Beating A Dead Horse:  An Analysis Of Formal and Motivic Significance In Martin Bresnick's Second String Quartet

Michael O'Brien

Matt Pickett

December 10, 1999

                 Although the inspiration for Martin Bresnick's second string quartet is literary, much of the actual composition itself is derived from mathematical processes.  The quartet, entitled "Bucephalus," alludes to the fabled war horse of Alexander the Great, as he appears both in Greek legend and within the Franz Kafka story "The New Advocate," which Bresnick uses as an inscription for the quartet.  Each of the movements is named thematically and alludes to either the horse or his owner.  Unlike the program music of Liszt or Wagner, however, Bresnick's work does not depend on programmatic narrative in order to determine the formal structure or small-scale compositional method of the piece.  Instead, "Bucephalus" is a structure based on rigorous mathematical manipulation of basic ideas which, when expanded to create the large-scale form of the work, relate themselves thematically to the work's literary subjects.  There are two principal numerically based unifying features of the piece:  First, palindromic structures appear throughout the work, and are used to derive aspects of melody, rhythm, and, to a lesser extent, formal design.  Also, the number eleven develops a nearly ubiquitous motivic significance, determining among other considerations rhythm, pairings between voices, aspects of form, and harmony.  Bresnick fully exploits numerous ways in which the number eleven provides fodder for development.   The number lends itself quite easily to creating palindromic structures:  as an odd number, it can be divided into two identical sections around central midpoint; it can be asymmetrically divided into groups of seven and four, five and six, or other groupings, which are reversible; even the numeral itself is a palindrome.  These bases serve to unify the work, allowing it to stand alone as a musical structure independent of any programmatic consideration.   Nonetheless, both the numerical manipulations that drive the compositional process and the literary context in which the work is offered must be considered in order to gain an adequate understanding of the work's significance.

            The first movement of the quartet, entitled "Bucephalus," is perhaps the least strictly defined by underlying numerical concepts.  It is instead characterized by a free flowing, rhythmically complex and irregular style devoid of consistent thematic unity.  The rhythmic repetition and unity between voices found in the more structured later movements are, by comparison, conspicuously absent here.  This can be perhaps attributed to the fact that the movement, named "Bucephalus," is a musical representation of the horse in the wild, prior to his domestication at the hand of Alexander.  The rigid order imposed by mankind and domestication would seem out of place in a movement depicting the horse's untamed state. 

            The movement is not, however, without a certain underlying logic.  Many of the seemingly arbitrary rhythmic and melodic figures seem reminiscent of a horse's whinny.  Furthermore, a dramatic shift in mood, characterized by a louder dynamic, increasing rhythmic activity, and a shift from a Lydian modality centered on A to the parallel Phrygian mode occurs in the eleventh measure of the movement.  This choice of mode to characterize the change in mood is appropriate; Renaissance theorists describing the classical Greek assumptions regarding modal ethos characterize the Lydian mode as "sanguine" and claim that it "averts quarrels, calms agitation, [and] fosters peace," while Phrygian "moves to choler and biliousness" (Powers, 399). Due to the frequent use of complex and irrational rhythms and deliberate masking of meter, the fact that this change takes place in the eleventh bar would not be apparent to a listener without a score.  Nonetheless, it is only reasonable to assume, within the context of a work so numerically obsessed, that such an intimation of the number's coming significance is deliberate.  Furthermore, it is worth noting that dividing the movement in two at the eleventh bar forms two sections whose lengths correspond to the ratio known as the "Golden Proportion," a formal division identified by Fibonacci and used to great significance in works of other composers as diverse as Monteverdi and Bartók.  While the actual ratio was not discovered until the early thirteenth century, its associations and implications to a contemporary composer include not only the perfection of Nature (where it was first identified) but also the ordered and harmonious aesthetic of Classical antiquity (Stolba, 249).  Thus, its inclusion in a work thematically linked to both concepts is especially appropriate.

            The second movement, entitled "Around to the Sun," contrasts sharply with the relative freedom and spontaneity of the first.  An allusion to Alexander's taming of Bucephalus by turning him toward the sun so he could not see his own shadow, the movement introduces a rigid and systematic formal design to correspond with Bucephalus' own domestication.  The material for the majority of the movement is based on a series of six distinct pitch collections.  Each of these pitch collections consists of six notes, which form the first five notes of a major scale with an added raised scale degree four.  The numerical basis of these pitch collections (six notes contained within the span of a fifth) suggests a recurrence of the eleven motive.  A further unifying feature is the use of B-flat, which appears in each of the pitch collections.  B-flat plays an important motivic role in the movement; the piece begins on a unison B-flat in all the voices, and the palindromes introduced throughout the movement invariably begin and end on a B-flat as well.  B-flat not only appears in each of the pitch collections, it is in fact the basis for them; the six collections used are all the permutations of this intervallic relationship in which a B-flat occurs.  Each pitch collection is used exclusively for a section of the piece, and the sections where the pitch collection changes mark small-scale formal divisions. Nearly all of these smaller sections are based on melodic palindromes.  The rhythmic basis for most of the movement is derived from various manipulations of the number eleven. 

            The first section of the movement, which employs the pitch collection ranging from B-flat to F (mm. 1-8), develops entirely from the eleven-note melodic palindrome introduced in the first violin in the first measure.  The base rhythmic value for this figure is the sixteenth note, while the central note of the palindrome receives an agogic accent.  This elongation of the midpoint is consistent throughout the work, and has a literary basis as well.  The alexandrine (which is alluded to in the third movement's title) is a metric scheme for a line of poetry named for the tradition of poems about Alexander the Great which first employed it.  It consists of six iambs, typically divided at the midpoint by a caesura, which serves the same function as Bresnick's rhythmic elongation.  For the first presentation of each palindrome, Bresnick further emphasizes the midpoint with a forte dynamic bordered by symmetrical crescendi and diminuendi (mm. 1-2). 

