Volume 2, Number 5, July 1996
Copyright © 1996 Society for Music Theory
Review/Article of Miguel A. Roig-Francoli’s article “Harmonic and Formal Processes in Ligeti’s Net-Structure Compositions,” Music Theory Spectrum 17/2 (Fall, 1995), 242–267.
KEYWORDS: Ligeti, harmony, form, proportions, golden section, Ramifications, Lux aeterna, Chamber Concerto, Second String Quartet
ABSTRACT: Miguel A. Roig-Francoli’s recent article, “Harmonic and Formal Processes in Ligeti’s Net-Structure Compositions,” Music Theory Spectrum 17/2 (Fall, 1995) discusses three compositions by Gyorgy Ligeti that use “net-structures.” This review summarizes and critiques several aspects of his article, including his choices of terminology, his analytical procedures, and his views of small- and large-scale form in these pieces.
 In “Harmonic and Formal Processes in Ligeti’s Net-Structure Compositions,” Miguel Roig-Francoli investigates three of Ligeti’s “net-structure” compositions: Ramifications (1968–69), the first movement of the Chamber Concerto (1969–70), and the fifth movement of the Second String Quartet (1968). He begins his article by introducing the concept of “net structures” and identifying their basic characteristics. He divides the net-structure techniques into four categories, each of which is illustrated by close analyses of small-scale harmonic features of sections of the works. He then considers longer-range connections in these movements, which he investigates using reductive techniques. He concludes with a consideration of symmetry and formal balance in the large-scale form of these compositions. In this review, I consider each element of his discussion in turn, providing additional background on some aspects and examining his methodology and his analytical conclusions.
 Like Pierre Boulez, Luciano Berio, Karlheinz Stockhausen, and other European composers of the 1960s, Ligeti has often commented publicly on his own works, with some of his remarks recorded in the context of interviews or program notes for performances or recordings.(1) Although a composer’s comments on his works should be examined critically (comments may be used to obfuscate rather than enlighten), Ligeti’s remarks provide a sense of how he thinks about his compositional processes, and have been useful to analysts who study his works. Ligeti uses a variety of terms to refer to his compositional techniques from the 1960s and 1970s; those most frequently used are “micropolyphony” and “meccanico.” Ligeti typically uses micropolyphony and meccanico as “umbrella” terms, referring to a wide range of pieces from the 1960s and early 1970s with specific shared characteristics.
 The term “net-structure,” which Roig-Francoli selects, is one used in the interviews in a more general sense than meccanico and micropolyphony. Roig-Francoli defines a net-structure as “a continuous web of finely-woven lines or repeated patterns in a constant, interactive process of transformation of one or more parameters, such as pitch, rhythm, texture, dynamics, or timbre” (page 243). Unlike the terms micropolyphony and meccanico, which Ligeti uses analytically, net-structure is not usually mentioned in the context of specific pieces or techniques. Rather, it is consistently used qualitatively to refer to a feeling Ligeti had about the textures of the pieces—an allusion to a childhood dream where Ligeti was caught in a room full of entangling webs.(2)
 Roig-Francoli does not include works such as Lontano and Lux
Aeterna in the “net-structure” group, although Ligeti’s own remarks
include them along with earlier works like Apparitions (the original
reference point for the web dream anecdote) and Atmospheres. In
footnote 5, Roig-Francoli acknowledges this point, but attempts to
distinguish these two works by saying that the process is “linear” in
one and “harmonic” in the other (page 243). This is a false
distinction—in both cases the changes in harmonies are created by
“chromatic fluctuation or intervallic expansion and contraction” as
Roig-Francoli confirms later: “The process of constant chromatic
transformation, a procedure which Ligeti has used widely both in
micropolyphonic and harmonic textures
 As Roig-Francoli notes, however, “Ligeti’s use of technical terms descriptive of his music is not always consistent” (page 244). In order to avoid terminological confusion, previous analysts have found it useful to coin terms of their own for groups of Ligeti’s compositions, using specific techniques rather than attempting to use one of his terms. In my own work, I use the term microcanon (invented as a subcategory of micropolyphony) to designate textures formed from a pitch succession set canonically in many voices at short time intervals, and pattern-meccanico (a subcategory of meccanico) for textures in which several linear strands, each constructed from small groups of pitches repeated mechanically, are overlaid to create a contrapuntal texture.(5) Compositions using microcanonic techniques include Lux aeterna (1966), Lontano (1967), and the ninth movement of the Ten Pieces for Wind Quintet (1968). Works with one or more passages of pattern-meccanico textures include Continuum (1968), Coulee (1969), and the fifth movement of the Second String Quartet. Later works, such as Ramifications, the Chamber Concerto, the “Selbstportrait” movement of Three Pieces for Two Pianos (1976), and the Drei Phantasien nach Friedrich Holderlin (1982) combine microcanon with pattern-meccanico in various ways.
