In the shadow of Brahms – Robert Fuchs and his First Piano Trio

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This article deals with the relatively little-known composer Robert Fuchs, who used developing variation – a composition technique associated predominantly with Johannes Brahms – in his First Piano Trio. More specifically, it traces the manifestation of goal-oriented developing variation in Fuchs’s Piano Trio in C major Op 22, a work which he dedicated to Brahms. After the noting of Fuchs’s reputation as Kleinmeister at the time Brahms was composing, the empirical section shows that the characteristic feature of the germ cell (G-A-G) that appears at the beginning of this composition, namely a movement away from and a return to the point of departure, manifests on micro (motivic), meso (thematic) and macro (structural) levels. Focusing on diastematic aspects of Fuchs’s trio, the article illustrates, firstly, how the germ cell grows teleologically within the four movements, and, secondly, how it develops from movement to movement to form new opening motives. Finally, the study shows that the replication of the germ cell’s basic structure on larger scales coincides with the mathematical property of self-similarity, a characteristic of the fractal geometry of Benoit Mandelbrot, thereby connecting brevity of motivic ideas with overall design.

Keywords: Developing variation, diastematics, Robert Fuchs, Piano Trio in C major Op 22, Johannes Brahms, Arnold Schoenberg.



In recent years, the emphasis on canonical composers has shifted to acknowledge the contribution of minor composers – the so-called Kleinmeister of music history and repertory (Proksch, 2013: 274). Many of these composers had outstanding reputations while they were alive. However, eclipsed by the works of the masters of their time, these composers and their music were quickly forgotten. According to Joseph Kerman, it is one of the main objectives of historical criticism to “explain and exemplify those evanesced reputations” (Kerman, 1983: 118).

Robert Fuchs (1847–1927) was one of the widely admired and better-known among Brahms’s followers (Dubins, 2007: 118). His compositional style developed toward economy and thematic unity during his career as a composer. With his being rivalled by Brahms, Bruckner, Wagner and Wolf, Fuchs’s music virtually vanished from the mainstream concert repertoire almost immediately after his death, although he was a teacher of note (Grote,1994: 21). One of the most prominent features of Brahms’s music is what Arnold Schoenberg called developing variation. Fuchs applied the principle of developing variation in his Piano Trio in C major Op 22, which he dedicated to Brahms, the composer who, according to Schoenberg, cultivated developing variation to its fullest potential (Frisch, 1982: 215‒6).


Developing variation

Developing variation is a term which defines a broad principle of thematic composition – a principle which Schoenberg describes in his essay Linear counterpoint (1931) as “the endless reshaping of a basic shape”.1 He added that “there is nothing in a piece of music but what comes from the theme, springs from it and can be traced back to it; to put it still more severely, nothing but the theme itself. Or, all the shapes appearing in a piece of music are foreseen in the ‘theme’” [our emphasis] (Schoenberg in Stein 1975: 290). He further defines developing variation as “variation of the features of a basic unit ... which provide[s] for fluency, contrasts, variety, logic and unity, on the one hand, and character, mood, expression and every needed differentiation, on the other hand – thus elaborating the ‘idea’ of a piece” [our emphasis] (Ibid, p 397).

Terms such as “motive”, “basic shape” and “basic unit” point to melodic gestures: “it is often an expressive gesture which provides the starting point for a work” (Dahlhaus, 1988: 131). Schoenberg’s concept of motivic content refers “exclusively to diastematics, not to rhythm and harmony”, so much so that it seems as if diastematicism (intervals, or complexes of intervals) “was the true substance” of his music (Ibid, p 131). Although rhythm is part of motivic design, because of lack of space we shall focus here on pitch.

The use of the word “diastema” synonymously with “interval” in music originates from the Pythagorean use of the word as the fractional length or ratio that determines a particular musical interval. In his book entitled The beginnings of Greek mathematics, Árpád Szabó more specifically defines diastema as an “interval between two successive sounds” because “the whole string was plucked first, followed by the unit” [our emphasis], which was a half in the case of the octave (Szabó, 1978: 133, 132). On the basis of Carl Dahlhaus’s view on composers of the late nineteenth century, Richard Taruskin states that they “did their thematic thinking-in-music in terms of motifs rather than full-blown melodies”. The crucial problem of musical composition of the latter part of the century was “the fundamental tension between ‘the brevity of the musical ideas and the monumentality of the formal designs’” (Taruskin, 2010: 735). The final conclusion of this article, which links the diastematic approach to developing variation in Fuchs’s piano trio with Benoit Mandelbrot’s theory of fractal structure, demonstrates a solution to this tension between brevity of motivic ideas and overall design.2

Schoenberg believed that the music of Brahms illustrates the most advanced manifestation of the technique of developing variation, in that he often starts to develop his motives from the very opening of a piece (Frisch, 1982: 216). The motive, according to Walter Frisch (1984: 11), is the composer’s primary tool. Frisch explains that the unity of a composition does not depend on the motive’s original form, but on the way it is treated and developed to provide material for the work as a whole. Ethan Haimo (1997: 355) describes the process of developing variation in its most basic form as the process whereby a motive undergoes significant changes, but retains enough of its characteristics to still be recognised.