The first violin's eleven-note palindrome is the basic material upon which the palindromes in all the other voices are based.  The texture is thick and complex, for each voice is actually presenting two melodic palindromes with different base rhythmic values simultaneously.  While the first violin's first eleven-note palindrome lasts three beats and has the sixteenth note as its base rhythmic value, it also presents the same melodic palindrome one octave lower with a base rhythmic value of two and one half beats, or ten sixteenth notes.  Thus, the sum of the rhythmic values of the two simultaneous palindromes in the first violin is eleven sixteenth notes.  The other voices also each contain two palindromes whose rhythmic values add up to eleven sixteenth notes; the second violin's primary palindrome is composed of eighth notes (equivalent to two sixteenth notes) while the secondary palindrome's base rhythmic duration is nine sixteenth notes; the viola's two palindromes have rhythmic values of three and eight sixteenth notes each; the cello's similarly are based on values of four and seven sixteenth notes.  

            The melodic construction of each of the palindromes is determined at first by its position relative to the primary palindrome in the first violin.  Each voice changes at its predetermined rhythmic value so that the next note in its sequence is a unison (or octave) with the pitch being played in the first violin.  Thus, the second violin's second note, which occurs on the second eighth note of the measure, is an E-natural to correspond with the third sixteenth note in the first violin's palindrome.  The second violin is the only voice to have the complete construction of its palindrome determined by the first violin; it reaches its midpoint on the C which is the penultimate note in the first violin's primary palindrome, and the remainder is merely a reversal of the previously occurring information.  The viola's primary palindrome is nearly complete when the first violin's primary palindrome reaches completion, so its midpoint, D, is derived by default as it is the only note within the six-note series which has yet to be used.  As the first violin's primary palindrome has already reached completion, the second violin then assumes the role of the first violin in determining the remainder of the pitches in the cello's primary palindrome.  The melodic structure of the secondary palindromes is identical to that of the primary palindrome in the same voice, so that each voice in effect plays what Messiaen terms a "rhythm canon" with itself.  While the melodic structure of the first violin's palindrome is not derived from any other material in the movement, it is worth noting that the first four notes of the palindrome, B-flat, C, E, and F, reappear in the third movement as well.  Using the German spellings, these become BCEF, which may be a contracted musical spelling of "Bucephalus."  Furthermore, the intervallic relationships within the first palindrome that begins with this melodic cell are preserved each time a new sixteenth-note palindrome is introduced in later pitch series, although there is some transformation to allow for the range limits of each respective pitch series.

The second section of the movement employs the pitch series ranging from A-flat to E-flat (mm. 8-15).  The rhythmic construction used in this section is an exact retrograde of the previous section, such that the combination of the two sections forms one large rhythmic palindrome.  Furthermore, within the section the rhythmic bases of the two palindromes in each voice, as in the first section, have the same ratio and their sum forms a total of eleven sixteenth notes.  As a result of this palindromic structure, the first violin's melodic palindrome, from which the other voices' pitches are derived (mm. 14-15), appears at the end of the section and is constructed in retrograde.  As this palindrome's rhythmic base is the sixteenth note, it is also melodically derived from the series of intervals from the first violin's primary palindrome from the first pitch collection.

The third pitch collection (mm. 15-22) extends from E to B, and according to the pattern established by the other two sections, would include the raised fourth scale degree, A-sharp.  This expectation is somewhat thwarted, however; Bresnick instead enharmonically re-spells the note as B-flat in order to maintain the consistent motivic appearance of this note.  As in the previous sections, the first violin introduces an eleven-note palindrome in sixteenth notes, from which the other voices' palindromes are derived.  Also, the pitch relationships in the new sixteenth-note palindrome are consistent with those in the previous palindromes in the first violin.  This section does differ in several ways from the previous material, however.  The first violin repeats its entire palindromic melody in sixteenth notes before proceeding to new material. Also, while the secondary voice in each instrument is derived from the same melodic palindrome with a base rhythmic value which is its complement (forming a sum of eleven sixteenth notes), its stronger articulation brings the secondary voices more immediately to the foreground of the texture.  Since the distance between secondary voice articulations in the first and second violin, viola, and cello are ten, nine, eight, and seven sixteenth notes respectively, the juxtaposition of these cycles creates a series of accents whose distance increases by one sixteenth note per cycle.  The secondary voices of this section also differ from their predecessors in that they are not sustained, with the exception of the sixth note in each palindrome, recalling the agogic accent on the midpoint of palindromes throughout the work.  The force of the dynamic accents also decreases and the overall dynamic gradually gets softer as the section progresses.  The section ends with a unison B-flat in the two violins; the first violin is ending the section while the second violin begins the next section.  This bridge into the next section by use of a unison B-flat reflects the same technique used to connect all the previous sections.

This fourth section uses the pitch collection ranging from E-flat to B-flat (mm. 21-29).  Here for the first time the sixteenth-note palindrome that determines the pitches of the other palindromes is conspicuously absent.  Instead, each instrument has an independent palindrome, which is only related to the others by virtue of the fact that they all begin and end on B-flat.  Within each instrument, however, the dual-voice structure which presents the same pitch series in a different but complementary rhythm remains.  The individual voices seem to enter erratically and do not immediately appear related to each other, but convene at the end of the section (m. 29); in fact it reflects a backward construction much like the second section of the movement (mm. 8-15).  Similarly, the dynamic directions of the two sections are related, as they both undergo a gradual crescendo to a fortissimo dynamic on the concluding B-flat. 

