Influence of the harmonic/functional analysis on the musical execution: representation and algorithm

MICHELE DELLA VENTURA

E-learning Assistant Conservatory of Music “A. Buzzolla”

Viale Maddalena 2 ADRIA (RO) 45011 Italy

dellaventura.michele@tin.it Abstract: The importance of the analysis in the execution of a musical piece is a concept that is shared by most of the interpreters and scholars: highlight hierarchical relations, deep structures, motivic - thematic aspects and the like, it is necessary for a better comprehension of the ”musical logic” of the piece. The composer gives a definite shape to his ideas, configuring and organizing the musical material. The executant reproduces that particular shape in order to convey the ideas of the composer, yet he fails in doing his task if he does not manage to understand how the shape is organized: it is the very structure of the music, re-sulting from its melodic, rhythmic, harmonic and dynamic component that determines at the same time shape and character. The identification of the melodic, rhythmic or harmonic cells is a first and fundamental step for the interpreta-tion of a musical piece, yet not enough in order to confer it the right character intended by the composer: it is the latter that gives us a significant indication in this respect, by means of the harmonic structure, particularly by means of the “function” that every single harmony (or chord) performs in the context in which it is inserted. In this article I will present a model of harmonic analysis able to explore progressively the symbolic level of the musical text, identifying its harmonic structures and by means of the “function” carried by every single one of these, provide useful information for the dynamics. The efficiency of the model was verified through the analysis of various musical pieces, by different authors and from different times trying to range over various styles. Key-Words: harmony, harmonic function, segmentation, analysis, Musical Object, Musical surface 1 Analysis and execution One of the recurrent themes within the framework of the musical execution is the necessity ratio exist-ing between analysis and execution [1]: the analysis has the task of making tangible (i.e. listenable) all the ratios, all the aspects of the musical nexus, of the contrast and of the construction lying at the heart of the sonorous image (deriving from the execution) [2].

Fig. 1: Prelude in E-flat Minor BWV 853 by J.S. Bach (the first four beats). Fundamental rhythmic and melodic cells generating thematic and timbral struc-tures within the entire piece. The identification of rhythmic or melodic elements (see fig. 1), also called pre-thematic because they

form together the various musical phrases, allows the executant to give a shape to the musical piece [3]. Melody and rhythm are, therefore, tightly knit with each other: the former represents meaning developed from a succession of sounds, while the latter is the shape and the proportions of that spe-cific succession [4]. Hence, the rhythm confers individuality and recog-nizability to the melody (see fig. 1) [5], but at the same time it characterizes the entire musical piece by means of rhythmic cells, that is groups of signs (or signs and rests) which create within the musical discourse recurrent rhythmic, even different at a me-lodic level, schemes (see fig. 2 and 3) [6].

Fig. 2: Prelude in E-flat Minor BWV 853 by J.S. Bach. Rhythmic cell C of the 4th beat repeated sev-eral times on different degrees

A

B

C

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Fig. 3: Prelude in E-flat Minor BWV 853 by J.S. Bach. Rhythmic cell (3) of the 4th beat repeated several times, contrariwise, on different degrees. How to interpret these rhythmic/melodic elements belonging to the musical piece? Do they all have the same importance? How can they give shape to the musical piece? One of the fundamental criteria, at the level of execution, is that of using different levels of inten-sity between elements or different voices [1]. With intensity we have already entered the field of val-ues that are not defined in absolute terms in the score: the sign “f”(forte) means, for example, that a certain sound must be executed with a superior dy-namics than the one marked with “p”(piano), but it does not give us precise information with respect to the actual decibels that need to be produced. Being precise in this field, relating to the dynamics is one of the main tasks of the interpreter who must scale its values by carefully studying the score and the semiographic habits of the com-poser [7]. The musical analysis provides, by means of the study of the harmony, a corpus of rules studied and extrapolated a posteriori by the operas of the authors [5]: it is a cognitive tool to be used in order to ana-lyze a particular musical piece. Although extremely useful, such knowledge basis is to be used with both thoughtfulness and caution, because of its intrinsic non-univocalness.

2 Functional harmony In the functional theory, the goal is to identify in a sound, a chord or a chord succession, the “intrinsic sonorous value” assumed, compared to a specific reference system polarized in a center, or the ca-pacity to establish organic relations with other sounds, chords or chord successions of the same sys-tem [8]. The functional theory tends to go beyond the sono-rous event as it manifests itself, to interpret what lies behind that which appears in a particular instant, to seize the meaning, the “role” that it covers in com-parison to other events that come before and after it,

therefore the “function” that it performs in the con-text within which it is immersed. In particular, as far as the chord is concerned, the functional theory tends to research, beyond what it represents by itself in comparison to a certain refer-ence system (for instance, the chord G-B-D, com-pared to the tonal system and the tonality of C Major, is the dominant chord), the harmonic func-tion performed, the organic relation established with the one that comes before and the one that comes after it. The pillars of the functional theory are the har-monic functions of tonic (T), subdominant (S) and dominant (D), that Riemann was the first to identify as the foundation and pivot of any type of chord succession, hypothesizing in the connection I-IV-V-I (fig. 4) the archetype of the tonal harmony and the model which any type of chord concatenation should be traced back to (fig. 5).

