The bowed string and its playability: Theory, simulation and analysis Hossein Mansour Department of Music Research McGill University Montr´eal, Qu´ebec, Canada June 2016 A dissertation submitted to McGill University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Music Technology c 2016 Hossein Mansour i Abstract This thesis describes the development of a highly refined physics-based model of a bowed string and its subsequent use to investigate several aspects of a bowed string, particularly its playability. Different components of a previously reported model are replaced by more accurate solutions, and several new features are added to make the model more realistic. Plucked-string simulations are described first, followed by enhancements to include the bow and the bowing process. Several properties of the simulated plucked response are compared to their designed values and to their experimental counterparts, which confirms the accuracy of the model and its parameters. An initial application of the model is focused on the minimum bow force above which the Helmholtz motion is sustainable. The Helmholtz motion is associated with the “speaking” sound of a bowed string, as opposed to its “whistling” or “crunching” sounds. An earlier theoretical relation for the minimum bow force is re-derived starting from a more robust assumption of an ideal stick-slip at the bowing point rather than a perfect sawtooth-shaped excitation force at the bridge. Simulations are used to evaluate and validate the improved accuracy of the new formulation. The revised relation makes some fundamentally different predictions that are confirmed by the simulations. Among those predictions is an upward shift in the frequency of the peaks in the minimum bow force plot with respect to the frequency of the body modes causing those peaks. The extent of that shift is shown to be a function of the bow-bridge distance which has important implications for the playability of an instrument close to its “wolf note”. Extending the scope of our analysis and acknowledging the fact that our model is still a few steps away from making predictions in quantitative agreement with measurements, it has been cautiously used to predict the trend and the relative strength of outcome when different physical details are included or excluded from the model. The features that are investigated are the minimum and the maximum bow force, the increase in the slip-to-stick ratio compared to its theoretical value, the spectral centroid, the pitch flattening, and the rate of occurrence for two special vibration regimes called the “ALF notes” and the “S- motion”. The initial transients of the bowed string are also analyzed in the bow force versus bow acceleration plane (a.k.a. the Guettler diagram). ii R´esum´e Cette th`ese d´ecrit le d´eveloppement et l’utilisation d’un mod`ele physique tr`es raffin´ede corde frott´ee qui permet l’´etude de plusieurs aspects de la corde frott´ee, en particulier de sa jouabilit´e. On a remplac´eles ´el´ements d’un mod`ele pr´ec´edemment d´ecrit par des solutions plus pr´ecises et de nouveaux ´el´ements ont ´et´eajout´es pour rendre le mod`ele plus r´ealiste. On d´ecrit tout d’abord des simulations de corde pinc´ee puis les am´eliorations permettant d’inclure l’archet et le processus d’excitation par l’archet. Plusieurs propri´et´es de la r´eponse simul´ee de la corde pinc´ee sont compar´ees aux valeurs d´esign´ees et exp´erimentales, ce qui confirme la pr´ecision du mod`ele et de ses param`etres. Une premi`ere application du mod`ele se concentre sur la force minimum fournie par l’archet au-dessus de laquelle le mouvement de Helmholtz est stable. Le mouvement de Helmholtz est associ´eau son “parl´e” d’une corde frott´ee, en opposition avec son “siffl´e” ou “craquant”. On reprend une relation th´eorique pr´ec´edemment ´etablie pour la force min- imum de fournie par l’archet, avec pour point de d´epart une hypoth`ese plus robuste de mouvement id´eal au point de contact entre la corde et l’archet plutˆot que l’hypoth`ese d’une excitation parfaitement en dents de scie au niveau du chevalet. Les simulations sont utilis´ees pour ´evaluer et valider les am´eliorations apport´ees par la nouvelle formulation en mati`ere de pr´ecision. La relation r´evis´ee permet de faire des pr´edictions fondamentalement diff´erentes, qui sont valid´ees par les simulations. Parmi ces pr´edictions on trouve une augmentation des fr´equences des pics dans le graphique de la force minimum fournie par l’archet par rapport aux fr´equences des modes de corps responsables de ces pics. On montre que l’ampleur de l’augmentation fr´equentielle est fonction de la distance entre le chevalet et le point de con- tact