The fifth section (mm. 29-35), like the section that immediately precedes it, lacks a sixteenth-note palindrome to determine the content of the others.  Instead, the base rhythmic values of each voice are maintained exactly from the section before.  The pitch series used in this section ranges from G-flat to D-flat.  Again, each instrument is functioning as two separate voices, each of which is a different, complementary rhythmic iteration of the same palindromic pitch series.  The primary voice in each instrument is more prevalent due to its higher register and fortissimo dynamic.  Both voices are characterized by short articulations, followed by rests to fill out the remainder of the rhythmic value.  The secondary voices remain mezzo forte until the primary palindromes have come to completion, and are preceded by a glissando from the note in the primary voice immediately before it. The only exception from these generally homogenous dynamics is the final B-flat of each palindrome, which is marked piano.  These terminal B-flats also disrupt the separated nature of the earlier articulations, as they are sustained until the end of the section, where they are articulated at intervals of eleven sixteenth notes, a motive which will be continued in the next section.

            The final pitch collection to be employed is that ranging between F and C, and lasts from measure 35 to measure 43.  For the first time in the movement, the dual-voice structure within each instrument disappears, and thinning the texture considerably. At the beginning of the section (mm. 35-40), a B-flat is articulated in one of the voices every eleven sixteenth notes, as in the end of the previous section.  The numerical implications of this material extend further, as (including the articulations from the previous movement) there are a total of eleven of these articulations. Except for this consistency, however, the section is significantly less ordered than all the previous sections.  It lacks not only the order of the pitch-generating sixteenth-note palindrome that disappeared in the fourth section, but the rhythmic regularity and complimentary pairings to generate groupings of eleven sixteenth notes, and even the palindromic pitch series.  The only unifying and ordered aspect of the section that is retained from previous sections is that it is comprised strictly of the last available pitch collection that contains a B-flat.

There is, however, within the more simplistic organization of the section, a new factor that does help to organize the less formally restricted nature resulting from the lack of rhythmic and palindromic regularity.  The voices are now paired not with another voice in the same instrumental line, but with another instrument; the first violin and the cello are usually playing octave doublings of the same material, while the second violin and viola are similarly paired.  This doubling of the registrally highest and lowest voices versus the inner two voices creates yet another structure that is symmetrical and palindromic.  It is the first violin and cello that first depart (m. 37) from the unison B-flat that begins the section.  In the following two measures, these voices undergo a series of one-note departures from and returns to the B-flat.  The collection of notes to which they briefly deviate eventually forms the first six notes of a pitch series that will ultimately occur as a palindrome (m. 43).  Meanwhile, the viola and second violin are not only paired, but evolve so as to coincide more closely with the violin and cello to the point where, immediately prior to the end of the section, they are playing nearly the same material (mm. 41-42).  Finally, after all the necessary notes to derive an eleven-note palindrome have been presented individually in the first violin and cello, an eleven-note palindrome beginning on a B-flat in the outer voices which preserves the same pitch relationships as the earlier sixteenth-note palindromes (presented simultaneously with a metrical shift to 11/16) emerges as the first signs of a return of rigorous order.  If the palindrome were to reach completion, it would end on a unison B-flat that would mark the end of the section and, since this would exhaust the possible permutations of pitch collections containing B-flat, presumably bring the movement to a close.  The anticipated B-flat, however, never arrives.  Instead an abrupt transformation occurs, introducing new material that is unrelated to anything yet presented in the movement. 

The sections that follow are characterized by a frequent juxtaposition of accented rhythmic ostinati of different lengths in all the voices, causing a complex, constantly evolving phase of interaction between the accents.  Furthermore, the B-flat which has been to this point ubiquitous is entirely avoided (in fact, B-natural is also avoided because of its close association with B-flat), and the varied palindromes and pitch collections of the first large section of the piece are replaced by one uniform eleven-note palindrome appearing in all the voices.  This abrupt stylistic contrast marks measure 43 as the most significant formal dividing point in the movement.  It occurs, like the corresponding point in the first movement, at the point that divides the movement into sections corresponding to the "Golden Proportion."

The pitch collection Bresnick uses in this section is effectively the Lydian mode centered on D.  While this differs significantly from the pitch collections employed earlier in the movement, the section is not without connection to earlier melodic material; the order of the pitch series which forms the palindrome used in this section corresponds intervallically with the numerical system used to determine the pitch order in previous sixteenth-note palindromes (mm. 1, 14-15).  The section also recalls earlier sections' treatment of rhythm as a manifestation of the eleven motive.  While in earlier sections this is achieved by the duration of the base rhythmic value for each voice, here all voices articulate each sixteenth note, but the pattern of accents in each voice indicates the numerological differences between the voices.  Every fifth sixteenth note receives an accent in the first violin, every sixth in the second violin, every seventh in the viola, and every fourth in the cello.  In each voice, the number of sixteenths between accents will change when the pattern has cycled through completely once and the accent falls again on the downbeat.  At this transformation, the number of notes between accents switches to its complement (such that the combined rhythmic value of the two patterns is eleven sixteenth notes).  Each instrument completes one complete cycle for each of its two rhythmic complements before all four voices complete their cycles simultaneously, and the accents uniformly coincide on the downbeat of measure 55.  At this point, the numerical assignments of duration between accents are completely changed, such that they fall every eight sixteenth notes in the first violin, every three in the second violin, every two in the viola, and every nine in the cello.  The voices go through the same process as before using the new rhythmic values, and finally coincide again to conclude the section in measure 66.  It is worth noting that this section begins in measure 44, undergoes its most significant transformation in bar 55, and concludes in bar 66 - all multiples of eleven. 

In the final section of the movement (mm. 66-72) the eleven-note Lydian palindrome which has been employed exclusively for the entirety of the last section is transformed and retained only by the first violin.  At the same time, the remaining three voices present a rhythmically synchronized accompanimental figure based on a number of non-triadic chords consisting of pitches from the D Aeolian scale.  The first violin repeats an eleven-note palindrome accenting every tenth note with a change in both dynamic and register. Thus, each note of the palindromic pitch series is accented in turn.  After the final note of the series has received the accent, the pitch is sustained and an E-flat is introduced in the cello.  The final measure of the movement is connective material that will reappear, slightly transformed, in the beginning of the third movement.  The doubling of the inner and outer voices appears again, and the melodic information in the first violin and cello is derived from the palindrome which comprises all of the material used between measures 44 and 65.

