To Bach, the Circle of Fifths was the language of the universe. Similar to the constellations used to understand the sky, the Circle of Fifths is a visualization device to understand the fundamental concepts of key signatures, which are the foundation for music. The Circle of Fifths is a guide for writing music because its structure helps compose and harmonize melodies, build chords, and move to different keys within a composition. The Octave is the most significant key signature because it completes the circle of fifths. The Octave, if in perfect tune will create an overtone, which is a tune unable to be created on its own. The movement through the Circle of Fifths led to a contemplation described by Pythagoras as "Music of the Spheres" or meeting between heaven and earth, between spiritual and material realms. The Well Tempered Building uses the Circle of Fifths as the underlying geometry for the foundation of the conservatory. Proportions from the Circle of Fifths, including the Octave, shaped the conservatory making the musicians, audience, sound, light, water and air tuned to each other. / Master of Architecture

The objects of this study are the thirty-four gigues of J. S. Bach. This corpus of pieces represents one musician's encounter with the most engrossing dance genre of his time, and by coming to terms with this repertory I develop analytical perspectives with wide relevance to music of Europe in the early eighteenth century. The dissertation has a clear analytical focus but it also speaks to methodological issues of the relationship between theory and analysis, and the problem of reading a creative practice out of fixed works. Its main theoretical commitment is to middle-out perspectives on musical process. The dissertation's main themes are form, hypermeter, and schema. Its primary contribution to music theory lies in setting out an original position on the analysis of hypermeter, and advancing approaches to form and schema that are consistent with that position. "Form" and "schema" refer to compositional formulas that associate with hypermeter on the larger and smaller scales respectively, with observational windows as wide as the first half of a binary movement and as narrow as a couple of bars. Chapter 2 addresses the form of Bach's cello gigues. I arrive at a complete model of formal functions and phrase rhythm by first considering the turning points in the rhetoric of Fortspinnung. Chapter 4 addresses the chain of fifths sequence in Bach's harpsichord gigues. I analyze over fifty sequence passages, develop a typology of their contrapuntal frameworks, and consider the connections from sequence passages to subsequent events. These substantive analytical case studies flank the discussion of hypermeter. Chapter 3 includes analyses of Bach's orchestral gigue and two chamber-sonata gigues but is the most purely theoretical chapter. Its arguments are relevant to the study of meter and hypermeter across the whole "common practice" period. Since Chapters 2-4 address subsets of the corpus, a comprehensive overview is entrusted to Chapter 1, which also introduces the dissertation. Chapter 1's overview anchors the more specialized chapters in a wider reflection on the ability of compositional technique to inflect different styles, idioms, genres, and affects. / Music

This document examines Rodney Friend’s approach to violin technique for the left hand, with specific emphasis on the execution of fifths. The application of this technique plays a valuable role in establishing correct hand positioning, creating consistent intonation, improving vibrato, and adding to the palette of colors available to the violinist. Mr. Friend is an esteemed violinist, a perceptive pedagogue, and a dedicated mentor in today’s music world, and, in those roles, he exemplifies the qualities of beauty, truth, and goodness that characterize meaningful engagement in the arts. His thoughtful teaching style is the product of decades of careful observation combined with a devotion to constant personal improvement. As a performer who has toured the world as concertmaster of classical music’s most renowned orchestras, Mr. Friend’s early and continued success could have easily led him to a justifiable sense of arrival. Instead, he continues to hone his craft daily--both as a performer and as a teacher--always striving to remove the roadblocks from the tricky terrain of violin playing. In Mr. Friend’s words, violin playing is “a whole complicated business that we need to make less complicated.” His keen insights into the mechanics of the trade prove that simplification is possible. I firmly believe that his pedagogical innovations will greatly serve the rising generation of violinists. Most recently, Mr. Friend has turned his attention to an area of left hand technique that is commonly avoided by violinists in their practice, and often glossed over by teachers: the study of fifths. For the past six years Mr. Friend has been systematically exploring the benefits of this ignored technique. Other pedagogues have had little to offer in specifically addressing this interval because there is a general lack of understanding about the benefits provided by mastering the execution of fifths. Examples of this omission can be seen in pedagogy books by some of the 20th century’s greatest violinists and teachers. For instance, when Leopold Auer discusses the left hand in his book, Violin Playing as I Teach It, he suggests practicing scales in thirds, fourths, sixths, and octaves, but completely skips over the interval of the fifth. The same omission is also made by Yehudi Menuhin in his instructional text entitled Violin and Viola. Mr. Friend has developed a systematic approach to mastering this difficult technique that is both simple and effective. He plans to publish his method later this year. Some of the benefits provided to the violinist from the consistent practice of this traditionally difficult and neglected interval include better hand positioning, improved vibrato, and more consistent intonation--the continual quest of every violinist. I have experienced the remarkable benefits of his method in my own playing, which motivated me to seek Mr. Friend’s permission to further explore and document his techniques, and with this monograph to share these benefits with the violin community.

