Dialectic in Miniature:
Arnold Schoenberg’s Sechs Kleine Klavierstücke Opus 19
If one were to ask, “is there a principle of contrast in Schoenberg’s music?” the answer, while always affirmative, would vary, depending on where a particular work fell in the chronology of his compositions. While the 12-tone works demand a certain set of answers to this question, Schoenberg’s transitional works - those that precede the twelve-tone system and follow the more-or-less tonal works - offer varied examples of dialectical processes.
The collection of miniatures, the Sechs Kleine Klavierstücke Opus 19, is an interwoven fabric of tonal artifacts and there are distinctions between the horizontal and vertical aspects that reveal a subtle connection to the tonal universe. What these aspects have in common is a high degree of symmetry. In the conflict between these aspects lies the dialectical process that governs the work as a whole and the six pieces individually.
The musical concentration of these pieces - combined with their brevity -creates a highly ambiguous musical language that defies conclusive analysis. Indeed, it is one of the attractions of Opus 19 that the longer one studies it, the more heterodox it seems.
The concept of symmetry as an organizing structural principle was not Schoenberg’s invention. Toward the end of the 19th century the idea of the tonal center became entwined with notions of inversional balance. The influential Hegelian music theorist, Moritz Hauptmann, provoked a re-evaluation of musical structure when he analyzed the tonic/subdominant relationship as an inversion of the tonic/dominant; David Lewin links this to Schoenberg’s tonal and post tonal practice, illustrating similar structures of inversion in the First String Quartet and the cruciform symmetries of Pierrot Lunaire.
The symmetrical implications of Opus 19 have been studied by Jonathan Dunsby and Arnold Whittall, who attempt to explain the second and sixth pieces as total symmetries, akin to the first movement of Anton Webern’s Symphony Opus 21. This requires a certain amount of manipulation; Dunsby and Whittall sometimes fill in incomplete symmetries with pitches that appear later in the pieces and certain octave displacements are necessary. Although their Example 29, an analysis of the second piece of Opus 19, shows tightly packed chromatic regions toward the center and relatively larger intervals at the extremes, similar results might well be attained from many chromatic pieces, given the general tendency toward larger intervals in the lower registers, and saturation of those intervals within the treble and bass staves.
Even within the confines of tonal music Schoenberg recognized a motivic kinship between like intervals (i.e., symmetries) in musical development. In Fundamentals of Musical Composition he writes: “Every element or feature of a motive or phrase must be considered to be a motive if it is treated as such, i.e., if it is repeated with or without variation;”  he also notes that dyads can be motivic details, or any “pitch succession or duration succession of two or three members …” This is significant when one considers the multitude of dyadic symmetries in Opus 19.
In Opus 19 the interval sound - its Klang - is tied up with replication of like intervals. Focus on the actual sound of a pitch rather than its relative meaning within a tonal structure was a decisive step in Schoenberg’s musical evolution. The often-noted 05s (perfect fourths) in the opening of the Kammersinfonie Opus 9 are a familiar example, and perfect fourths play an important structural role in Opus 19 as well. Indeed the Kammersinfonie opening foreshadows a texture completely dominated by this one interval (end of measure 360 to 368; end of 371 to first half of 373). Such a concentration on a single interval is probably the first appearance of such an idea in the history of music. The whole-tone content of the first theme of the Kammersinfonie is similarly composed out (Schoenberg was proud of the work’s integration of themes and harmonies). By Opus 19 the interval was liberated and was projected into the free musical space of which Edgard Varèse and Stefan Wolpe would later speak.
The differentiated function of steps and skips in Opus 19 is a relic of tonality. From the perspective of Schoenberg’s twelve-tone works, which banished the distinction between the horizontal and vertical, it seems out of character to find just such a distinction in Schoenberg’s earlier music. And yet, verticals are manifested in Opus 19 by specific intervals deployed in symmetrical configurations: in the entire six pieces there are no simultaneous pairs of adjacent half steps or whole steps - or pairs of adjacent half-steps and adjacent whole-steps; these interval combinations are always expressed horizontally.