The third movement, entitled "Alexandrine," is not as concerned with palindromic structures as the preceding movement.  It does, however, make frequent and varied use of the eleven motive, particularly in its rhythmic ramifications.  It also reintroduces, both horizontally and vertically, the B-C-E-F motive from the beginning of the second movement, and, through a series of alterations of pitch collection, transforms it into a tone cluster and ultimately, through the integration of long glissandi, anticipates the microtonality of the fourth movement. 

The first measure of the movement is an alteration of the last measure of the previous movement, only the doublings have switched so that the violins are paired in unison, as are the viola and the cello. While there are transitional elements shared between all of the movements, this is the only pair of movements that contain an exact replication of the same material.  The second measure contains the first appearance of the B-C-E-F motive in this movement; each of the constituent notes appears in each of the six articulations in the measure, although the voicing changes several times.  The symmetrical and palindromic pairing of the inner and outer voices established in the second movement also returns here; while the voices do not double each other they alternate between playing two notes of the B-C-E-F collection such that all four notes are always present.  Rhythmically, the six articulations are reminiscent of the six iambs in an alexandrine line, to which the title of the movement refers.  The rhythmic figure in the third measure is a more direct allusion to the numerical motive elaborated in the first two movements; there are eleven sixteenths. The pitch collection used through the twenty-fifth bar of the movement is restricted to only the pitches B-flat, C, E and F.  The number eleven appears in several different rhythmic manifestations in this section; there are eleven distinct articulations, spaced four sixteenth notes apart, and separated from the following material by a fermata (mm. 7-10), and eleven articulations grouped together in measure 12.  The rhythmic unison disappears in measure 15, where each of the voices articulates a portion of the B-C-E-F chord at regular intervals.  These asynchronous ostinati recall the cyclical, out-of-phase rhythmic cells in the second movement.  The first violin repeats its chord every four sixteenth notes, the second violin every five, the viola every six, and the cello every seven.  This further reinforces the pairing of the outer and inner voices established earlier in the movement, for the sums of the rhythmic values of these cycles for each pair is eleven sixteenth notes.  Once each voice has completed its complement cycle of iterations (i.e. the first violin articulates its chord seven times, etc.), the pattern is interrupted by a figure of repeated thirty-second notes, which each voice joins as it completes its cycle.  While the voices enter and leave the repeated thirty-second note figure at separate times, all the voices coincide for exactly eleven thirty-second notes (mm. 14-15).  Following this interruption, the ostinati return, although the time between articulations quickly alters; once every eleven thirty-second notes one of the voices' base rhythmic value for its ostinato shifts to form its complement such that the sum of the values is eleven thirty-second notes.  This section ends with each voice articulating a series of consecutive thirty-second notes corresponding with the length of its base rhythmic duration.  The rhythmic unison is quickly reestablished, and a brief hemiola follows which involves eleven groups of two thirty-second notes, which all are comprised vertically of the complete pitch collection of B-flat, C, E, and F.  This section concludes with the same chord sustained for eleven sixteenth notes.

The next section is an elaboration of a four-note melodic cell by means of assigning one specific rhythmic value to each pitch (regardless of the voice in which it appears).  The four notes of this cell, B-flat, E, F, and C, always appear in this order, although each voice begins the section at a different point within this series.  The first five sixteenth notes of measure 24 make explicit each voice's starting point within this series.  After the original five thirty-second note declamation of this series, each pitch is assigned a distinct and consistent rhythmic value; all B-flats are four thirty-second notes long; C's last five thirty-second notes; E's (which quickly transform into E-flats) last six thirty second notes, and F's last seven.  Each voice cycles through the complete pitch series four times, followed by which there is a lift and a timbral change as all voices are marked sul tasto (m. 29).  This same pattern continues through one and one half more cycles of the pitch series in each voice, at which point each voice sustains its final note.  

             It is also in this section that Bresnick first alludes to the collapsing of the pitch set which will continue throughout the movement, ultimately ending in the next movement in microtonality.  The first note in the pitch collection to shift downward is the E natural, which becomes an E-flat at the first repetition of the pitch series (m. 24).  The F-natural follows, hinting at its coming transformation in temporary substitutions in each of the voices in turn before finally settling on the E-natural (mm. 26-31). 

            The strict relationship between pitch and rhythm is removed in measure 32, where a tutti unison sforzando lasting eleven thirty-second notes begins a new section and the E-flat has been transposed down a half step to form a D-natural.  All four voices are now in rhythmic unison, and form a rhythmic palindrome that extends from measures 32 to 40. Clearly, in this section pitch now functions independent of rhythm.  The melodic content does, however retain the same order from the original pitch series, although the corresponding notes have started to shift downward.  Throughout this section (mm. 32-41) the pitch collection gradually collapses downward toward the B-flat that forms its base.  The transformation occurs chromatically, with one note at a time shifting downward by a semitone.  In each case, the altered note is first introduced in the first violin, and echoed in score order immediately thereafter.  This collapsing finally reaches its conclusion when the pitch collection has been reduced to a tone cluster, immediately after which a new section begins (m. 42). 