 After his introductory comments, Roig-Francoli’s attention turns to the analysis of “harmonic processes” in Ligeti’s net-structures. While Roig-Francoli is careful to try to distance himself from previous analysts of Ligeti’s music, his analytical approach is built on foundations provided by other writers. Most of the analytical comments are directed toward Ramifications, one of several works in which Ligeti combines pattern-meccanico techniques with microcanon (illustrated in Roig-Francoli’s Example 1).
 Roig-Francoli begins his analysis of Ramifications by partitioning each instrument’s flowing melodic line of pitches with brief durations into small repeated units, or patterns. He does not state the criteria he uses for partitioning, but seems to be following the segmentation procedures established by previous analysts for pattern-meccanico textures. Analytical procedures for segmentation in twentieth-century music typically draw on discontinuities in one or more aspects of the musical texture—rests, abrupt changes of range, separation of melodic strands, durations markedly longer or shorter than those proceeding or following, and changes in timbre or articulation. Repetition of a sequence of pitches or durations can also be used for partitioning. In Ligeti’s pattern-meccanico compositions, long rests (more than the notated beat unit), abrupt changes of range, markedly longer or shorter durations, and noticeable variation in articulation or timbre are rare, but the formation of the texture by weaving of individual instrumental parts is typical, making the initial partition the separation of individual instrumental parts. Unlike Continuum, which has continuously flowing lines, the instrumental lines in Ramifications (and other later works like the “Selbstportrait” movement of Three Pieces for Two Pianos) do include brief rests, but their duration is generally less than the notated beat unit and they do not interrupt the continuous flow of the lines. The criteria for partitioning this type of texture depends on recognition of repeated melodic units or patterns. Each unit is a series of ascending (or descending) steps or skips, which is separated from its repetitions by a skip from the highest to lowest (or lowest to highest) boundary pitches of the pattern (the largest skips in the melodic line). Roig-Francoli’s segments fit these criteria.
 After partitioning, Roig-Francoli then examines the patterns. One
innovative aspect of his analysis of the opening patterns of
Ramifications (in Example 2a) is the emphasis on the inclusion of
various partitions of the pitch interval  in the melodic
 In order to classify various types of net-structures in Example 2 and
those that follow, Roig-Francoli introduces four types of
net-structures: 1) “chromatic fluctuation of
 In his analysis of Example 3, Roig-Francoli represents his partitioning of the patterns in a “pitch reduction” graph, in which the pitches of each pattern are “stacked” in harmonies. As in the previous example, his interest lies in the partitioning of the outer span into pitch intervals rather than the voice-leading from one pattern to the next. Drawing on Bernard’s theory of trichordal relationships,(8) Roig-Francoli isolates several trichords related by infolding and unfolding (, , , and ) that he considers significant in the “middleground” structure of this passage (page 248–249). His criteria for significance are “main points of structural articulation”: beginnings and endings of points of “textural transformation,” “relatively stable sonorities,” harmonies that initiate or close “processes of chromatic expansion or contraction,” and “intervallic and spatial symmetrical designs” (page 248). Roig-Francoli defines “textural transformation” as “progressive changes of patterns or figuration in all instruments” (page 248, note 18)—in other words, a textural transformation is a passage with frequent pattern shifts or a rapid “harmonic rhythm.”
