Toward an analysis of polyphonic music in the textual symbolic segmentation

Toward an analysis of polyphonic music in the textual symbolic segmentation MICHELE DELLA VENTURA Department of Technology Music Academy Studio Musica Via Terraglio, 81 TREVISO (TV) 31100 Italy dellaventura.michele@tin.it Abstract: The listener is able to identify a melody (motif) within a "sonorous image" created during the performance of music, be it generated by a single voice (monody) or by several simultaneous voices (polyphony). The "sonorous continuum" of a musical piece, in fact, does not exhaust itself in the mere acoustic - perceptive fact, yet it has a meaning that emerges from the structural properties of the musical piece, cognitively processed by the listener and linked back to conceptual structures. This article presents a model of analysis developed so as to identify a melodic line inside a polyphonic musical structure. Study highlights the manner in which the horizontal (monody) and vertical (polyphony) dimensions of a given musical piece must be pondered, on the basis of recent considerations on a harmonic and cognitive level developed respectively by D. de la Motte and D. Huron. The algorithm conceived to this effect was tested on a set of data (scores) of small dimensions serving as a sample. Key-Words: analysis, monody, musical object, musical surface, polyphony, segmentation 1 Introduction The concept of melody or motif is well known to musicians, yet there is no theory to describe how a certain motif may be identified. In Fremdwörter- Duden 1 the item Motif reads: "1) foundation, impulse, cause... 3) the smallest musical unit". Hugo Riemann defines motif as a "melodic fragment"... which forms by itself a minimal unit possessing autonomous expressive meaning". Whereas Herman Grabner believes that: "Motif (ultimately from Latin movere) means the smallest autonomous and characteristic unit of melodic movement... It gives a decisive motor impulse to the consecutive evolution". Generally speaking, motif may be defined as a melodic fragment possessing individuality and recognizability [1]. D. de la Motte [2] valorizes this aspect by highlighting the absence of rules concerning the structural characteristics that the motif must have, even with respect to the simplest element that is its length intended as its number of bars (see fig. 1). 1 A dictionary of foreign words that is part of a German dictionary family which is well-known in Germany. a) b) c) Fig. 1: Examples of melodies. The first one is an excerpt from Die Schöne Müllerin by Schubert: here the singer is involved in three bars in which melody flows without returning elements. The second melody, made up of only three bars, is an excerpt from a Piano Sonata by Haydn. The third melody, made up of 10 bars, is an excerpt from Sonata No. 1 op. 2 by Beethoven. A second equally important aspect that D. de la Motte highlights in his analyses is the one concerning the theory of harmony that draws its foundation from the polyphonic aspect of the musical text. In this field, certain rules (in western music) remain valid throughout time: many aspects of these rules change, upgrade, get old, get modified, but their ISBN: 978-1-61804-194-4 19

bases, their general principles remain virtually unchanged [3]. Harmonic analysis 2 allows one to see the musical piece divided into movements every single one of which characterizes a degree of the scale (depending on the existing tonality) [4]: inside every single movement, all the sounds existing in the various voices (taken into consideration simultaneously), may belong to the chord built on that particular degree or may be melodic figurations [5] which in such case may be ignored precisely because of the role that is assigned to them [6]. The harmonic structure is going to be one of the basic aspects of the method of analysis of polyphony proposed in this article, together with the cognitive aspects that characterize human perception. D. Huron presents a set of principles [7] in this respect, specifying that the polyphonic rules lead one to perceive voices as distinct and therefore play an important perceptive role that introduces the alternation of entrance turns and the difference of the interval and rhythmic profiles among the various voices. This paper is organized as follows. Section 2 describes the modality of representation of the score. Section 3 explains the harmonic and cognitive principles used for the proposed method. Section 4 shows an experimental test that illustrates the effectiveness of the proposed method. Finally, conclusions are drawn in Section 5. 2 Representation of the score The representation of the score is a fundamental point in the field of assisted musical analysis: which is seen as a list of numbers, every single one of which corresponds to a sound, based on its actual pitch [8]. Taking into consideration the piano keyboard, which is the instrument having the largest extension, the lowest note (A) is assigned the value 1 and every single successive sound is assigned an increasing value (A# = 2, B= 3, C= 4, and so on) (see fig. 2). 2 In music, harmonic analysis is the study of vertical sonorities and their connections, and is paramount to the understanding of tonal compositions. The analyst may find the chord roots, label sonorities with proper chords names (such as A minor ), and identify their relationships using roman numerals. Fig. 2: Numerical representation of sounds. Most of the systems of automatic analysis of the score for the identification of a melody or a motif, with a view to simplifying the problem, limit the field of study to monody: a non simultaneous note sequence. The musical sequence may in this case be represented under the form of a sequence S m of N notes n i indexed on the basis of the order of occurrence i: S ) m = ( n i i N [ 0, 1] A polyphonic score that presupposes the simultaneity of several notes may therefore be seen as an overlapping of multiple monodic sequences S m1, S m2, S m3,... Ergo the possibility to represent the score as a monodic sequence S p in which the notes of the various voices, are placed, in succession, one after another (see fig. 3). S p = ( S m1 + S m2 +...) Fig. 3: Representation of a polyphonic score. Another way to represent a polyphonic score, that lies at the basis of the analysis model presented in this article, is by means of a matrix [5] that contemplates simultaneously the different structural aspects of the musical piece, that is: - the number of voices, every single one of which will be represented as a sequence S m of notes; - the number of movements, a default value in the concept of musical rhythm. It is basically a number that depends on the initial tempo of the musical piece (4/4, 6/8,...) and therefore on the total number ISBN: 978-1-61804-194-4 20