corde/archet, ce qui a d’importantes cons´equences sur la jouabilit´ed’un instrument au voisinage de la “note du loup”. En ´elargissant notre analyse et en restant conscient qu’il reste des am´eliorations `aap- porter pour que notre mod`ele puisse faire des pr´edictions quantitatives en accord avec les mesures, on a utilis´eavec pr´ecautions le mod`ele pour pr´edire des tendances et les l’ampleur relative des r´esultats lorsqu’on ajoute ou retire du mod`ele certains d´etails physiques. On ´etudie la force minimum et la force maximum de l’archet, l’augmentation du rapport en- tre glissement et adh´erence compar´ee `asa valeur th´eorique, le centre de gravit´espectral, l’abaissement de la hauteur de jeu et le taux d’occurrences de deux r´egimes sp´eciaux de vibration appel´es “les notes ALF” et le “mouvement-S”. Les transitoires d’attaque sont ´egalement analys´es dans le plan force de l’archet / acc´el´eration de l’archet (diagramme de Guettler). iii Acknowledgements First and foremost, I would like to thank my supervisor Gary Scavone. His openness for new ideas gave me freedom to explore a wide range of topics while he was always there to give me constructive feedback when it mattered most. The better part of my PhD studies was done remotely and Gary did a lot to make it as smooth as it could be. This work would have not been possible without his continued support, technical and otherwise. I’m tremendously grateful to Jim Woodhouse whose contribution to this work has been invaluable. I had the pleasure of working with him over two multi-month trips to Cambridge which set important milestones in my academic career. I would like to extend my gratitude to Colin Gough, and his wife Sophie, for their hospitality over a week of dense and very fruitful discussions in Birmingham. Special thanks to Anders Askenfelt for carefully reviewing my thesis and for his insightful comments on the draft. My sincere thanks to Fan Tao for providing various data on string properties, and for patiently responding to my questions with great insight. A big thanks is due to Paul Galluzzo for his encouragements and for sharing with me a large amount of data. Many thanks to Darryl Cameron for his technical support and to Yves M´ethot, Julien Boissinot, Jacqueline Bednar, and Harold Kilanski for their assistance in conducting my experiments in the Centre for Interdisciplinary Research in Music Media and Technology (CIRMMT). I had the pleasure of serving as the student representative to the board in CIR- MMT for a year, during which I had close collaborations with CIRMMT director, Marcelo Wanderley. I would like to thank him for his guidance during that period, and for his support and advice regarding motion capturing and high-speed video recording. In the course of this study I had the fortune of receiving valuable input from many people including, but certainly not limited to: Julius Smith, George Bissinger, Gabriel Weinreich, George Stoppani, Claudia Fritz, Evan Davis, Frank Fahy, Knut Guettler, Chris Waltham, and my colleagues at the Music Technology area, Charalampos Saitis, Esteban Maestre, Bertrand Scherrer, Erika Donald, and Vincent Fr´eour. I would like to thank them all for iv their contribution to this work. Special thanks to Arthur Pat´eand to Indiana Wollman for helping with the French version of the thesis abstract. This research has been funded by the Natural Sciences and Engineering Research Council of Canada (NSERC Vanier scholarship, Michael Smith foreign study scholarship, NSERC En- gage), Dr. Richard H. Tomlinson (R. H. Tomlinson doctoral fellowship), CIRMMT (student award, exchange student scholarship), and Schulich School of Music (graduate excellence fellowship, mobility award, GREAT award). The generous support by these organizations allowed me to focus on my studies without being worried about financial concerns, for which I’m extremely grateful. And above all, I would like to thank my mother Parvin, my father Ali, my sister Leila, and my wife Nazanin for their continued support and encouragement. v “[...] The trouble is that the nature of research is forever to be doing something that we do not know how to do and, as soon as we have learned how to do it, to stop doing it and look for a new problem; this means that a researcher’s mind is forever fixed on what has not been achieved which, by the standards of the world, means being condemned to a life of perpetual discouragement. That this is not the way that we researchers perceive it is one of the great miracles of human creativity, and the primary reason that we love our work as much as we do.” Gabriel Weinreich [1] To Nazanin for understanding my obsessions and for her support in times of frustration.