            This new section is characterized by a change in tempo, a decrease in rhythmic activity, and an abandonment of the collapsed pitch collection for a more expanded melodic and harmonic palette that is heavily based on the pairing of E-flat and G-flat.  The pairing of the outer and inner voices is restored, and the section begins with the first violin and the cello providing accompanimental material consisting entirely of E-flats and G-flats, while the inner voices present melodies which are inversions of one another.  The duration of the notes in the inner voices, and the time between them in the outer voices, are also mathematically related.  The first violin's articulations begin spaced four sixteenth notes apart, while the cello's are seven sixteenth notes, for a total of eleven sixteenth notes.  Each of these durational values switches to its complement (i.e. the first violin's base rhythmic value becomes seven sixteenth notes, and the cello's becomes four) on the eleventh articulation.   Similarly, the second violin's and viola's lines begin with note values of five and six sixteenth notes respectively (m. 43).  The rhythmic values in these lines are somewhat less consistent than those in the outer voices in order to allow the voices to line up rhythmically each time the pitches E-flat and G-flat appear in them simultaneously.  The inner and outer voices exchange functions in measure 56, such that the second violin and viola are articulating the accompanimental E-flat and G-flat figures in rhythmic complements which exchange durational values on the eleventh articulation, while the first violin and cello assume the melodic lines, which are inversions of one another and begin on E-flats and G-flat. 

            All four voices line up rhythmically in measure 70, where two fortississimo quarter notes consist entirely of tutti E-flat and G-flats.  This marks the emergence of a second independent voice in each instrumental line; one voice in each instrument will function as a pedal E-flat or G-flat until the end of this section (mm. 70-85). Vertically, the alignment of voices also forms a palindrome, as the moving line in the violins is the upper voice, while the moving line in the viola and cello is the lower voice.  Furthermore, the moving lines of the first violin and cello are melodic inverses, as are the moving lines of the second violin and the viola.  The pedal voices in each of the instruments maintains its E-flat or G-flat until measure 79, where each voice assumes the opposite note.  The final measure of this section consists of three chords in rhythmic unison.  The first chord consists exclusively of the pitches E-flat and G-flat, while each successive chord adds a pitch (E-natural followed by F-natural) to result in a four-note tone chromatic tone cluster at the end of the bar. 

            The eight-voice structure and tone cluster are immediately compressed into a unison F at the downbeat of the next measure, which marks the beginning of the next section (m. 86).  This section consists of 110 consecutive thirty-second notes (a multiple of eleven) played in rhythmic unison in all four voices.  The unison F gradually expands upward by extending the range chromatically until it spans an entire fifth by the end of the section (m. 92). This expansion is in essence the reverse of the process used to collapse the original pitch collection of B-flat, C, E, and F down into a tone cluster, and also anticipates a similar expansion of pitch range which precedes the movement's final collapse into microtonality. 

            Before this second expansion of the chromatic pitch collection based on F, a four- measure interruption appears which recalls the movement's earlier technique of assigning specific rhythms to specific pitches (mm 93-97).  This segment is bordered on either side by a period of silence lasting eleven thirty-second notes.  Immediately following, the F-centered chromatic pitch collection returns in constricted form; the harmonic range of the downbeat of measure 98 is only a minor third, but the pitch collection rapidly re-expands to span a fifth within two measures before beginning its final contraction.  Here, for the first time, the manipulation of pitch collection extends beyond chromatic transformation to include microtonal movement.  The first violin introduces the first of a series of glissandi which gradually collapse the pitch collection toward F, the lowest note (mm. 99-100).  The glissandi are all quite gradual, spanning only a semitone over the course of eleven sixteenth notes.  After the first violin, the cello is the next to present a glissando, reflecting the typical pairing of the two outside instruments (mm. 100-101).  Eventually, the second violin and viola also pick up the glissando motive (mm. 105-108) and the voices continue to trade glissandi with exactly one taking place at any time, as the pitch collection gradually narrows, collapsing on its base.  The final glissando appears in the second violin and is twice as long as the others, with a duration that totals eleven eighth notes.  When the glissando reaches its endpoint, the pitch collection has entirely collapsed into a chromatic tone cluster (mm. 110-112).  This tone cluster is repeated in thirty-second notes for a rhythmic duration totaling eleven sixteenth notes.  The section comprised of the nine glissandi and this final repetition contains a total of eleven groups of eleven sixteenth notes.  Following this section, each voice maintains its pitch from the final tone cluster, and repeats eleven times in rhythmic unison a figure of two thirty- second notes, the distance between which increases by one sixteenth per articulation such that the final two groups are eleven sixteenth notes apart (mm. 114-122).

            The title of the fourth movement, "At Jhelum," refers to the site of the battle in which Bucephalus died, and features several quasi dirge-like viola solos as well as a renewed use of palindromic structures.  The movement begins on a tone cluster comprised of the same four pitch classes which ended the third movement, forming a unifying link between the two movements. This opening chord has a length of eleven thirty-second notes, immediately introducing the numerical motive  While the texture at the opening of the movement is quite homogenous, as all four voices are in rhythmic unison and in a very small range of pitches, the viola quickly separates itself from this texture, moving first to a dynamic louder than the other three voices (m. 3) and eventually changing tessitura such that the lower range of its pitch collection extends nearly an octave below the narrow range of the pitch collection employed in the other three voices (m. 5).  The range in the other voices, meanwhile, is actually contracting, such that it eventually spans only one and one quarter steps.  The microtonality which was alluded to in the glissandi of the previous movement finally comes to full fruition here, as the full range of quarter steps within this range is drawn upon.  The section ends on a tone cluster which consists of stacked quarter steps based off of an F.  The rhythmic basis for this information is the same palindrome that was employed in the third movement (mm. 32-40) during the first collapsing of the B-flat-C-E-F pitch collection. 

            In the ninth measure the rhythmic unity is fragmented, and each of the voices except the viola begins an eleven-note palindrome, which never reaches its final note, based on a different base rhythmic duration.  These palindromes are all based on a pitch series consisting of a collection of quarter tones within the span of a minor third, but here the series is based on C instead of F.  This collection of quarter tones is nearly comprehensive, although the quarter tone between C-sharp and D is conspicuously absent.  The viola, meanwhile, presents a line which is effectively a composite of the other three lines' information, but at a louder dynamic and with a more forceful accent.  The viola line simultaneously includes one note every seven sixteenth notes that is not doubled in any other line.  This effectively completes the collection of base rhythmic values to result in two pairings whose sum is eleven sixteenth notes; each note in the violin's palindrome lasts four sixteenth notes, the second violin's last five, and the cello's last six.  Each of the palindromes in the other three voices never actually reaches its final note; they all begin on an E-flat but fade away to niente before reaching the final E-flat which would complete the series.  The E-flat is finally delivered in the solo viola in measure 14, which marks the beginning of the next section, a viola solo.

