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 Holy Trinity, Matlock Bath, Derbyshire cast by Warners in 1899:

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 detailed investigation of the modes of vibration was done for one hemispherical bell and confirms the identification of the partials. The relationship of the frequencies of the ‘rim’ partials (those most stimulated by a blow at the rim and which create the strike note) is similar to the relationship in bells of conventional shape. Investigation of hemispherical bells produced by Mears & Stainbank and Warners proves they tuned using strike notes, not partial frequencies, and that shifts in strike pitch due to upper partial spacing occur as they do in bells of conventional shape.

A model based on the modified Chladni law proves to be a good fit to the rim 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 |

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, and as will be seen below, these partials have an antinode at the rim. There are other frequencies visible, and these are also investigated in detail below.

## The pitch of hemispherical bells

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.

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.

## Modes of Vibration

The modes of vibration for all the significant partials were investigated for the biggest bell at Matlock Bath, the bottom left bell in the picture at the head of the page. The investigation used the technique described at https://www.hibberts.co.uk/identifying-bell-partials/. The bell was marked up in chalk at 65 equally-spaced points around the rim, and 24 points from rim to the crown of the bell. Recordings of the bell struck at each point were taken, and doublets then used to identify antinodes around the rim and up to the crown for each partial. The complete set of partials investigated and their antinodes are as follows. The frequency is the average of the doublet pair.

Freq (Hz) | Rim antinodes | Waist antinodes | Freq (Hz) | Rim antinodes | Waist antinodes | |
---|---|---|---|---|---|---|

131.1 | 4 | Rim | 2640.6 | 12 | 1 | |

336.1 | 6 | Rim | 2945.7 | 10 | 4 | |

598.3 | 8 | Rim | 2978.6 | 8 | 3 | |

906.9 | 10 | Rim | 3057.4 | 14 | 1 | |

1259.9 | ++12 | Rim | 3075.6 | 20 | Rim | |

1312.1 | 4 | 0 | 3473.3 | 12 | 2 | |

1407.3 | ++ | 0 | 3514.6 | 10 | 2 | |

1618.0 | 6 | 1 | 3625.1 | 22 | Rim | |

1652.4 | 14 | Rim | 4034.8 | 18 | 2 | |

1753.5 | 4 | 2 | 4209.8 | 24 | Rim | |

1934.2 | 8 | 1 | 4599.2 | 22 | 2 | |

2042.0 | 6 | 1 | 4641.1 | 16 | 1 | |

2087.2 | 16 | Rim | 4689.4 | 12 | 3 | |

2269.4 | 10 | 1 | 4822.4 | 26 | Rim | |

2462.0 | 8 | 2 | 5464.7 | 28 | Rim | |

2562.1 | 18 | Rim | 6131.8 | 30 | Rim |

++ These partials had no doublets, so the antinodal patterns couldn’t be observed. The number of antinodes for the rim partial was established from its frequency.

The rim antinodal patterns for the lowest four partials appear as follows:

Examples of the lip to crown patterns for a rim partial (336.1Hz) and a partial with two antinodes (3473.3Hz) are as follows. The horizontal axis is the fraction of the distance from rim to crown.

## Frequencies of Rim Partials

The bells investigated are of different founders and dates, with differences in shape and thickness, and this might be expected to affect the spacing of the rim 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 |

## Temperament and pitch shifts in hemispherical bells

Many of the bells investigated in this study were cast by Mears & Stainbank and Warners. At the time these hemispherical bells were cast, the practice of these two foundries was to tune bells of conventional profile by tuning strike notes, i.e. the aural impression of the pitch of the bell, rather than a particular partial frequency. In bells of conventional shape, the exact pitch heard is affected by the upper rim partials. If these are stretched further apart, the strike note sharpens, and vice versa. So the hypothesis for hemisperical bells is that the founders tuned them by strike note, and that the strike note is similarly affected by upper rim partials.