 A comparison of his reduction with the score reveals some inconsistencies. Some of the trichords he selects for his “middleground reduction” are highlighted in the musical context—for example,  as the end of an expansion in Group 2, measure 20 and  as a resting point in Group 1, measures 21–22. Others occur in the midst of an ongoing process: the “textural transformation” identified by Roig-Francoli in measures 20–21 does not stop at the  in Group 1 which he selects, but continues uninterrupted to ; likewise the s in Group 2, measures 22 and 23, and the  in Group 2, measure 24. He does not show the eventual goal of the textural transformation in Group 2 in his example—the rapid pattern shifts continue unabated until a reiterated pattern E4–B4–G5 in measure 25, a span of .(9) In addition, trichords , , and  are significant in this passage as ending and resting points (Roig-Francoli recognizes them as significant in his Example 3b), yet are not accounted for in the trichordal constellation that he discusses.
 Roig-Francoli delays consideration of his third type of net-structure (chromatic transformation of triadic units) to pages 257 and 258, and then only mentions it briefly. This “short shrift” is appropriate since Ligeti does not prioritize “triadic units” but treats harmonies with three pitches as one of the possible types of “harmonic cells.” Roig-Francoli’s use of the term “triadic” is questionable here in any case—these are trichords made from stacks of s, s, and s (perfect fifths and major and minor sixths), but they have no tonal implications, are not derived from chords stacked in thirds in this context, and are in no way functional.(10) Although Roig-Francoli’s division of Ligeti’s voice-leading techniques into the first three categories is problematic, the step-wise (but not always “chromatic”) voice-leading that he observes in each of these examples is typical of Ligeti’s style in the late 1960s.
 The fourth type of Roig-Francoli’s net-structures, “progressive change in dynamics, timbre, or rhythm,” is certainly an interesting aspect of Ligeti’s music. Unfortunately, Roig-Francoli does not deal with these elements in any detail; he provides only a brief surface description of “rhythmic layers” and changes in prevailing durations in two passages from Ramifications to illustrate this “category” (pages 250–252, 257–258). The examination of non-pitch elements can be a complex process, due to changes in the prevailing duration in various instrumental parts, harmonic rhythm (the rate of change in the pitch content of patterns), alignment of starting points of patterns (pattern interaction), dynamics, or timbre. Additionally, these elements can interact, supporting—or contradicting—the shaping of a section by range, pitch content, or other factors. One approach to the interaction of pitch and non-pitch elements is to separate out each contributor and consider its effect in the sound of the piece. But this solution does not capture the full richness of the interaction. This aspect of Ligeti’s compositions warrants further investigation.
 In addition to Ramifications, Roig-Francoli locates “net-structures” in the first movement of Ligeti’s Chamber Concerto (pages 252, 262–263) and the fifth movement of his Second String Quartet (pages 252–253, 256–257). His comments on the details of these two pieces add little to the work of previous analysts, aside from the emphasis on harmonic stacks and registral and associational links between specific pitches. Roig-Francoli’s representation of the harmonies of the closing section of the first movement of the Chamber Concerto in Example 5 suffers from a lack of precision in the distinction between pitch and pitch class, and a lack of specification of his segmentation criteria. Since the final section incorporates octave doublings of the canon in six octaves, Roig-Francoli’s Example 5 is mislabeled: those are not “pitch reductions” or “pitch collections” as he states in his text, but pitch-class collections.(11) Unlike the canon melodies of Ramifications, the canonic line here is not divided into recurring patterns. It is unclear how Roig-Francoli arrives at the scalar segments shown in his Example 5: they are not patterns or segments of the canon melody. Presumably, they derive from temporal segments, but they only roughly correspond to the pitch-class content within the segments I examined. The harmonic process is one of “subtraction and addition of single pitches” (page 252); however, close inspection of the canon melody and harmonies it creates in this section reveals that the process is not as orderly as is indicated in his example. In his discussion of this example, Roig-Francoli is interested in the “alternation of symmetrical and asymmetrical states” (page 252). As previously noted, chromatic pitch collections are always symmetrical—therefore the symmetricality in the first three segments that he shows is trivial. The alternation between symmetry and asymmetry is a natural by-product of a gradual, one-pitch-at-a-time expansion outward of chromatic clusters around a “hollow” center. That type of expansion is most logically constructed by moving one side out a little, compensating by expanding out the other side, then repeating the process. This type of systematically expanding wedge is common in others of Ligeti’s compositions, including the “Christe” settings from the “Kyrie” of the Requiem. Like this example, the Requiem does not expand with a regular alternation, while the ninth movement of the Ten Pieces for Wind Quintet does.