The current study examines the behaviors of four adverse impact measurements: the 4/5ths rule, two tests of significance (ZD and ZIR), and a newly developed AI measurement (Lnadj). Upon the suggestion of the Office of Federal Contract Compliance Program Manual about the sensitivity of the assessment of AI when the sample size is very large (Office of Federal Contract Compliance Programs, 2002), Lnadj is a new statistic that has been developed and proposed as an alternative practical significance test to the 4/5ths rule. The results indicated that, unlike the 4/5ths rule and other tests for adverse impact, Lnadj is an index of practical significance that is less sensitive to differences across selection conditions that are not supposed to affect tests of adverse impact. Furthermore, Lnadj decreases Type I error rates when there is a small d value and Type II error rates when there is moderate to large d value.

Obwohl die genaue Kenntnis der Tonartenverwandtschaften für die Modulationslehre wesentlich ist, verschwand dieses Thema im 19. Jahrhunderts allmählich aus den Harmonielehren, bis sich die Messung von Verwandtschaftsgraden in der Musiklehre des 20. Jahrhunderts fast nur noch auf das Kriterium des Quintenzirkels beschränkte. Der Blick zurück ins 18. und 19. Jahrhundert ergibt ein weitaus differenzierteres Bild in der Beurteilung von Verwandtschaftsgraden. Zwei Tendenzen lassen sich feststellen: Erstens führt die Orientierung der Theorie an der kompositorischen Praxis zu einer stetig sich erweiternden Legitimierung entfernter Tonarten als nahe Verwandte. Zweitens ist bis Jacob Gottfried Weber eine Zunahme an Verwandtschaftskriterien zu konstatieren, deren Differenzierung in der Folgezeit zugunsten einer wachsenden Integration fremder Tonarten wieder zurückgeht. Differenzierung der Kriterien und Integration von Tonarten verhalten sich gegenläufig zueinander. / Despite the intrinsic significance of key relationships on the conceptualization of modulation, this topic gradually disappeared from treatises in the nineteenth century until the measurement of key relationships was eventually restricted almost exclusively to the circle of the fifths in twentieth-century teachings. A glance back at the eighteenth and nineteenth centuries reveals a far more differentiated picture regarding the judgement of key relationships. Two tendencies can be discerned: first, the orientation of theory on compositional practice led to a continually expanding legitimization of distantly related keys as more closely related ones; second, until Jacob Gottfried Weber one can detect an increase in relational criteria whose differentiation in the following years declines again in favor of an expanding integration of foreign keys. Thus differentiation of criteria and integration of keys exist in opposition to one another.

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Die rätselhaften Mensuren 35–41 des Petrarca-Madrigals O voi che sospirate a miglior’ note von Luca Marenzio (Il secondo libro de madrigali à 5 voci, Venedig 1581) haben in der Theoriegeschichte seit dem 19. Jahrhundert immer wieder für Gesprächsstoff gesorgt. Im Zentrum eines im dritten Ton disponierten Stückes finden sich im Bassus acht sukzessive Quintfälle, die schließlich im melodischen Intervall Ges–h kulminieren. Der Text umkreist den Topos des ›Fallens in tiefste Tiefen‹ mit zahlreichen Beispielen aus Theorie- und Kompositionsgeschichte und zeigt verschiedene Lösungsansätze zur intonatorischen Realisierung der Passage jenseits modernen gleichstufigen Zirkel-Denkens auf. / The enigmatic seven measures 35–41 of Luca Marenzio’s madrigal O voi che sospirate a miglior’ note (Il secondo libro de madrigali à 5 voci, Venice 1581) have repeatedly provoked discussions in the history of music theory since the 19th century. At the centre of a piece in third mode, there is an episode with eight successive descending fifths in the bassus, culminating in the melodic interval G-flat–b. The paper revolves around the topic of ‘falling in deepest depths’, citing numerous examples from the history of theory and composition, and offers several approaches to an intonational realisation beyond modern thinking in equal-tempered circles.