INHARMONICITY IN THE NATURAL MODE FREQUENCIES OF OVERWOUND STRINGS
Pochaman Chumnantas
PhD University of Edinburgh 1995 DECLARATION
I declare that this thesis has been composed by me and that the work is my own.
Pochaman Chumnantas ACKNOWLEDGEMENTS
I would like to express my thanks to Dr. Murray Campbell for his teaching, support, encouragement and patience. He gave me both freedom and direction. I would also like to thank to Professor Clive Greated and Dr. Raymond Parks for the guidance and facilities afforded.
I acknowledge the scholarship received from the Thai Government for the whole period of this work. ABSTRACT
The natural frequencies of piano strings depart somewhat from the harmonic series and the degree of inharmonicity has important implications for tone quality, tuning and the electronic synthesis of piano sounds. Apart from effects due to the finite compliance of the supports, the stiffness of the steel wire from which piano strings are made accounts almost entirely for the inharmonicity of the plain wire strings. It has been shown, however that the string stiffness is not the only source of inharmonicity of the overwound piano strings. Not only the effects of wave-reflection at the terminations of the various copper covering layers of overwound strings, but also the effects of nonuniformity may contribute weak partials that cannot be explained by string stiffness alone. Some discussions on the stepped string have appeared over the last few years by Levinson, Sakata and Sakata, and Gottlieb, but their analyses have not incorporated the stiffness of the stepped string. In this thesis, an expression for the frequencies of vibration of a stepped overwound string was described, and numerical calculations have been undertaken to compute theoretical mode frequencies for strings with varying degrees of overwinding. The numerical results of the frequency equation were compared with data from experimental measurements of the inharmonicities of overwound strings on a rigid monochord. The rigid monochord has been designed in order to control the parameters and to reduce external effects disturbing the vibration of the strings. It is evident from the comparison that the theory presented here gives a better fit to measured inharmonicities than Fletcher's analysis for a uniform string. The original motivation for this study was to determine the extent to which the non-uniformity of the overwinding on a bass piano string affected the inharinonicity of its mode frequencies. To examine the extent to which this work was relevant to the behaviour of overwound piano strings with the end support conditions typical of normal use, a series of measurements was performed on the bass strings of a Broadwood grand piano. It is evident from the results that the major cause of the discrepancy between the Fletcher prediction and the measurement is indeed the non-uniformity of the winding. CONTENTS
DECLARATION ACKNOWLEDGEMENTS ABSTRACT CONTENTS
CHAPTER 1 INTRODUCTION 1 1.1 Outline of research programme. 1 1.2 The history and physics of the piano. 5
1.3 The Theory of Inharmonicity. 13
CHAPTER 2 THEORETICAL CONSIDERATIONS. 28 2.1 Transverse wave equation for a string. 28
2.2 Bending waves in a bar. 31 2.3 Vibrations of a stiff string. 35 2.4 Vibrations of a nonuniform stiff string. 38
CHAPTER 3 NUMERICAL ANALYSIS. 42 3.1 Numerical Root Finding. 42
3.2 Numerical parameters. 45 3.4 Numerical results. 50
CHAPTER 4 EXPERIMENTAL TECHNIQUES AND RESULTS. 59 4.1 Experimental apparatus. 59 4.2 Analogue to digital converter and
sampling process. 62 4.3 Fourier Analysis. 67 4.4 Spectrum analysis. 68 4.5 The experimental uncertainty. 72 4.6 Experimental results. 84
CHAPTER 5 COMPARISON. 92 5.1 Comparison between the theoretical and experimental Inharmonicity. 92 5.2 Comparison of the B-coefficients from the theoretical results, experimental results
and Fletcher's results. 97 5.3 Uncertainty of experimental fundamental
frequencies. 98
CHAPTER 6 MEASUREMENT ON GRAND PIANO STRINGS. 106 6.1 Experimental technique on the grand piano. 106 6.2 The experimental and theoretical results
for the grand piano strings. 108
CHAPTER 7 SUMMARY AND CONCLUSION. 118 7.1 Conclusion. 118
APPENDIX A EXAMPLES OF NUMERICAL METHOD. 122 APPENDIX B THEORETICAL MODE FREQUENCIES. 131
APPENDIX C THEORETICAL INHARMOMCITY IN CENTS. 138
APPENDIX D EXPERIMENTAL MODE FREQUENCIES. 145
APPENDIX E EXPERIMENTAL INHARMONICITY IN CENTS. 152
APPENDIX F THE PIANO STRINGS' THEORETICAL AND EXPERIMENTAL MODE FREQUENCIES. 159
APPENDIX G THE PIANO STRINGS' THEORETICAL AND EXPERIMENTAL INHARMONIC1TY IN CENTS. 164