Music review — Federal University of Bahia
[email protected] – Universidade Federal do Rio de Janeiro
In this article, we investigate a compositional methodology called systemic modeling, which allows us to propose hypothetical compositional systems for musical intertexts through analysis and generalization of parametric properties. We will use this methodology to plan and compose the first movement of a trio for violin, clarinet, and piano, entitled Brazilian Landscapes No.15, based on the composition systems of Guarnieri’s Ponteio No.6.
Research on systemic modeling applied to music composition has been extensively conducted by the author and his undergraduate and graduate students since 20111. In this paper, the Ponteio No.6 by Brazilian composer Camargo Guarnieri (1907-1993) is modeled, producing as a result a compositional system, from which the first movement of a trio for violin, clarinet, and piano, entitled Brazilian Landscapes No.15, will be planned.
In the first part of this article (section 2), we define the concepts of systemic modeling, compositional system, and parametric generalization. In the second part (sections 3, 4, and 5), after a brief historical and theoretical contextualization on Guarnieri’s Ponteios, we analyze the Ponteio No.6 in order to propose the hypothetical compositional system underlying it. Finally, based on this system we plan and compose the new work.
“A model is defined as a simplified representation of a real system with the aim of studying this system” (Mororó, 2008: 27). For engineers, a model consists of a prototype and a mathematical description of the properties and operation of the modeled system. For the musical field, we propose that a model of a specific work is the set of its musical objects (lexical domain) and their interrelationships (syntactical domain). We are particularly interested in those relationships, i.e., our focus is on a type of non-exhaustive model that, even though describes perfectly the way a piece of music works under certain parametric perspectives, is generalized enough to produce new works, which are similar to the original exclusively in a deep level. This deep level is what we call a compositional system. Formally, a compositional system can be defined as “a set of guidelines, forming a coherent whole, which coordinates the use and interconnection of musical parameters, in order to produce works” (Lima, 2011:65). This concept loosely derives from the studies in General Systems (Bertalanffy, 2008; Klir, 1991; and Meadows, 2008). A compositional system derived from systemic modeling is always hypothetical (since we can hardly be aware of the composer’s intentions) and can be expressed as a series of definitions, diagrams, and/or computational algorithms.
Besides the theory of compositional systems, another fundamental field for the methodology of systemic modeling is the theory of intertextuality, which studies the various levels of influence of a text (or texts) onto another text. Julia Kristeva proposed that a “text is (…) a permutation of texts, an intertextuality: in the space of a given text, several utterances, taken from other texts, intersect and neutralize one another (1980:36). Even though having coined the term, she recognizes that this idea is “in fact an insight first introduced into literary theory by Bakhtin: any text is constructed as a mosaic of quotations; any text is the absorption and transformation of another” (1980:66). Kristeva also mentions the similarity between musical and literary universes, which makes it possible to apply the theory of intertextuality to the musical field. This has been significantly examined by Klein (2005), Straus (1990), and Korsyn (1991).
Systemic modeling is, therefore, an epistemological convergence of the theory of compositional systems, as proposed by Lima and Pitombeira (2010), with the theory of intertextuality. Central to this methodology is the concept of parametric generalization, which consists of disregarding particular objects, associated with specific musical parameters, and taking into consideration only the relationships amongst those objects. Systemic modeling has three methodological steps: in the first step, called parametric selection, a prospective assessment is undertaken in order to determine the best parameters to be examined; in the second step a comprehensive analysis takes place, which aims at identifying the objects and relationships; and, finally, in the last step, called parametric generalization, only the relationships identified at the analytical step remain.
The excerpt of Figure 1 illustrates measures 4 to 6 of Bartók’s Fourth String Quartet. The horizontal set class2 A, with prime form , is spread through out-of-phase entrances of the four instruments (starting with the cello) resulting in the vertical set class B, with prime form , which is already the vertical set formed by the first notes of each appearance of A. Set class A is a subset of set class B, and Set class C, at the beginning of measure 6, is the result of a Rahn multiplication3 of A by the constant 2. This would be a possible model of the excerpt as a result of an analysis that only takes into consideration the pitch parameter.
Figure 1: Excerpt from Bartók’s Fourth String Quartet.
