Let $A \subseteq \Z_n$ with $|A| = k$ for some $k \in \Z^+$. We consider the metric space $(\Z_n,\delta)$ in which $\delta$ is the distance metric on $\Z_n$ defined as follows: for every $x,y \in \Z_n$, $\delta(x,y) = |x-y|_n$ where $|z|_n = \min(z,n-z)$ for $z \in \{0,\ldots,n-1\}$. We say that $A$ is \emph{Erd\H{o}s-deep} if, for every $i \in \{1,2,\dots,k-1\}$, there is a positive number $d_i$ satisfying $$|\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}|=i.$$ Erd\H{o}s-deep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k-1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$. Erd\H{o}s-deep sets have primarily been considered in metric spaces $(\Z_n,\delta)$ and $(\R^d,\norm{\cdot})$ for $d = 2$, but some exploration for $d > 2$ has been done as well. We introduce the notion of an \emph{Erd\H{o}s-deep family}. Let $\mathcal{F}=\{A_1,A_2,\dots,A_s\}$, where $A_1,\ldots, A_s \subseteq \Z_n$. Then we say $\mathcal{F}$ is Erd\H{o}s-deep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k-1\}$ there is exactly one positive number $d_i$ satisfying $$\sum_{j=1}^s |\{\{x,y\} \subseteq A_j: \delta(x,y)=d_i\}|=i,$$ and no such $d_i$ for any $i \ge k$. We provide a complete existence theorem for Erd\H{o}s-deep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}s-deep families of three arithmetic progressions. Using an identity on triangular numbers, we show a general construction for larger families whose size $s$ is the square of an integer. This construction suggests the existence of Erd\H{o}s-deep families often relies on such number-theoretic identities. We define an extremal case of the Erd\H{o}s-deep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k-1\}$; such families are called Winograd families. We conjecture that Winograd families of arithmetic progressions do not exist in the metric space $(\Z,|\cdot|)$. Erd\H{o}s-deep sets in $(\Z_n,\delta)$ correspond to a class of interesting musical rhythms. We conclude this work with a variety of musical demonstrations and original compositions using Erd\H{o}s-deep rhythm families as a creative constraint in composing multi-voiced rhythms. / Graduate

La musique et la parole possèdent toutes deux un degré d’organisation temporelle i.e. de régularité dans le temps. Les stimuli de nature rythmique ont la particularité de pouvoir être anticipés par le cerveau et des études en linguistique et neurosciences ont montré que plus le cerveau est capable d’anticiper les évènements auditifs, meilleure est la qualité du traitement des stimuli. Les enfants sourds, bien que bénéficiant d’un input auditif de plus en plus précis grâce aux implants cochléaires et d’une prise en charge précoce, n’atteignent pas des niveaux de langage homogènes et souffrent de difficultés de perception en milieux bruyants ou lors de conversations. La situation conversationnelle présente un contexte complexe, nécessitant l’activation de la voie audio-motrice pour anticiper et s’adapter aux variations de la parole de son interlocuteur notamment au niveau temporel. Dans ce travail de thèse, nous avons cherché à analyser, grâce à des mesures électrophysiologiques et comportementales, si un entrainement rythmique actif de 30 minutes, pouvait avoir un effet sur les capacités de perception et d’accommodation temporelles de l’enfant sourd dans une tâche de dénomination en alternance avec un partenaire virtuel. Nous avons également testé les capacités rythmiques de ces enfants à différents niveaux de complexités. Les résultats montrent que les enfants sourds souffrent de difficultés à structurer les événements acoustiques selon différent niveaux de hiérarchie mais qu’un entrainement rythmique de 30 minutes versus une stimulation auditive, permet d’améliorer leurs compétences de perception et de production temporelles de la parole dans une situation d’interaction. / Music and speech both possess a certain degree of temporal organization i.e. a certain degree of regularity across time. Studies in linguistics and neuroscience have shown that the brain can extract regularities and use them to anticipate the forthcoming stimuli. It is furthermore established that the better the brain is able to anticipate auditory events, the better the quality of stimulus processing. Deaf children benefit from more and more precise auditory inputs due to advances in cochlear implants development, together with early rehabilitation interventions. However, a great majority of them do not achieve consistent language levels and have strong difficulties in noisy environments or conversations. The conversational situation presents a complex context, requiring the activation of the audio-motor path to anticipate and adapt to the variations of the speech of its interlocutor notably at the temporal level. In this thesis work, we have investigated the temporal perception and accommodation capacities of deaf children in a naming task alternating with a virtual partner, at both behavioral and electrophysiological levels. We have also tested whether an active rhythmic training lasting 30 minutes, could enhance these conversational abilities. Then, we have investigated the rhythmic abilities of these children at different levels complexities. The results show that deaf children suffer from difficulties in structuring acoustic events according to different levels of hierarchy but that a rhythmic training of 30 minutes versus an auditory stimulation, makes it possible to improve their skills of temporal perception and production of speech in a situation of interaction.