The concept of developing variation, as understood by Schoenberg, embodies the metaphor of organicism within a musical composition. A distinctive trait of an organism is its dynamic form, continuously undergoing development and mutation. “It is a form of becoming rather than being” (Broyles, 1980: 355). Developing variation allows new melodic material to stem from the original thematic material, thus demonstrating direction and growth in a teleological, or goal-oriented, manner (Grimes, 2012: 130). Composers of nineteenth-century Romanticism regarded musical form “as a process rather than a mold” (Longyear, 1973: 57). This kind of forward motion allows growth through new or contrasting, but interrelated, ideas.

Despite Schoenberg’s clarification of the concept of developing variation, this article does not rely on his terminology, because in his writing, “the terms ‘theme’, ‘basic shape’ and ‘idea’ tend to overlap”, and in 1950, he “dropped the terms ‘basic shape’ and ‘theme’ in favour of ‘idea’ and ‘basic unit’” (Dahlhaus 1988: 128–9). He also used different definitions for “motiv” (Neff, 1993b: 416). The meaning of terms such as “cell”, “motive” and “(basic) idea”, as they are used in this article, will be explained below.


Robert Fuchs and his First Piano Trio

Robert Fuchs achieved immediate popularity with his serenades, which granted him the nickname Serenaden Fuchs. The catalogue of Fuchs’s works is extensive, and his works were championed by notable conductors, such as Arthur Nikisch, Hans Richter and Felix Weingartner. Fuchs was even called “the master of small art” by Eduard Hanslick, the famous Viennese music critic (Grote, 1994: 18‒9, 21, 72). Johannes Brahms, who rarely handed out compliments, praised Fuchs with the following words: “… everything is so fine, so skilful, so charmingly invented that one always derives pleasure from it” (Ibid, p 26).3 His compositional style is described by his close friend and biographer, Anton Mayr, in the following words:

The strength of Fuchs’s art lies in the fact that it applies the most modest means, is only interested in faithfulness and truth of expression and denies itself any merely outward flashy effect. This art carries its worth in itself and its modesty touches and convinces all the more, because the strict form and economy of the means used offer a wealth and depth that many other works of art fail to attain (Mayr, 1934: 110).4

Fuchs’s Piano Trio Op 22 consists of four movements. The first movement is marked allegro moderato, and the second, adagio con molto espressione, followed by a lively scherzo marked allegro. The finale, allegro risoluto, is full of energy, with a contrasting expressive legato theme. At the end, a heroic mode emerges. Due to the limited scope of this article, we shall concentrate primarily on the way in which developing variation manifests itself in the main theme of each movement.5


Developing variation in the four movements

This discussion focuses on diastematic aspects of Fuchs’s piano trio, beginning with three-note melodic patterns in the main theme of the first movement. We use the term “germ cell” (GC) for the opening pattern (G–A–G), because most of the thematic material of Fuchs’s First Piano Trio develops from this cell.6 The discussion below will show how the basic feature of the germ cell is also reflected in the work as a whole – “in a truly organic work the opening already presents the form of the whole” (Neff, 1993b: 418). When the upper auxiliary note pattern changes direction, and/or the whole tone is replaced by other intervals, the three-note formation will be described as a “cell”, and not a “germ cell”. When the extension of the germ cell forms a well-defined melodic pattern of four or more notes, we use the term “motive”.

The discussion will demonstrate how the characteristic of the germ cell, namely a movement away from and a return to the point of departure, manifests on micro (motivic), meso (thematic) and macro (structural) levels. The germ cell (G–A–G) is labelled here as GC 1–2–1, number 1 referring to the first note of the melodic three-note formation. The subsequent numbers represent the distance, expressed in number of steps, from the first note of the cell (always indicated as 1). The numbers do not refer to scale degrees or key.

