Essa pesquisa tem como foco principal explorar como obras musicais de Villa-Lobos são passíveis de serem lidas ou analisadas por meio de uma racionalidade matemática. O intuito é buscar um enfoque didático alternativa didática para a abordagem de conceitos oriundos da Teoria dos Conjuntos, baseados nos estudos do matemático Georg Cantor (Teoria Ingênua dos Conjuntos) e nos estudos de Allen Forte (Teoria dos Conjuntos aplicada à Música). Busca-se trazer para o universo da Música e da Matemática ambas as teorias, por meio de um enfoque transdisciplinar, e situar o saber em regiões em que o aspecto afetivo já adquiriu níveis capazes de dar sentido ao conhecimento e propiciar a assimilação de significados relacionados à outra área. Em busca desses objetivos, e ainda estudar possíveis indicações das relações entre Matemática e Música em um cenário didático/pedagógico, essa obra lança mão da afetividade, transdisciplinaridade e pensamento analógico como forma de articular áreas aparentemente distantes, mas com forte semelhança em suas estruturas. Esse estudo pretende explorar (1) trabalhos que usaram a Teoria dos Conjuntos em análises de obras de Villa-Lobos, (2) processos criativos e composicionais presentes em obras musicais de Villa-Lobos, (3) técnicas matemáticas de análise musical, (4) tipos e estruturas matemáticas que possam auxiliar em análises musicais e verificar de que maneira a racionalidade matemática está presente na composição musical. Este estudo ao pesquisar trabalhos que usaram a Teoria dos Conjuntos em análise musical de obras de Villa-Lobos preenche uma lacuna na teoria musical; evidencia estruturas matemáticas que auxiliam na análise musical, mostrando a presença da racionalidade matemática. Uma das grandes contribuições desse trabalho é estabelecer relações de analogia entre conteúdos do currículo da matemática, frequentemente traduzidos por códigos numéricos, e aspectos da área musical, reconhecidos por sons. / This research is mainly focused on exploring how musical works by Villa-Lobos are likely to be read or analyzed by a mathematical rationality. The aim is to seek a didactic approach a teaching alternative in order to deal with concepts from the Set Theory, based on studies by mathematician Georg Cantor (Naive Set Theory), and from studies of Allen Forte (Set Theory applied to Music). It intentsthe following: to bring both theories into the world of Music and Mathematics through a transdisciplinary approach; to situate knowledge in areas where the affective aspect has already acquired levels able to make sense of such knowledge; to encourage the assimilation of related meanings from area to the other. In the pursuit of such goals, and still researching possible indications of the relationship between Mathematics and Music in a didactic/pedagogical scenario, this work makes use of affection and transdisciplinarity analogical thinking as a way of articulating seemingly distant areas with yet strong similarities in their structures. This research therefore explores (1) studies that used the Set Theory in analysis of works by Villa-Lobos, (2) creative and compositional processes present in musical works by Villa-Lobos, (3) mathematical techniques of musical analysis, (4) types and mathematical structures that can assist in musical analysis, and it verifies how the mathematical reasoning is present in the composite musical work. The present study, by researching papers that used the Set Theory in musical analysis of works by Villa-Lobos, fills a gap in music theory; it shows evidence of mathematical structures that can assist in musical analysis, showing the presence of mathematical reasoning. A major contribution of this work is to establish relations of analogy between the mathematical content of the curriculum, often translated by numerical codes, and aspects of Music recognized by sounds.