Fig. 4: Archetype of the tonal harmony according to Riemann.

Fig. 5: Riemann’s analysis of Beethoven’s Piano Sonata n° 1 op. 2

It follows that all the chords will have a harmonic function of relaxation or of tonal center T, or of tension towards such center D, or of breakaway from it S. The three harmonic functions of I, IV and V de-gree are called main because they are linked by a relation based on the interval of the perfect 5th that separates the keynotes of the three corresponding chords; the chords relating to the rest of degrees on the scale are considered “representatives” of the I, IV and V degree (with which there is an affinity of the third - two sounds in common - because the 3rd is actually the interval that regulates the distance be-tween the respective keynotes) and secondary har-monic functions rest with it.

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The identification of these harmonic functions within the various rhythmic/melodic cells and the rela-tionship that every single one of them has within the context in which it is inserted (with what comes be-fore and after it), allow the executant to have an im-age of the character of the piece so as to better define the field of dynamics.

3 Harmonic operator In order to be able to compare the different musical phrases with one another, so as to determine the im-portance of every single one, it is necessary, first of all, to locate the modulations within the musical piece, that is the passages from one tonality to an-other tonality (identifying the characteristic notes, or, to be more precise, the notes that make the two tonali-ties different, the one that is being left behind and the one towards which the executant modulates) [9], so as to define the various tonal areas. Afterwards it is possible to identify the tonal func-tions of the various chords contained in every single one of them. To this purpose, an operator is used for the recogni-tion of the tonal function (TFR): such operator, given a certain chord, should interpret the degree of the scale on which it is built, and draw conclusions with regard to the reason for which it was built in that specific position [10] [11]. This last stage, although expressed synthetically, re-quires a global countercheck, therefore this opera-tor will be asked to perform the demanding task of recognizing the harmony represented by the argu-ment and synthesize it in a musical meta-language: T (tonic), S (Subdominant) and D (dominant). It is defined as a tonal function recognition opera-tor and it is designated by

{ }DSTPTFR ,,)(: →∑

an operator defined as follows:

TFR(A) = T if A is a Tonic chord TFR(A) = S if A is a Subdominant chord TFR(A) = D if A is a Dominant chord

As it can be inferred, once in possession of a note set, resulting from the vertical disposition of the score, the work of the TFR operator consists in eliminating the doubled sounds [12] and then order the degrees existing in the chord in order for a well-defined degree, that will be recognized as Harmonic Bass, to follow and interval of a third, a fifth, a sev-enth.... (Fig. 6).

In order to decide which note is to be considered a Harmonic Bass, a simplest harmonic rule is there to help us, a rule that decrees how every single harmony leans on the lowest note from an acoustical perspective, therefore, of a minor pitch.

Fig. 6: The first system shows the first two beats of the chorale “Es ist das Heil uns kommen her” by J.S. Bach; in the second system the same chords are represented without the doublings of the sounds; in the third system the chords in the fun-damental state are represented. Immediately afterwards, all the intervals generated by other notes against the bass, thus building the various chords that are correlated to their functional significance, relating them to the tonality of the tonal area they are in [13]. 4 The results obtained The analysis model exposed in this article, was veri-fied by realizing an algorithm, the structure of which takes into consideration every single aspect of the ones described above, as well as the nature of the data: the results of the analysis are indicated by means of charts representing the course of the tonal functions of the various chords, that allow one to have a view of the character of the piece in order for it to be interpreted. The results of the analysis of two musical pieces are illustrated below: - Musical Moments Op. 94 No.3 by F. Schubert, - Romance without words n° 3 op. 30 by F. Men-

delssohn.

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Musical moments n° 3 op. 94 by F. Schubert The musical moment n. 3, is part of a collection of 6 delectable miniatures sprinkled over the last five years of life of Schubert, in which a multitude of elements that seem to emerge from the past under the form of images and sonorous memories. Melodic fragments, rhythms and gestures are not only matched in a kind of collage, but completely reinvented as well. The shape is free and it proceeds by associations of ideas, just as they emerge to cognizance, without the least attempt to build a formal logic, a syntax. The tonalities are touched now and then depending on their color, so as to create changes in the ambience, determining commendably arduous modulations. Below there are the graphic representations of two distinct phrases (fig. 7 and 8), in which all the above-mentioned elements can be gathered, through functional analysis.