Adjacent half steps often act as connectors between symmetrical events, recalling similar moments in Debussy’s Des pas sur la neige. The dialectic between the vertical skip and the horizontal step, long present in Western music - where ”Pythagorean” perfect intervals oppose the melodic line and its dissonances - is still faintly present in Opus 19. One can think of symmetrical simultaneities as points of relative stasis and of steps as destabilizing transitions. Melodic shapes - particularly in I, III, IV and V - can be heard as surface manifestations of the interplay of symmetries.
The following analysis is based on an Occam’s razor approach; it considers only two elements: like-interval sets (3/3, 4/4, 5/5, etc.) and horizontal symmetries. While possessing no specific tonal function in the work, like-interval sets appear throughout, sometimes alone and often interpenetrating. Horizontal symmetries, in contrast, form connective links between these cells.
The 3/3/3, “diminished seventh,” and 4/4, the “augmented triad” cells, have a special property not shared by sets other than the total chromatic: they are infinitely repeating (i.e., C E F# A/C E ……/C E G# C E …..). Besides its philosophical suggestiveness, this property helps to form the specific Klang of the work; these subsets are more “redundant” than other subsets: their registral displacement merely produces more of the same pitches, forming a subtle timbral backdrop less determined by registral distinctions than the asymmetrical elements that create the specific environment of tonal music. The frequent 5/5 details imply a fourth cycle which would require 11 iterations to return to its initial pitch class, like the chromatic scale. However, it is still an infinite series, and to the extent that it plays a role in the piece (particularly in pieces I and VI) it also suggests boundlessness; indeed, the quartal, ic5 collection, which fully emerges only in the last movement, is prepared from the first measure of the work, as will be discussed in relation to piece VI.
Every pitch in Opus 19 can be understood as a member of at least one symmetry, although nearly all pitches are members of at least two. Besides the like-interval cells and linear symmetries, there are contour symmetries and convergence symmetries, which will be discussed below. A special chart is devoted to the emergence of ic5 symmetry over the course of the pieces.
In the following analyses, the four possible augmented triads are indicated as AUG0, 1, 2, 3 (G B D# = AUG0 as in measure 1 of I, see Example 1 below). The three diminished seventh chords are identified as DIM0, 1, 2, where DIM0 =A C D# F#. Diminished triads are included in the DIM forms to which they belong. Other symmetries are bracketed and expressed as interval pairs, with additional interval projections indicated as well. AUGs and DIMs in which a symmetry is disturbed by registral displacement are indicated with a slash line through the circle indicating the symmetry.
Consecutive symmetries in Opus 19 often share a common tone. Their combination and linkage determine the narrative of each piece. To whatever degree the pitch-class content is subset-related, these relationships are of a lower order of structural importance than registral symmetries.
Schoenberg, in the radio address quoted above, notes that “a minor third can become a major third.” If the remark has any meaning - and I think it is very significant - it must refer to such events as the opening right hand collection in the first piece B D# F F#, compared to the subsequent B A F# D# in measure 2. Or, in measure 3 the converging figure B E A G G# (-7 +5 -2 +1). Since the pieces are composed of ic3 and ic4 compounds, the entirety of the work can be heard as a play of expansion and contraction, signaled on the larger scale by convergent figures such as the one in measure 3.