O presente trabalho procura descrever e avaliar possibilidades de reconhecimento de ambiguidades rítmicas a partir da perspectiva do ouvinte musical. Adota como principais referenciais: a análise de cenário auditivo, de Bregman (1990); a percepção por categorias, de Clarke (1987); o conceito de relógio interno, de Povel e Essens (1985) e a indução do tempo musical, de Povel e Okkerman (1981); o modelo de regras de preferência, de Lerdahl e Jackendoff (1983), depois desenvolvido por Temperley (2001); os critérios de padrões inerentes e o padrão da linha do tempo, de Kubik (1962); conceitos da psicologia da expectativa, de Huron (2007) e a aplicação de princípios gestálticos a processos cognitivos de percepção do ritmo. Com base neste referencial teórico, propõe parâmetros organizados como fatores endógenos e exógenos para a verificação de aspectos relacionados à maneira como ouvintes médios percebem as articulações rítmicas propostas nos exemplos musicais propostos. Por considerar que os compassos iniciais de uma obra constituem uma fase onde ocorre o processo de indução da percepção métrica, os exemplos musicais são constituídos de análises de trechos iniciais de obras musicais potencialmente ambíguos nomeadamente, Sinfonia n. 5, op. 67 (I), Sonata para piano, op. 14, n. 2 (I), Sonata para piano, op. 109 (I), de Beethoven; Quarteto de cordas, op. 51 n. 1 (I), de Brahms; Sinfonia n. 5, op. 64 (III), de Tchaikovsky. Traz, ainda, um exemplo de ambiguidade entre Ijexá e Drum n´Bass. Na conclusão, defende a ideia de que alguns modelos cognitivos atuais são capazes de justificar percepções auditivas ambíguas. / This study describes and evaluates the recognition of possibilities of rhythmic ambiguity from the perspective of the musical listener. Adopts as main references: the Auditory Scene Analysis, by Bregman (1990); the Categorical Rhythm Perception, by Clarke (1987); internal clock concept, by Povel and Essens (1985), and induction of musical time, by Povel and Okkerman (1981); the preference rule model, by Lerdahl and Jackendoff (1983), later developed by Temperley (2001); Inherent patterns and Timeline pattern criteria, by Kubik (1962); Psychology of Expectation concepts, by Huron (2007), and application of Gestalt principles to cognitive processes of rhythm perception. Based on this theoretical framework it proposes parameters organized as endogenous and exogenous factors to verify aspects related to the perception of proposed articulations by averages listeners, in the proposed rhythmic musical examples. Considering that the initial measures of a work constitute a stage where the induction process of meter perception occurs, the musical examples are made up of initial excerpts analysis from musical works potentially ambiguous namely, Symphony no. 5, op. 67 (I), Piano Sonata, op. 14, no. 2 (I), Piano Sonata, op. 109 (I), by Beethoven; String Quartet, op. 51 no. 1 (I), by Brahms; Symphony no. 5, op. 64 (III), Tchaikovsky. This paper also provides an example of ambiguity between Ijexá pattern and Drum n\'bass pattern. In the conclusion, defends the idea that some current cognitive models are able to justify ambiguous auditory perceptions.