of movements. Every single movement will be divided in its turn into several parts according to the smallest duration existing in the musical piece [1]. The duration is intended as the time interval during which the sound can be perceived, regardless of whether this is referred to a single sign or several signs joined to one another by a value tie (ligature). This latter principle allows us to surpass musical concepts such as "accent", "meter" or "measure". Given a score A, it may be represented, by means of a matrix, in the following manner: A x, y where x (number of rows) equals the number of voices and y (number of columns) equals the number of rhythmic movements existing in the musical piece (fig. 4): every element of the matrix shall represent a sound, expressed by means of a number that identifies the pitch of the sound. certain models of the functioning of the mind involved in analysis and in creation of sonorous structures, submitting them to empirical control [11]. While listening to a musical piece, the listener performs a segmentation of the "sonorous continuum", seizing its contrasts and assonances: the groupings of sounds are enclosed in rhythmic and melodic schemata within which themes, motifs, repetitions and variations emerge [1]. The instruments of polyphonic analysis, summarized below, were developed on the basis of all these considerations: 1) a sound that is repeated several times in succession (without it being alternated with other sounds), is perceived as a single sound: the mind automatically starts to search in the set of sounds a more pleasant melodic line (fig. 5); Fig. 5: Intermezzo op. 119 n. 3 by J. Brahms Fig. 4: Example of a polyphonic score and its related representative matrix. 2) an obstinate, that is a melodic figure repeated incessantly without changes of pitch and rhythm, is considered in the same way as repetition (fig. 6); 3 The principles of musical analysis Musical analysis always presupposes a process of segmentation 3 of the composition [9], but the criteria of such an operation are often laborious. Musical analysis and psychology traditions propose different solutions to this issue. The comparison between the two traditions leads to reflection on a wider subject: namely the relationships between musical theory and cognitive psychology. Music psychology has significantly evolved, overcoming nowadays the angst of the psychoacoustic paradigm and updating certain intuitions of the Gestalt school of psychoanalysis, even by means of new conceptual instruments made available by cognitivism. The attention was drawn, particularly, on certain thought processes implied by the fruition of sounds: perception, memory, hierarchization, abstraction, organization [10]. The cognitive psychology of music outlined Fig. 6: Little piece by L. Ledenev 3) the accompaniment of a melodic line, often created by an arpeggio, is assimilated to an obstinate even if unlike the latter it may be subject to variations of sounds (and therefore of pitch) because of the modulations (transitions from one tonality to another): the rhythmic element, as in the obstinate, remains unchanged and therefore the perception is of a repetitive thing (fig. 7); Fig. 7: Sonata K545 by W.A. Mozart. 3 I. Bent: Musical analysis is the division of a musical structure into relatively simpler constitutive elements and the study of the functions of these elements within this structure. 4) the melodic figurations are not enough (in most cases) to make an accompaniment different: on a ISBN: 978-1-61804-194-4 21

cognitive level and based on the interpretation of the musical piece, the listener succeeds in recognizing a previous accompaniment even if it is modified by means of melodic figurations (fig. 8); the basis of the characteristics of the analyzed musical piece. The results of the analysis are indicated in diagrams that facilitate their evaluation. The results of the analysis of two musical pieces are illustrated below: - Intermezzo op. 119 No. 3 by J. Brahms: - Rhapsody op. 79 No. 1 by J. Brahms. Fig. 8: A melodic passage excerpted from "Knecht Ruprecht" belonging to R. Schumann's Album for the Young. 5) the harmonic structure characterizes the entire musical work and therefore its every single movement: hence the sounds that may be written by a composer in a movement must belong exclusively (unless melodic figurations are used) to the chord built on the degree of the scale that is represented in that specific movement (fig. 9). It follows that if in a movement all the sounds belong to the harmonic structure of the related degree of the scale, they represent an accompaniment and may be treated as it can be seen above. Intermezzo op. 119 n. 3 by J. Brahms This composition (fig. 10) exhibits a type of writing that is entirely chordal: the perceived melody is located in the middle of the polyphony, almost hidden by a continuous repeat of the exact same note in the high-pitched voices (right hand) and accompanied by an arpeggio on the bass (left hand). Fig. 10: Intermezzo op. 119 No. 3 by J. Brahms with its associated matrix representing the score. V Fig. 9: Rapsodia op. 79 n.1 by J. Brahms The simplification of the score is performed by eliminating, first of all, the repeated notes (fig. 11). The score, seen as a big matrix, is gradually simplified by eliminating the sounds that are not necessary on the cognitive and harmonic level (as shown above): by examining the melodic evolution of the notes left in the various voices it is possible to identify (even graphically) a motif (or melody). 4 The results obtained The model of analysis set forth in this article was verified by realizing an algorithm the structure of which takes in consideration each and every single aspect described above: the algorithm does not provide any limitation with respect to the dimensions of the matrix representing the score that will be automatically dimensioned in every single analysis on Fig. 11: Elimination of the sounds of the repeated notes and the associated score. Afterwards the degrees of the scale are identified (Fig 12a), then the sounds that belong to the chord eliminating the ones that occur several times (Fig. 12b). ISBN: 978-1-61804-194-4 22

