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.
 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
 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" (p.
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.
 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 (p. 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 . . ." (p. 246). In the
following paragraph, after he has eliminated pieces from Ligeti's
broad category of "net-structure," Roig-Francoli does the same with
the term "meccanico" (p. 244), a term which Ligeti uses in reference
to an anecdote about a story of a widow with a house full of ticking
clocks. Ligeti uses this term in a general way, to describe music
with a sense of mechanical action that is reminiscent of
 As Roig-Francoli notes, however, "Ligeti's use of technical terms
descriptive of his music is not always consistent" (p. 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
 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 Ex. 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 Ex. 2 and
those that follow, Roig-Francoli introduces four types of
net-structures: 1) "chromatic fluctuation of. . . short melodic
patterns," 2) "chromatic transformation of harmonic cells," 3)
"chromatic transformation of triadic units" and 4) "progressive change
in dynamics, timbre, or rhythm" (p. 246). However, his examples show
that the first three of his categories actually involve the same
compositional procedure: step-wise voice-leading (usually involving
half-steps) that achieves gradual transformations in the harmonic
content of the music. In regard to his first two categories, the
example he cites for his Category 2 (his Ex. 3) has a microstructure
of rapidly reiterated short melodic patterns just like his example for
Category 1 (his Ex. 2). The "melodic patterns" are the source of the
harmonies involved in the "chromatic transformation of harmonic
cells." The only difference in the two is the size of the pitch
intervals involved in the patterns: Roig-Francoli limits Category 1 to
patterns spanning no more than  with no more than  between
adjacent pitches. Although he does not explain the reasons for these
limitations on Category 1 (nor does he explain the lack of specificity
regarding acceptable intervals for Categories 2 or 3), it is possible
that Roig-Francoli makes the distinction because the patterns with
larger intervals between successive pitches are more like
"traditional" harmonic arpeggiations. However, there is fluidity
between the harmonic dimension and the melodic dimension in many of
Ligeti's compositions, and the boundaries between perception of melody
and harmony were elements that Ligeti was exploring in his
compositions at the time these pieces were written.
 In his analysis of Ex. 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
 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, m. 20 and
 as a resting point in Group 1, mm.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, mm.22 and 23, and the  in Group
2, m.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 m.25, a span
 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.
 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" (pp.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
(pp. 252, 262-263) and the fifth movement of his Second String
Quartet (pp. 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 Ex. 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 Ex. 5 is
mislabeled: those are not "pitch reductions" or "pitch collections" as
he states in his text, but pitch-class collections.
 The most significant contribution of this article is the discussion of possible types of longer-range harmonic structures in Ligeti's music (pp. 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 Ex. 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" (p. 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.
 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" (p. 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
 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 . . . such as instrumentation and dynamics" (p. 260). From examining Figures 2 and 3, it seems that formal balance and golden section calculations have influenced his choices. For example, a comparison of Roig-Francoli's discussion of the formal outline of Ramifications with Figure 2 and the score, reveals that his comments acknowledge elisions, overlapping processes, the precise locations of events, and the continuous nature of each main section, but his chart glosses over those "messy details" to present a parallelism between sections and a correspondence of events to golden section proportions that is more tenuous in the music than it appears on the chart (p. 260-262). Roig-Francoli's chart of the subdivisions of the second large section in the Chamber Concerto suffers from the same flaw (pp. 262-263). The precision of golden section calculations in that work is further disturbed by the presence of varying measure lengths and tempi and by unmeasured cadenza-like passages. Comparing Roig-Francoli's calculations to those made by approximating the length of time that each measure would last in seconds indicates that some of his locations may be "off" by as much as two measures.
 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 (p.
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
 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 (p. 265): "I detest both absolute geometrical precision and total openness. I want a certain order, but an order slightly disorganized . . .I love irregularities" and "prima la musica, dopo la regola."
1. The most significant sources for the works considered in this
article are cited in Roig-Francoli's footnote 1, p. 242.
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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).
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3. Ligeti in Conversation, 16-17 and 21-23.
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4. Ibid., 16.
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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.
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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.
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7. As Ligeti says in Ligeti in Conversation, 86: "polyphony is
written; but harmony is heard."
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8. Bernard, Varese, 74-76.
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9. There is an error in graph 3b--the final sonority should be
D4-C#5-A5, as shown in graph 3a, making a .
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10. See Roig-Francoli's own discussion of the lack of "tonality" and
related issues in Ligeti's music on pages 253-256.
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11. His earlier examples so labeled are actually "pitch reductions"
because all instrumental parts are playing within the same limited
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12. See Clendinning, "Contrapuntal Techniques," Vol. I, 221-229, and
Vol. II, 120-126.
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13. Ligeti in Conversation, 134-135. Ligeti categorizes some of
these same works differently in other interviews.
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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
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