O presente trabalho procura descrever e avaliar possibilidades de reconhecimento de ambiguidades rítmicas a partir da perspectiva do ouvinte musical. Adota como principais referenciais: a análise de cenário auditivo, de Bregman (1990); a percepção por categorias, de Clarke (1987); o conceito de relógio interno, de Povel e Essens (1985) e a indução do tempo musical, de Povel e Okkerman (1981); o modelo de regras de preferência, de Lerdahl e Jackendoff (1983), depois desenvolvido por Temperley (2001); os critérios de padrões inerentes e o padrão da linha do tempo, de Kubik (1962); conceitos da psicologia da expectativa, de Huron (2007) e a aplicação de princípios gestálticos a processos cognitivos de percepção do ritmo. Com base neste referencial teórico, propõe parâmetros organizados como fatores endógenos e exógenos para a verificação de aspectos relacionados à maneira como ouvintes médios percebem as articulações rítmicas propostas nos exemplos musicais propostos. Por considerar que os compassos iniciais de uma obra constituem uma fase onde ocorre o processo de indução da percepção métrica, os exemplos musicais são constituídos de análises de trechos iniciais de obras musicais potencialmente ambíguos nomeadamente, Sinfonia n. 5, op. 67 (I), Sonata para piano, op. 14, n. 2 (I), Sonata para piano, op. 109 (I), de Beethoven; Quarteto de cordas, op. 51 n. 1 (I), de Brahms; Sinfonia n. 5, op. 64 (III), de Tchaikovsky. Traz, ainda, um exemplo de ambiguidade entre Ijexá e Drum n´Bass. Na conclusão, defende a ideia de que alguns modelos cognitivos atuais são capazes de justificar percepções auditivas ambíguas. / This study describes and evaluates the recognition of possibilities of rhythmic ambiguity from the perspective of the musical listener. Adopts as main references: the Auditory Scene Analysis, by Bregman (1990); the Categorical Rhythm Perception, by Clarke (1987); internal clock concept, by Povel and Essens (1985), and induction of musical time, by Povel and Okkerman (1981); the preference rule model, by Lerdahl and Jackendoff (1983), later developed by Temperley (2001); Inherent patterns and Timeline pattern criteria, by Kubik (1962); Psychology of Expectation concepts, by Huron (2007), and application of Gestalt principles to cognitive processes of rhythm perception. Based on this theoretical framework it proposes parameters organized as endogenous and exogenous factors to verify aspects related to the perception of proposed articulations by averages listeners, in the proposed rhythmic musical examples. Considering that the initial measures of a work constitute a stage where the induction process of meter perception occurs, the musical examples are made up of initial excerpts analysis from musical works potentially ambiguous namely, Symphony no. 5, op. 67 (I), Piano Sonata, op. 14, no. 2 (I), Piano Sonata, op. 109 (I), by Beethoven; String Quartet, op. 51 no. 1 (I), by Brahms; Symphony no. 5, op. 64 (III), Tchaikovsky. This paper also provides an example of ambiguity between Ijexá pattern and Drum n\'bass pattern. In the conclusion, defends the idea that some current cognitive models are able to justify ambiguous auditory perceptions.

The purpose of this study is to present a personal interpretation of progressive education and its function in "teaching" primary music. A few supplementary aids are provided to correlate with the Amarillo, Texas, Course of Study for Primary Grades to lend aid and encouragement toward a rhythmic approach to child learning.

Nesta tese de doutorado abordamos modelos cognitivos de percepção da métrica musical e entrainment a partir de questões musicológicas, como a ambiguidade métrica decorrente de exemplos da literatura. Partindo de uma verificação do estado da arte em pesquisas rítmicas que envolvem o estudo de anacruses, realizamos um percurso experimental que investiga a efetividade da chamada percepção da acentuação subjetiva revelada por Povel e Okkerman (1981), por acreditarmos que o fenômeno da acentuação subjetiva esteja envolvido na desambiguação da percepção de referenciais métricos dúbios. Para tanto, desenvolvemos uma metodologia de quantificação das similaridades entre os padrões de acentuação coletados em grupo universitário da cidade de São Paulo e os padrões de referência da literatura, sobretudo de Povel e Essens (1985). Estes experimentos revelaram que há uma tendência significativa à percepção da acentuação subjetiva predominantemente em grupo sem estudo formal de música. Por outro lado, os estudantes de música revelaram uma tendência de acentuação mais relacionada à pulsação musical. / In this doctoral thesis we address cognitive models of perception of musical meter and entrainment from musicological issues, such as the metric ambiguity arising from examples of literature. Based on a state-of-the-art check on rhythmic researches involving the study of anacruses, we conducted an experimental study that investigates the effectiveness of the so-called subjective accent revealed by Povel and Okkerman (1981), because we believe that the phenomenon of subjective accent is Involved in the disambiguation of dubious beat references perception. Therefore, we developed a methodology to quantify the similarities between the accentuation patterns collected in university group of São Paulo and the literature reference standards, especially Povel and Essens (1985). These experiments revealed that there is a significant tendency to subjective perception of accent predominantly in people without formal music study. On the other hand, the students of music revealed a tendency of accentuation more related to the musical beat.