This article focuses on the shape and contour of melodic patterns. In order to distinguish between different variants of the germ cell, the variant of the germ cell where the whole tone is replaced with a semitone (for example, E–F–E) is labelled as c 1–2–1. The contour of this cell remains the same as that of the germ cell. The inversion of the germ cell (for example, A–G–A) is labelled as C 1=2=1, and the inversion of c 1–2–1 (for example, F–E–F) is labelled as c 1=2=1.

Table 1 shows the labels, the descriptions of the germ cell and its most frequent variants in Fuchs’s First Piano Trio. It also summarises the development of the germ cell through its various permutations to reach the pentatonic cell at the climax, near the end of the fourth and last movement.7 The development of the germ cell is, in other words, goal-oriented.

Table 1: Labels, descriptions and examples of the germ cell and its variants




GC 1–2–1

Original germ cell; an auxiliary-note figure

G–A–G, Ex 1

C 1=2=1

Inversion of the germ cell

D–C–D, Ex 3

c 1–2–1

A variant of the germ cell, where the whole tone of the auxiliary-note figure is replaced with a semitone

The tonalities of C, C♯, C represent the macro structure of the first movement.

c 1=2=1

Inversion of c 1–2–1

A♭–G–A♭, Ex 2


Ascending whole-tone segment of the germ cell

C–D, Ex 2


Descending whole-tone segment of the germ cell

B♭–A♭, Ex 2


Ascending semitone segment of the semitone variant of the germ cell

E–F, Ex 2


Descending semitone segment of the semitone variant of the germ cell

D♭–C, Ex 2


Extended germ cell, namely cell 1–2–3–2–1

C–D–E♭–D–C, Ex 4


First segment of the extended germ cell

C–D–E♭, Ex 4


Second segment of the extended germ cell

E♭–D–C, Ex 4

Main motive

First four notes of the fourth movement

G–A♭–C–B, Ex 6

Pentatonic motive

When the A♭ of the main motive changes to A♮ in the second theme, the first three notes relate to the pentatonic scale.

G–A–C‒B, Ex 7

Pentatonic cell

Climax near the end of the fourth movement

C–D–F, Ex 8


First movement, allegro moderato (348 measures)

Because of the absence of a contrasting second theme, it is almost impossible to analyse the first movement in terms of a traditional formal scheme, such as sonata form. This correlates with the notion that developing variation extends over the entire structure of a composition to transcend the conventional principle of contrasting themes that is basic to many structural schemata. However, by investigating the development of the germ cell, the movement can still be divided into three sections, creating a non-traditional A–B–A structure. The first section, which presents the germ cell and the development of new cells, ends in m 108.1. The middle section (mm 108.2–203.3) begins in C♯ minor and consists mainly of repetitions, variants and transpositions of the germ cell, which is developed in various ways.8 The repetition of the first section starts with the main theme in the original key of C major (m 203.4). Other thematic material is derived from the first theme, more specifically, from the germ cell.

The development of the germ cell already starts within the first theme, thus from the very beginning of the piece. The broken slurs in all the examples indicate arpeggiated secondary material that connects one cell with the next cell. It can be seen as connective or prolonging material. The square brackets in Example 1, marked with numbers 1 to 7, indicate the germ cell and its variants. Only the germ cell is marked with a solid square bracket. Cells (variants of the germ cell) are marked with broken square brackets.

Example 1: Piano Trio Op 22, first movement, mm 0.4–12.2

Cell 1: G–A–G is the germ cell, labelled as GC 1–2–1. The auxiliary-note figure has a slight angular shape formed by the rising contour of the first segment and the falling contour of the second segment.

Cell 2: The pitch names of this cell correspond to those of cell 1 (thus also GC 1–2–1), but the melodic pattern appears an octave higher and the rhythm is varied. In the first appearance of the germ cell, the accent falls on the second note, but here the first note is emphasised, varying the metrical accent and, therefore, the character of the pattern.

In the first two-measure phrase of the work, two germ cells frame connecting material, creating an a–b–a pattern, where b represents a movement away from the point of departure a. This a–b–a pattern, therefore, reflects the same idea on meso level (thematic level) as the germ cell on micro level.

Cell 3: The original GC 1–2–1 appears an octave higher, with corresponding rhythm and metrical placement.

Cell 4: The opening GC 1–2–1 is transposed one step higher (A–B–A).

Cell 5: This cell (GC 1–2–1) is a transposition of cell 2, one step higher.

The distance between the two notes of both segments of the germ cell is augmented conspicuously from m 6.4 onwards.