a) b) I I III VI IV The analysis operations performed by the algorithm are summarized below: - harmonic analysis by grouping the sounds of every single movement (for the construction of the chords related to the represented harmonic degree) (fig. 14a), - elimination of the sounds that, in one and the same voice and (rhythmic) movement, make up the representative chord of the degree (fig. 14b); - identification of the melodic line (fig. 14c). a) Fig. 12: Identification of the harmonic degrees and elimination of the sounds on the basis of the harmonic structure of every degree. Finally, the evolution of the remaining notes is analyzed: Musically, a motif or a melody is characterized (as mentioned in the introduction) by a non horizontal evolution which nevertheless creates a sense of motion (Fig. 13). Fig. 13: Graphic representation of the two melodic lines that may be identified after the simplification of the score. The second melodic line contains more movement and is identified as melody. Rapsodia op. 79 n. 1 by J. Brahms This composition presents 2 equal sections (A and A1) separated by a central lyric section (B) and a conclusive coda (fig. 14) in which there emerges, concealed, the melody presented in the central part (B). b) c) Fig. 14: Stage of analysis of the Rhapsody op. 79 No. 1 by Brahms. 5 Conclusion The tools presented in this article, developed on the basis of specifically musical and didactic objectives, represent a means of support for the melodic analysis within the field of automatic segmentation of a musical piece: the high degree of complexity of musical phenomena imposes certain forms of achievement that must be adequate and that, for completeness' sake, must cope with the problems under a sufficiently large number of angles. Although the results obtained are very satisfying, it is appropriate to draw attention to a limit of this method due to the fact that it may be applied only to tonal compositions and not to contemporary music, on which harmonic analysis by degrees is impossible to perform. Further developments of this methodology might allow a synthetic analysis among different compositions, for the study and the identification of the ''musical form'', that is the schema according to which a certain composition is articulated and that reflects its divisions, its succession and the development of the themes, the rhythmic and harmonic structure. Fig. 14: Rapsodia op. 79 n. 1 by J. Brahms.. ISBN: 978-1-61804-194-4 23

References: [1] M. Della Ventura, Rhythm analysis of the sonorous continuum and conjoint evaluation of the musical entropy, in Proceedings of the International Conference on Acoustic & Music: Theory & Applications (AMTA 2012) (pp. 15-21), Iasi, Romania. [2] D. de la Motte, La Melodia, Astrolabio MMX, Roma, 2010. [3] M. Baroni, R. Dalmonte e C. Acoboni, Le regole della musica: indagine sui meccanismi della comunicazione, E.D.T. Edizioni di Torino, 1999. [4] B. Coltro, Lezioni di armonia complementare, Zanibon, 1979. [5] D. de la Motte, Manuale di armonia, Bärenreiter, 1976. [6] M. Della Ventura, The concealed motif: analysis and identification (AIKED 2013) (pp. 44-48), Cambridge, UK. [7] D. Huron, Tone and Voice: A Derivation of the Rules of Voice-leading from Perceptual Principles, Music Perception, Vol. 19, No. 1 (2001) pp. 1-64. [8] M. Della Ventura, Analysis of algorithms implementation for melodical operators in symbolical tectual segmentation and connected evaluation of musical entropy, in Proceeding of The International Conference on Models and Methods in Applied Sciences, Drobeta Turnu Severin, 2011, pp. 66-73. [9] I. Bent, W. Drabkin, Analisi musicale, EDT, Turin, 1990. [10] E. Narmour, The analysis and cognition of basic melodic structures, The University of Chicago Press, Chicago and London 1990 [11] F. Lerdahl, R. Jackendoff, The generative theory of tonal music, The MIT Press, Cambridge Mass. 1983. ISBN: 978-1-61804-194-4 24