Series of whole and half steps form connections between harmonic events throughout the opus. In piece I, the chromatic descent from the F in measure 4 down to the A at the end of measure 5 foreshadows the major third descent at the end of piece II, the linear ascent and descent in piece III, measures 1 to 4 (lh), and the stepwise ascent in IV, measures 4 to 6 and 7-8. Note that the chromatic descent in I is also clearly referenced in IV, measures 2-3; both begin with an F in the same register:
I: F (G) E E-flat D D C B B A
IV: F E D# D (C# preceding F) C A# B
Figure 1: Comparison of Scalar Motions in Piece I, m. 4 and IV m. 2-3 in Op. 19
Symmetrical step-wise motion permeates Opus 19. Note, for example, in I m. 6 - 7 (lh) the dyad pair D E/B A (+1 -1). It connects the AUG2 in measure 6 with the DIM2 in measure 7, or the -5/+5 symmetry between measures 16 and 17 (lh). This ic5 cell recalls the D and F# in measure 2, where they occur in the same register:
Measure 2: rh A B / lh F E D B
Measures 16-17: lh B A E (rh) F# C# F
Figure 2: Comparison of Scalar Motions in I, m. 4 and I. m. 4-6, 7-8 of Op. 19
Convergent motion - a pattern of decreasing interval sizes - is a most interesting surface detail in Opus 19; it is essentially highly developed contrary motion and often in the form of a compound melody. It occurs a number of times in I (see Example 1 below): the first across measures 2 and 3, fills in a minor third, the second is in measure 3 and can be understood either as a strand of shrinking interval sizes or as contrary motion by steps (B-A-A / E-G-G#). A third, in measure 7, is very close to conventional voice leading, and the convergence in measure 13 (lh) answers measure 7 and extends into the next measure. The very significant convergence at measure 11 is echoed in another prominent convergence in V, measures 1 to 3. The cadence point - F# D# E - is common to both figures. Secondly, the passages share the same pitches:
I measure 11 rh: D/B F G (C#) F# D# E
V measure 1 lh: D F G (B) rh: F# D# E
C.f. III m. 7: D# E
Figure 3: Comparison of Convergent Motions in I, m. 11 and V m. 1, of Op. 19
Finally, as we shall see, convergence in the final measure of I is of crucial importance in the unity of the work.
Contour symmetries, i.e., brief arch-forms, are ubiquitous in the six pieces (indicated by V-shaped lines). These brief oscillatory moments are a bit nostalgic - similar shapes might occur in Schumann - but they also reinforce the thoroughgoing symmetrical structure of the work. While contour symmetries occur earlier in the first piece - e.g., measure 2 A B A - the most prominent is the concluding gesture, which foreshadows events in subsequent movements and signals closure in the first piece. The uppermost contour symmetry D# E D# is a very significant foreshadowing of the last piece.
Opus 19, I
The first piece in the Opus 19 collection is very suggestive of a prelude: it is the longest of the pieces and most varied in texture. The arpeggiations in the first three measures signal its prelude quality, while the melodic arc of the first two measures is a model for both the conclusion of the movement and for similar melodic shapes in the others.
The sudden intricacy of measures 4-6 is governed by a descending chromatic line (discussed below), supported by a series of AUG and DIM cells and concluding with the contrapuntal measure 6. A second melodic arc can be traced in the left hand of measure 6, passing to the right hand A in measure 9, then t o the G in measure 10, and finally converging on the E in measure 12. This shape encapsulates the movement’s emergent moment - the tremolo/sustained figure in measures 8-12 - and connects it with the contrapuntal music. Two further melodic arcs (13-14, 15-end) close the movement, in a gently lingering decay anticipating the conclusion of the entire set.
Example 1: Schoenberg, Sechs Kleine Klavierstücke, I
All transpositions of the third-cells are exposed by measure 5, which concludes with an unadorned first statement of AUG3, a cell that also occurs in the last measure. AUG0 has cadential implications in three of the other pieces as well:
First Measures Last Measure
I AUG0 AUG0
I AUG0 AUG0
III AUG0 AUG0
IV AUG0 (penultimate measure)
Figure 4: Use of Opening and Closing AUG0 Collections in Op. 19
In addition, nearly every pairing of third-cells in the entire set first appears in this movement. As noted above, the DIM0 and AUG0 in measures 8 to 10 are first paired in measure 1. These pairings - often, interweaving - create the quicksilver harmonic interplay so characteristic of this movement.
The AUG0 tremolo/sustained figure in measures 9 to 13 is the longest event in I, and its emergence is carefully prepared in the opening measure where this trichord first occurs together with the DIM0 collection (notes G and B appear in the same registers in both places). The tremolo/sustained figure decelerates the harmonic motion, but its bonding with the mercurial events accompanying it assert the dialectic of vertical and horizontal symmetry.
Measure 2 of I contains a brief but significant contrapuntal moment: the superimposed imitation of B A F# D# and F E D B. Measures 4-6 present a dense web of contour- and inversional symmetries linked with the chromatic thread running through the upper voices. Symmetries are formalized by inversion in measures 5-6:
measure 5-6: A G B/D F# F
measure 6: D F# F/E D# G
Figure 5: Contrapuntal Symmetries in mm. 5-6 o op.19, I
These measures also contain less formally related cells, such as the A A C# in the inner voice of measure 6 and the lowest voice F D E, both comprised of thirds and half-steps.
