Abstract
In this paper we propose the systemic modeling of Camargo Guarnieri’s Ponteio No.1 with the aim of identifying a hypothetical compositional system that gave rise to this work. From this compositional system we will plan a new work for woodwind trio. The model, specifically related to the harmonic syntax and the melodic gestures, is encoded into two algorithms written in Python and MATLAB.
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Notes
- 1.
It is important to mention that we are considering here an expansion of the concept of parameter: instead of being associated to surface level elements, which are closely related to a specific aesthetic profile, a parameter can be as abstract as an inversional axis, for example, which disregards tonal or atonal biases.
- 2.
Numbers 10 and 11 are represented by their hexadecimal equivalent, A and B, to avoid ambiguity.
- 3.
Peer-reviewed papers blindly evaluated by researchers in the fields of composition and theory. All these papers contain at least one piece created with the systemic modeling of another piece. Some of the new pieces were already premiered.
- 4.
The formula for the calculation of arrangements with repetitions is: \(A_{n,p} = np\) [18].
- 5.
INV(C), Inversion: inverts the sign of each element of C; RET(C), Retrogradation: realizes the retrogradation in C; ROT(C, n), Rotation: rotates the set Cn times; SUBROT(C, n), Subrotation: rotates the last three elements of C, n times; COMP(C, n), Compression: subtracts n from each element of C; MULT(C, n), Multiplication: multiplies n to each element of C; and SOMA(C, D), Concatenation: concatenates the sets C and D.
- 6.
Open-source application for editing music scores, available in http://www.lilypond.org/, visited in 02.22.2015.
- 7.
The generator set is identified as \(C_{x,0}\), in which \(x = {3,4,5,6}\), i.e., the set can be a trichord, a tetrachord, a pentachord, or a hexachord. The first value (x) indicates the set’s cardinality (how many elements has the set) and the second value (0) is simply a label to differentiate the set from the other sets used in the system.
- 8.
For example, \(T_{4}I(024) = (024)\).
- 9.
This cardinality depends on the number of common pitch-classes between the generator set and its transposed inversion. In the first gesture of Guarnieri’s Ponteio No.1 the concatenation of two trichords produced a tetrachord because there are two common pitch-classes (0 and 11). If there is no common pitch-classes the result is a hexachord, as we see in the compositional planning of the new work, in which the generator trichord (45A) will produce the hexachord (456AB0) from the same operations used in Guarnieri’s first gesture.
- 10.
This is the first movement of a piece entitled Vientos Tejanos, Op. 203 (2016), dedicated to the Vientos Tejanos Trio, from Texas (USA). The other two movements—Tejido and Siluetas—were also composed with the methodology of systemic modeling.
- 11.
See in Fig. 4 the passing \(G\flat \) connecting the G of measure 2 to the F of measure 3 (if one considers the structural harmony to be formed by the chord A-C-G-B), and the \(D\sharp \) between G and E in the fourth measure (considering the harmony to be A-C-E-G).
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Pitombeira, L. (2017). Determination of Compositional Systems Through Systemic Modeling. In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_23
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