PhD Thesis

In April 2008, I was awarded a doctorate by theĀ Open University following submission of a thesis on the strike note of bells. The aim of the research work was to quantify how the pitch or strike note of a bell is determined by the frequencies of it’s partials.

Here are three introductory articles which cover some key aspects of the work:

The work presented in the 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.

The full thesis can be downloaded here.

The abstract of the thesis is as follows:

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.

If you have comments or questions on the articles or thesis contents, please do not hesitate to get in touch.