The form of each piece is to a great degree determined by the location of its cells. AUG0 is of particular importance in piece I and throughout the set, especially its pairing with DIM0, as in I mm. 1, 4 and 9. In measure 4, the pairing precedes the semi-cadential AUG3; in measure 8 the pair makes up the “emergent moment.” In the final measure, a brief AUG0 is encapsulated in AUG3, which now forms a full cadence (in phrasing rather than tonality), as opposed to its earlier half-cadence.
Opus 19, II
Opus 19 II is a study of ic3 and ic4, placed in various symmetrical relationships with each other and underpinned by broken linear patterns. There are no ic5s or ic7s in the movement. The obsessive concentration on ic3 and ic4 is signaled by the B/G ostinato figure that runs throughout. Following the diminished triad “half-cadence” in m. 6, the conflict between major and minor thirds is resolved definitively in the last measure in favor of ic4 with the superimposition of two augmented triads.
In contrast to the orderly situation of ic4s in specific registers, ic3s are rather “vagrant” (to use Schoenberg’s anthropomorphism). The dyad D#/C, first occurring within measure 3, appears an octave higher as an interpolation in the B/G ostinato. It reappears as a simultaneity in its original register in measure 6 as one of a pair of such repetitions, the other being C/A, also in its original register. The reappearance of these dyads opposes the third-types: a pair of ic4s (C/A, C#/A) is succeeded by a pair of ic3s (E/C, D/B). The F#/D# ic3 between measures 2 and 3 appears an octave higher between the two augmented triads in the final chord, encapsulating the interval in a dialectical end point.
Particularly noteworthy is the “half cadence” of superimposed diminished triads at the end of measure 6, and their “resolution” in the concluding superimposed augmented triads. The diminished-triad verticality is an adumbration of a similar event in the previous measure in which DIM0 and DIM1 are superimposed, with the G out of place.
The ostinato B/G that runs through the movement is symmetrically completed by the E B G triad in the final chord, and foreshadowed at measure 4. Similarly, the upper triad of the final chord - D B F# - is foreshadowed by the B/G in measure 5 and the high D in measure 2.
The descending line of whole steps (measures 7 to end) is unique in the opus, although echoed in V, measures 8-10. Harder to parse is a first line of descending thirds from measure 3 to measure 6. The ic3 A/C and ic4 C/A dyads in measure lead to ic4 A# /F# in measure 5, followed in the next measure by C/A (as in measure 3), C#/A and E/C (rh) (ic4, ic4, ic3/ ic3), culminating in the D/B dyad in the “half-cadence.” The strikingly literal descent of ic4s prefigures other more or less obvious scalar passages throughout the opus.
The first four measures of II may almost be thought of as a re-composition of measure 1 of I. Note that the G/B dyad retains its registral position (rh), as does the F# in measure 2 and the E in measure 4 (spelled D# in measure 1). Only the D in measure 2 of II, is absent in the opening measure of the first piece. Indeed, the pair of AUG0 and DIM0 are featured in both. In its terseness and concentration, piece II offers the greatest number of symmetrical relationships since every event is intervallically mapped onto every other.
Example 2: Schoenberg, Sechs Kleine Klavierstücke, II
The third piece contains the densest texture in the opus; yet it dwindles to a single note - G - by measure 8. The dialectical process is a reductive one. A thinning-out begins in measures 5 and 6, signaled by a converging symmetry in the lower voices B /C A/C# A/ D. Note that the group B E F A begins and ends the piece (c.f. pitches marked 1-4 in measures 1 and 9). Closure is further asserted by the reversal of the descending B -E octaves in the first measure in a mirror echo in the last measure. In addition, the group of four pitches in measure two - B C D E (numbered 5-8) - is partially retrograded in measures 3-4 (see pitches numbered 8, 6 and 5).
A long convergence of decreasing interval size can be traced through the bass pitches in measures 1-3, (note also the contrary voice leading in the rh in m.2) and is reflected further in measure 4, before the convergence by contrary motion in measure 6 - B A A (descending) / C C# D (ascending) – thus encompassing virtually the entire movement (see Example 3).