Essa pesquisa tem como foco principal explorar como obras musicais de Villa-Lobos são passíveis de serem lidas ou analisadas por meio de uma racionalidade matemática. O intuito é buscar um enfoque didático alternativa didática para a abordagem de conceitos oriundos da Teoria dos Conjuntos, baseados nos estudos do matemático Georg Cantor (Teoria Ingênua dos Conjuntos) e nos estudos de Allen Forte (Teoria dos Conjuntos aplicada à Música). Busca-se trazer para o universo da Música e da Matemática ambas as teorias, por meio de um enfoque transdisciplinar, e situar o saber em regiões em que o aspecto afetivo já adquiriu níveis capazes de dar sentido ao conhecimento e propiciar a assimilação de significados relacionados à outra área. Em busca desses objetivos, e ainda estudar possíveis indicações das relações entre Matemática e Música em um cenário didático/pedagógico, essa obra lança mão da afetividade, transdisciplinaridade e pensamento analógico como forma de articular áreas aparentemente distantes, mas com forte semelhança em suas estruturas. Esse estudo pretende explorar (1) trabalhos que usaram a Teoria dos Conjuntos em análises de obras de Villa-Lobos, (2) processos criativos e composicionais presentes em obras musicais de Villa-Lobos, (3) técnicas matemáticas de análise musical, (4) tipos e estruturas matemáticas que possam auxiliar em análises musicais e verificar de que maneira a racionalidade matemática está presente na composição musical. Este estudo ao pesquisar trabalhos que usaram a Teoria dos Conjuntos em análise musical de obras de Villa-Lobos preenche uma lacuna na teoria musical; evidencia estruturas matemáticas que auxiliam na análise musical, mostrando a presença da racionalidade matemática. Uma das grandes contribuições desse trabalho é estabelecer relações de analogia entre conteúdos do currículo da matemática, frequentemente traduzidos por códigos numéricos, e aspectos da área musical, reconhecidos por sons. / This research is mainly focused on exploring how musical works by Villa-Lobos are likely to be read or analyzed by a mathematical rationality. The aim is to seek a didactic approach a teaching alternative in order to deal with concepts from the Set Theory, based on studies by mathematician Georg Cantor (Naive Set Theory), and from studies of Allen Forte (Set Theory applied to Music). It intentsthe following: to bring both theories into the world of Music and Mathematics through a transdisciplinary approach; to situate knowledge in areas where the affective aspect has already acquired levels able to make sense of such knowledge; to encourage the assimilation of related meanings from area to the other. In the pursuit of such goals, and still researching possible indications of the relationship between Mathematics and Music in a didactic/pedagogical scenario, this work makes use of affection and transdisciplinarity analogical thinking as a way of articulating seemingly distant areas with yet strong similarities in their structures. This research therefore explores (1) studies that used the Set Theory in analysis of works by Villa-Lobos, (2) creative and compositional processes present in musical works by Villa-Lobos, (3) mathematical techniques of musical analysis, (4) types and mathematical structures that can assist in musical analysis, and it verifies how the mathematical reasoning is present in the composite musical work. The present study, by researching papers that used the Set Theory in musical analysis of works by Villa-Lobos, fills a gap in music theory; it shows evidence of mathematical structures that can assist in musical analysis, showing the presence of mathematical reasoning. A major contribution of this work is to establish relations of analogy between the mathematical content of the curriculum, often translated by numerical codes, and aspects of Music recognized by sounds.

The task of finding the beat in music is simple for most people, but surprisingly difficult to replicate in a robot. Progress in this problem has been made using various preprocessing techniques (Hitz 2008; Tomic and Janata 2008). However, a real-time method is not yet available. Methods using a class of oscillators called relay relaxation oscillators are promising. In particular, systems of forced Hopf oscillators (Large 2000; Righetti et al. 2006) have been used with relative success. This work describes current methods of beat tracking and develops a new method that incorporates the best ideas from each existing method and removes the necessity for preprocessing.

Diante da afirmação recorrente entre alunos e pesquisadores de que a aprendizagem escolar dos conteúdos matemáticos são carentes de sentido e significado, temos elaborado propostas contextualizadoras para o ensino dessa disciplina. Na primeira parte deste trabalho, procuramos esclarecer a significação das palavras sentido, significado e contextualização, do modo como têm sido por nós interpretadas e a relação que essas palavras têm entre si. Para isso, nos embasamos em teorias da aprendizagem, as quais nos apontam caminhos para acreditar que a aprendizagem escolar pode ser relevante e significativa para os estudantes. A partir dessas teorias, argumentamos que a aprendizagem de um conteúdo escolar deve estar conectada às necessidades do indivíduo e a outros conteúdos, os quais compõem os contextos do conteúdo principal. Na segunda parte, através de uma pesquisa-ação, buscamos atingir o nosso objetivo de investigar se determinada sequência didática, elaborada a partir de uma concepção específica de contextualização, contribui para conferir sentido e significado para a aprendizagem de um conteúdo matemático específico. O conteúdo matemático escolhido foi Progressões Geométricas e a sequência didática contextualizadora foi a oficina Matemática e Música, aplicada a alunos do Ensino Médio numa escola estadual de São Paulo. Utilizamos, como ferramentas de investigação, entrevistas com os alunos e avaliações diagnósticas. As análises dos resultados contribuem para exemplificarmos a nossa concepção de práticas contextualizadoras, bem como atingirmos o objetivo da pesquisa-ação, acima enunciado. / Faced with the recurring affirmation between students and researchers that school learning of mathematical contents is short of sense and meaning, it was elaborated contextualized teaching purposes for that subject. In the first part of this work, we sought to clear the signification of the words sense, meaning and contextualization, the way that we have been making interpretations about them and the connection among those words. Thus, we based on learning theories that point us out some ways to believe that school learning can be relevant and meaningful for the students. Originating from those theories, we discuss that learning a school content must be connected to the needs of an individual and other contents that compose the contexts of the main content. In the second part, through an action research, we tracked down our objective of investigating if specific didactical sequence, elaborated from a particular conception of contextualization, contributes for giving sense and meaning for learning an explicit mathematical content. The chosen mathematical content was Geometrical Progression, and the didactical sequence was a workshop about Mathematics and Music, applied for high school students of a state school of Sao Paulo. We used, as investigations tools, interviews with students and diagnostic evaluations. The analysis of the results contributes to exemplify our conception of contextualized practices, as well as to reach the aim of action research, as proposed above.

Music Learning and Mathematics Achievement: A Real-World Study in English Primary Schools Edel Marie Sanders Abstract This study examines the potential for music education to enhance children's mathematical achievement and understanding. Psychological and neuroscientific research on the relationship between music and mathematics has grown considerably in recent years. Much of this, however, has been laboratory-based, short-term or small-scale research. The present study contributes to the literature by focusing on specific musical and mathematical elements, working principally through the medium of singing and setting the study in five primary schools over a full school year. Nearly 200 children aged seven to eight years, in six school classes, experienced structured weekly music lessons, congruent with English National Curriculum objectives for music but with specific foci. The quasi-experimental design employed two independent variable categories: musical focus (form, pitch relationships or rhythm) and mathematical teaching emphasis (implicit or explicit). In all other respects, lesson content was kept as constant as possible. Pretests and posttests in standardised behavioural measures of musical, spatial and mathematical thinking were administered to all children. Statistical analyses (two-way mixed ANOVAs) of student scores in these tests reveal positive significant gains in most comparisons over normative progress in mathematics for all musical emphases and both pedagogical conditions with slightly greater effects in the mathematically explicit lessons. This investigation addresses concerns that UK and US governments' quests for higher standards in mathematics typically result in impoverished curricula with limited access to the arts. In showing that active musical engagement over time can improve mathematical achievement, as hypothesised, this work adds to a growing body of research suggesting that policy-makers and educationalists should reconsider curriculum balance.

In der Arbeit wird ein fächerverbindender Zugang zur Sinusfunktion entwickelt. Periodische Funktionen werden über die Analyse und das Aufzeichnen von Instrumenten untersucht. Die Sinusfunktion wird als Modell für Töne eingeführt. Auf Basis dieses Modells werden Synthesizer entwickelt an denen mathematische und musikalische Fragestellungen behandelt werden. Das Konzept wird exemplarisch erprobt und reflektiert.:1. Einleitung 2. Vergleich verschiedener Lehrbuchansätze 3. Unterrichtskonzept 3.1. Motivation 3.2. Fachliche Grundlagen und erste didaktische Überlegungen 3.2.1. Der Funktionsbegriff 3.2.2. Der Begriff Sinus und die zugehörigen Schüler*innenvorstellungen 3.2.3. Grundbegriffe zu periodischen Prozessen und Schwingungen 3.2.4. ModellierungvonInstrumenten 3.2.5. Schüler*innenvorstellungen zu Schwingungen, Wellen und Tönen 3.2.6. AnwendungderentwickeltenModelle 3.2.7. Diagramme und damit verbundene Schwierigkeiten im Unterricht 3.3. Didaktische Grundlagen 3.3.1. Rahmenbedingungen und Curriculumsbezug 3.3.2. DasKonzeptunddieKMKBildungsstandards 3.3.3. Vorwissen 3.4. Einführung: Instrumentenanalyse - Töne als Wahrnehmung von Schwingungen 3.4.1. Didaktisches Konzept Instrumentenanalyse 3.4.2. Verlaufsplan Instrumentenanalyse 3.5. Mathematische Modellierung der Töne - Die Sinusfunktion 3.5.1. Didaktisches Konzept mathematische Modellierung 3.5.2. Verlaufsplan mathematische Modellierung 3.6. Anwendung des Modells von Tönen als harmonische Schwingungen 3.6.1. Didaktisches Konzept Anwendung des Modells von Tönen als harmonische Schwingungen 3.6.2. Verlaufsplan Anwendung des Modells von Tönen als harmonische Schwingungen 3.7. Verwendete Software 3.7.1. Audacity 3.7.2. Geogebra 3.7.3. Viana 3.7.4. SonicVisualiser 3.7.5. VCV-Rack 4. Durchführung des Konzepts 4.1.Rahmenbedingungen, Lerngruppe und Vorwissen 4.2. Betrachtung der Einzelstunden 4.2.1. Erste Stunde 4.2.2. Zweite Stunde 4.2.3. Dritte Stunde 4.2.4. Vierte Stunde 4.2.5. Fünfte Stunde 4.2.6. Sechste Stunde 4.2.7. Siebte Stunde 4.2.8. Achte Stunde 5. Evaluation des Konzepts 5.1. Auswertung des Tests 5.2. Evaluationsgespräch mit SuS 6. Entwicklung eines Synthesizers für den Unterricht auf Basis eines Mikrocontrollers 6.1. Konzept 6.2. Umsetzung A. Arbeitsblätter 114 A.1. Einführung: Instrumentenanalyse A.1.1. Die Begriffe periodisch, Periode, Periodendauer und Amplitude A.1.2. Parameter einer periodischen Schwingung A.1.3. Parameter einer periodischen Schwingung (bearbeitet) A.2. Mathematische Modellierung der Töne - Die Sinusfunktion A.2.1. Modell einer harmonischen Schwingung A.2.2. Die Sinusfunktion A.2.3. Parameter der Sinusfunktion A.2.4. Die Sinusfunktion als Modell für Töne A.3. Anwendung des Modells von Tönen als harmonische Schwingungen A.3.1. Entwicklung eines Synthesizers auf Basis des Modells für harmonische Schwingungen A.3.2. Amplitudenverlauf A.3.3. Intervalle A.3.4. Obertoene B. Test C. Präsentationen C.1. Parameter der Sinusfunktion und Zeitabhängigkeit C.2. Ausblick zum Abschluss der Erprobung D. Verlaufspläne der Erprobung D.1. Erste Stunde D.2. Zweite Stunde D.3. Dritte Stunde D.4. Vierte Stunde D.5. Fünfte Stunde D.6. Sechste Stunde D.7. Siebte Stunde D.8. AchteStunde E. Programmcode Synthesizer F. Quellenverzeichnis G. Abbildungsverzeichnis H. Verzeichnis der Erklärboxen I. Selbstständigkeitserklärung

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This thesis covers pretty much everything about the 4’th movement of “I Ching” by Per Nørgård (“IV. Towards Completion. Fire over Water”). I have delved deep into an analysis, that helped me develop my own interpretation and understanding of the piece.  While I was doing my research, I discovered fascinating links between music and mathematics, that showed me how much we don’t know and that there are interesting subjects left for us to find. My exploration did not stop there just yet, I also found out about the “I Ching”, an ancient Chinese book, that covers a broad topic, which can be summed up in two words: Yin & Yang. This, combined with a bit of mathematics contributed to the creation of a unique vocabulary that Per Nørgård pioneered, called: “infinity series”. My thesis aim is to promote Per Nørgård’s music, that has not yet been discovered by enough percussionists.