Fig. 7: The first eight beats (plus two introductory beats) of the Musical Moment n° 3 op. 94. It can be noticed how the ambience is of relaxation (or of rest) inasmuch as the tonal function of tonic is pre-dominant, with the exception of the 7th, 8th and 9th beats in which the dominant function (in cadence) that “seems to be representing the re-emergence of a memory from the past which, however, finds im-mediate satisfaction in the present situation” (with the return to the tonic).

Fig. 8: The beats from 19 to 26 of the Musical Mo-ment n° 3 op. 94. In this phrase, there is an alterna-tion between the dominant function and the tonic function, that is between tension and relaxation (between past and present). The contrast is created by means of the repeat of the dominant chord pre-sented in the cadence of the first phrase (9th beat) alternating with the rhythmic and melodic cell of the 3rd beat (contrariwise). Romance without words n° 3 op. 30 by F. Men-delssohn As a second musical piece for analysis I chose Ro-mance without words n° 3 op 30 by Mendelssohn which is part of a collection of 48 piano “songs” able to express, without making use of verse and rhymes, the feelings that the human soul can experi-ence: a kind of an intimate diary that accompanies the author from the age of 19 until his death.. The intent is clearly expressed from the very title: it is about regaining from the vocal art one of the most eloquent forms, though liberating it from the word and from the weight of the voice, thus turning the pianoforte into a singer. Below there is the graphic representation of an ex-cerpt from Romance without words n° 3 op.30 (fig. 9), in which all the above-mentioned elements can be gathered, through functional analysis.

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Fig. 9: Excerpt from Romance without words n° 3 op. 30 (from beat 1 to beat 14). The constant pres-ence of the dominant function grants the musical piece a strong sense of tension that it only resolves in the final phrase in which the introduction is repre-sented unaltered. 6 Conclusions The tools presented in this article, developed on the basis of specifically musical objectives, are not meant in any way to be considered a system for the in-terpretation of a musical piece, but rather represent means of support to the didactic activity: a useful tool in order to allow specific in-depth examinations, stimulate the recovery of abilities not entirely ac-quired or as a simple tool of consultation and support to the explanations of the professor. The extension of this methodology to the me-lodic and rhythmic analysis, within the framework of the automatic segmentation of a musical piece, could help us analyze Jazz and Blues musical pieces, in which it is more difficult to recognize the informa-tion conveyed by the author because of the frequent usage of swing. Although the results obtained are satisfactory, the question still remains: what may be a limit of the automatic segmentation of a musical piece: Much as one attempts to find a solution to the many

problems relating to the analysis, is it possible to en-visage everything or is it that “genius” may nevertheless elude the “mathematical eye”? References: [1] T.W. Adorno, Zu einer Theorie der musikali-

schen Reproduktion, hrsg. von Henri Lonitz, Frankfurt am Main, Suhrkamp, 2001.

[2] I. Bent, W. Drabkin, Music Analysis, EDT, Tu-rin, 1990.

[3] E. Stein, Form and Performance, with a for-ward by Benjamin Britten, New York, Lime-light Editions, 1989.

[4] Moles, Abraham, Teorie de l’information et Perception esthetique, Paris, Flammarion Edi-teur, 1958.

[5] P. Fraisse, Les structures rythmiques, Erasme, Paris, 1958.

[6] M. Della Ventura, Rhythm analysis of the “so-norous continuum” and conjoint evaluation of the musical entropy, In Proceedings of the In-ternational Conference on Acoustics & Music: Theory & Applications (AMTA '12).

[7] W. Berry, Musical structure and performance, Yale University Press, New Haven, 1989.

[8] D. de la Motte, Manuale di armonia, Bärenrei-ter, 1976.

[9] B. Coltro, Lezioni di armonia complementare, Zanibon, 1979.

[10] G.Haus, Elementi di informatica musicale, Ja-ckson, 1984.

[11] M. Pancini, Problematiche e modelli formali per la segmentazione automatica/interattiva di partiture musicali simboliche, Rapporto Tecni-co CNR-PFBC-MUS-TR, 2000.

[12] F.Lerdhal, R. Jackendoff, A Generative Theory of Tonal Music, The MIT Press, 1983.

[13] Drabkin W. Pasticci S. Pozzi E. (1995), Analisi schenkeriana. Per un’interpretazione organica della struttura musicale, Quaderni di Musi-ca/Realtà 32, LIM, Lucca, (rist. 1999).

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