this sonata enough for a change.  May and June should be better, since the semester is ending.

After the massivness of “Hammerklavier’s” approach, this little gem in E major just blows me away.  It strikes me that what Beethoven is really doing in the first two movements is preludizing, and that the meat of this sonata is in the set of variations of the last movement.

I will have to dig deeper some day and do some research on the first movement, because there are aspects of it that suggest to me that it is also a variation on the theme from the last movement.  Beethoven isn’t the only composer to have put variations before theme, but I’m not aware of an earlier instance.  This movment is related formally to the second variation (Leggieramente) in the last movement, although the first movment features an additional reprise of the opening material.  The textures of the opening sections also seem to parallel each other.  As a non-pianist, I find myself thinking linearly in much of my instrumental writing, while Beethoven (and other great composers for the piano) are able to draw melody from texture in ways that I often don’t initially perceive by a glance at the score.  This is really the point of this survey of Beethoven’s piano sonatas–to help me understand the approach to composition of a man with whom I believe I share some stylistic traits, but whose life as a musician was completely different than mine.  Another way I heard the first movement is an an extended cadenza or fantasia, much like the beginning of the Choral Fantasy.  This only extends as far as the character of the piece, of course, because a true fantasia would probably not bear so much repetition.

The second movement, Prestissimo, brings to mind some of Beethoven’s bagatelles in both character and design.  I’m thinking particularly of the Opus 119 set in this instance.  It also has the feel of a prelude, and I’m beginning to wonder if this sonata isn’t purely a set of preludes.  More on that in the third movement.

I’m always a little taken aback when I see the title “variations,” because it inevitably brings to mind lightweight, virtuosic pieces by Rossini or Weber for clarinet.  The variation form is, of course, much richer than this, and I wouldn’t trade Bach’s Goldberg Variations for anything.  Beethoven’s Diabelli set is a close second, and he certainly knew what he was doing in this form.  I have also used variation form on occasion, and my first published piece, due out this summer, is a set of Variations on a French Carol for concert band.

I’ve decided that this set of variations–masterful, of course–continues the series of preludes in this sonata.  I’m thinking of the preludes of Bach or Chopin, which explore a texture to its fullest degree.  Some of these variations have very clear parallels in the Well-Tempered Clavier.

Theme–a homophonic chorale, in binary form, with an augmented-sixth chord placed ahead of the half-cadence.

Variation 1–a slow waltz or landler?  If the tempo marking were removed, it could look like Chopin.

Variation 2–I’ve mentioned the similarities to the first movement.

Variation 3–Ingenious use of invertible counterpoint… he only had to write half the variation.  In this sense, some similarities to WTC I, C# major prelude.  The texture is related to that of WTC 1, D major prelude.

Variation 4–This sort of counterpoint is almost a cliche of Bach’s style, but WTC 1, Eb major and A minor preludes come the closest, with G# minor not far behind.  One of my teachers, Gregory Proctor, mentioned Beethoven’s habit of opening a window, harmonically, letting the listener peek through it, and then abruptly drawing the curtains.  This happens at the end of the first section of this variation, where the German augmented-sixth chord is spelled enharmonically to resolve to F-major instead of to the expected dominant-seventh on B, but is immediately snapped back to the home key.  Beethoven is playing with equal temperament here in a way that Mozart or Haydn would never have dreamed of.

Variation V–I’ve studied the book by Beethoven’s counterpoint teacher, Albrechtsberger, and it’s clear that quite a bit rubbed off on his pupil.  This variation begins with a fantastic little canon in four parts, with entrances at the second.  The parts don’t all continue, but the effect is quite fun.    Again, Beethoven opens the window to F-major, but only lets us look out for a moment.  Bach’s Goldberg Variations make use of canon, so why should Beethoven not do the same?  There are similarities here to WTC 1, B-minor prelude in texture and form.

Variation VI–I am completely in awe of the compositional prowess on display here.  There is no parallel to this in Bach that I am aware of, but the idea of creating a sort of accelerando and building the tension through faster and faster note values is so simple as to be genius.  Absolutely fantastic.  Bringing the theme back at the end is a clear homage to the Goldberg Variations, in my opinion.

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Opus 101

Friday, January 30th, 2009

Another month and not nearly enough time spent with Beethoven.  Many composers tell me that their New Years resolutions are to get more score study in, but that it never seems to happen.  I guess I’m in that club, too.  With that in mind, I’m going to confine myself to what has become by favorite movement of Op. 101 this month:  the second, Lebhaft.

I suppose this movement falls into the Scherzo-and-Trio category, although it isn’t particularly schero-like in its character.  It has the ternary form that one expects, and some other very interesting aspects.  I’m going to skip over to the trio–the B-flat major section.  Canon is the name of the game here.  Hadyn and Mozart occasionally wrote minuet movements in their sonata cycles that were strictly canonic in construction, and Beethoven once again reveals himself to be a classical composer in outlook by doing the same thing.  This two-page trio is filled with interesting exercises in canon and invertible counterpoint.  There are no fewer than four canons–beginning in the 6th, 11th, 16th and 25th measures–and two uses of invertible counterpoint (the same material, appearing in the 3rd and 23rd measures). 

For all this, Beethoven still manages to make music.  There is both craft and art here, and one need not notice the canonic stucture to appreciate the good work that has gone on.  An especially interesting moment is in the second and third canonic sections, when, rather than the very static harmony often generated in this type of piece, Beethoven uses the canon to move, first, away from the home key, and then, back to it–from Bb to C by falling thirds, then through a funny little progression back to the dominant-function. 

Meanwhile, the economy of motive is staggeringly brilliant–only three or perhaps four motives account for the material of the trio, and they are mostly derived from the head-motive, which itself is derived from the material found in the march.  In the fall, I will be teaching form and analysis, and I can promise my students that they will be looking into this piece.

Next month is the big one–the piece that I has loomed over me since the start of this project.  The next sonata is No. 29, Opus 106, the “Hammerklavier.”

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