A parametric generalization of this model will produce a compositional system shown in Table 1. As one can see, it consists of a set of three definitions that contemplates only the relationships amongst three generic set classes. In other words, the specific values associated with the pitch parameter were generalized.
Table 1: A possible compositional system for the excerpt of Bartók’s Fourth String Quartet shown in Figure 1.
From that compositional system (Table 1) one can plan and compose a new fragment, which will keep the same internal relationships of the original Bartók’s fragment. For set class A we will chose , which is a subset of set class B, . Set class C, , is the result of B times the constant k = 2. Figure 2 shows a possible musical realization.
Figure 2: New fragment composed with the composition system derived from the excerpt of Bartók’s Fourth String Quartet.
Guarnieri’s Ponteio No.6 is part of a series of fifty Ponteios organized into a five-volume collection. This is the sixth Ponteio of the first book, published in 1955. It is a short piece of 39 measures, whose first gestures are shown in Figure 3, with an ABA’+Coda structure (Table 2). Such structure can be roughly understood as the overlapping of three separate layers: an upper layer with the main melody, predominantly in octaves; an intermediate layer consisting of a moto perpetuo mostly in sixteenth notes plus an intermittent auxiliary line; and a lower layer consisting of a bass line that appears mostly as parallel perfect fifths in section A. If we break down the texture into separate lines, some passages reveal themselves as fairly complex, presenting sometimes a density-number 8, as we can see in the last measure of Figure 3.4 Figure 4 shows this breaking-down procedure for phrase a1.
Figure 3: First gestures, corresponding to phrase a1 of Guarnieri’s Ponteio No.6.
Figure 4 shows our tonal harmonic analysis for the passage considering an expanded version of the Bb minor scale, which includes inflections on the sixth and the seventh scale degrees. It is a tonal passage ending with a deceptive cadence in VI, or R.5 The notes with a red + sign below them do not belong to this scale. As one can see, the great majority of the pitches used in the melodic lines (first and third) and in the moto perpetuo line (fourth) belong to the referential scale.6 The passages within the red rectangles (second and fifth lines) will not be considered in the modeling because they just complement harmonically and rhythmically the other lines. Another important information is the interval between the lowest and highest pitches of the sixth line and also the interval between the first pitch of the moto perpetuo line and the lowest pitch of the sixth line. For the entire section A these intervals are predominantly third (major or minor) and perfect fifth, which means they produce major and minor triads. The harmonic analysis of the entire A section is shown in the Table 3. The last line of this table (harmonic profile) will be very significant to determine the system.
Figure 4: Phrase a1 of Guarnieri’s Ponteio No.6 broken down into separate layers and analyzed.
Figure 5 shows an excerpt of section B, from measures 11 to 18. The texture is much more simple compared with section A. Here, we basically have three lines: 1) Melodic line, in the right hand of the piano, mostly in octaves7; 2) A moto perpetuo line in the upper part of the left hand; and 3) A bass line, consisting of only a single note. For this section, we propose a reduction considering only the first notes of each measure, except for the last four measures (25—28), which clearly will have a more defined tonal design if we consider the second note. The result of this reduction is shown in Figure 6, which also has detailed indications of the intervallic structure. Such structure, summarized in Table 4, represents the harmonic-melodic model for the passage.
Table 2: Structure of Camargo Guarnieri’s Ponteio No.6.
Table 3: Harmonic analysis of section A of Camargo Guarnieri’s Ponteio No.6.
Figure 5: Excerpt of section B of Guarnieri’s Ponteio No.6.
Section A’ is, for the first three measures (29—31), equals to Section A, except for one different pitch in the moto perpetuo line. Measure 32 does not correspond to measure 4. Instead, the melodic line is the same of measure 31, except for the last note, which is a minor second up. Section A’ ends with a grace ascending arpeggio built on consecutive thirds (major and minor), ending with a note that corresponds to the root of R (Fá), enharmonically spelled.
The Coda consists basically of two gestures. The first one is a descending arpeggio built on the juxtaposition of a Db minor triad and a  trichord. The arpeggio’s cell is transposed octave down four times and the last one is incomplete. The second gesture is basically an augmentation of the first five melodic notes, with a passing note before the last note. A two-voice homorhythmic passage built on parallel major thirds accompanies the melodic line. The piece end in a Bb minor chord.
Figure 6: Reduction of section B of Guarnieri’s Ponteio No.6.
