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 105 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).

A model based on the modified Chladni law proves to be a good fit to the partials of hemispherical bells.

Sets of hemispherical bells investigated

The investigation included the following sets of hemispherical bells:

Location	No. of bells	Founder	Date	Recordings
Barkingside, Essex	8	Warner / Mears & Stainbank	1892 / 1925	Youtube / Matthew Higby
Buckhurst Hill, Essex	8	Mears & Stainbank	1907	Kye Leaver
Gawler, South Australia	8	Mears & Stainbank	1921	Youtube / Gawler History Team
High Beach, Essex	12 (of 13)	Mears & Stainbank	1873	Youtube / Lucas Owen
High Lane, Greater Manchester	6	Mears & Stainbank	1870	Youtube / Great British Bells
Loughborough Bell Museum	8	Warner	1911	Taken by me
Madonna of the Miracles, Malta	2	Unknown	Unknown	Youtube / Rayden Mizzi Media Productions
Mariinsky Theatre, St Petersburg	5 (of 33)	Grassmayr	2017	Youtube / Grassmayr bellfoundry
Matlock Bath, Derbyshire	8	Warner	1899	Taken by me
Newtown, Sydney, Australia	8	Mears & Stainbank	1877	Youtube / Laurie Alexander
Orwell Park School, Suffolk	5 (of 16)	Unknown	Unknown	Youtube / Sarah Kirby Smith
Shaftesbury, Dorset	8	Mears & Stainbank	1928	Youtube / now deleted
Surbiton, Greater London	8	Warner	Unknown	William Allberry
Union College, Schenectady, USA	8 (of 11)	Meneely	1925	Youtube / Union College
Whitby, North Yorkshire	3	Mears & Stainbank	1891	Nick Bowden / Tim Jackson

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:

Partial	P1	P2	P3	P4	P5	P6	P7	P8
Slope	0.479	1.158	1.996	2.980	4.097	5.349	6.720	8.231

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 P5 to P3 was used as a proxy for differences in shape and thickness.

Here is a plot of the cents of each partial to P3, against the cents of partial P5 to P3, for each bell:

Clearly all the partials track together across changes in shape and thickness – there is a slight gradient on all the lines. 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:

Partial	P1	P2	P3	P4	P5	P5	P7	P8
Cents to P3	-2,486	-947	0	694	1,245	1,706	2,100	2,450
Cents to strike	-1,286	253	1,200	1,894	2,445	2,906	3,300	3,650

Chladni’s Law and Hemispherical Bells

Ernst Chladni originally proposed the law named after him while investigating the vibration of flat circular plates in 1802. Vibration patterns in flat plates have nodal diameters (extending from the centre to the edge) and nodal circles concentric around the centre. Chladni observed that the addition of a nodal circle raised the frequency of a mode of vibration by about the same amount as adding two nodal diameters. If f is the frequency of vibration, m the number of nodal circles and n the number of nodal diameters, this suggests that f is proportional to (m + 2n)² , a formulation first suggested by Lord Rayleigh. Examples of the vibration patterns in circular plates are given in Fletcher and Rossing page 79.

When this law is applied to instruments with more complex shapes, the exact formulation no longer holds. If the number of nodal circles is ignored, which proves to be a valid assumption for the rim partials of bells of normal profile and hemispherical bells, and also tubular bells, a revised formulation known as modified Chladni’s law has f proportional to (2n + c)^p where c is a constant to be determined by experiment, and p (also to be determined by experiment) is no longer exactly 2. Fletcher and Rossing explore the utility of the modified model for cymbals and bells on pages 650 and 680. Bob Perrin and I applied the modified Chladni law to 2,752 church bells in a paper published in 2014 and saw a moderately good fit. The constants c and p had to be adjusted for very thin and very thick bells. The work I did recently on tubular bells achieved a much better fit to the modified Chladni law.

For hemispherical bells, the law suggests that for a rim partial with n nodal diameters (i.e. 2n nodes around the rim):

f_n = K * (2n + c)^p

where K is a constant depending on the diameter, shape and physical properties of the bell. The partial frequencies of all the bells can be normalised by relating them to the frequency of partial P3, chosen because it determines the pitch of the bell. This partial has 4 nodal diameters. Then we have:

f_n / f₃ = (2n + c)^p / (2*4 + c)^P

which removes the dependency on K. Taking logs:

log (f_n / f₃) = p * log{ (2n + c) / (8 + c) }

A least-squares regression for the measured partials for all the bells investigated gives values p = 1.5530, c = -1.3400 and an intercept very close to zero. A plot of log (f_n / f₃) against log{ (2n + c) / (8 + c) } is as follows:

The orange line is the regression line, and the bars show the maximum and minimum values across all the bells. There is clearly a very good fit with the modified Chladni law. There are slight variations in the slope of the line for different bells. As seen in the previous section, the interval of each partial to P3 rises or falls more or less linearly with the interval of P5 to P3 in each bell. To investigate this, the bells were divided into three groups with a low, middling and high interval between partials P5 and P3. The least-squares regression was rerun on these three sets of data with the following results:

P5 to P3 cents range	c	p	correlation
1174 to 1227	-1.2372	1.5322	0.99979
1230 to 1263	-1.3209	1.5590	0.99982
1264 to 1313	-1.5070	1.5524	0.99975
All bells	-1.3400	1.5530	0.99951

The parameters c and p both depend slightly on the relative dimensions and shapes of the bells. The individual groups of bells have slightly better correlation coefficients that that for all bells taken together – the correlation for all bells is a compromise across the different shapes of bells.

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 poor so the effect is barely 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.

References

Fletcher N H, Rossing T D, The Physics of Musical Instruments, Springer 1988

Hibbert W A, Sharp D B, Taherzadeh S, Perrin R, Partial Frequencies and Chladni’s Law in Church Bells, Open Journal of Acoustics, 2014, 70-77