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This thesis describes an exploratory application of a statistical analysis and modeling technique (Markov chains) for the modeling of jazz improvisation with the intended subobjective of providing increased insight into an improviser's style and creativity through the postulation of quantitative measures of style and creativity based on the constructed Markovian analysis techniques. Using Visual Basic programming language, Markov chains of orders one to three are created using transcriptions of improvised solos by John Coltrane in his song Giant Steps. Still considered as statistical data, the Markov chains are examined and information is extracted from them through the development of several statistical tools for musical analysis. Two general categories of tools for analysis are developed: Subtraction matrices and graphical comparisons of distributions. Using these tools and the raw Markov chain data for musical analysis, quantitative measures for creativity and style are postulated. These measures are based on previously developed models and definitions of creativity and style taken from the literature. The information acquired from the implementation of the analysis tools is applied to the models in order to provide a theoretical basis for the development of the quantitative measures and a framework for the interpretation of the information. Guilford's Structure of Intellect model is used for developing creativity measures and Heen's model of the constructs of style analysis is used for determining measures of style. Overall, this research found that Markov chains provide distinct and useful information for musical analysis in the domain of jazz improvisation. Many examples of Markov chains are enumerated and tools for analysis are developed that implement the Markov chains. It is then explained how Markov chains and the tools for their analysis can be interpreted to determine quantitative measures of creativity and style. Finally, this thesis presents conclusions on Markov chain portrayals, new analysis tools and procedures, quantitative measures of creativity and style, and, in sum, that Markovian modeling is in fact a reasonable and useful modeling approach for this application. / Master of Science

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Cette thèse décrit et analyse les caractéristiques ethnologiques, musicales et acoustiques de la lyre bagana des Amhara d’Ethiopie. L’étude des données ethnologiques montre que le bagana incarne de nombreuses valeurs de la société traditionnelle amhara. Instrument considéré comme un don de Dieu et qui fut joué par des rois, le bagana est sacré. C’est un instrument intime, dont le jeu (toujours en solo ou accompagné seulement de la voix) est considéré comme un acte de prière ou une méditation à caractère religieux. Le musicien ne se donne pas à voir, ni par une dimension spectaculaire de sa performance, ni par une dimension phatique. L’analyse des accords utilisés dans le répertoire du bagana montre que cet instrument est essentiellement accordé selon deux échelles modales pentatoniques, tezeta et anchihoye. L’organisation temporelle des chants de bagana est fondée sur des pulsations discrètes très rapides. La pulsation apparente se compose d’un multiple de cette pulsation discrète qui change en fonction du motif joué, ce qui crée une sensation de rythme libre ou de rubato. Les chants de bagana sont fondés au niveau mélodique sur des unités qui se composent de paires de notes. L’analyse musicale du répertoire du bagana montre que celui-ci est fondé sur la répétition variée d’un ostinato musical assez court couplé à des paroles qui changent sans se répéter (à l’exception du refrain) selon les lois de la poésie amharique orale traditionnelle. Les procédés de variations mis en œuvre sont en général assez subtils car ils doivent préserver la sensation de répétition tout en apportant des éléments nouveaux. L’étude des propriétés acoustiques du bagana permet de déterminer que celle-ci produit des sons très graves (jusqu’au sol 1). Le dispositif chevalet large-obstacles modifie tous les paramètres du son. L’analyse de la facture traditionnelle montre que l’instrument est conçu pour produire un son grésillant, long et intense sans avoir recours à une caisse de résonance volumineuse. Le bagana est un instrument puissant, qui permet l’établissement d’une relation directe avec des entités surnaturelles via une transe légère. La voix et l’instrument sont dans un rapport de fusion et de renforcement mutuel. Les modes phonatoires utilisés sont « breathy » et « harsh ». La présence de la voix agit comme un guide perceptif, qui intervertit le rapport fond-forme dans la perception de l’instrument.

A presente tese se ocupa, do ponto de vista didático-pedagógico, em estudar a presença da racionalidade matemática na criação musical. A linguagem matemática é uma poderosa ferramenta que pode ser utilizada para compreender estruturas subjacentes às composições. Com o intuito de defender essa característica, são apresentados, neste trabalho, conceitos e estruturas matemáticas passíveis de analisar algumas obras musicais, como a teoria de conjuntos de Forte, a qual permite, por exemplo, tratar translações e inversões por meio do conceito de função matemática. Essa mesma teoria possibilitou ainda analisar algumas composições do século XX, como as de Almeida Prado e Rodolfo Coelho de Souza. A presença da razão áurea é pesquisada na obra de Mozart, Villa Lobos, Bartók e Debussy. Exemplos de autossimilaridade na música são apresentados analisando-se composições de Bach e Rodolfo Coelho de Souza. São estudados diversos tipos de simetria e feitas algumas aplicações em correlação com a música. É verificado que funções transposição (T) e inversão (TnI) formam um grupo com a operação composição. São definidas as funções P, L e R que têm como elementos do domínio e da imagem acordes maiores e menores e é mostrado como essas funções, juntamente com a operação composição geram o grupo PLR. São analisados alguns Choros de Pixinguinha e algumas Canções dos Beatles, como Octopuss Garden e verifica-se que tais composições apresentam este grupo PLR de funções no seu encadeamento. Demonstra-se que os grupos T/TnI e PLR são isomorfos ao grupo diedral D12, oferecendo aos graduandos em matemática um exemplo representativo do rico potencial da interface matemática/música, no caso via uma aplicação em música da Teoria de Grupos. O forte caráter interdisciplinar do presente trabalho se fundamenta, do ponto de vista didático-pedagógico, em textos de Olga Pombo e Ivani Fazenda. Uma tentativa de reintegrar a música à educação pode ser verificada pela aprovação do Projeto de Lei 2732/2008, o qual determina a obrigatoriedade do ensino musical na Educação Básica. Assim sendo, um importante resultado deste trabalho é a proposta de uma disciplina, a ser oferecida na graduação, voltada tanto para estudantes de música como de matemática, que contribua, de alguma maneira, com a formação desses profissionais, oferecendo-lhes subsídios para atuar no ensino médio ao integrar essas duas disciplinas. Tal disciplina tem o intuito de gerar um vasto campo de trocas de experiências entre os alunos, os quais poderão se apropriar de novos conhecimentos proporcionados pela união dessas áreas do conhecimento. / This thesis deals, in didactic-pedagogical terms, with the study of the presence of mathematical reasoning at the musical creation. The mathematical language is a powerful tool that can be used to understand structures underlying the compositions. In order to defend this characteristic, concepts and mathematical structures capable of analyzing some musical compositions, as the theory of sets of Forte are presented in this work, allowing, for example, treating translations and inversions through the concept of mathematical function. This same theory enabled the detailed analysis of particular twentieth centurys compositions, such as works by Almeida Prado and Rodolfo Coelho de Souza. The presence of the Golden Ratio is investigated in the works of Mozart, Villa Lobos, Bartók and Debussy. Examples of self-similarity in music are presented through the analysis of compositions by Bach and by Rodolfo Coelho de Souza. Specific types of symmetry are studied and some applications in correlation with music are realized. The fact that transpositions (T) and inversions (TnI) functions form a group with the compositions operation is verified. The functions P, L and R, whose domain and image elements are major and minor chords, are defined, and a detailed description is given on how these functions generate the PLR group through the composition operation. Cries by Pixinguinha and Beatles songs such as Octopuss Garden are analyzed and the fact that these compositions have the PLR group in their chaining can be verified. According to the demonstration, the groups T/TnI and PLR are isomorphic to the dihedral group D12, offering to the undergraduate mathematics students an example of the rich potential of the mathematics/music interface, in this case, through an application of the Groups Theory in music. The strong interdisciplinary character of this work is based, in didactic-pedagogical terms, on Olga Pombo\'s and Ivani Fazenda\'s texts. An attempt to reintegrate music to the standard education can be verified through the approval of the Law project 2732/2008, which stipulates the mandatory teaching of music in Basic Education. This way, an important result of this work is the proposal of a subject, to be offered at an undergraduate level to both students of music and mathematics, which contributes to the their professional training, offering them tools to integrate these two subjects when acting in high school. This subject aims to generate a wide range ofexperience exchange between students, who can expand their knowledge through the combination of these two subject matters.

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This dissertation explores multi-modal, symbolic, and embodied strategies for navigating musical periodicity, or “meter.” In the first half, I argue that these resources and techniques are often marginalized or sidelined in music theory and psychology on the basis of definition or context, regardless of usefulness. In the second half, I explore how expanded notions of metric experience can enrich musical analysis. I then relate them to existing approaches in music pedagogy. Music theory and music psychology commonly assume experience to be perceptual, music to be a sound object, and perception of music to mean listening. In addition, observable actions of a metaphorical “body” (and, similarly, performers’ perspectives) are often subordinate to internal processes of a metaphorical “mind” (and listeners’ experiences). These general preferences, priorities, and contextual norms have culminated in a model of “attentional entrainment” for meter perception, emerging through work by Mari Riess Jones, Robert Gjerdingen, and Justin London, and drawing upon laboratory experiments in which listeners interact with a novel sound stimulus. I hold that this starting point reflects a desire to focus upon essential and universal aspects of experience, at the expense of other useful resources and strategies (e.g. extensive practice with a particular piece, abstract ideas of what will occur, symbolic cues) Opening discussion of musical periodicity without these restrictions acknowledges experiences beyond attending, beyond listening, and perhaps beyond perceiving. I construct two categories for various resources and strategies: those which involve dynamic symbolic encoding (such as conducting patterns and tala gestures) and those which utilize static theoretical information (such as score-based knowledge and calculation of abstract relationships). My primary means of revealing and exploring these additional resources involves instances of “metric multi-tasking,” in which musicians keep track of multiple non-nested periodicities occurring simultaneously. One of the reasons these situations work so well at revealing additional resources is that attentional entrainment offers no explanation for how one might be able to do such a thing (only that attention is insufficient for the task). I do not make these moves in an attempt to significantly alter the theory of attentional entrainment. Rather, I frame that model as but one mode of temporal perception among many. I also leave room for types of temporal knowledge which may not be perceptual at all, but are nonetheless useful in situations involving musical periodicity. Pedagogical systems already make use of dynamic symbols and theoretical knowledge to help with temporally difficult tasks, and generally not virtuosic feats of metric multi-tasking. With these ideas in mind, I return to more straightforward “mono-metric” contexts and reconsider what to do with the concepts of “meter” and “perception.”

Franz Schubert (1797-1828) left many musical scores containing his earliest compositional efforts. Here 'earliest' is taken, for convenience, to refer to the works from the first extant (1810) up to and including the lied Gretchen am Spinnrade (1814), his first generally recognized masterpiece. This dissertation tells the story tracing those efforts in a chronological series of analytical essays. The essays mention only incidentally the external evidence of the home environment, lessons received, concerts attended, and so on, but refer instead primarily to the internal evidence of the compositions themselves, that is, the notes on the page. That story has not previously been told in these terms. The dissertation is thus a contribution to musical analysis applied to a quite important and certainly instructive but very little-known repertoire. An essential feature is that the story proceeds chronologically, to the (fairly large) extent that the exact chronology is known. Over a hundred works are involved, some containing several movements, so the story is necessarily long. Further, music is by no means a simple phenomenon, so the story is necessarily detailed. Another feature contributes to the tracing of the skein of anticipations of resources used in Schubert's later and more famous works, as well as to the evidence of derivation from models of other composers' works. Each work studied is provided with identifying information; musical incipits are also provided in view of the unfamiliarity of the repertoire. This identifying information, though necessary, is merely auxiliary to the story being told, and is accordingly set off from the latter. After the chronological story has been completed, a series of summaries is presented under the various categories