Generalized trapezoidal words

The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor complexity $C_w(n)$ as a function of $n$ (for $0 \leq n \leq |w|$) is that of a regular trapezoid (or possibly an isosceles triangle); that is, $C_w(n)$ increases by 1 with each $n$ on some interval of length $r$, then $C_w(n)$ is constant on some interval of length $s$, and finally $C_w(n)$ decreases by 1 with each $n$ on an interval of the same length $r$. Necessarily $C_w(1)=2$ (since there is one factor of length $0$, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of $\mathbb{R}^2$ made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet $\mathcal{A}$, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when $|\mathcal{A}| \geq 3$, but we can describe all those that are rich.Comment: Major revisio

Directive words of episturmian words: equivalences and normalization

Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. Thus, an episturmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing a common episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.Comment: 15 page

Detecting Episodes with Harmonic Sequences for Fugue Analysis

International audienceFugues alternate between instances of the subject and of other patterns, such as the counter-subject, and modulatory sections called episodes. The episodes play an important role in the overall design of a fugue: detecting them may help the analysis of the fugue, in complement to a subject and a counter-subject detection. We propose an algorithm to retrieve episodes in the fugues of the first book of Bach's Well-Tempered Clavier, starting from a symbolic score which is already track-separated. The algorithm does not use any information on subject or counter-subject occurrences, but tries to detect partial harmonic sequences, that is similar pitch contour in at least two voices. For this, it uses a substitution function considering "quantized partially overlapping intervals" [Lemström and Laine, 98] and a strict length matching for all notes, except for the first and the last one. On half of the tested fugues, the algorithm has correct or good results, enabling to sketch the design of the fugue

Rhythm extraction from polyphonic symbolic music

International audienceWe focus on the rhythmic component of symbolic music similarity, proposing several ways to extract a monophonic rhythmic signature from a symbolic poly- phonic score. To go beyond the simple extraction of all time intervals between onsets (noteson extraction), we select notes according to their length (short and long extractions) or their intensities (intensity+/− extractions). Once the rhythm is extracted, we use dynamic programming to compare several sequences. We report results of analysis on the size of rhythm patterns that are specific to a unique piece, as well as experiments on similarity queries (ragtime music and Bach chorale variations). These results show that long and intensity+ extractions are often good choices for rhythm extraction. Our conclusions are that, even from polyphonic symbolic music, rhythm alone can be enough to identify a piece or to perform pertinent music similarity queries, especially when using wise rhythm extractions

Towards Modeling Texture in Symbolic Data

International audienceStudying texture is a part of many musicological analyses. The change of texture plays an important role in the cognition of musical structures. Texture is a feature commonly used to analyze musical audio data, but it is rarely taken into account in symbolic studies. We propose to formalize the texture in classical Western instrumental music as melody and accompaniment layers, and provide an algorithm able to detect homorhythmic layers in polyphonic data where voices are not separated. We present an evaluation of these methods for parallel motions against a ground truth analysis of ten instrumental pieces, including the first movements of the six quatuors op. 33 by Haydn

Quasiperiodic and Lyndon episturmian words

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their "directive words". Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular "normalized" directive words, which we introduced in an earlier paper. Also of importance is the use of "return words" to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.Comment: 33 pages; minor change

Structures en combinatoire des mots et en analyse musicale

Ce mémoire présente certains de mes travaux de recherches de ces dernières années,en particulier des problèmes de structure sur les mots et sur la musique.En combinatoire des mots, l'étude de la structure peut être simplement de regarder la manière dont les lettres sont positionnées dans un mot, ou comment les facteurs (suites de lettres juxtaposées) sont répétés.Le problème auquel nous allons nous intéresser ici est celui de la quasipériodicité, c'est-à-dire la structure des mots constitués d'un motif répété avec ou sans chevauchement. En musique, la structure peut désigner l'agencement de différentes parties musicales (similaires ou distinctes) pour former un schéma global, apportant un certain équilibre à une oeuvre. Nous allons nous intéresser ici aux différents éléments entrant en jeu dans l'analyse de la structure "horizontale" d'une pièce musicale : répétition de motifs, harmonie, rythme, texture... et au moyen de les combiner pour aboutir à une analyse musicale par ordinateur proche d'une analyse musicale humaine. Nous verrons aussi que l'on peut étudier localement la structure "verticale" d'une pièce lorsque celle-ci est polyphonique, en analysant l'agencement des différentes voix entre elles

Vers une analyse informatique des textures dans le quatuor classique

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Vers une analyse informatique des textures dans le quatuor classique

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