Cell 6: A–D–C (cell 1–4–3) is a variant of the germ cell. Beginning with the opening note of the previous cell, the original ascending major second is enlarged to create a perfect fourth, followed by a descending whole tone, as in the germ cell.

Cell 7: Both segments of the germ cell enlarge to create a climactic cell 1–6–4 (G–E–C), with the largest opening interval yet. This conspicuous development is accentuated by the tenuto on G, as well as by the higher dynamic level of mezzo forte, in contrast with the mezzo piano of the first cells. The outline of cells 1 to 5 is a major second, the outline of cell 6 is a perfect fourth and the outline of cell 7 is a major sixth. In the process, the shape of the original germ cell becomes significantly angular.

The basic feature of the germ cell is also recognisable in the greater tonal scheme of the first movement, that is, on macro level. The first theme is in C major. From m 114, the key of C♯ minor dominates, until the original key of C major returns in m 203.4, where the first theme appears in its totality for the first time since its first statement. On a macro level, the tonal scheme (C major – C♯ minor – C major) represents c 1–2–1, the semitone variant of the germ cell. The main feature of the germ cell, therefore, also represents the tonal design of the first movement, embodied in the macro structure of this movement.


Second movement, adagio con molto espressione (110 measures)

The second movement, as a whole, is based on the thematic material in mm 1–4, which is also derived from the germ cell and variants of it. The three themes in mm 1–44 are followed by a modulating bridge passage, which leads to a repetition of mm 1–44 that starts in m 50, followed by a codetta. Some of the themes appear in different keys. In this movement, as a whole, the basic idea of the germ cell manifests itself in the manipulation of texture, as will be shown later.

In the first measure of the movement, the two segments of the c 1–2–1 semitone variant of the germ cell function independently (Example 2). The first segment of the cell (an ascending minor second) is labelled s1, and the second segment (a descending minor second) s2, indicated by broken square brackets. The solid square brackets marked as S1 and S2 (in Example 3) indicate, respectively, the ascending and descending whole-tone segments of the germ cell.

Example 2: Piano Trio Op 22, second movement, mm 113

The main thematic material starts with a juxtaposition of s1 and s2. The descending s2 (D♭–C), rhythmically varied, appears in the right hand of the piano part in m 1. The slur over D♭–C accentuates the sigh effect, and embodies the con molto espressione stated at the beginning of this movement. The left hand of the piano part shows the ascending s1 (E♮–F), creating a mirror image of the descending semitone interval in the right-hand part. The juxtaposition of these two segments creates a vertical variant, which appears only in the second movement. In m 2, the two segments are inverted. The vertical variant, with its contrary motion, forms the basis of the main thematic material of the second movement.

In the third measure, the violin and cello repeat the s2 (D♭–C of m 1) in octaves against the left hand of the piano part repeating s1 (E♮–F). In the next measure, s2 is transposed to F–E♮, the retrogression of the original s1. The vertical variant of m 1, repeated in mm 2–3, crystallises out into the horizontal thematic material of mm 3–4, played by the violin and cello. The connecting secondary material (m 4.1), framed by s2, is a diminished triad, with the interval of a diminished fifth between the lowest and highest notes. This interval and the secondary material9 also develop throughout the second movement. The two-note slur and diminuendo accentuate the sigh effect in m 4.2.

The thematic pattern in mm 3–4 corresponds with the a–b–a pattern in the first two measures of the main theme of the first movement, where the germ cell is connected to another germ cell by means of secondary material. This a–b–a pattern reflects the same idea on meso level as the germ cell on micro level and the C–C♯–C tonal scheme of the first movement on macro level. In the second movement, the secondary material also develops when its outline, the diminished fifth (B♮–F) in m 4, changes to an outline of a minor sixth (G–E♭) in the violin part in m 8. This variation by means of intervallic expansion recalls the systematic intervallic expansion of cells in the first theme of the first movement (see Table 1).

The succeeding bridge passage (mm 9–14) further develops the secondary material into scale passages, where the diminished fifth forms the outline of both arpeggiated and scale variants. When the stringed instruments move in contrary motion with the right hand of the piano part in m 9 (here only the violin) and in m 11 (violin and cello), the secondary material creates a mirror image, imitating the contrary motion of the vertical cell. All three instruments alternate this secondary material, thereby drawing attention to its significance. The scale passage that fills up the ascending diminished fifth in m 11 leads to a dotted rhythmic pattern, in which the descending semitone (D♭–C) with the emphasis on D♭, sounding three times in mm 12‒13, is reminiscent of the opening semitone.