            The viola solo iterates a slightly ornamented eleven-note palindrome in quarter notes with a rhythmically elongated central pitch (mm. 14-17), echoing palindromic structures from the second movement.  The series of pitches used in this palindrome is essentially the same series used in the previous section, only the pitch collection expands slightly to include the E-flat inflected one quarter tone upward and fills in the previously absent quarter tone between C-sharp and D.  The ornaments to the palindrome are a rare allusion to musical material outside of the work, as they evoke the dotted rhythms of a funeral dirge; this seems to be a form of lament for the deceased Bucephalus. 

            The viola solo continues with a new eleven-note palindrome (mm. 17-21) using a new pitch collection which contains all of the quarter tones between an upwardly-inflected D and an upwardly-inflected F except the quarter tone between E and F.  The base rhythmic value is now seven sixteenth notes, which forms a complementary total of eleven with the quarter note value from the immediately preceding palindrome.  The ornamentation continues as in the previous palindrome.  As the palindrome is presented, each of the other voices enters in turn presenting a new palindrome beginning on E-flat.  The first and second violins employ the same palindromic sequence of pitches in different complementary values.  The viola concludes its palindrome and begins a new palindrome based on the same pitch sequence as the new palindrome in the cello with a base rhythmic value of a quarter note, which is the complement of the cello's base rhythmic value of seven sixteenth notes.  The voices all align for what would be a unison E-flat to conclude all four palindromes simultaneously.  Once again, however, the expectation is thwarted and the E-flat arrives only in the viola, which forms a transition into the movement's second viola solo (m 25).

            The second viola solo contains another ornamented eleven-note palindrome beginning and ending of E-flat, with a basic rhythmic value of six sixteenth notes.  Before the series reaches completion, each of the other voices enters in turn with its own palindromic melody with base rhythmic values which are complementary, reflecting a return of the pairing of the outside and inside voices. Each pair also shares the same palindromic pitch sequence.  At the same time, a secondary voice appears in each instrumental line in notes of shorter duration (single sixteenth notes).  These secondary voices also reflect the instrument pairings, for while they are not palindromic, they contain the same pitch series in reverse order such that the cello's secondary line is the retrograde of the first violin's and the viola's is the retrograde of the second violin's.  The palindromes coincide at their conclusion, and again set up an expected unison E-flat.  Following the established pattern, this E-flat arrival is once again subverted, and instead a new section begins on a unison G (m. 33). 

            In the following section, the idea from the prior section of separate and complementary base rhythmic values in the separate voices is retained, but the function of these values changes.  Instead of marking the duration between notes of a sequence, the values define a basic rhythmic cell for each voice which, over a series of bars, becomes more rhythmically active until each of the sixteenth notes within the cell are articulated within each repetition (mm. 33-39).  The first violin's rhythmic cell lasts four sixteenth notes, and as it is the shortest, is the first to complete its rhythmic elaboration; all of the sixteenth notes are articulated beginning in measure 34, although due to bowing they maintain a grouping of one and three within each beat until the downbeat of the first 11/16 bar (m. 36), where an eleven-note palindrome is introduced which will eventually be adopted by all four voices.  Each of the other voices similarly articulates an increasing number of sixteenth notes within each rhythmic cell (the values of the second violin, viola, and cello's rhythmic cells are five, six, and seven sixteenth notes respectively), until the cell lines up with the final sixteenth note of a measure.  When this occurs, the voice adopts the eleven-note palindromic pitch sequence first introduced in the first violin.  This gradual increase in rhythmic activity is ultimately leading to the point where all four voices coincide with the unison palindrome, an increase which is accentuated by an accelerando and which finally reaches its arrival in measure 39.  As this is the most anticipated arrival in the movement and all the material following it shares a strong motivic relationship, it is logical to consider this measure as the most significant formal dividing point of the movement.  It is not surprising that this division again yields two sections whose ratio in length corresponds to the "Golden Proportion." 

            The new section marked by this arrival is unusual in its simplicity in some senses; all four voices are in unison (allowing for some octave switching due to registral differences between the instruments) and play the same rhythm.  The distinction between the voices lies entirely in the way in which the notes are grouped and the different pattern of accents in each voice which results; each voice has a different number of sixteenth notes per grouping which is the complement of the value of the rhythmic cell in the same voice from the previous section.  Thus, the eleven motive is maintained both within each individual voice and within the instrument pairings.  The first time at which all four voices' groupings begin on the downbeat of a measure is after each of the voices has articulated its grouping eleven times (m. 43).  At this point, each instrument's rhythmic value switches with that of its pair, again maintaining the eleven sixteenth-note relationship both horizontally and vertically.  This cycle, which exchanges base rhythmic values every eleven statements of each rhythmic cell, continues throughout the remainder of the movement.   

            The pitch series utilized in this section are largely modally based and mainly comprised of new collections not previously employed in the piece.  The unison palindrome is based on a G Aeolian scale. When each of the voices completes its eleventh iteration of the palindrome, the melodic material alters and the pitch collection changes to G Dorian.  The new palindrome in this new mode is repeated eleven times, after which the melodic material is transformed. As the first violin has the shortest note grouping, it is the first to undergo this transformation, which shifts the mode to C Aeolian.  The new melodic information does not present a palindrome of consecutive notes; instead the first notes of each seven-note groups form a larger scale palindrome while the following notes in each group merely elaborate and re-articulate the group's primary note (mm. 47-64).  Each of the other voices follows in turn, and the pairing of inside and outside voices is maintained, as the pairs again share the same palindromic pitch sequence but articulate it in different rhythmic durations. 