Table 4: Harmonic profile of section B of Camargo Guarnieri’s Ponteio No.6.
The analysis of Guarnieri’s Ponteio No.6 carried out in the previous section will allow us to define a suitable compositional system for the piece. This system, shows in Table 5, is designed in the form of a series of definitions derived from generalizations of the parametric properties observed in the piece.
Table 5: A compositional system for Guarnieri’s Ponteio No.6.
Based on the compositional system shown in Table 5, we will plan the first movement of Brazilian Landscapes No.15, entitled Incelença. We started by planning the harmonic skeletons of both Sections A and B. Following the 4th definition, we created the harmonic progression showed in Figure 7, and following the 5th definition, we generated the harmonic profile for section B, considering the amount of 40 chords. The resulted harmonic progression derived from Table 6 is shown in Figure 8. With these materials at hand, we started composing Incelença, the first movement of Brazilian Landscapes No.15. The first eight measures of section A is shown in Figure 9.
Figure 7: Harmonic scheme for Section A of Incelença.
Table 6: Harmonic profile for Section B of Incelença.
Figure 8: Harmonic scheme for Section B of Incelença.
Figure 9: First eight measures of section A of Incelença.
In this article, we have introduced the basic concepts of a methodology called systemic modeling, which has demonstrated to be a possible and effective method for the manipulation of intertexts. Through formal and harmonic analysis, we were able to propose a hypothetical compositional system, i.e., a model that regulates the original text in terms of objects and their relationships. This system consisted of a series of definitions exclusively associated with formal design and the manifestations of the pitch parameter, in terms of vertical and horizontal sonorities. Having defined the compositional system for Ponteio No. 6, by Camargo Guarnieri, we were able to plan the first movement of a new piece, keeping relevant deep features from the original text. One important conclusion of this experiment is the aesthetic dissimilarity between the two pieces (the intertext and the new composed work), even though their systemic correspondence in the deepest level. The aesthetic dissimilarity can be perceived especially in the harmonic and textural levels: while Guarnieri’s piece is more harmonically and texturally complex, the new piece sounds more tonal-related and its layers are more clearly defined. We should emphasize that the systemic modeling neither has the intention of describing a general model for the entire output of a composer nor has interest in exhaustively describe a specific work, which is already established. The intention of this methodology is to reach deep levels under specific parametric perspectives (pitch, contour, texture, etc.). Another conclusion is the pedagogical use a composition teacher can make of this methodology. On one hand it encourages acquaintance with the language of other composers and, on the other hand, it allows a steady growth of the students’ own compositional voice through the exercise of pre-compositional reflection.
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1 (Moraes, Pitombeira, Lima, Castro-Lima, Mesquita, Oliveira, Silva, Usai e Kühn, 2011, 2012, 2013, 2014, 2015 e 2016). These papers can be freely downloaded at http://ufrj.academia.edu/LiduinoPitombeira. The author presently develops two research projects on systemic modeling at the School of Music of the Federal University of Rio de Janeiro, as a member of the MusMat Research Group (http://musmat.org).
2 A set class is a set of pitch-classes sets related to each other through transposition (Tn) or inversion (TnI). For a more detailed explanation see Straus (2002). In this paper, instead of Forte’s number (STRAUS, 2000) we identify set classes by their prime form inside brackets.
3 Rahn multiplication (RAHN, 1987:53-55), in contrast with Boulez multiplication (STRAUS, 2000:197-202), is just the arithmetic multiplication of a pitch-class set by an integer.
4 The concept of density-number comes from Berry (1987:204) and refers to the number of components involved in a texture. In the last measure of Figure 3 there are eight individual components, some of them independent and others interdependent, with respect to rhythm. This is easily visible in the last measure of Figure 4: first and third lines are interdependent between each other, second line has two interdependent components, fourth and fifth lines are independent, sixth line has two interdependent components.
5 This nomenclature comes from Kopp (2002:146) and refers to both diatonic mediants: the root of R, which is a major triad, is a major third (down in minor keys and up in major keys) from the tonic and the root of r, a minor triad, is a minor third (up in minor keys and down in major keys) from the tonic.
6 From the 57 pitches, only 11 do not belong to the referential scale.
7 Some notes are not doubled in octaves for easiness of execution.