A presente tese investiga os processos composicionais na obra Zé Barbeiro (José Augusto Roberto da Silva, n. 1952). O autor desenvolveu um estilo de escrita bastante intuitivo e essencialmente orientado pela sua prática enquanto violonista de Choro. Não obstante, observamos que suas obras diferem-se sonoramente daquela dos compositores chorões do final do século XIX e do século XX, quer pela modificação de aspectos harmônicos, melódicos, rítmicos, timbrísticos, métricos e formais, culminando no que tem se chamado por Choro contemporâneo, dentre outras denominações. Assim, num primeiro momento, discutimos acerca destas terminologias, além de analisarmos as múltiplas acepções que a palavra \"Choro\" tem recebido ao longo dos anos, baseando-nos em fontes primárias e no discurso verbal e textual dos próprios chorões. Apoiando-nos na hipótese de que é a organização formal o principal elemento responsável por caracterizar a obra de Zé Barbeiro enquanto Choro, procedemos, na segunda etapa desta pesquisa, por uma investigação acerca dos recursos empregados pelo compositor responsáveis por inovar ao modelo formal tido como padrão. Como conclusão inicial, verificamos que o Choro tem exercido múltiplas funcionalidades, as quais legitimam o seu enquadramento enquanto manifestação cultural. Constatamos também que a profissionalização dos músicos, aliada à desvinculação do Choro com as funções da dança (que outrora exigiam uma construção musical baseada na previsibilidade), são os principais fatores responsáveis por proporcionar as mudanças que o Choro tem sofrido. Finalmente, concluímos que a expansão evidenciada na obra de Zé Barbeiro tem o seu germe na atuação composicional e interpretativa de seus antecessores. / This thesis investigates the compositional processes in the work of Zé Barbeiro (José Augusto Roberto da Silva, b. 1952). The author has developed an intuitive writing style, essentially oriented by his practice as a Brazilian Choro guitarist. Nevertheless, it\'s well-known that his works differs auditorily from those of the Choro composers of the late nineteenth and the twentieth centuries. This is caused by the modification of harmonic, melodic, rhythmic, timbristic, metric, and formal aspects, and culminates in what has been called contemporary Choro, among other denominations. Initially I discuss these terminologies and I analyse the multiple meanings that the word \"Choro\" has received over the years. For this purpose, I examined primary sources and also the verbal and textual discourse of the chorões. In the second stage of this research, I investigated the compositional resources responsible for the innovations on the formal model considered as a pattern. For this, I worked from the hypothesis that formal organization is the main element responsible for chraracterising Zé Barbeiro\'s work as Choro. As a first conclusion, I verified that Choro has exercised multiple functionalities, which legitimise its framing as a cultural manifestation. I also identified that the professionalization of the musicians and the untying of Choro to dance functions (which erstwhile demanded a musical construction based on predictability) are the main factors responsible for providing the changes that Choro has undergone. Finally, I concluded that the musical expansion evidenced in the work of Zé Barbeiro has its germ in the compositional and interpretative performance of his predecessors.

A presente tese investiga os processos composicionais na obra Zé Barbeiro (José Augusto Roberto da Silva, n. 1952). O autor desenvolveu um estilo de escrita bastante intuitivo e essencialmente orientado pela sua prática enquanto violonista de Choro. Não obstante, observamos que suas obras diferem-se sonoramente daquela dos compositores chorões do final do século XIX e do século XX, quer pela modificação de aspectos harmônicos, melódicos, rítmicos, timbrísticos, métricos e formais, culminando no que tem se chamado por Choro contemporâneo, dentre outras denominações. Assim, num primeiro momento, discutimos acerca destas terminologias, além de analisarmos as múltiplas acepções que a palavra \"Choro\" tem recebido ao longo dos anos, baseando-nos em fontes primárias e no discurso verbal e textual dos próprios chorões. Apoiando-nos na hipótese de que é a organização formal o principal elemento responsável por caracterizar a obra de Zé Barbeiro enquanto Choro, procedemos, na segunda etapa desta pesquisa, por uma investigação acerca dos recursos empregados pelo compositor responsáveis por inovar ao modelo formal tido como padrão. Como conclusão inicial, verificamos que o Choro tem exercido múltiplas funcionalidades, as quais legitimam o seu enquadramento enquanto manifestação cultural. Constatamos também que a profissionalização dos músicos, aliada à desvinculação do Choro com as funções da dança (que outrora exigiam uma construção musical baseada na previsibilidade), são os principais fatores responsáveis por proporcionar as mudanças que o Choro tem sofrido. Finalmente, concluímos que a expansão evidenciada na obra de Zé Barbeiro tem o seu germe na atuação composicional e interpretativa de seus antecessores. / This thesis investigates the compositional processes in the work of Zé Barbeiro (José Augusto Roberto da Silva, b. 1952). The author has developed an intuitive writing style, essentially oriented by his practice as a Brazilian Choro guitarist. Nevertheless, it\'s well-known that his works differs auditorily from those of the Choro composers of the late nineteenth and the twentieth centuries. This is caused by the modification of harmonic, melodic, rhythmic, timbristic, metric, and formal aspects, and culminates in what has been called contemporary Choro, among other denominations. Initially I discuss these terminologies and I analyse the multiple meanings that the word \"Choro\" has received over the years. For this purpose, I examined primary sources and also the verbal and textual discourse of the chorões. In the second stage of this research, I investigated the compositional resources responsible for the innovations on the formal model considered as a pattern. For this, I worked from the hypothesis that formal organization is the main element responsible for chraracterising Zé Barbeiro\'s work as Choro. As a first conclusion, I verified that Choro has exercised multiple functionalities, which legitimise its framing as a cultural manifestation. I also identified that the professionalization of the musicians and the untying of Choro to dance functions (which erstwhile demanded a musical construction based on predictability) are the main factors responsible for providing the changes that Choro has undergone. Finally, I concluded that the musical expansion evidenced in the work of Zé Barbeiro has its germ in the compositional and interpretative performance of his predecessors.