Before presenting the results of the analysis, it is legitimate to ask whether Mears & Stainbank and Warners tuned their hemispherical bells. My assumption is that they did: the bells I inspected at Matlock Bath had machined crowns, and tuning a hemispherical bell to a pitch could have been achieved by machining the bell on its lip. The Matlock Bath bells had flat lips as if they had been cut on a lathe. Whether the bells were tuned, or just judged as being in tune after casting, this part of the analysis is investigating the hearing of the people who worked on them in the foundry.

To investigate the hypothesis that these bells were tuned by strike pitch, I took most of the bells listed above, excluding the bells from Malta (which are by an unknown founder), Surbiton (which have recently been retuned by Taylors) and Grassmayr (which I assume were tuned by partial frequencies rather than by ear). To the remaining 87 bells I applied the technique used to investigate strike pitch tuning of historic bells of normal profile explained at https://www.hibberts.co.uk/stretch-tuning-of-old-style-bells/.

In a hemispherical bell partial P3 is the primary determinant of pitch, the interval between P5 and P3 represents the scale of thickness of the bell, and partial P1 (roughly an octave below the strike pitch) might affect the pitch heard. A temperament for the sets of bells was assumed, and then a regression done of the deviation of the cents of partial P3 from the temperament against the intervals of P5 to P3 and P1 to P3. As all the partial intervals to P3 are related, adding extra partials as independent variables would not improve the regression. An iteration adjusted the temperament to minimise the deviation of all the bells from temperament. There is an assumption here that Mears & Stainbank and Warners would both use a similar temperament.

Initial trials showed that the effect of the P1 to P3 interval was very small, with a large p-value, i.e. not statistically significant. Therefore the P1 to P3 interval was removed as an independent variable. The final regression coefficients were:

Coefficient | Effect | p-value |
---|---|---|

Constant | 179.66 | 0.017 |

Cents P5 to P5 | -0.1447 | 0.017 |

The low p-values indicate that the null hypothesis should be rejected, and that the strike pitch is definitely affected by the P5 to P3 interval. Bells with a larger P5 to P3 interval have flatter P3, i.e. the strike pitch relative to P3 is sharper when the P5 to P3 interval is larger.

This result means two things: these hemispherical bells were definitely tuned by strike pitch, not partial frequency. Also, the effect of upper rim partial spacing on the strike pitch of hemispherical bells is similar to the effect in bells of normal profile. This is to be expected, as the mechanism that forms the strike pitch in the ear is the same. The standard deviation of the residuals is 15.9 cents, which suggests moderately coarse tuning of the strike notes.

Here is a plot of the regression data, showing how P3 changes with the P5 to P3 interval. The orange line is the regression line. The change in strike pitch is in the opposite direction to the change in P3, i.e. the strike pitch gets sharper as the P5 to P3 interval increases, so that P3 is lower for the bell to sound in tune. For every 1 cent increase in the P5 to P3 interval, the strike pitch rises relative to P3 by 0.14 cents.

The best fit temperament in cents across a set of 8 bells is as follows, compared with equal temperament. The bells are numbered with 1 as the biggest / deepest:

Bell | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

These bells | -4.6 | 182.1 | 373.9 | 494.3 | 700.3 | 880.5 | 1088.1 | 1219.4 |

Equal temp. | 0.0 | 200.0 | 400.0 | 500.0 | 700.0 | 900.0 | 1100.0 | 1200.0 |

The small negative number for bell 1 arises because, on average, the biggest bell in the sets investigated is slightly flat. The temperament of these bells is similar to those used for bells of conventional shape in the 19th century, with bells 3, 6 and 7 flattened compared with equal temperament. The octave is stretched by 19 cents. This amount of stretch is commonly seen in sets of bells from the 19th century.

## 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* + 2*n*)^{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 (2*n + 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. 2*n* nodes around the rim):

*f*_{n} = *K* * (2*n* + *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} / f_{3} = (2*n* + *c*)* ^{p}* / (2*4 +

*c*)

^{P}which removes the dependency on *K*. Taking logs:

log (*f*_{n} / f_{3}) = *p* * log{ (2*n* + *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} / f_{3}) against log{ (2*n* + *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.

## 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. I am especially grateful to Sheila Laming, Churchwarden at Holy Trinity, Matlock Bath, for permission to examine and experiment on the bells at that church.

## 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