Hemispherical bells

Hemispherical bells were produced by a number of bellfoundries (including Mears and Stainbank, Warners and Taylors) in the second half of the 19th and the first half of the 20th century. Used as chiming or clock bells, they have the advantage of smaller size and weight for a given note. They can be stacked together meaning that very little space is taken by a set of bells. Here is a set of hemispherical bells from St Giles, Ollerton, Nottinghamshire cast by Mears and Stainbank in 1879:

The purpose of this investigation, prompted by a suggestion from the Dove team, was to find out which partial determines the note which is heard when a hemispherical bell is struck.

Investigation of recordings of 88 hemispherical bells shows that the partial frequencies have a very regular structure, like tubular bells. In all but the smallest bells, the pitch, i.e. the note heard, is an octave below the third partial in sequence from the lowest. In small hemispherical bells, just as with bells of traditional shape, the pitch is determined by the lowest partial.

In his 1890 paper on bell acoustics, Lord Rayleigh describes a brief investigation of a single Mears & Stainbank hemispherical bell. The note of the bell he examined, as given by the founder, corresponded with the third partial in order. This result is confirmed by my investigation for all founders and sizes of bells (apart from the very smallest).

Sets of hemispherical bells investigated

The investigation included the following sets of hemispherical bells:

Almost all the recordings are from Youtube, typically of bells rung in sequence together. This limits the accuracy to which frequencies can be measured. This does not substantially affect the outcome of the investigation.

As with the work on tubular bells, the investigation involved estimating the strike pitch of each bell by comparing its note to a tone produced by my pitcher program, and then measuring the partials for each bell. Estimating the pitch was easier than it was for tubular bells, and there were no cases where the pitch was ambiguous.

Here is the spectrum of bell number 2 at St Ninian’s, Whitby:

Nine significant partials are named. Their relative frequencies are similar to the rim partials of a bell of normal profile. There are other frequencies visible. Investigation of these is planned, using my technique for examining nodal patterns in bells.

The pitch of hemispherical bells

As explained above, the pitch of each bell was estimated by comparison with a pure tone. The partial frequencies of each bell were then measured from the recording. The range of pitches from the smallest to the largest bell was around two and a half octaves. The two smallest bells, from Malta, pitched by the lowest partial P1 and were excluded from this part of the analysis. Here is a plot of all partial frequencies against the estimated pitch.

The gradient of each line, i.e. the ratio of partial frequency to pitch frequency, was established by regression and the results are as follows:

Partials P3, P4 and P5 form a harmonic series with frequencies 2 x, 3 x and 4 x the pitch frequency. So as with bells of normal shape and tubular bells, the pitch heard is a virtual pitch effect and P3, the third partial in order from the lowest, determines the note heard when the bell rings. The p-values for all the regressions were very small indeed, so statistically the results are very highly significant, despite measurement difficulties (shown by the scatter in the plot) caused by the short sound segments in recordings of bells rung in rapid sequence.

There is nothing to distinguish bells by Grassmayr and Meneely, they show the same relationship as bells by Mears & Stainbank and Warner.

Partials of hemispherical bells

The relationship between the partial frequencies in the bells was investigated. The bells are of different founders and dates, with differences in shape and thickness, and this might be expected to affect the spacing of the partial frequencies in different bells. As partial P3 has been established as the partial that determines pitch, equivalent to the nominal in a bell of normal shape, the cents from each partial to P3 was calculated. The cents of partial P7 to P3 was used as a proxy for differences in shape and thickness. P7 was the highest partial measured across all the bells.

Here is a plot of the cents of each partial to P3, against the cents of partial P7 to P3, for each bell:

Clearly all the partials track together across changes in shape and thickness. This plot is very similar to the equivalent plot of the rim partials in bells of normal shape – for example, see the plots on this page. Again, there is nothing to distinguish bells by Grassmayr and Meneely. The average intervals in cents of the various partials to the strike pitch are as follows:

Pitch shift due to partial spacing

If the strike pitch of hemispherical bells is a virtual pitch effect based on partials P3, P4 and P5, we would expect that when these partials move further apart due to changes in the bell shape and thickness, the strike pitch would rise. However, with this particular set of recordings, because often the bells are rung in rapid sequence, the pitch estimates are approximate, based on a fraction of a second of sound.

A regression was done between the cents of the estimated pitch frequency to the frequency of partial P3, against the cents of the frequencies of partial P5 to P3. The regression shows a small positive effect, i.e. the strike pitch does increase if the ratio of P5 to P3 increases. However, the P-value from the regression is very poor so the effect is not statistically significant. The standard deviation of the pitch estimates is 34 cents, showing the inaccuracy of the estimates due to the short sound segments.

With a better set of recordings this analysis could be repeated to confirm or disprove the effect.

Acknowledgements

Grateful thanks are due to all those who recorded their bells and posted the recordings to Youtube, making this investigation possible without visits to multiple churches.