Posts Tagged ‘harmonic structure’

Mahler, Symphony No. 6, third movement

Wednesday, June 16th, 2010

Eric Knechtges, a colleague at Northern Kentucky University, recently sent out a survery to university composers.  One of the questions was,  “3) Any advice for potential composition students concerning the college application process, and/or constructing a portfolio?”

Here’s my answer:

In our portfolios, I like to see three compositions which demonstrate the student’s stylistic preferences, ability to pursue a project to completion, and interest in various media.  In general, it is not necessary to include a large-ensemble piece, especially if performance recordings are available of smaller-scale works.  MIDI realizations can do more harm than good.  I would rather hear or see short-to-medium length pieces that demonstrate technical mastery of compositional skills such as motivic development, phrase and phrase group organization, variation technique, harmonic and rhythmic coherence and ability to pursue an idea to its conclusion in a fully-formed piece (with beginning, middle and end).  Submitted scores should have a professional appearance, with attention to the details and standard practices of manuscript or digital score preparation–dynamics, tempi, articulation.  There should be a clear sense that I am not looking at a “first draft,” and that significant effort has been put into revision and the “polishing” phase of work.

Perhaps these are merely my personal prejudices (particularly about making a score look good), but some of these traits are evident to me in the great music of the past.  One of these, motivic development, is the main idea behind the third movement of Mahler’s Sixth, and I want to explore that today.

When I’m teaching basic composition to my students, I always stress economy of material, because emphasizing a single motive or a small group of motives throughout a piece builds unity while also providing opportunities for variety.  Unity is essential because it makes the piece sound like itself and not like a string of melodies or harmonies.  Variety, however, is very important in most styles, because very few listeners want to hear a great deal of exact repetition.

Mahler has set this movement in the key of E-flat major, a key that is somewhat removed from the symphony’s key of A minor.  On closer inspection, though, it is the relative major of the parallel minor of the relative major of the home key (a minor to C major to C minor to E-flat major), so there is a relation here, although it’s somewhat tenuous.

The music begins with a theme, stated in the violins, that introduces much of the material with which Mahler concerns himself throughout the movement.  As Russell Mikkelson frequently states, composers are like bad poker players, because they show you their cards at the beginning of each hand.  In addition to the head-motive of this theme, with its distinctive sol-mi-sol rising and falling sixth, there are motives in the second half of the first full measure (motive a, four eighth-notes, descending by third, then by seconds) and the second half of measure 3 (motive b, a written-out “turn”).  In measure 8, the oboe presents a final important motive, motive c, a figure which alternatively rises falls and rises, with sixteenth-notes on the second half of each beat to give the impression of hesitancy.

The a motive reappears in the violin melody in m. 13, first implying a IV triad, then a borrowed iv on its repetition.  Immediately thereafter, the c motive appears in the violins and woodwinds, again as part of the melody.  In m. 16, the a motive reverses its earlier trick, outlining iv and then IV (the entire passage is constructed over a tonic pedal point).   Measures 20-27 present a fascinating woodwind accompaniment texture, based on the c motive and its inversion.  The melody is assigned to the English horn, and begins in m. 22 with an inversion of the head-motive of the first theme–a falling and rising fifth instead of the sixth from before.  The key of g minor is suggested here, but it does not last, with a return to E-flat major in the next section of music, beginning in m. 28 with a horn melody that incorporates all the important motive material so far.  In m. 31, Mahler extends the dissonant Db5 in the solo horn by two beats, requiring a 2/4 bar (m. 34) to put the next cadence on the downbeat.

There follows a chromatic passage (mm. 36-41) that appears to lead toward C major, but then at the last moment returns to E-flat.  The next passage is based solely on the motives (a and c) from the first theme, with the c motive dominating the music in mm. 42-52, with a making its appearances in mm. 45-46, again highlighting an alternating major-minor chord.  While the overt major-triad-turning-minor motive that has characterized the previous movements of the symphony does not appear in this third movement, there seem to be more sublte, buried echoes of it in this particular use of the a motive, which occurs several times.

Measures 53-56 present a fascinating common-tone modulation, where the pitch G changes from mi in E-flat major to me in E minor.  First the c motive and then the a motive introduce the “second theme,” this time in the horn.  As this theme dissolves (it never really becomes a full-fledged theme, but its certainly too long to be simply a motive), Mahler begins to expand upon the a motive–first in the clarinets by inversion and rhythmic displacement, then in the bass instruments by expanding the third into a fourth, allowing two repetitions of the motive to cover an octave (in m. 65).  In mm. 68-70, a chromatic sequence that maintains the contour of the a motive is heard against the c motive (modified) in the trumpet and oboe.

In mm. 76-77, an almost Baroque-sounding descending-fifths sequence appears–extremely familiar in Common Practice styles, but realtively rare in Mahler, who simply doesn’t seem to have harmonic rhythms that move this quickly.  In the following measure (m. 78) is an early appearance (although not the first, but the first significant one) of the a motive transformed by both retrograde motion (the third at the end instead of the beginning) and the displacement of the third note up an octave, putting dramatic leaps of a seventh and a tenth into the texture.  The c and then the a motives pull the music to the next key, E major, at m. 84.

A note to my students, a spectacular example of the technique known as “horn fifths” appears in m. 85, introducing a trumpet melody that relies on the c motive.  It seems that the tendency is for the c motive to be spun out into some variation of the a motive at many points in this piece, such as in mm. 89-92.  In mm. 95-99, the c motive, and then the a motive create a monophonic modulation (based on the diminished seventh chord) to return to the main theme and the home key.

