Rhythm analysis of the sonorous continuum and conjoint evaluation of the musical entropy

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1 Rhythm analysis of the sonorous continuum and conjoint evaluation of the musical entropy MICHELE DELLA VENTURA E-learning Assistant Conservatory of Music A. Buzzolla Viale Maddalena 2 ADRIA (RO) Italy dellaventura.michele@tin.it Abstract: Music is the meeting place par excellence for arts and sciences and within this meeting, mathematics plays an essential and central role. The relations between Music and Mathematics are very close both at elementary and more complex levels. These relations are easily comprehensible even for the uninitiated, for instance with respect to note durations (double or half, quarter and so on): rhythm is a number, it is the precise measure of time. Rhythm is an element that confers a musical piece a special physiognomy and while listening to it, it allows the listener to perform a ation of the sonorous continuum, seizing in it contrasts and consonances: sound clusters are collocated in rhythmic and melodic schemata, in which themes, motifs, repetitions, variations emerge. The musical content, therefore, does not exhaust itself into the mere acoustic and perceptual fact, but has a meaning that emerges from the structural properties of the piece, cognitively elaborated by the listener and traced back to conceptual structures. In this article, I will present a model of rhythm analysis able to explore progressively the symbolic level of the musical text, identifying the rhythmic cells on the basis of the information that every single one of them carries. The efficiency of the model was checked analyzing various musical pieces by different authors and of different times, trying to range over different styles by means of a single analysis methodology. Key-Words: musical surface, ation, entropy, rhythmic extraction, Musical Object 1 Melody and rhythm: two inseparable dimensions Melody and rhythm are two fundamental components as far as musical structuring is concerned, two nearly inseparable components: a melody evolves along the rhythm in the absence of which it does not exist [1]. Melody in itself is weak and quiescent, but when it is joined together with rhythm it becomes alive and active [2]. Melody and rhythm are therefore tightly joined: the former stands for the meaning evolved from a succession of sounds, while the latter is the form and the proportions of the succession [4]. Rhythm, therefore, bestows individuality and recognizability (see fig. 1), yet, in the mean time, it characterizes the entire musical piece by means of rhythmic cells, that is groups of signs (or signs and rests) that create, within the musical discourse, recurring rhythmic schemata that could also be different at a melodic level (see fig 2 and 3) [3]. Fig. 1: Excerpt from the score of Ravel s "Bolero". The initial notes of the theme are represented on the first staff without any indication with respect to rhythm; the same notes are represented of the second staff together with the rhythm assigned by the composer. Fig. 2: J.S. Bach s Two voice invention in E major BWV 777. The rhythmic cell of the 4 th beat is represented on the first staff, while on the second and third staff there is the same cell that we can find respectively at the 16 th beat and at the 18 th beat. ISBN:

2 Fig. 3: String Quartet Op. 18 no.1, 1st Movement, by L. van Beethoven (the first four beats of the first Violin). The fundamental rhythmic cell which generates thematic and timbral structures inside the whole first movement. The core of rhythmic analysis is to define what is meant by rhythmic repetition; more specifically which ones among the textual repetitions existing in the score can be associated to the notion of rhythmic cell and, in case there are several such repetitions, what criterion to adopt in order to select only the part that can be deemed interesting by an expert [6]. 2 Information theory In the various algorithms realized for the rhythmic ation, the final choice of the is performed by comparing every single fragment found with a specially drawn up list of rules. The proposed model intends to present a new choosing methodology, based on the assessment of the information that every rhythmic fragment identified carries within. The analysis based on the information theory sees music as a linear process supported by its own syntax [7]. However, it is not a syntax formulated on the basis of grammar rules, but rather based on the occurrence probability of every single element of the musical message compared to their preceding element. The fact that music is assimilated to a message transmitted by a sender to a receiver, or that information is mentioned, may give the impression of an analytical approach of the interpretive type, interested in the meanings of communication. In fact, here, the interest goes exclusively to the manner in which the expectations of the receiver are spurred, fulfilled or let down. From the definition of message as a chain of discontinuous units of meaning, follows that the musical units of meaning coincide with the minimal events of a composition: usually isolated notes, chords. Any event of such a chain demands a prediction as to the event that will follow it: there is information transmission when the prediction turns out to be disregarded, there is none when it turns out to be confirmed. In other words, the greater the unpredictability 1 degree of the content of a certain message, the larger the amount of information contained in the message. Moreover, the fact that the events of a composition are modularly organized indicates the possibility to calculate, using a formula, or to express, using an index, the total information transmitted by a certain musical module. According to W. Weaver and C. Shannon, the concept of information [8] includes: - the symbolic apparatus through which it manifests. Information is an abstract concept that materializes through a set of symbols, while the meaning assigned to the symbols that represent it allows the information to be interpreted. - the semantic content conveyed by it. The set of symbols that convey a certain piece of information contains a precise meaning that characterizes its value; and it is this very meaning the one that constitutes the semantic part of the piece of information. In this regard, the manner in which information is sensitive to the state of the receiver: information loses value if it is already known to the receiver. - the pragmatic effect produced by it. Every single piece of information carries meanings with the intent to produce a certain effect. In a communication which takes place by way of a given alphabet of symbols, information is associated to every single transmitted symbol [9]. Therefore, information can be defined as the reduction of uncertainty that would have been obtained a priori on the transmitted symbol. The wider the range of messages that the source can transmit (and the greater the uncertainty of the receiver as to the possible message) the larger the amount of information transmitted and along with it, its extent: the entropy. In the information theory, the entropy is a positive value and not a negative one as it is originally in physics. From a mathematical perspective, the extent of the content of a certain piece of information (I) is obtained using Shannon s formula: ' p I = log 2 p where p is the probability for the message to be transmitted, p' is connected to the use that the observer makes of the message and it coincides with the probability to materialize of the content of the information expected by the observer after the message has been transmitted. 1 Unpredictability here means simply the difficulty to guess the term. ISBN:

3 3 Entropy In information theory [8] (in connection with the signal theory [10]) entropy measures the amount of uncertainty or of information existing in a random signal. For every single symbol (of a message) that we transmit, we have a certain amount of information associated to that specific symbol. In most practical applications of information theory a choice among the messages of a set must be made and every message has its own probability of being transmitted. Shannon provided the definition of entropy of such a set, identifying it as the information content that the choice of one of the messages will transmit. If every single message has a probability p' of being transmitted, the entropy is obtained by summing all the set of the functions p' log2 p', every one relating to a message, that is: H ( X ) = E n [ I ( xi )] = I ( xi ) P( xi ) = n P( x i= i= i) log P( xi ) The term entropy, borrowed from thermodynamics, designates, therefore, the average information content of a message. In light of everything said so far, the musical message can be defined as a sequence of signals organized according to a code. 4 Rhythm and information In order to be able and compare the different s with one another, so as to determine which one is more important [11], the entropy of every single one was calculated: the lower the entropy value is, the greater the information that the carries. In order to calculate the entropy one must refer to a specific alphabet: it is specific to the language, and as it can be deduced immediately from the formula (based on the probability for both symbols, rather than other symbols, of being transmitted) it turns out to be associated to language. In the case of musical language, considering that what characterizes the melody (as mentioned in the first paragraph) is the duration that every single sound assumes within and that confers the melody a specific physiognomy, the durations of the various sounds were considered symbols of the alphabet: duration intended as the time interval in which sound becomes perceptible, regardless of whether it is due to a single sign or to several signs joined together by a value connection. This principle allows us to go past musical concepts such as accent, meter or bar. 1 Fig. 4: J.S. Bach s Two Voice Invention in E Major BWV 777 (the first three beats of the first staff). If we were to analyze a score, the sound duration will not be expressed in seconds but calculated (automatically by the algorithm) on the basis of the musical sign (be it sound or rest) with the smallest duration existing in the musical piece. The duration of every single sign will therefore be a (integer) number directly proportional to the smallest duration. The rhythmic analysis imposes the necessity to consider rigorously the rests as well as elements of the musical continuum. Rests often assume and expressive value of expectation of what is to come, yet, it is also true that this same sense of expectation becomes loaded with a multitude of meanings, that go all the way from simple "suspension" of the sonorous flow (acquired within the full autonomy of an interval of silence considered in itself) to the perception of a kind of virtually given musical structure, which, at the exact same time when it expresses waiting, it also expands into the two dimensions of the present and of the past: as if the current moment of silence predisposed towards the expectation of what would be, by way of the mental echo of what has just passed [2]. In order to distinguish between sound and rest, the latter is associated to a duration value directly proportional to the smallest duration, but negative. In the example shown in figure 4, the smallest duration sign is represented by the thirty-second note to which the value 1 is associated (automatically): it follows that the sixteenth note shall have the value 2, the eighth note the value 4 MI FA# Fig. 5: J.S. Bach s Two Voice Invention in E Major BWV 777 (the first three beats if the first staff). Graphic representation of the sounds based on duration. ISBN:

4 FA DO Fig. 6: String Quartet Op. 18 no. 1, 1st Movement, by L. van Beethoven (the first three beats of the first staff). Graphic representation of the sounds based on duration. From every musical piece is (afterwards) extracted the characteristic of the language on which the piece was developed, analyzing a posteriori its distribution of sound durations. For every single musical piece a table, which represents its own alphabet, is filled in, distinguishing between sounds and rests: Figure 7 shows an example drawn out of J.S. Bach s Two Voice Invention in E Major BWV 777. chose to deduce the transition probability that determines the passage from a state of the system to the next uniquely from the immediately preceding state. This entails the loss of a historical vision of the process focused on a greater number of transitions, but it there results a good compromise with the necessity to simplify the analysis, besides observing the compositional rules that see the concatenation of a string of durations as being based on the generation of a duration starting from the preceding duration. Obviously, the characteristic of human inventiveness considers a certain number of durations preceding the generated one as exerting influence, but we thought the generated model is sufficient. Furthermore, since the language based on which rules are to be synthesized is confined to a specific composition, a stochastic process with multiple transitions would extrapolate its probabilities from a number of cases too limited to be relevant. Therefore the transition matrix is created. It is made up of the transition probabilities between the states of the system (conditional probability). In our case, the matrix represents the probabilities for a kind of duration to resolve to another kind of duration. In order to better clarify these concepts, let us consider the following example. The first duration of the score is taken into consideration: this corresponds to a sixteenth note rest to which the value -2 is assigned, which precedes the duration of an eighth note to which the value 4 is assigned. The transition matrix shall be updated incrementing by one unit the corresponding square where the two durations cross. Fig. 7: Example of an alphabet. The duration is indicated in the first column, the distinction between sound and rest in the second column and the number of durations of that specific kind found in the musical piece is indicated in the third column. Determining the alphabet thus identified is not enough in order to calculate the entropy of a musical : it is necessary to consider the manner in which the durations succeed one another inside the musical piece. In order to do this, the Markov process (or Markov s stochastic process) is used: we ISBN:

5 Now, considering this last duration, one can notice that the next duration is an eighth note (value 4), therefore the matrix is updated incrementing by one unit the corresponding square where the duration of an eighth note duration crosses an eighth note. This procedure is repeated all the way to the end of the score. (Figure 8). the algorithm does not provide any limit with respect to the dimensions of the table representing the alphabet and of the matrix of the transitions, but, on the contrary, these are automatically dimensioned after every analysis on the basis of the features of the piece of music analyzed. Finally, another important aspect considered in order to define the logical functional bases of the algorithm is the nature of data: the results of the analysis are indicated by numeric representative values of the entropy value of every single element. On the basis of the definition of entropy described in the preceding paragraphs, one deduces that there will not be values above zero, but, on the contrary, very low values, therefore it is important to evaluate the number of decimals that can influence, even slightly, the identification of the most important s: the greater the number of decimals, the higher the accuracy of the analysis. Various pieces by different authors and of different times were analyzed, trying to range over different compositional techniques. The results of the analysis of two musical pieces are shown below: - J.S. Bach's Two Voice Invention in E Major BWV 777, - the first movement of L. Van Beethoven s Pastoral Symphony No. 6, op. 68. In all the cases, the fundamental themes existing in the piece and the rhythmic cells that characterize the theme and the entire musical piece were identified. Fig. 8: Example of a matrix of the transitions drawn out of J.S. Bach s Two Voice Invention in E Major BWV 777. J.S. Bach's Two Voice Invention in E Major BWV 777 The Two Voice Invention BWV 777 (that was taken as a model from the very start of this article for musical examples), is part of a collection of fifteen pieces of which every single piece contains two voices, that is two melodies that move independently, which interweave, communicate in a jocular manner with each other, conferring unity and consistency to the musical piece. First Voice 5 The results obtained The model of analysis exposed in this article was checked by realizing and algorithm, the structure of which takes in consideration, in the first place, all the single aspects described above: Second Voice Fig. 9: The first four beats of Bach s two voice invention BWV 777. ISBN:

6 The two distinct voices are presented in the first four beats: the first (the right hand) presents two distinct rhythmic elements, that is an irregular syncopation in the first three beats and one rhythmic cell made up of a sixteenth note plus two thirtysecond notes, repeated for the entire fourth beat; the second voice (the left hand) presents a regular rhythm where the fundamental rhythmic cell is represented by 3 eighth notes ( ). An excerpt of the final table (Figure 10) restored by the algorithm is shown below, where in the first two columns there are the initial position of the so that it can be identified on the score and its length (how many notes it is made up of) whereas the entropy value of that particular is represented in the third column. Position beginning Length Entropy , , , , Fig. 10: Final table restored by algorithm. Second Voice First Voice First Voice Second Voice Fig. 11: Rhythmic s identified by the algorithm. The s corresponding to the table in figure 10 are represented in figure 11. The first that starts with note 93 (beat 21) is made up of 34 notes, corresponds to the exposition of the second voice followed by the First Voice; the second (that starts with note 1 and is 34 notes long) is exactly the opposite: exposition of the first voice followed by the second voice; the third (from beat 28) highlights the rhythmic cell of the fourth beat of the first voice that is repeated five time; the fourth is characterized by two beats, repeated too for several times, which contain two rhythmic cells: the syncopation (in the first beat) and the rhythmic cell of the fourth beat of the first voice. The first movement of L. Van Beethoven s Pastoral Symphony No. 6, op. 68 As a second piece of analysis, we chose the first movement of Beethoven s Pastoral Symphony No. 6, in order to check the solidity of the algorithm when applied to a polyphonic piece. The entire first movement is nothing less than a constant metaphorical transformation of the material contained in the first four opening beats. Fig. 12: The first four beats of Beethoven s "Pastoral Symphony No. 6. An excerpt of the fragment summarizing table is illustrated below. Position beginning Length Entropy , , , , Fig. 13: Final table restored by algorithm. It is interesting to notice that in this case too, the first identified is the initial rhythmic cell existing in beat 2 that is stubbornly exposed by the Violins from beat 16 for 11 beats. Instead, the second represents the main theme exposed by the oboe in beat 29. Fig. 14: Rhythmic s identified by the algorithm. ISBN:

7 6 Conclusions The instruments illustrated in this article follow an innovative point of view that enables the discovery and the exploitation of the information which are consistent with our perception. These instruments have been developed on the basis of specifically musical objectives. For this reason we tried to find, first of all, a unique system of analysis, valid to whatever musical form; then we tried to look at the results from a strictly scientific perspective using statistics. The extension of this methodology to the melodic analysis of a piece of music could assist us in assessing the single fragments found and, therefore, enable the extension of the study to pieces of contemporary music as well, where it is more difficult to recognize the information conveyed by the author. In spite of the very satisfying results obtained, there is still the open question with respect to what could be a limit of the automatic ation of a piece of music: no matter how hard one struggles to find the solution to the many problems related to analysis, is it possible to predict everything or can "genius" nevertheless escape the "mathematical eye? [11] O. Lartillot-Nakamura, Fondements d un système d analyse musicale computationnelle suivant une modélisation cognitiviste de l écoute, Doctoral Thesis, University of Paris, [12] R. Monelle, Linguistics and Semiotics in Music, Harwood Academic Publisher, [13] J.J. Nattiez, Fondements d'une sémiologie de la musique, Union Générale d'éditions, Paris, [14] U. Eco, A Theory of Semiotics, Bloomington & London: Indiana University Press, References: [1] C. Orff, Schulwerk, elementare Musik, Hans Schneider, Tutzing, [2] P. Fraisse, 1974, Psychologie du rythme, Puf, Paris, [3] P. Fraisse, Les structures rythmiques, Erasme, Paris, [4] Moles, Abraham, Teorie de l information et Perception esthetique, Paris, Flammarion Editeur, [5] F. Lerdhal, R. Jackendoff, A Generative Theory of Tonal Music, The MIT Press, [6] L. Camilleri, Musical Grammars and Computer Analysis, Leo Olschki Editore, Florence, [7] I. Bent, W. Drabkin, Music Analysis, EDT, Turin, [8] W. Weaver and C. Shannon, The mathematical theory of information, Urbana, Illinois Press, [9] F. Lerdhal, R. Jackendoff, A Grammatical Parallel between Music and Language, New York, Plenum Press, [10] E. Angeleri, Information, meaning and universalit, UTET, Turin, ISBN: