PHYSICS AND PSYCHOACOUSTICS OF PLUCKED-STRING INSTRUMENTS Wiliam H. Roberts A thesis submitted to Cardiff University for the degree of Doctor of Philosophy 4/8/2015 Acknowledgements Firstly I must thank my parents for their tremendous support, guidance and patience throughout my education, particularly during this doctorate. Thanks to Bernard for engrossing conversations and guidance over the past seven years or so, for introducing me to the fascinating world of musical acoustics and for having faith in my abilities. Many thanks too to Ian Perry for his help with admittance and wire-break measurements, for the use of his sound-field data and for general chats on the subject of musical acoustics. Thanks to Rob Mcleod for various chats on the subject of psychoacoustics and for a few pointers on conducting listening tests. I would also like to express my gratitude to everyone who took listening tests, especially to those who sat through a number of them. This work would not have been made possible were it not for your kindly giving up your time to listen to my sounds. Thanks to Rhys for the use of his folk guitar, and Eugene, Ben and Rhodri for the use of their banjos. A special mention goes to the musicians I’ve played alongside over the past few years, who have constantly fuelled my interest in this subject; cheers Sam, Heledd, Tara, Rhys, Huw, Daf, Laura, Guto, Rhodri, Rhys, Robin, Simon, Jimmy and Winskill! Also special thanks to Tara for making the final year of my doctorate the best it possibly could have been. Yn olaf, diolch arbennig i fy holl deulu am fod yn gyfeillion a chefnogwyr mor frwd, ac yn enwedig i Nain am edrych drosta i. i ii Summary The overall goal of this work is to help classical guitar makers to have better control over the potential sound of their instruments. This is done by correlating changes in the vibrational behaviour of an instrument with perceivable changes in its radiated sound. Three strategies have been employed in this thesis in an attempt to accomplish this goal. An investigation is carried out on the vibrational properties and radiated sound of a classical guitar, steel-string folk guitar and a five-string banjo. The aim here is to demonstrate how large constructional differences in plucked-string instruments lead to their unique acoustical characteristics. Results demonstrate that effective masses of low-order body modes relative to higher-order ones, internal damping of strings and amount of coupling between strings and the body of an instrument are responsible for the main characteristics of these instruments’ acoustical signatures. The problem of over-coupling a string to the body of a classical guitar is then addressed. Over-coupling creates an uneven tone quality between notes, an effect known as wolf notes. Knowledge of the perceptual threshold of over-coupling, which is found using psychoacoustical tests, can help instrument makers build a strongly radiating instrument without wolf notes. Simple remedies are suggested to minimise the effects of wolf notes on existing guitars in the most effective manner. Finally, the smallest perceivable changes are found in the parameters of a physical model of a classical guitar. The parameters that are responsible for the most perceptually obvious changes are likely to be the most important ones for the maker to control because these are the parameters to which our ears are most sensitive. The effective mass and effective area of low-order modes are studied in detail because these parameters have been identified previously as having a strong influence over the radiated sound from the instrument. iii iv Contents Chapter 1 Introduction ................................................................. 1 1.1 Aims ................................................................................ 1 1.2 Brief history and anatomy of the guitar and banjo ......................... 2 1.2.1 Guitar ......................................................................... 2 1.2.2 Banjo ......................................................................... 5 1.3 Basic function of a plucked-string instrument ............................... 7 1.4 Previous work ..................................................................... 9 1.4.1 Modelling ..................................................................... 9 1.4.2 Psychoacoustics ............................................................ 10 1.5 Structure of thesis .............................................................. 14 Chapter 2 Background Theory ........................................................ 17 2.1 String vibration .................................................................. 17 2.1.1 Dispersion and damping ................................................... 17 2.1.2 Peak heights of string partials ........................................... 18 2.1.3 Characterising string decays ............................................. 19 2.2 Instrument body ................................................................. 21 2.2.1 Dynamics of the body and the oscillator model ....................... 22 2.2.2 Mode shapes ................................................................ 23 2.3 String-body interaction ......................................................... 24 2.3.1 Local coupling strength ................................................... 25 2.3.2 Global coupling level ...................................................... 26 2.4 Sound radiation .................................................................. 26 2.4.1 Monopole radiation ........................................................ 27 2.4.2 Dipole radiation ............................................................ 29 Chapter 3 Experimental Methods .................................................... 31 3.1 Extracting parameters of exponentially decaying sinusoids .............. 31 3.2 Error in amplitude measurement of decaying sinusoidal signals ......... 33 v 3.2.1 Parseval’s theorem ........................................................ 35 3.2.2 Leakage error .............................................................. 35 3.2.3 Comparison of methods ................................................... 37 3.3 Characteristic impedance of a string ........................................ 38 3.4 Mechanical response ............................................................ 40 3.4.1 Measuring input admittance ............................................. 41 3.4.2 Simple peak picking ....................................................... 41 3.5 Studying mode shapes .......................................................... 42 3.6 Sound-field measurement ..................................................... 43 Chapter 4 Physical Measurements ................................................... 47 4.1 Radiated sound .................................................................. 47 4.1.1 Waterfall plots ............................................................. 48 4.1.2 String parameters ......................................................... 52 4.2 Characteristic impedance results ............................................. 58 4.3 Input admittance ................................................................ 60 4.3.1 Comparing the response at different positions along the bridge of a classical guitar ............................................................................. 60 4.3.2 Adding mass to the anti-nodal region of a mode ...................... 64 4.3.3 Input admittance of a banjo ............................................. 66 4.3.4 Comparing input admittances of different instruments.............. 68 4.3.5 Modal parameters of plucked-string instruments ..................... 71 4.4 Global coupling level ........................................................... 73 4.5 Mode shapes ..................................................................... 74 4.5.1 Classical guitars ............................................................ 75 4.5.2 Steel-string guitars ........................................................ 77 4.5.3 Banjo ........................................................................ 80 4.5.4 Discussion ................................................................... 81 Chapter 5 Psychoacoustics ............................................................ 83 5.1 Signal detection theory ........................................................ 83 5.2 Adaptive listening tests ........................................................ 85 5.2.1 Test procedure ............................................................. 85 5.2.2 Relating test result to psychometric function ......................... 87 5.2.3 Choice between two-down and three-down tests .................... 88 5.3 Masking and critical bands ..................................................... 89 vi 5.4 Basic JND tests .................................................................. 90 5.4.1 JNDs in parameters of the fundamental of note A2 ................... 91 5.4.2 Extending investigation ................................................... 92 5.4.3 Results and discussion ..................................................... 93 Chapter 6 Wolf notes on the classical guitar ....................................... 95 6.1 Modelling coupled string-body modes ........................................ 95 6.1.1 Previous model ............................................................. 95 6.1.2 Developing the model ..................................................... 96 6.1.3 Validating the