The overall effect of the second movement is the reverse of that of the first movement, creating a mirror pattern on macro level. Where the first movement starts with the germ cell, which is then varied and developed throughout the movement, the second movement starts with the two isolated segments of the semitone variant of the germ cell. Thus the main theme of the second movement starts with fragmented material, gradually developing into the inversion of the original germ cell (marked =C), heard at the beginning of the second theme in the piano part, marked molto espressivo in m 26, followed by a crescendo (Example 3).

Example 3: Piano Trio Op 22, second movement, second theme, mm 24‒27

The way in which musical texture is manipulated on macro level also reflects the a–b–a pattern of the thematic material. The fragmented thick texture covering more than four octaves in the opening measures, leads to a more open texture when flowing parallel sixths extend over three octaves in the second section. The original thick texture and the broad sound spectrum of the opening measures, frame the thinner texture of the second and third themes when they return in m 41 (compare Example 3 with Example 4).


Third movement, scherzo: allegro (303 measures)

The third movement, as a whole, also reflects the basic idea of the germ cell, namely moving away from and returning to the point of departure. It consists of three sections, the third being a repetition of the first 108 measures. Although the middle section (mm 113–264) does not consist of contrasting thematic material, as in traditional ternary form (ABA), it contrasts with the outer sections, regarding the following aspects:

  • Tonality/mode: section A and its repetition are in C minor; section B is in C major;
  • Dynamic level: section A is marked piano; section B is marked sempre forte;
  • Tempo and character: section A is marked allegro; section B is marked largamente, preceded by two measures marked pesante;
  • Articulation: section A alternates staccato and legato, combined with trills; section B is played extremely legato, as the performance instruction (largamente) indicates;
  • Texture: the texture of section A is transparent; the texture of section B is thick.

This description of the way in which contrast is created emphasises the necessity of considering aspects other than contrasting themes as indicators, when determining the structure of a piece of music.

The entire third movement is thematically derived from one motive, an extended variant of the germ cell that appears at the beginning of the movement (Example 4).

Example 4: Piano Trio Op 22, third movement, mm 0.3–8

The main theme of the third movement is based on an extension of the original germ cell to form a new five-note motive, namely cell 1–2–3–2–1 (C–D–E♭–D–C) in mm 1–2. This motive is referred to as the extended germ cell (abbreviated as EGC). The extended germ cell is preceded by an upbeat of 5–1 (G–C). Because the EGC, more often than not, appears without the upbeat in the scherzo, it will not be regarded here as part of the extended germ cell.

The contour of the extended germ cell corresponds with that of the original germ cell, in the sense that it represents a movement away from and a return to the point of departure. The first segment of the extended germ cell (1–2–3) is labelled sm1, and the second segment (3–2–1) sm2, indicated by broken square brackets marked EGC (in Example 4). The first extended germ cell is extended even further with the repetition of sm2, rounded off with an ascending melodic C minor triad. The violin repeats the first four measures a minor third higher in mm 5–8.2, rounded off with a melodic E♭ major triad. From m 28b.3 onwards the extended germ cell develops by means of diminution, augmentation, prolongation and modulation.

The main theme of the contrasting section B (mm 113–264) is based on the extended germ cell, as well. It has developed from section A to section B in such a way that the character of the five-note motive has changed. The indication largamente, at the beginning of section B, is emphasised by the prolonged note values (Example 5). The repetition of the sm2 segment in the opening phrase of section A falls away in section B.

Example 5: Piano Trio Op 22, third movement, mm 113–124

The extended germ cell, now in C major, is varied rhythmically through the extension of the first note of the first segment and the last two notes of the second segment. The ascending triad that followed the original EGC in the first theme is replaced here by a descending leap of a minor third (C–A) in m 116. The contraction of the arpeggiated material from the first phrase of the third movement to this minor third, connects one statement of the EGC with another. The a–b–a pattern, therefore, also appears on meso level in section B of the third movement. Because the secondary material (melodic triad) shrinks to a minor third, it becomes part of the thematic material, as in the first and second movements.


Fourth movement, finale: allegro risoluto (418 measures)

On a macro level, the overall structural scheme of the fourth movement also represents the basic idea of the germ cell, namely that of a movement away from and back to the point of departure: the middle section begins in m 104.2, the repetition of the first section in m 231.2b, and the coda in m 335.2. The way in which the main thematic material of the fourth movement develops from the germ cell to culminate in the pentatonic cell at the climax of this movement, and also of the work as a whole, will be discussed later.

While the semitone segments of cell 1–2–1 function independently on two different levels in a two-part framework in the second movement, in the fourth movement the two semitone segments are linked diastematically to form a new four-note motive (Example 6). The ascending semitone segment and the descending semitone segment frame an ascending major third, to create cell 1–2–4–3 (G–A♭–C–B), the first four notes of this movement. The necessity of proceeding from the term “cell” (as used in the first movement) to the term “motive” (in the third and fourth movements) further demonstrates the concept of developing variation in the work as a whole.

Example 6: Piano Trio Op 22, fourth movement, mm 0.2b–16.1

The broken square brackets, marked with numbers 1 to 9, indicate the main motive and variants of the germ cell.

Motive 1: In this trio, the motive at the beginning of the fourth movement, namely G–A♭–C–B (cell 1–2–4–3), is the first melodic opening pattern with four different pitches.

Motive 2: C–D–E♭–D–C is the extended germ cell of the third movement. The accents on the descending segment 3–2–1 (E♭–D–C) are significant, as they accentuate the material that is developed further in this movement.

Cell (We use “cell” for three notes and “motive” for more than three notes) 3: D–C–D is an inversion of the original germ cell, thus C 1=2=1.

Motive 4: The extended germ cell (motive 2) is developed by means of sequential continuation of the descending segment 3–2–1 (E♭–D–C followed by B♭–A♭–G). This motive is referred to as the scalar extended germ cell. This scale passage recalls the succession of sm1 segments in the third movement (mm 121–124; see Example 5). Scale passages are regarded as secondary material in the first and second movements, but have now developed from the germ cell in such a way as to become thematic material.

Motive 5: B♭–A♭–G is a rhythmically varied repetition of the descending segment of the extended germ cell.

Cell 6: C 1=2=1 (F–E♭–F)

Motive 7: Transposition of motive 4

Motive 8: B–D–B represents the outline of the extended germ cell of the third movement. The contour of this melodic movement corresponds with the angular contour of the original germ cell of the first movement.

Motive 9: B–F♯–B is a variant of motive 8 by means of intervallic expansion, but here, the leap of a minor third is expanded to a perfect fifth. In the process, the shape of the motive becomes significantly angular, similar to the increased angularity of the germ cell in the main theme of the first movement. Starting with the leap of a minor sixth, from m 13 onward, the main theme disintegrates through two-note melodic patterns, to dissolve into the prolongation of the dominant seventh chord of C major.


Goal-oriented developing variation in the work as a whole

From a diastematic point of view, the opening patterns of the four movements demonstrate a systematic development of the original germ cell:

  • First movement: the opening germ cell consists of three notes (G–A–G) that represent two different pitches;
  • Second movement: the vertical variant in m 1 consists of four different pitches, but its two-part framework (D♭–C / E–F) covers the temporal space of two pitches;
  • Third movement: the germ cell develops into a five-note motive, the extended germ cell (C–D–E♭–D–C) that represents three different pitches;
  • Fourth movement: the opening motive (G–A♭–C–B) consists of four different pitches.

With regard to the development of the germ cell in the work as a whole, the pentatonic cell constitutes the ultimate point of arrival through developing variation, thereby demonstrating a kind of development that is goal-oriented. The germ cell develops from three notes (first movement) to two notes (second movement), to five notes (third movement) and to four notes (fourth movement), to arrive again at a three-note cell, but now with a clear pentatonic structure. In the fourth movement, the systematic disintegration of the main motive starts at the beginning of the second theme, when a semitone shift in the opening motive of the movement produces a new motive (G‒A‒C‒B in m 16.2b). The violin, here, plays in its lowest register on piano level (Example 7), contrasting with the tutti effect at the beginning, marked allegro risoluto on forte level. The first three notes of this quiet theme represent a pentatonic scale.

Example 7: Piano Trio Op 22, fourth movement, second theme, mm 16.2‒20.2

Four measures after the beginning of the B section, a three-note pentatonic cell is isolated (slurred) in the violin and cello parts (m 108), followed by accentuated notes G♯‒A♯‒C♯ (mm 180.2‒181) and C♯‒D♯‒F♯ (mm 184.2‒185), marked ff con fuoco, played by the violin and cello. In the coda, the pentatonic cell is played by the violin and cello in slurred parallel sixths (mm 339.2‒340).

The climax, near the end of the last movement, marked largamente ff sempre(m 393.2), also constitutes the climax of the work as a whole, when ten measures are saturated by pentatonic cells, and an isolated C‒D‒F is framed by minim rests (mm 401.2‒402.2) (Example 8).

Example 8: Piano Trio Op 22, fourth movement, mm 393–402


Because the thematic material of Robert Fuchs’s First Piano Trio is based on the germ cell and its variants, contrast in the overall structure is created by means of various textures, registers, tempos, modes, tonalities, expressivity, dynamics and articulation. The fact that these musical characteristics could guide the determining of the overall structure emphasises the necessity of considering aspects other than themes as indicators, when trying to understand musical structure on micro level as well as on macro level.

Considering the development of the germ cell within the work as a whole, it shows a kind of transformation that moves towards a goal, namely the pentatonic cell. The germ cell and its variants represent the systematic development from a three-note melodic cell into various kinds of melodic patterns and motives that combine with one another to create new, but always related, ideas on micro level.

When one cell connects to another cell by means of secondary material, it replicates the basic idea of the germ cell on meso level, namely moving away from and returning to the point of departure. Most of the themes in this piano trio follow this a–b–a pattern, but it also develops throughout the work. In the second theme of the second movement, the connecting material does not connect one germ cell with another germ cell, but connects one segment of the germ cell with another segment. Whereas the secondary material in the previous movements is framed by cells or motives that are similar, the secondary material in the first theme of the fourth movement is framed by two different motives, namely the main motives of the second and third movements.

The basic feature of the germ cell also manifests itself on macro level, because it represents the overall design of all four movements. The non-traditional A–B–A structure of the first, third and fourth movements reflects the structure of the germ cell on macro level. A similar effect is created in the second movement, but here through the manipulation of texture: the thick texture of the first thematic section frames the thinner texture of the second and third themes.

The replication of a small-scale design (the basic feature of the germ cell on micro level) on the meso and macro levels results in three distinct levels of scaling, giving the composition a fractal structure. The term “fractal” was introduced by Benoit Mandelbrot in 1975 to describe his celebrated computer-constructed visualisations of scale-invariant, self-similar patterns in nature (Mandelbrot, 1983). A fractal object can be compared to Chinese boxes or Russian dolls, where structures are nested within one another (Kanadoff, 1986: 6). “The comparison serves to analyse the procedure of structural changes from the key notes to the melodies and to the whole” (Hsü and Hsü, 1991: 3509). Structural scaling also appears in art, architecture and literature, but, in music, fractal analysis is still “in its infancy” (Brothers, 2007: 94).

By tracing the way in which Robert Fuchs used developing variation in his First Piano Trio, it was found that it coincides with Jack Boss’s description of developing variation, namely: “variation of the features of a unit” produces “successions of forms, which in turn manifest the idea of the whole piece” (Boss, 1991: 1). This finding coincides with fractal geometry as a study of structures in which the form of a small unit is similar to the form of the structure as a whole (Shiffman, 2012: 356), a phenomenon which creates in the listener the sensation that the music has its own self-awareness.

The concepts of structural scaling and developing variation were both formulated in the twentieth century. But, this article shows that these phenomena had already appeared in Fuchs’s First Piano Trio, which was written in the nineteenth century, highlighting the significance of intuition in the shaping of music in time and space. This study connects micro and macro structures, linking a diastematic approach to developing variation in Fuchs’s piano trio with Mandelbrot’s theory of fractal structure. In resolving the tension between short motivic gestures and a composition’s overall design, this article addresses a crucial problem that composers experienced at the end of the nineteenth century.



Boss, Jack F. 1991. An analogue to developing variation in a late atonal song of Arnold Schoenberg. New Haven: Yale University.

Brothers, Harlan J. 2007. “Structural scaling in Bach’s cello suite no 3”. Fractals 15/1: 89‒95.

Broyles, Michael. 1980. “Organic form and the binary repeat”. The musical quarterly 66/3: 339–60.

Dahlhaus, Carl. 1988. “What is developing variation?” in Schoenberg and the new music,transl Derrick Puffett and Alfred Clayton. Cambridge: Cambridge University Press:128–33.

Dubins, Jerry. 2007. “Brahms and his contemporaries, vol 1”. Fanfare 31/2: 118–9.

Frisch, Walter. 1982. “Brahms, developing variation and the Schoenberg critical tradition”, 19th-century music 5/3: 215–32.

Frisch, Walter. 1984. Brahms and the principle of developing variation Berkeley: University of California Press.

Fuchs, Robert. 1879.,%20Op.22%20(Fuchs,%20Robert).

Grimes, Nicole. 2012. “The Schoenberg/Brahms critical tradition reconsidered”. Music analysis 31/2: 127‒75.

Grote, Adalbert. 1994. Robert Fuchs: Studien zu Person und Werk des Wiener Komponisten und Theorielehrers. München: Emil Katzbichler.

