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It is well established that musical sounds comprising multiple partials with frequencies approximately in the ratio of small integers give rise to a strong sensation of pitch even if the lowest or fundamental partial is missing—the so-called virtual pitch effect. Experiments on thirty test subjects demonstrate that this virtual pitch is shifted significantly by changes in the spacing of the constituent partials. The experiments measured pitch by comparison of sounds of similar timbre and were automated so that they could be performed remotely across the Internet. Analysis of the test sounds used shows that the pitch shifts are not predicted by Terhardt’s classic model of virtual pitch. The test sounds used were modelled on the sounds of church bells, but a further experiment on seventeen test subjects showed that changes in partial amplitude only had a minor effect on the pitch shifts observed, and that a pitch shift was still observed when two of the lowest frequency partials were removed, so that the effects reported are of general interest.

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

The rim partials of a church bell (those with an antinode at the soundbow) generate the strike pitch or perceived note of the bell. The spacing in frequency of the higher rim partials has an important effect on the tonal quality of the bell. Investigations into the partial frequencies of 2752 bells, both bronze and steel, of a wide variety of dates, founders and sizes, show a simple and unexpected relationship between the frequencies of the rim partials. This relationship explains why attempts to tune the higher rim partials independently have failed. A modified version of Chladni’s law provides insight into the musical relationship of the partials, and predicts the partials of individual bells well, but fails to give a simple model of the spacing between the partials seen in bells with different profiles.

It is well established that musical sounds comprising multiple partials with frequencies approximately in the ratio of small integers give rise to a strong sensation of pitch even if the lowest or fundamental partial is missing—the so-called virtual pitch effect. Experiments on thirty test subjects demonstrate that this virtual pitch is shifted significantly by changes in the spacing of the constituent partials. The experiments measured pitch by comparison of sounds of similar timbre and were automated so that they could be performed remotely across the Internet. Analysis of the test sounds used shows that the pitch shifts are not predicted by Terhardt’s classic model of virtual pitch. The test sounds used were modelled on the sounds of church bells, but a further experiment on seventeen test subjects showed that changes in partial amplitude only had a minor effect on the pitch shifts observed, and that a pitch shift was still observed when two of the lowest frequency partials were removed, so that the effects reported are of general interest.

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

The rim partials of a church bell (those with an antinode at the soundbow) generate the strike pitch or perceived note of the bell. The spacing in frequency of the higher rim partials has an important effect on the tonal quality of the bell. Investigations into the partial frequencies of 2752 bells, both bronze and steel, of a wide variety of dates, founders and sizes, show a simple and unexpected relationship between the frequencies of the rim partials. This relationship explains why attempts to tune the higher rim partials independently have failed. A modified version of Chladni’s law provides insight into the musical relationship of the partials, and predicts the partials of individual bells well, but fails to give a simple model of the spacing between the partials seen in bells with different profiles.