Nesta tese de doutorado abordamos modelos cognitivos de percepção da métrica musical e entrainment a partir de questões musicológicas, como a ambiguidade métrica decorrente de exemplos da literatura. Partindo de uma verificação do estado da arte em pesquisas rítmicas que envolvem o estudo de anacruses, realizamos um percurso experimental que investiga a efetividade da chamada percepção da acentuação subjetiva revelada por Povel e Okkerman (1981), por acreditarmos que o fenômeno da acentuação subjetiva esteja envolvido na desambiguação da percepção de referenciais métricos dúbios. Para tanto, desenvolvemos uma metodologia de quantificação das similaridades entre os padrões de acentuação coletados em grupo universitário da cidade de São Paulo e os padrões de referência da literatura, sobretudo de Povel e Essens (1985). Estes experimentos revelaram que há uma tendência significativa à percepção da acentuação subjetiva predominantemente em grupo sem estudo formal de música. Por outro lado, os estudantes de música revelaram uma tendência de acentuação mais relacionada à pulsação musical. / In this doctoral thesis we address cognitive models of perception of musical meter and entrainment from musicological issues, such as the metric ambiguity arising from examples of literature. Based on a state-of-the-art check on rhythmic researches involving the study of anacruses, we conducted an experimental study that investigates the effectiveness of the so-called subjective accent revealed by Povel and Okkerman (1981), because we believe that the phenomenon of subjective accent is Involved in the disambiguation of dubious beat references perception. Therefore, we developed a methodology to quantify the similarities between the accentuation patterns collected in university group of São Paulo and the literature reference standards, especially Povel and Essens (1985). These experiments revealed that there is a significant tendency to subjective perception of accent predominantly in people without formal music study. On the other hand, the students of music revealed a tendency of accentuation more related to the musical beat.

Syftet med den här studien är att analysera vilka initiala och operationella normer som har använts vid översättningar till tyska av sångboksuppslag till visor skrivna av Astrid Lindgren. Modellerna som har använts är hämtade från Franzon (2009), Franzon (2016) och van Meerbergen (2010). Frågeställningarna berör musikalisk rytm, retoriska stilmedel inom texterna samt interaktionsgraden i text och bild med läsaren. Syftet är även att se hur aktiva respektive passiva huvudpersonerna i böckerna är. Resultaten visar att översättningarna av beskrivande visor med långa verser tenderar att bli kortare och mycket förändrade i jämförelse med översättningarna av visor med korta verser och en mindre beskrivande natur. Interaktionsgraden med läsaren är för det mesta låg i både text och bild i käll- och måltexterna. Huvudpersonerna i de tyska översättningarna är även mer aktiva i det tyska översättningarna än i de svenska originalen. / The aim of this study is to analyse which initial and operational norms that were used in the German translation of pages from a songbook written by Astrid Lindgren. The models that are used are taken from Franzon (2009), Franzon (2016) and van Meerbergen (2010). The research questions concern musical rhythm, rhetorical methods within the text, the interaction level between images and text and the reader. The purpose is also to see how active or passive the main characters in the books are portrayed. The results show that translations of describing songs with long verses tend to become shorter and very changed compared to translations of songs with short verses and with a less describing nature. The interaction level with the reader is mostly low in both source and target texts and images. The main characters are also more active in the German translations than in the Swedish originals.