Tone-Metric Analysis: A Novel Approach to the Study of Form and Performance in North German Baroque Organ Music by Federico Andreoni A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Faculty of Music University of Toronto © Copyright by Federico Andreoni 2020 Tone-Metric Analysis: A Novel Approach to the Study of Form and Performance in North German Baroque Organ Music by Federico Andreoni Doctor of Philosophy Faculty of Music University of Toronto 2020 Abstract This dissertation studies the principles and applications of tone-metric analysis, a novel analytical approach based on the assumption that beat (or tactus) is a combinatorial set of strong and weak beats. As such, tactus can be studied using mathematical tools such as Pascal’s and Sierpinsky’s triangles, which help to quantitatively describe and model, through graphic constructs such as tone-metric waves and trees, the relationship between rhythmic, melodic, and harmonic designs as unfolding functions of the tactus’s regularity. The specificity of these functions, and their lengths and positioning within a piece, have consequences for the performance and perception of music, the analysis of which is at the core of this dissertation. Therefore, waves and trees can be used to describe phenomena such as the perception of tension and relaxation in music, and to initiate the analysis of how agogic elements can be integrated and rendered in performance. Because of its ability to describe and graphically model the elusiveness of music’s unfolding in time, tone-metric analysis is an ideal tool for the analysis, performance, and perception of repertoires that display a high degree of phrasal and formal variety, and are improvisatory in character. This dissertation studies primarily North German organ music, focusing on stylus ii phantasticus preludes and fantasias, chorale-based pieces, and fugues by Johann Sebastian Bach, Nicholaus Bruhns, Dietrich Buxtehude, Nicholaus Hanff, and Johann Pachelbel. For comparison, the dissertation also explores brief passages from later repertoires (classical, romantic, and extended tonal) by Wolfgang Amadeus Mozart, Frédéric Chopin, and Alban Berg. A major novelty of tone-metric analysis is its approach to the combinatorial and geometrical nature and features of tactus, and the consequent generation of an algorithmic basis for the analytical discourse. This approach raises questions about the nature of the relationship between tactus and other musical parameters such as melodic, harmonic, and rhythmic elements, and about their role in shaping phrasal and formal designs in music governed by a steady beat. The thesis thus recommends a new look at the fundamental nature of tactus as a conceptually combinatorial and fractal space for music. iii Acknowledgments First and foremost, I would like to thank my supervisor Don McLean for his invaluable insights and encouragement throughout my doctoral studies at the University of Toronto. Without him as a mentor, this journey would have not been as fascinating, productive, and inspiring. I would also like to thank the members of my doctoral committee, Kevin Komisaruk, Harald Krebs (External Examiner), Ryan McClelland, and Mark Sallmen, for their insightful suggestions and careful guidance during the dissertation’s writing process. A special thank you to my wife Afra Saskia Tucker for her wonderful presence and unceasing support; to my family and the many students, colleagues, and friends who have contributed to my musical and intellectual journey; and to all the composers and performers who have enriched my life through the beauty of their art. iv Table of Contents Acknowledgments.......................................................................................................................... iv Table of Contents ............................................................................................................................ v List of Tables ................................................................................................................................ vii List of Figures .............................................................................................................................. viii List of Examples ............................................................................................................................ ix Chapter 1 Introduction to a new theory of meter and form ......................................................... 1 1.1 Tactus, geometry, and performance ............................................................................................ 3 1.2 Research components .................................................................................................................. 4 1.3 Terminology ................................................................................................................................. 7 1.4 Repertoire .................................................................................................................................. 11 1.5 Corollary to Chapter 1 (a primer on tone-metric graphs) ........................................................ 12 1.6 Outline of the dissertation ......................................................................................................... 41 Chapter 2 Historical and theoretical perspectives ..................................................................... 42 2.1 From tactus to meter and beyond in analytical methodologies across the centuries ................ 43 2.2 Meter, form, and hierarchies ..................................................................................................... 44 2.3 Roman numeral analysis and form ............................................................................................ 46 2.4 Theorizing perception ................................................................................................................ 48 2.5 The origin of efflorescence and tone-metric waves ................................................................... 51 2.6 Scientific developments and studies in beat perception ............................................................ 56 2.7 Historic-analytical contextualization of the repertoire analyzed .............................................. 60 2.8 Perspectives on the relationship between performance practice and music analysis ............... 62 2.9 Conclusions................................................................................................................................ 66 Chapter 3 Conceptualizing the musical space ........................................................................... 67 3.1 Tactus, combinatorics, and fractal spaces ................................................................................ 68 3.2 Using fractals to model tactus ................................................................................................... 82 3.3 Conclusions................................................................................................................................ 84 Chapter 4 Analytical method ..................................................................................................... 87 4.1 Modeling purely binary and mixed binary-ternary types of metric organizations .................... 88 4.2 Pure binary type of metric organization .................................................................................... 88 4.3 Mixed binary-ternary type of metric organization .................................................................... 94 4.4 Pivots and moments of articulation ........................................................................................... 97 4.5 Tone-metric trees ..................................................................................................................... 101 v 4.6 Summary and further thoughts ................................................................................................ 103 Chapter 5 Analytical applications............................................................................................ 108 5.1 Durch Adams Fall, BuxWV 183 ............................................................................................... 109 5.2 Chorale-fantasia on Nun freut euch (mm. 45–80) ................................................................... 117 5.3 Magnificat primi toni (mm. 1–11; 76–91; and 139–141) by Buxtehude ................................. 125 5.4 Magnificat primi toni (first verse; mm. 1–17) by J. Praetorius ............................................... 132 5.5 Fugue on the Magnificat primi toni by Pachelbel ................................................................... 136 5.6 The Prelude in e minor by Bruhns. Mm. 1–5 and 6–10. .......................................................... 140 5.7 Conclusions.............................................................................................................................. 145 Chapter 6 Expanding the repertoire ......................................................................................... 146 6.1 Chorale harmonization on O Haupt by Bach .......................................................................... 146 6.2 Piano sonata in C major, K. 279/189d, iii, by Mozart. Mm. 1–10. ......................................... 152 6.3 Prelude in E minor, Op. 28 no. 4, by Chopin. Mm. 1–13. ....................................................... 155 6.4 Sonata, Op. 1, by Berg. Mm.