Example 3: Schoenberg, Sechs Kleine Klavierstücke, III
The planar structure of the opening of piece III is an unusual combination of opposing textures and dynamics, suggestive of polytonality. The symmetries in the first four measures are manifested in the bass in octaves and its distinctive scalar motion together with its engagement with the upper voices. The upper voices in these measures are rich in symmetrical motions involving thirds and fourths. Especially important to the melodic shape of the opening is the symmetry between measures 1 and 2 (+3 -3: C# E/ A F#).
The importance of contour symmetries in Opus 19 is often reinforced by subtle echoes between movements. For example, the B D# B in measure 5 is anticipated in II, m.4 (B E B); the little contour symmetry E E D# in measure 7 is a registral displacement of pitches that conclude piece I (upper voice) and which re-emerge prominently in VI.
As in II, important cadence points in III are determined by superimposed AUG and DIM cells. AUG0 and AUG2 form a half-cadence at measure 4, followed by DIM0 and DIM2 intertwined in measure 6
Densely packed cells AUG0, DIM0, AUG2, DIM2 and AUG3 dominate III, nearly all of them overlapping with the bass motion. AUG0 and DIM0 cells appear at the opening, as they did in the previous two movements, and the AUG0 in measure 5 is a direct reference to the opening of both I and II: the G/B dyad appears again, protrusively, in the same register as measure 1 of I, and together with the D# revisits measure 4 of II. In III, m. 6, this continuity is reinforced by the pairing of AUG0 and DIM2 - again, as in II, m. 6 - in which pitches G, E D and A share registral identities with III, m. 6.
The premise of this movement is the exposure and compression of a melodic line, supported by occasional sharp chord-like punctuations. The melody unfolds registrally in jagged skips through the opening, then as smooth ascending lines in measures 4 to 6 and 7 to 9. A parallelism is maintained in the legato responses to the angular opening in measures 2-3 and 7, where the C# E D# figure is transposed to F# A G#.
The octave displacement and temporal compression of measure 1 into the first beats of measure 10 frames a similar relationship between measures 3-6 and 6-8. The scalar D C F# G# A# B C (measures 3-6) - with all but F# and G# in a lower octave – appears, also time-compressed in mm. 6-8. The initial D C in the former group, after its fleeting grace-note reflection in m. 5, appears at the end of the latter group, thus breaking up the whole-tone content of the pitch collection.
IV is primarily scalar, with references to scalar passages in other movements. The descending chromatic line in measures 2-5 (see Example 4) derives, as mentioned, from that of mm. 4-5 in the first piece (see above). This derivation is reinforced by the appearance of the same AUG0 and AUG2 cells in measure 4, supporting a descending scale, just as they do in I, m. 5. This is further emphasized by the subtle reappearance of the 5/5 set (D) F G C in IV, m. 4, which appears at the beginning of I and also comprises a major element of VI.
One of the most interesting structural details of the piece is the shifting function of the A# B dyad that concludes the piece, where it completes the linear F F# A# B symmetry and recalls the same tones in identical registration within the unstable and interweaving melodic line of mm. 7-8.
Example 4: Schoenberg, Sechs Kleine Klavierstücke, IV
Opus 19, V
The fifth piece - more or less a waltz - is based on accumulation of intervals. The increasing density of the thirds - inner voices in measure 1 to pairs of thirds in all voices at the end - suggest that the dialectical structure of the fifth piece is in the transformation of single intervals into a texture composed of these intervals, recalling the saturation of ic5s in the Kammersinfonie (i.e., transformation of quantity into quality).
The opening measure is characteristic Schoenberg; the contrary motion between the hands encloses an inversional relation between two diminished triads: C A G (rh) and B D F (lh), offset by a sixteenth note. This is echoed in measure 2 in a linear inversion between the left hand (G A B) and the right (A G# F#), the latter in augmentation and echoing the descending contour of I, m.2 (B A F# D#). The right-hand convergence in the opening two measures is paired with another in contrary motion in the left hand (see Example 5) and in m. 4, IV also recalls the polyphonic texture of I, m. 6 although with more structured organization of trichordal retrogrades and inversions.
The fifth piece is particularly referential. The AUG2 and DIM0 in the opening two measures is a reinterpretation of the first three measures of IV: F C and B are registrally invariant, and pitches D B (bass clef) and A are reiterated. Also note the quasi-tonal transposition of the B D C A figure (IV, mm. 1-3) to G B A (V, opening) within a pseudo-F-major context.
Three occurrences of AUG0 (measures 8, 9 and 12) are part of a network of references that spans all of I, II, III, IV and VI. In particular, its occurrence in measure 12 is in the same registration as that of the opening of the work (I, m.1). The interesting superimposition of AUG1 and AUG2 at the conclusion, in which one pitch in each is shifted to the other, recalls the conclusion of II and its superimposed AUG0 and AUG3. Note the identical interval of separation (ic1) for both pairs and the reversal in the lower left hand of the voice leading of the exposed E-D of m.11.
The contour symmetry in measures 7-8 recalls I, m. 3, in its shape if not in its pitch content, and the series of descending ic4s in the left hand in measures 9-10 - as blatant as any of the internal references in the work - clearly suggests the descending ic4s at the end of II. The half-step motion of the contour symmetry also predicts the descending ic4 pairs in measures 9-10, which in turn anticipate the cadence. Pairs of thirds also connect V and II:
measure 5 measure 9
G B/F# A# G B/F# A#
measure 6 measure 10
A C/A C# B D/A C# (inversion)
measure 5 measure 12
G B/F# A# G B/G B
measure 6 measure 14-15
A C/A C# rh A C/A C#
measure 7 measure 14-15
F A lh F A/E G#
Figure 6: Schoenberg Comparison of Third Pairings in Op. 19, II and V
Example 5: Schoenberg, Sechs Kleine Klavierstücke, V
Op. 19, VI
A tolling figure runs through the brief final movement, which Schoenberg composed after hearing of Mahler’s death. It owes its retrospective and pensive character not only to the sound quality of the tolling intervals, but also because of strands of reference to previous movements. In fact, there is very little in the movement that cannot be found in earlier ones.
Tones B and A in the upper half of the tolling figure are the initial notes of the entire piece; the F# occurs at the end of measure 1. (Note that the three pitches appear an octave higher in VI, but in the same 6/9 registral relation to each other.) The other element of the tolling figure - F C G - appears in I, m, 1 as well, and in the same register. In VI the DIM0 in measures 4-5 linking the two halves supports an important long-range connection between the first and last movements.
The DIM0 in measures 4-5 is accompanied - unusually - by a D# in octaves. The upper note is part of the contour symmetry D# E D#, which first occurs in the last measure of I (upper voice, with adjacent D# octave). This detail is also one of the most telling and poignant recollections in the set of pieces (also see III, m. 7: E E E, as well as in the inner-voice E-E-E in m. 9 at the close of this piece).
The final movement also affirms the quartal process and connects it with the opening of the work. The second half of the tolling figure - F G C - is transposed to B F C in measure 6, and shares two pitches with the F C G cell. In fact, the same quartal hexachords can be found in both movements:
VI 6 m. 6
D G C F B (D# in previous measure)
I, m. 2
D G C F B (rh) (D# in previous measure)
Figure 7: Schoenberg, Comparison of ic5 Collections in Op. 19, I and VI
This emergence is based on the accumulation of a number of details throughout the work (excepting the third-obsessed piece II).
- The left hand pitches in III, mm. 1-2 can be expressed as a fourth cycle (A E B F C), which shares three pitches with the quartal material in VI, (B F C); another quartal figure follows in measure 3 (A D G A C).
- The F C G D subset appears in IV, m. 4 in the left hand, and also in mm. 11-12.
- A nearly complete quartal hexachord con be found in V, m. 8: A# C/G F/D, where the G C and F are in their original register in I.
The chart in Figure 8 traces the emergence of perfect-fourth intervals with triangular note heads indicating those in their original registral positions in. I.
Measure 9 contains the pitches C F# G G# E E, all of which (except for C# and D) appear in I, measure 1. Note that the measure 9 pairing of AUG0 and AUG1 also references I, m. 1. The pair of symmetries 5/5 and 10/10 at measures 6-7 point toward the final measure - and the end of the work - which closes with a symmetrical 14/14 (B-A / BA) echo wie ein Hauch: “like a breath.”
Figure 8: IC5 Collections in Op. 19
Example 6: Schoenberg, Sechs Kleine Klavierstücke, VI
 For example, the opening of Schoenberg’s Second String Quartet Op. 10 could be mistaken for Brahms, but gives way to a developmental chromaticism that reaches an epiphany in the final movement - a setting of the Stefan George poem “Ich atmete den Luft ferne Planeten” (“I breathe the air of distant planets”). A transitional work and a direct predecessor of 12-tone composition, the Serenade op. 24 offers a more concentrated example in its Tanzscene movement. The motif of the parodic first section of the Tanzscene is a hexachord whose symmetrical complement is the thematic basis of the movement’s more lyrical trio sections. Hexachordal opposition produces a polarity in the fabric of the work, but these hexachords are not treated identically: the second hexachord is treated as a conventional theme, complete with homophonic texture and “vagrant” harmonies.
 David Lewin, “Inversional Balance as an Organizing Force in Schoenberg’s Music and Thought,” Perspectives of New Music, 6/2 (Spring/Summer 1968) 2-5.
 Canonic Studies, ed. Ronald Stevenson (New York: Crescendo, 1977).
 Jonathan Dunsby and Arnold Whittall, Music Analysis in Theory and Practice (New Haven: Yale University Press, 1988) 125-130. See especially the charts on pages 125, 127 and 129.
 Style and Idea: Selected Writings of Arnold Schoenberg (London: Faber & Faber, 1975), 222.
 Fundamentals of Musical Composition. 2nd ed. Gerald Strang and Leonard Stein (London: Faber & Faber, 1970) 8-9.
 Jack Boss, “Schoenberg’s Op. 22 Radio Talk and Developing Variation in Atonal Music.” Music Theory Spectrum, 14/2 (Autumn, 1992) 134. Also note that Schoenberg uses only conventional terms (major second, etc.) to analyze his Opus 22, and remarks that a “minor third becomes major third,” a comment to which I will return below.
 See Greenbaum, “Debussy, Wolpe and Dialectical Form.” Contemporary Music Review: Stefan Wolpe Issue (2/2008)
 Leonard Stein, “Schoenberg: Five Statements,” Perspectives of New Music 14/1 (Autumn, 1975) 165.
 A similar perfect-fourth stack is concealed in the first and last of the Klavierstücke: see below.
 Except perhaps for the opening of Beethoven’s King Stephen Overture and the Devil tuning his violin in Liszt’s first Mephisto Waltz.
 Walter B. Baily, ed. The Arnold Schoenberg Companion (Greenwood Press: Westport, 1998) 74.
 The doubled octaves in III, as well as the octave displacement in the fourth movement (cf. measures 1 and 10) also suggests the transitional nature of these pieces in respect to spatial equality.
 See Greenbaum “Debussy, Wolpe and Dialectical Form,” in volume XI/2 of this journal.
 The accompanying examples indicate many symmetries of different kinds; too many to discuss in detail, despite the brevity of the pieces. Please see examples 1 to 6 for a complete inventory.
 For a further discussion of contour symmetry, see Greenbaum, ”The Proportions of Density 21.5: Wolpe and Symmetries in the Music of Edgard Varèse” in On the Music of Stefan Wolpe, Austin Clarkson, ed. Pendragon (Hillsdale, New York: 2003). Contour symmetries dominate the surface and structure of Density 21.5.
 See for example, in piece I, upbeat to measure 1, where DIM0 and AUG0 share the note D#.
 -7 +5 -2 +1: i.e. a descending perfect fifth, an ascending perfect fourth, a descending major second, and an ascending semitone.
 There may be a suite-like aspect to Op. 19 that justifies the use of the term “prelude.” All the other pieces are briefer, and self-contained. Most have a dance-like character: No. 2 is perhaps indefinable, but No. 4 opens with a gigue-like figure, and No. 5 is patently a waltz.
 Perfect fourths are outlined in measure 2, B-F#, measure 4, C-G, and in the descent of major thirds at the end FA to C/E.
 It is possible that the lower triad should read F# A# D#, although this would go against the symmetrical logic of the piece. Dunsby 126.