            This same pattern of changing modes and palindromes every eleventh repetition of each voice's note grouping continues until the end of the movement, cycling through palindromes in F Aeolian, beginning in measure 54, and then E Aeolian, which begins in measure 58 in the first violin and continues until the movement's close.  The movement ends abruptly, as all four voices coincide and appear about to conclude their respective palindromes on a unison G.  The arrival of this resolution, however, reflecting the trend throughout the movement, never comes, and the movement is suddenly over.  The only evidence that the movement is approaching its conclusion is a change in tempo to coincide with the beginning of the F Aeolian section (m. 54); otherwise the texture and rhythmic organization remains unchanged from the beginning of the second large section (m. 39) through the end of the movement. 


The fifth movement takes its title, "The New Advocate," from the Kafka story, which is included as an inscription in the score.  The form of the movement is also suggestive of the theme of the story, which removes Bucephalus from his historical context and places him in a modern setting as a lawyer.  While Bucephalus is clearly the same horse from earlier stories, the associations with Alexander and with his historical life are here only accessible as memories of things past, and lack the immediacy they had in the previous setting.  Kafka muses that perhaps it is really best to do as Bucephalus has done and absorb oneself in law books.  In the quiet lamplight, his flanks unhindered by the thighs of a rider, free and far from the clamor of battle, he reads and turns the pages of our ancient tomes (Kafka, 149-150).

The fifth movement similarly incorporates material from previous movements, but in a dissociated context in which the material seems more reminiscent than immediate.  Sections of the movement are derived in various ways from material in each of the previous movements.  One might expect such a juxtaposition of previous disparate material to act as an affirmation of the concepts' validity and combined significance, but in fact the effect achieved here is more ironic.  As Bucephalus' legendary status is dwarfed by placing him in a mundane, modern setting, so the collage of fragments and motives only serves to undermine the significance they once had within the more consistent context of the movements from which they are taken.

            The first section of the movement (mm. 1-5) divides each of the instruments into two voices, a technique used in the second and fourth movements.  Here this double structure is even more explicit than in any prior occurrence, as Bresnick uses separate staves for each voice, and there is a dramatic timbral difference between the voices as each instrument plays one voice arco and the other pizzicato.  As in movements two, three, and four, each voice is assigned a base rhythmic value which forms one half of a pair of values which add up to eleven sixteenth notes.  Not only do the voices within each instrument reflect this complimentary paired relationship, but also the overall arrangement of base rhythmic values forms a palindrome vertically.  The pitch content of each voice also reflects this palindromic relationship; each pair of voices (i.e. the first violin arco voice and the cello pizzicato voice, etc.) has duplicate material, allowing for octave differences. The pitch collection which is employed is the Lydian mode centered on A, which recalls, among other material, the opening to the first movement.  Furthermore, this choice of mode reveals a programmatic motivation as well.  The restful, sanguine mood ascribed to the Lydian mode by ancient theorists is as appropriate for depicting Bucephalus' solemn academic study as it was to represent his innocent, undomesticated state in the first movement.  

            The restful mood is disrupted with an abrupt change of pitch collection (m. 6), which resolves to a sustained unison A in all four voices. This gesture lasts for a total of eleven sixteenth notes and ushers in melodic material which is reminiscent of the first movement (mm. 6-12).  Another emphatic borrowing from the first movement in this section is the metronome marking and the instruction "tempo rubato," which exactly replicate the instructions which begin the work.  For the majority of this section, there are never more than two voices playing at any time, recalling the orchestration of the first movement which is scored for the two violins alone.

            The importance of the melodically exposed palindromes which appear throughout the second and fourth movements is undermined in the following, very brief section (mm. 12-14) in which a long palindrome formed by repetitions of a shorter, eleven-note palindrome quickly loses the dramatic weight with which it appears (marked sul ponticello and forte) as the dynamic decreases and the phrase eventually dissipates.  The tie between this ironic statement and the second movement is strengthened by the use of polymodality, which recalls the ending section of the second movement. 

            Following a silence that separates this material from the coming section, all four voices articulate a chord based on a pitch collection which suggests either a B-flat Lydian or A Phrygian modality (the mode's center is not clear), which is sustained for a duration of eleven sixteenth notes.  The next appearance of material directly derived from an earlier movement is the tutti articulation of consecutive identical thirty second note chords that recalls the opening of the third movement.  This link is reinforced by the concordance of their tempo, dynamic, and articulation markings.  This appearance of the material does differ from its derivative in one important way, however; rather than the strident, constricted tone clusters used in the fourth movement, the pitch collection here is A Lydian, which is retained until the conclusion of the piece.  The juxtaposition of this mode, with its connotations of restfulness and repose, and the thematic material from the previous movements echoes the images of Kafka's Bucephalus recalling the drama of previous glories within the placid setting of a modern library.  After the eleventh articulation of this chord, the cello disrupts the pattern with different material (m. 17).  Each of the other voices enters in turn (in reverse score order), assuming a base rhythmic value which determines the duration of each of the chords in the following section.  The assignment of rhythmic values is such that they form sums of eleven thirty-second notes between the outer and inner voices.  Both this rhythmic arrangement and the voices' staggered entrances to the section directly recall the information immediately following the viola solos in the fourth movement.  As a result of this arrangement of entrances, the voices are aligned rhythmically on each voice's eleventh articulation of the same chord (m. 22).  At this point, the voices also fall into an alignment of pitches; each voice has a double stop consisting of a C-sharp and a D-sharp.  This chord is sustained, and a new chord consisting of all of the pitches in the A Lydian scale is articulated eleven thirty-second notes later.  Following this chord, a relationship is established between each pitch and a specific rhythmic value; for example all of the F-sharps in all of the voices last five thirty-second notes; each of the other pitches also has a corresponding exclusive rhythmic value.  This technique is derived from a section in the third movement (mm. 93-96).  Each of the voices also employs the same pitch sequence in this section, although they begin on different notes within the sequence.  Since these same pitches will necessarily have the same rhythmic value in each voice, the four voices are essentially functioning canonically (mm. 23-24).  After one completion of the pitch sequence, the base rhythmic value assigned to each note is altered.  The pitch sequence does appear a second time, although in the cello and second violin the sequence is somewhat altered toward the end to allow for a more effective transition into the next section.  While the durations assigned to each pitch in this following segment (mm. 25-27) differ from the first appearance of the pitch sequence, they retain the original proportion relative to one another; they only differ in that the durations consist of equivalent numbers of twenty-fourth notes rather than thirty-second notes. 

