Visual Insight in Geometry

According to a traditional rationalist proposal, it is possible to attain knowledge of certain necessary truths by means of insight—an epistemic mental act that combines the 'presentational' character of perception with the a priori status usually reserved for discursive reasoning. In this dissertation, I defend the insight proposal in relation to a specific subject matter: elementary Euclidean plane geometry, as set out in Book I of Euclid's Elements. In particular, I argue that visualizations and visual experiences of diagrams allow human subjects to grasp truths of geometry by means of visual insight. In the first two chapters, I provide an initial defense of the geometrical insight proposal, drawing on a novel interpretation of Plato's Meno to motivate the view and to reply to some objections. In the remaining three chapters, I provide an account of the psychological underpinnings of geometrical insight, a task that requires considering the psychology of visual imagery alongside the details of Euclid's geometrical system. One important challenge is to explain how basic features of human visual representations can serve to ground our intuitive grasp of Euclid's postulates and other initial assumptions. A second challenge is to explain how we are able to grasp general theorems by considering diagrams that depict only special cases. I argue that both of these challenges can be met by an account that regards geometrical insight as based in visual experiences involving the combined deployment of two varieties of 'dynamic' visual imagery: one that allows the subject to visually rehearse spatial transformations of a figure's parts, and another that allows the subject to entertain alternative ways of structurally integrating the figure as a whole. It is the interplay between these two forms of dynamic imagery that enables a visual experience of a diagram, suitably animated in visual imagination, to justify belief in the propositions of Euclid’s geometry. The upshot is a novel dynamic imagery account that explains how intuitive knowledge of elementary Euclidean plane geometry can be understood as grounded in visual insight.

Selected Works from Character Pieces in the 19th - Early 20th Centuries

During the Romantic Era, the character piece flourished and became one of the most important genres. Even though character pieces existed as early as the eighteenth century, it was not until the nineteenth century that Romantic Era literary figures discovered that music was as powerful as any other artistic medium in expressing unspeakable emotions. The character piece allowed composers to express a definite mood, a programmatic idea, or a pictorial association. Character pieces often have titles that suggest briehess or casualness, such as a Beethoven Bagatelle, a Schubert Impromptu or Moment musicaux. Others have expressive programmatic titles, such as Schumann's Kreisleriana and Liszt's Apres une lecture de Dante. Ths dissertation was based on selected works from the nineteenth century and early-twentieth century in order to demonstrate the great variety of musical thought and stylistic development. I have examined composers from different countries and of different nationalities to establish what and how they contributed to the remarkable diversity in this genre. The selected works allowed for very personal programs. These concerts, in addition to being good combinations of pieces, are representative of the genre. The first program consisted of Chopin's Preludes, Op. 28; three pieces from Debussy7s Images Books I and 11; and Liszt7s Apres une lecture de Dante. The second program included seven of Beethoven's Bagatelles, Op. 33; Schumann's Kinderscenen, Op. 15; Poulenc's Mouvements perpetuels and Intermezzo in A-jlat Major; Faure7s Nocturne in B Major, Op. 33, No. 2 and Impromptu No. 2 in F Minor; three of Mompou's Canciones y Danzas; four of Villa-Lobos Prole do bebe Suite No. 1; and Albkniz's Navarra. The third and final program comprised four selections from Rachrnaninoff s Moment musicaux, Op. 16; five pieces from Grieg's Lyric Pieces; and Arensky7s Suite No. 1 for Two Pianos and Suite No. 2 for Two Pianos. I wanted to bring out nuances specific to the individual composers and show how their character pieces helped in shaping the Romantic Era. Even in the twenty-first century, an era of highly specialized musical tastes, the popularity and accessibility of character pieces stand out.

THE PIANO MUSIC OF ROBERT SCHUMANN (1810-1856) AND JOHANNES BRAHMS (1833- 1897): A SURVEY OF CONCERTI, VARIATIONS, AND CHARACTER PIECES

Robert Schumann (1810-1856) and Johannes Brahms (1833-1897), in some ways Robert Schumann's artistic descendant, are the most important and representative German piano composers during the Romantic period. Schumann was already a mature and established musician in 1853 when he first met the young Brahms and recognized his talents, an encounter that had a long-lasting affect on the lives and careers of both men. After Schumann’s mental breakdown and death, Brahms maintained his admiration of Schumann’s music and preserved an intimate relationship with Clara Schumann. In spite of the personal and musical closeness of the two men, Schumann’s music is stylistically distinct from that of Brahms. Brahms followed traditions from Baroque and Classical music, and avoided using images and expressive titles in his music. Brahms extraordinarily intermingled earlier musical forms with multicolored tones of German Romanticism. In contrast, Schumann saw himself as a radical composer devoted to personal emotionalism and spontaneity. He favored programmatic titles for his character pieces and extra-musical references in his music. While developing their own musical styles as German Romantic composers, Schumann and Brahms both utilized the piano as a resourceful tool for self-realization and compositional development. To investigate and compare the main characteristics of Schumann and Brahms’s piano music, I looked at three genres. First, in the category of the piano concerto, I chose two major Romantic works, Schumann’s A minor concerto and Brahms’s B-flat major concerto. Second, for the category of piano variations I included two sets by Brahms because the variation framework was such an important vehicle for him to express his musical thoughts. Schumann’s unique motivic approach to variation is displayed vividly in his character-piece cycle Carnaval. Third, the category of the character piece, perhaps the favorite medium of Romantic expression at the piano, is shown by Schumann’s Papillons and Brahms’s sets of pieces Op.118 and Op.119. This performance dissertation consists of three recitals performed in the Gildenhorn Recital Hall at the University of Maryland, College Park. These recitals are documented on compact disc recordings that are housed within the University of Maryland Library System.

BEYOND THE LIBRETTO: HOW THE COMPOSER GUIDES THE CREATION OF CHARACTER THREE CASE STUDIES

When performing in opera, a singer portrays a character. A libretto is used as the principal resource for the research. Music can also reveal insights into the composer’s ideas regarding characterization. This performance dissertation examines how musical devices such as genre, texture, meter, melody, instrumentation and form can be used to inform choices of characterization. Three roles from diverse operas were examined and performed. The first role, Estelle Oglethorpe in Later the Same Evening (2007) by John Musto (b 1954) was performed November 15, 16, 17, 18 2007. The second role, Dorabella in Così fan tutte (1789) by Wolfgang Amadeus Mozart (1756-1791) was performed April 20, 25, 27, 2008. The third role, Olga in Eugene Onegin (1878) by Pyotr Ilyich Tchaikovsky (1840-1893) was performed on April 19, 2009. All operas were presented by the University of Maryland Opera Studio at the Ina and Jack Kay Theater in the Clarice Smith Performing Arts Center, University of Maryland College Park. DVD recordings of all performances can be found in the University of Maryland library system.

SELECTED CHARACTER PIECES OF FRÉDÉRIC CHOPIN (1810-1849), FRANZ LISZT (1811-1886) AND JOHANNES BRAHMS (1833-1897)

During the 19th century, Frédéric Chopin (1810-1849), Franz Liszt (1811- 1886), and Johannes Brahms (1833-1897) were among the most recognized composers of character pieces. Their compositions have been considered a significant milestone in piano literature. Frédéric Chopin (1810-1849) did not give descriptive titles to his character pieces. He grouped them into several genres such as Mazurkas, Polonaises. His Mazurkas and Polonaises are influenced by Polish dance music and inspired by the polish national idiom. Franz Liszt (1811-1886) was influenced in many ways by Chopin, and adopted Chopin’s lyricism, melodic style, and tempo rubato. However, Liszt frequently drew on non-musical subjects (e.g., art, literature) for inspiration. “Harmonies poétiques et religieuses” and “Années de pèlerinage” are especially representative of character pieces in which poetic and pictorial imagination are reflected. Johannes Brahms (1833-1897) was a conservative traditionalist, synthesizing Romantic expression and Classical tradition remarkably well. Like Chopin, Brahms avoided using programmatic titles for his works. The titles of Brahms’ short character pieces are often taken from traditional lyrical or dramatic genres such as ballade, rhapsody and scherzo. Because of his conservatism, Brahms was considered the main rival of Liszt in the Romantic Period. Brahms character pieces in his third period (e.g., Scherzo Op.4, Ballades of Op.10, and Rhapsodies of Op.79) are concise and focused. The form of Brahms’ character pieces is mostly simple ternary (ABA), and his style is introspective and lyrical. Through this recording project, I was able to get a better understanding of the styles of Chopin, Brahms and Liszt through their character pieces. This recording dissertation consists of two CDs recorded in the Dekelboum Concert Hall at the University of Maryland, College Park. These recordings are documented on compact disc recordings that are housed within the University of Maryland Library System.