of musical analysis; these summaries naturally refer back to the individual works. The ferreting out and telling of the story is itself the aim; no hypothesis is entertained. A review of the story yields several main results concerning the various elements of musical composition. (1) Schubert's attitude to the important matter of sonata form ranged from initial rather extreme experimentation possibly combined with some degree of misunderstanding to a clearly demonstrated ability to handle it convincingly first shown perhaps in his First Symphony D082 (October 1813). (2) Melody and text setting also showed early extremes as in the long and wild ballad Der Taucher D077 (first version September 1813 - April 1814), subsequently settling down, from about his first Opera D084 (first version October 1813 - May 1814), to a more suitable range of expression which was to serve him so well. (3) Counterpoint remained something of a weakness throughout, being used often but only in simple manifestations. (4) Harmony and orchestration were in general well handled throughout and many experiments were noted in methods of modulation. (5) An important factor to be found not in the notes but in the text of the score contributed to the mastery shown in Gretchen am Spinnrade (among other factors which are explored): Schubert's coming into contact with the inspiring poetry of Goethe. Three conclusions are offered on the broadest level. (1) The extent of Schubert's progress as a composer over the period studied was on the whole slight, because of the wealth of resources already at his disposal at the starting point at age 13. (2) The time at which Schubert wrote his first Symphony and first Opera - about October 1813 - is proposed as marking a settling down from earlier extravagance to more acceptably controlled writing. That applies to the two genres, instrumental and vocal music, on the one hand, as well as to the techniques of form and expression, on the other. It thus divides the period studied into two stages. (I naturally hope here to avoid oversimplification and acknowledge that the division is by no means watertight.) (3) By October 1814, the end of the present investigation, it was only in vocal music and specifically the lied, thus not also in instrumental or stage music, that real mastery can be recognized.

This thesis examines the relations among emotions and musical features and their changes with time, based on the assertion that there exist underlying, culturally specific, quantifiable rules which govern these relations. I designed, programmed and tested a computer controlled Two-Dimensional Emotion Space (2DES) which administered and controlled all aspects of the experimental work. The 2DES instrument consisted of two bipolar emotional response (ER) dimensions: valence (happiness-sadness) and arousal (activeness-sleepiness). The instrument had a test-retest reliability exceeding 0.83 (p &gt 0.01, N = 28) when words and pictures of facial expressions were used as the test stimuli. Construct validity was quantified (r &lt 0.84, p &gt 0.01). The 2DES was developed to collect continuous responses to recordings of four movements of music (N = 67) chosen to elicit responses in all quadrants of the 2DES: &quotMorning&quot from Peer Gynt, Adagio from Rodrigo???s Concierto de Aranjuez (Aranjuez), Dvorak???s Slavonic Dance Op 42, No. 1 and Pizzicato Polka by Strauss. Test-retest reliability was 0.74 (p &gt 0.001, N = 14). Five salient and objectively quantifiable features of the musical signal (MFs) were scaled and used for time series analysis of the stimuli: melodic pitch, tempo, loudness, frequency spectrum centroid (timbral sharpness) and texture (number of different instruments playing). A quantitative analysis consisted of: (1) first order differencing to remove trends, (2) determination of suitable, lagged MFs to keep as regressors via stepwise regression, and (3) regression of each ER onto selected MFs with first order autoregressive adjustment for serial correlation. Regression coefficients indicated that first order differenced (???) loudness and ???tempo had the largest correlations with ???arousal across all pieces, and ???melodic pitch correlated with ???valence for Aranjuez (p &gt 0.01 for all coefficients). The models were able to explain up to 73% of mean response variance. Additional variation was explained qualitatively as being due to interruptions, interactions and collinearity: The minor key and dissonances in a tonal context moved valence toward the negative direction; Short duration and perfect cadences moved valence in the positive direction. The 2DES measure and serial correlation adjusted regression models were, together, shown to be powerful tools for understanding relations among musical features and emotional response.