            A brief pause and a tempo change mark the abrupt transition to a new section derived from material in the third movement.  The ubiquitous pairing of E-flat and G-flat returns, although the notes have been enharmonically re-spelled in order to maintain the A Lydian modality.  The section (which restores the metronome marking from a parallel section ranging from measures 42 through 85 in the third movement) begins with two unison articulations of chords consisting exclusively of the pitches D-sharp and F-sharp.  The total duration of these two chords is eleven sixteenth notes (mm. 27-29).  The pairing of the inside and outside voices from the parallel section in the third movement is distorted here; instead the viola and cello are paired in a set of lines that are each other's melodic inverses in rhythmic unison.  The violins are also paired thematically, although each instrument takes up a two-voice structure.  In the first violin's upper voice, the pitch F-sharp is re-articulated regularly every four thirty-second notes, while the lower voice re-articulates a D-sharp once every seven thirty-second notes, thus forming within the instrument's line a complete set of eleven and the motivic dyad.  The second violin line is structured similarly, although the time between articulations of the F-sharp and D-sharp are five and six thirty-second notes respectively (mm. 28-32).  The section concludes with a chord comprised of all of the pitches in an A Lydian scale except B, and whose duration is eleven sixteenth-notes.  After eleven thirty-second notes of silence, a new section begins, recalling the end of the third movement. 

            The next segment is a brief recollection of the extended section involving descending glissandi that concluded the third movement.  The idea is dramatically constricted in its reappearance here, however; the first violin begins a glissando that spans a minor seventh over the course of forty-four thirty-second notes, a radical expansion of the range of the glissandi in their original context.  The second violin and viola each enter at intervals of eleven thirty second notes with glissandi of their own, concluding at the same point (m. 37).  This brief section is punctuated by a gesture eleven sixteenth notes long, recalling the similar transitory material from measures 33 and 34 (mm. 38-39). 

            The next brief segment recalls the viola solos of the fourth movement. While the dotted rhythmic gestures that evoke a dirge-like quality are retained, the quarter-tone inflections are absent, and the melody's range is greater (mm. 39-42).  Furthermore, the material is altered timbrally and registrally as it occurs in the cello rather than in the viola.  Two brief transitory chords (whose combined duration, not surprisingly, is eleven sixteenth notes) lead into another section derived from material in the fourth movement (mm. 47-64).  Each voice has a base rhythmic value which determines the number of sixteenth notes grouped together under each slur, and the rhythmic values again combine (following the pairing of inside and outside voices) to form sums of eleven sixteenth notes.  In the section from which this material is derived, a palindrome is presented in each voice based on the first note of each grouping.  In this new context however, this symmetry is undermined and the pattern is interrupted before a complete palindromic set can develop (m. 45).  Instead, yet another transitional segment lasting eleven sixteenth notes leads into the final section of the piece. 

The violins are doubled throughout this last section, as are the viola and the cello, essentially creating a two-voice texture which recalls the first movement of the piece.  Otherwise, however, the section consists principally of original material.  The sustained A and E in the violins which mark the beginning of the section indicate that the piece has reached its conclusion; as the first and fifth degrees of the mode, the pair establishes a stability which continues until the movement's end.  The violins begin a series of long sustained chords whose durations (thirty-three sixteenth notes and three sets of twenty-two sixteenth notes) are all multiples of eleven.  The lower voices fall into rhythmic unison after the first chord, and by the final chord the A-E pairing has moved to the cello, resulting in the most final and stable permutation of the Lydian pitch group to bring the piece to its conclusion.

            The movement's string of disconnected fragments derived from other movements ironically undercuts the significance these fragments had in their original contexts; the ideas are constricted and often interrupted before they are allowed to complete themselves, such that structures that were extended and elaborate in the earlier movements here seem almost trite.  This progression reflects the movement's relation to the Kafka story that shares its title and completes the work programmatically.  While Bresnick is interested in musically reflecting this distinction between the historical Bucephalus and the modern Bucephalus, he does attempt to achieve this by abandoning the motivic structures that form the basis of the first four movements.  Instead, he retains these motives and formal designs within a transformed musical context to gain the advantage of both narrative development and an overall thematic unity.



Bresnick, Martin.  String Quartet No. 2, "Bucephalus."  New Haven, Conn.:  Self-published.  1984. 

Kafka, Franz.  "The New Advocate."  Selected Short Stories.  Trans. Willa and Edwin Muir.  New York:  Random House. 1952.

Powers, Harold S.  "Mode."  The New Grove Dictionary of Music and Musicians.  Vol. 12.  Ed. Stanley Sadie.  Washington D.C.:  Groves Dictionaries of Music Ltd.  1980.

Stolba, K. Marie.  The Development of Western Music.  3rd ed.  Boston:  McGraw Hill.  1998. 


While the pitch classes only span the range of a fifth, the aurally perceived range is actually a major seventh as the C is transposed down the octave.

Such a technique would not be unprecedented; composers from Bach to Schumann, Shostakovich and Berg have employed  musical pitches to spell verbally.

This admittedly idiosyncratic nomenclature seems to be the most efficient way we can refer to one-third of a sixteenth note triplet set, i.e. one twenty-fourth of a measure in 4/4.