 The most significant contribution of this article is the discussion of possible types of longer-range harmonic structures in Ligeti’s music (pages 253–257). Roig-Francoli confronts Ligeti’s assertion that his musical forms are non-teleological and “object-like” (rather than “process-like”). He argues persuasively that Ligeti’s music is teleological, with forward-directed linear motion created by extended harmonic processes with step-wise voice-leading between contextually-established local harmonic goals. He observes that the concept of prolongation is problematic in compositions like Ligeti’s that do not exhibit the characteristics of tonal function and voice-leading, pitch organization using centricity, or other large-scale means of predicting specific goals for linear motion. In Example 6b, Roig-Francoli introduces a “long-range pitch reduction,” a type of “middleground sketch” which he states “displays the voice-leading connections between major points of formal articulation” (page 253). His graph is not intended to imply prolongation or directed motion, but instead to display associational links between specific pitches and intervals.
 As with other systems of reductional analysis, such as Schenkerian reductions, the parsing and interpretation of foreground events are crucial to middleground decisions. Details of Roig-Francoli’s graph are difficult to evaluate since the presentation of this graph is not supported by detailed analysis of each of the sections represented in the longer-range graph. For the most part, his comments (and his graph) parallel the observations made by previous analysts who provide a more detailed foreground analysis.(12)Roig-Francoli’s graph provides a means for highlighting specific pitches and locations in the piece—and this is difficult to accomplish with detail-rich range graphs (plots of pitches sounding over time). Unfortunately, the task of comparing his comments with his graph (and with the score) is made more difficult by his use of square-bracket notation to indicate both the size of intervals in semitones and the reduction of compound intervals.
 In the concluding section of this article, entitled “Formal Processes,” Roig-Francoli lists four main types of form that he asserts Ligeti “identifies among his compositions” (page 260). This is a case in point about the dangers of extracting information from interviews without careful consideration of the context. In Roig-Francoli’s presentation, these formal categories seem to be definitive and clear-cut. However, the context from which they were extracted is a wandering, informal, and internally-contradictory conversation from Ligeti’s relatively early (1971) “self-interview”—hardly the type of text from which one derives indisputable categories.(13) (Ligeti is uninclined in “interviewing himself” to insist he clarify points that are vague or unclear).
 After identifying the net-structure compositions as among
Ligeti’s “balanced, static forms,” Roig-Francoli lists the factors
that he intends to use to identify formal and sectional divisions:
“harmonic, intervallic, and spatial processes”; “rhythmic processes”;
“textural changes”; “formal articulation”; and “auxiliary factors
 This is not to imply that sectional balance (including golden section proportions at the level of the entire movement) is absent in Ligeti’s music. Roig-Francoli’s locations of the golden section proportions at the level of the entire movement are more convincing than his smaller-scale ones, and, as he notes, other analysts have located golden section proportions overarching entire movements (page 264, note 45). Lux aeterna provides further evidence that proportional balance is important to Ligeti. At the end of the third section, Ligeti has notated seven measures of rests, the precise meaning of which is not explained in the score. When Ligeti was asked about them, he replied they “depend on proportions of the durations of the piece.”(14) The addition of this “silent coda” changes the total number of measures from 119 to 126, a difference which affects the calculation of the golden section (similarly to the silent measures at the end of Ramifications), but neither calculation of the golden section corresponds to a significant structural point in the piece. One explanation of the additional measures is the creation of a large-scale symmetry in the length of sections: from the beginning to the end of each canonic section, the three sections are 37, 50, and 37 measures long (including the seven silent measures), with each pair of the canonic sections separated by one complete measure without canon. These may be the proportions that Ligeti had in mind when he added the seven measures.