Haimo, Ethan. 1997. “Developing variation and Schoenberg’s serial music”, Music analysis 16/3: 349–65.

Hsü, Kenneth J and Andrew Hsü. 1991. “Self-similarity of the ‘1/f noise’ called music”. Proceedings of the National Academy of Sciences of the United States of America 88/8: 3507–9.

Kanadoff, Leo P. 1986. “Fractals: Where’s the physics?”. Physics today 39/2: 6–7.

Kerman, Joseph. 1983. “A few canonic variations”. Critical inquiry 10/1: 107–25.

Levy, Sharon G. 1991. Developing variation, Mozart and the classical style. Chicago: University of Chicago.

Longyear,Rey M. 1973. Nineteenth-century Romanticism in music. 2nd ed. Englewood Cliffs, New Jersey: Prentice-Hall.

Mandelbrot, Benoit B. 1983. The fractal geometry of nature. San Francisco: WH Freeman.

Mayr, Anton. 1934. Erinnerungen an Robert Fuchs. Graz: Leuschner & Lubensky.

Neff, Severine. 1993a. “Schoenberg’s theoretical writings after the harmonielehre: A study of the published and unpublished manuscripts”, College Music Symposium 33. 2105:schoenbergs-theoretical-writings-after-the-harmonielehre-a-study-of-the-published-and-unpublished-manuscripts&Itemid=124 (accessed 24 August 2017).

Neff, Severine. 1993b. “Schoenberg and Goethe: Organicism and analysis”, in Music theory and the exploration of the past, eds Christopher Hatch and David W Bernstein. Chicago: University of Chicago: 409‒33.

Proksch, Bryan. 2013. “Jan Ladislav Dussek (1760–1812): A Bohemian composer ‘en voyage’ through Europe, ed Roberto Illiano and Rohan H Stewart-MacDonald” (review), Notes 70/2: 274–7.

Shiffman, Daniel. 2012. The nature of code. uploads/graficacion/nature_of_code-005_shiffman_12.4.12.pdf (accessed 24 August 2017).

Stein, Leonard (ed). 1975. Style and idea: Selected writings of Arnold Schoenberg. New York: St Martin’s Press.

Strang, Gerald and Leonard Stein (eds). 1967. Arnold Schoenberg: Fundamentals of musical composition. London: Faber and Faber.

Szabó, Árpád. 1978. The beginnings of Greek mathematics,transl AM Ungar, Anfänge der Griechischen Mathematik, Akadémiai Kiadó. TheBeginningsOfGreekMathematics/Szabo-TheBeginningsOfGreek Mathematics#page/n13/mode/2up (accessed 19 June 2018).

Taruskin, Richard. 2010. Music in the nineteenth century: The Oxford history of Western music. New York, Oxford: Oxford University Press, Vol 3.



1 Schoenberg first wrote about the technique of developing variation in an unpublished manuscript of 1917, entitled Zusammenhang, Kontrapunkt, Instrumentation, Formenlehre (Neff, 1993a).

2 We thank Philip Brown, associate professor of mathematics at Texas A&M University at Galveston (USA), for his assistance with regard to diastematics and Mandelbrot’s concept of fractal structure.

3 “… alles ist so fein, so gewandt, so reizend erfunden! Man hat immer seine Freude daran.”

4 “Die Stärke der Fuchsischen Kunst liegt darin, daß sie mit den bescheidensten Mitteln arbeitet, auf jeden nur äußeren Effekt verzichtet und im Ausdruck treu und wahr ist. Sie trägt ihren Wert in sich und ihre Schlichtheit rührt und überzeugt umso mehr, da sie in strenger Form und Knappheit eine Fülle und Tiefe bietet, an die manches andere Kunstwerk kaum heranreicht.”

5 For a discussion of developing variation in Fuchs’s piano trio as a whole, see Engelbrecht (2014). The score is available at Trio No.1, Op.22 (Fuchs, Robert).

6 Schoenberg used the term “germ” when referring to the basic idea of a composition (Strang and Stein, 1967: 8).

7 See Engelbrecht (2014) for a comprehensive list of the germ cell and its most frequent variants in this trio.

8 Measures and beats are referred to in the following manner: m 2.4 indicates the fourth beat of the second measure. Further subdivisions are conveyed by the addition of the letters a, b, c or d after the beat, for example m 5.3b refers to the second half of the third beat of m 5.

9 Secondary material, whether it be ornamental, accompaniment or non-thematic, can be subjected to developing variation as well (see Levy, 1991: 121).

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