Measure 100 and the following passage suggest a recapitulation, but Mahler has other plans in mind.  The last chord in m. 114 acts as an augmented sixth chord which points to C major (an interesting use of the augmented sixth to point to a tonic function instead of the dominant, in this case to a key a minor third below the original key).  All three motives (a, b and c) appear in this C-major section, which ends in an unprepared modulation to A major  (mm. 124-145, again, down a minor third).  In this section, Mahler employs the a motive in the bass with the c motive in the horns against a violin melody that reaches higher and higher, to a C#7.  In m. 137, A major turns to A minor, without a key signature, as the oboe gives the “second theme” material. 

A slightly less abrupt key change leads to C-sharp minor in m. 146, as the full orchestra begins to enter with with climactic material based on the a motive (at first, to m. 156 or so), then on a chromatic version of the c  motive, this time in eighth notes instead of the alternating eighth-sixteenth-sixteenth-rest pattern.  By m. 160, the music has abated, leaving music in B major based on the b motive, followed by the c motive to set up an entrance of the a motive on the Neapolitan chord (C major, m. 164).  The major-minor motive is impled by the a motive in mm. 167-8, again extending the phrase by two beats, which are then rectified by the other 2/4 bar in the movement, m. 171.

The a motive takes over the texture in m. 173, as the music returns to E-flat.  Then in m. 176, the descending fifths sequence appears in a moment that is reminiscent of nearly every Hollywood love theme.  A note on the scoring here–one of the interlocking voices here is given to the 1st and 2nd violins, and the other to the violas with the oboes and clarinets, and the effect is very strong (of course, it seems to require seven woodwinds to allow the violas to balance.

The remainder of the movement is coda material, dependent mostly on the a motive and some of its modifications.  Mahler’s use of dynamics in m. 188 allows an effective color change, and there is an itneresting use of rhythmic augmentation of the a motive (with octave displacement, and modified to suggest harmonic closure) in the flute in m. 196ff.  Overall, the tautness of this piece seems to outdo everything Mahler has presented so far.  Despite the sprawling length and scoring of this symphony, the motivic clarity allows it to be highly managable in a way that hasn’t always been the case in these works.

On, then, to the highly-charged, tense finale.  I hope to be able to concentrate on aspects of compositional structure rather than any supposed autobiographical content (a study of how much of this is authentic and how much simply mythological would be very interesting; one day, I hope to tackle Henri-Louis de la Grange’s massive biography of Mahler.  Until then, my biographical understanding of these pieces comes largely from Kurt Blaukopf’s shorter work).

A final note, I’ve recently become aware of a similar project to my own, done much better, I must say, and by a composer of vastly greater experience than myself.  Anyone reading this blog should head over to YouTube to see Don Freund’s videos analyzing Book I of Bach’s Well-Tempered Clavier.  Great stuff!

Mahler, Symphony No. 5, movement 4

Sunday, April 11th, 2010

This Adagietto captures, to my hearing, a version of Mozart’s Romanza design, appearing, for example, in that composer’s Concerto, K. 466 (D-minor).  Here is Mahler at his most reflective, most concentrated, with each note seemingly imbued with meaning.

Again, I’m somewhat confused by Mahler’s labelling of this symphony in C-sharp minor, as only the first movement is in that key.  This movement is in F major.  The Romanza plan calls for a basically ternary structure, with the outer sections being closely related to each other in the home key and a faster middle section in a contrasting key.  In this case, the sections divide at measures 39 and 72, with the middle section in G-flat major, the lowered second scale-degree.

Mahler’s melodic material in the first section is centered around two ideas, a three eighth-note anacrusis followed by a retardation.  At times, the anacrusis motive is augmented to three quarter-notes, as in the first celli in m. 10.  This second appearance of the theme leads to A minor in m. 19, which then pulls back to F major.  From here, the music builds to a climax on the dominant in m. 30.  The next few measures are “after-the-ending” music for the first section.

The middle section of this Romanza begins with the tempo indication “Fliessender,” in F major.  The introduction of E-flat starts to suggest that F is now the dominant instead of the tonic, and a deceptive resolution in m. 46 establishes the next key of G-flat major.  Where the first theme was centered on the tonic pitch, this second thematic material tends to descend from the dominant in something of an inversion of the original motive.  The register of the melody rises throughout this section, until a written key change to E major in m. 60.  This would appear to be a transposition of convenience, as it lasts only three measures before D major appears in m. 63.  D major is never fully established as the tonic, but the entire nine measures in this key are given over to a long dominant chord.  Instead of D major, the music shifts down another step to give C, the first note of the piece, and the dominant of the home key. 

In m. 72, the original music returns, giving the second “A” section of the Romanza form.  Mahler states this section in abridged form, with only one appearance of the first section theme and no modulation to A minor.  Measure 87 is roughly parallel to measure 23, but the approach to the dominant relies on V/V instead of the Neapolitan, and the climax of the movement in m. 95 employs the highest register of the violin.  I feel that the moment to which the movement has been building comes very late in the overall structure.  An obvious comparison is Samuel Barber’s Adagio (in its various incarnations), which seems to me to have a more proportional denouement.  This is not to say that Mahler’s music is ineffective in any way, but it is intriguing to see to very different approaches to much the same musical idea.

The fortissimo lasts until m. 100, whereupon a final suspension brings the music to the expected tonic chord, strangely, strangely voiced without the middle strings (although in a wondrously sonorous open voicing that would get a good mark from me on on orchestration assignment).

What then is the compositional lesson one can take from this movement?  I hope that I can learn from Mahler’s approach to tension and release, to slow unfolding, to harmonic variety within tonal coherence.

This leaves the rest of the month for the final movement.  I’ve given myself a little extra time, as my wife and I are anticipating the birth of our son–if my posts become less frequent, that may be the reason, but I’m going to try to keep to my schedule.

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.