 In summary, this article provides an introduction to
Ramifications, the first movement of the Chamber Concerto, and the
fifth movement of the Second String Quartet—three of Ligeti’s
compositions written shortly after Lux aeterna, Lontano, and
Continuum that have not received as much attention in the analytical
literature. Roig-Francoli makes some interesting observations about
these pieces, but his analysis would benefit from a clearer statement
of the criteria he used for partitioning, delineating categories, and
selecting significant harmonies. More attention to analytical details
would also strengthen his arguments. In prioritizing harmonies that
fit trichordal constellations and in his search for symmetries, golden
sections, and sectional parallelisms, Roig-Francoli seems willing to
overlook elisions or other continuities in the musical surface,
canonic processes and voice-leading details, and the exact content or
locations of events in his desire for regularity and order. He should
heed Ligeti’s remarks that he quotes (page 265): “I detest both absolute
geometrical precision and total openness. I want a certain order, but
an order slightly disorganized
2. The most complete versions of the dream of webs are in Ligeti’s
article “Zustande, Ereignisse, Wandlungen,” Melos 34 (1967):
165–169, translated by Jonathan Bernard as “States, Events,
Transformations” in Perspectives of New Music 31/1 (Winter 1993),
164–265 and in a footnote to an interview by Peter Varnai translated
in Ligeti in Conversation, 25 (with comments on the dream on
preceding and following pages).
5. Jane Piper Clendinning, “Contrapuntal Techniques in the Music of
Gyorgy Ligeti” (Ph.D. dissertation, Yale University, 1989); “The
Pattern-Meccanico Compositions of Gyorgy Ligeti,” Perspectives of
New Music 31/1 (Winter 1993), 192–234; “Structural Factors in the
Microcanonic Compositions of Gyorgy Ligeti” in Concert Music, Rock,
and Jazz Since 1945, ed. Elizabeth West Marvin and Richard Hermann
(Rochester: University of Rochester Press, 1995), 229–256.
6. I will use the convention (established by Jonathan Bernard in his
The Music of Edgard Varese (New Haven: Yale University Press,
1987)) of using integers in square brackets to represent pitch
intervals measured in semitones.
14. Jan Jarvlepp, “Pitch and Texture Analysis of Ligeti’s Lux
aeterna,” ex tempore 2/1 (1982), 26. Jarvlepp observes that the
silent measures are not present on commercial recordings of Lux
aeterna and would be covered in a live performance by the audience’s
The most significant sources for the works considered in this article are cited in Roig-Francoli’s footnote 1, page 242.
The most complete versions of the dream of webs are in Ligeti’s article “Zustande, Ereignisse, Wandlungen,” Melos 34 (1967): 165–169, translated by Jonathan Bernard as “States, Events, Transformations” in Perspectives of New Music 31/1 (Winter 1993), 164–265 and in a footnote to an interview by Peter Varnai translated in Ligeti in Conversation, 25 (with comments on the dream on preceding and following pages).
Ligeti in Conversation, 16–17 and 21–23.
Jane Piper Clendinning, “Contrapuntal Techniques in the Music of Gyorgy Ligeti” (Ph.D. dissertation, Yale University, 1989); “The Pattern-Meccanico Compositions of Gyorgy Ligeti,” Perspectives of New Music 31/1 (Winter 1993), 192–234; “Structural Factors in the Microcanonic Compositions of Gyorgy Ligeti” in Concert Music, Rock, and Jazz Since 1945, ed. Elizabeth West Marvin and Richard Hermann (Rochester: University of Rochester Press, 1995), 229–256.
I will use the convention (established by Jonathan Bernard in his The Music of Edgard Varese (New Haven: Yale University Press, 1987)) of using integers in square brackets to represent pitch intervals measured in semitones.
As Ligeti says in Ligeti in Conversation, 86: “polyphony is written; but harmony is heard.”
Bernard, Varese, 74–76.
There is an error in graph 3b—the final sonority should be D4–
See Roig-Francoli’s own discussion of the lack of “tonality” and related issues in Ligeti’s music on pages 253–256.
His earlier examples so labeled are actually “pitch reductions” because all instrumental parts are playing within the same limited range.
See Clendinning, “Contrapuntal Techniques,” Vol. I, 221–229, and Vol. II, 120–126.
Ligeti in Conversation, 134–135. Ligeti categorizes some of these same works differently in other interviews.
Jan Jarvlepp, “Pitch and Texture Analysis of Ligeti’s Lux aeterna,” ex tempore 2/1 (1982), 26. Jarvlepp observes that the silent measures are not present on commercial recordings of Lux aeterna and would be covered in a live performance by the audience’s applause.
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