The Quantification of Strike Pitch and Pitch Shifts in Church Bells Thesis submitted by William A. Hibbert M.A. M.Sc. for the degree of Doctor of Philosophy February 2008 Updated April 2008 Department of Design, Development, Environment and Materials Faculty of Mathematics, Computing and Technology The Open University Milton Keynes United Kingdom Abstract The primary objective of the work reported in this thesis was to quantify how the pitch or strike note of a bell is determined by the frequencies of its partials. Pitches of bells are generally virtual pitch or missing fundamental effects, generated in the ear rather than present as a frequency in the radiated sound. The exact pitch is shifted from that expected for the missing fundamental by changes in the frequency of various partials. This can cause bells whose partials are in theory tuned precisely, to sound out of tune by considerable fractions of a semitone. The pitch shifts were quantified at frequencies across the audible spectrum by a set of experiments carried out on 30 subjects. Subsidiary experiments established which partials create a bell’s pitch or pitches at different frequencies, and showed that partial amplitude does not significantly affect bell pitch. A simple model of pitch shift was devised from the test results which gave good agreement with the stretch tuning in a number of peals of bells. Stretch tuning has not previously been satisfactorily explained. The pitch shifts were also compared against Terhardt’s algorithm for virtual pitch, which did not predict the shifts seen in practice. To prepare for these experiments, a comprehensive investigation was done of the partial frequencies of over 2,000 bells with a wide range of dates, weights and founders. An unexpected and straightforward relationship was found between the frequencies of the upper partials which generate virtual pitches, which seems to apply to all bronze and steel bells of Western shape. The relative frequencies of these partials are in turn determined by the thickness of the bell’s wall near the rim. This relationship between the partials has not been previously reported, and explains previous failed attempts by bellfounders to tune these partials independently. The modified version of Chladni’s law proposed by Perrin and Rossing for these partial frequencies was found not to give as good a fit to their frequencies as the relationship discovered in this research. The work presented in this thesis is important for at least two reasons: • It provides new practical guidance for the design and tuning of bells • The shifts in virtual pitches observed as a result of upper partial changes support current research into pitch generation mechanisms in the human ear. Acknowledgements I am grateful to my supervisor Dr Shahram Taherzadeh and his colleague Dr David Sharp, both of the Open University Acoustics Research Group, for their constant insight, encouragement and support over the past five and a half years. Robert Perrin, Ernst Terhardt and the late André Lehr have proved good teachers and correspondents, willing to impart knowledge and make suggestions for areas of fruitful investigation. Ernst Terhardt kindly allowed me to translate his 1984 paper into English. I am also grateful to David Kelly and Stephen Ivin who took part in many discussions about bell acoustics in the early stages of my investigations, and acted as sponsors when I applied to The Open University to embark on the formal research reported in this thesis. My thanks are due to the staff of The Whitechapel Bellfoundry and Taylors, Eayre and Smith Ltd, and in particular to Nigel Taylor, Andrew Higson and David Marshall for many interesting discussions on bell tuning and acoustics. The subscribers to the Bell Historians Yahoo Group have provided historical insights, details of peals of bells, and descriptions of historical tuning practices which are not documented elsewhere. I wish to thank the many bellringers and church authorities who allowed me to record and analyse their bells. Over 50 people participated in one or more experiments and I am grateful to them for their efforts and test results. Thanks are due to my son Christopher who was a trial subject for all the experiments done as part of this research and made many helpful suggestions. He also helped take many of the bell recordings in my collection. Finally, this work would not have been possible without the loving and patient support of my wife Margaret, who believed in the project from the start, even though she understood how much of my time it would take. Her encouragement and co-operation have been essential and invaluable. Table of Contents 1 INTRODUCTION.............................................................................................. 1 1.1 Core proposition of this thesis ............................................................................. 4 1.2 Overview of chapter contents .............................................................................. 5 1.2.1 Chapter 2 Historical Perspective and Literature Survey................................ 5 1.2.2 Chapter 3 Introduction to Bell Acoustics....................................................... 5 1.2.3 Chapter 4 Investigation into the Pitches of Bells........................................... 6 1.2.4 Chapter 5 Partial Classification Schemes and Frequencies ........................... 7 1.2.5 Chapter 6 Practical Issues in Pitch Measurement .......................................... 8 1.2.6 Chapter 7 Statistical Design of Virtual Pitch Tests........................................ 8 1.2.7 Chapter 8 Further verification of virtual pitch tests....................................... 8 1.2.8 Chapter 9 Virtual Pitch Test Results.............................................................. 8 1.2.9 Chapter 10 Validation of Pitch Shift Model .................................................. 9 1.2.10 Chapter 11 Concluding Remarks ................................................................... 9 1.2.11 Appendices................................................................................................... 10 1.3 Frequency measurements in this thesis.............................................................. 10 2 HISTORICAL PERSPECTIVE & LITERATURE SURVEY.................... 13 2.1 Introduction........................................................................................................ 13 2.2 Studies into the partial structure of bells............................................................ 13 2.3 Theories of pitch................................................................................................ 16 2.4 Pitch shifts and the dominance region for virtual pitch ..................................... 18 2.5 Pitch of bells....................................................................................................... 21 2.6 Measurement techniques for partial frequencies and pitch................................ 27 2.6.1 Partial frequency measurement.................................................................... 27 2.6.2 Strike pitch measurement............................................................................. 29 3 INTRODUCTION TO BELL ACOUSTICS................................................. 31 3.1 Introduction........................................................................................................ 31 3.1.1 Acoustics of a typical bell............................................................................ 31 3.2 Production and tuning of bells ........................................................................... 35 3.2.1 The shape or profile of bells ........................................................................ 35 3.2.2 The numbering and names of bells in peals and carillons ........................... 39 3.2.3 Casting of bells............................................................................................. 39 3.2.4 Tuning of bells ............................................................................................. 40 3.2.5 The problem of doublets .............................................................................. 42 3.2.6 True-harmonic tuning.................................................................................. 43 3.3 Partial amplitudes............................................................................................... 44 3.4 Formation of strike pitch.................................................................................... 47 3.4.1 Pitches of bells with different nominal frequencies..................................... 48 3.5 Strike pitch shifts............................................................................................... 49 3.5.1 Effect on upper partials of thickness and relative weight of bells ............... 52 3.6 Effect of bell shape and weight on upper partial tuning .................................... 53 3.6.1 Weights, diameters and partial frequencies of 553 bells ............................. 53 3.6.2 Weights and partial frequencies of 1,757 bells ............................................ 59 4 INVESTIGATION INTO THE PITCHES OF BELLS ............................... 63 4.1 Introduction........................................................................................................ 63 4.2 Secondary strike in bells .................................................................................... 63 4.2.1 Secondary strike experiment design ...........................................................