The Sound of Bells

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Great Paul, St Paul's Cathedral, London, 17,001kg (334-2-19)

Recorded: Stephen Ivin
Analysed: WAH 24/11/01

This bell is the largest in the UK. It was cast by Taylors of Loughborough in 1881 and is hung for swing chiming in the South-West tower of St Paul's. The bell is rung for five minutes every day at 1.00pm. A bell of this size and grandeur ought to metaphorically frighten the horses, and literally stop the traffic. However, due to its position in the tower it is not particularly loud in the street. The bell is a maiden casting, i.e. has not been tuned, because its size exceeded the capacity of any UK tuning machine of the date. As part of their design process for Great Paul, Taylors conducted a series of experiments including casting a half-size trial bell weighing just over 37 cwt which was later broken up. The details above were taken from 'The Story of Great Paul' by Trevor S. Jennings. This book has a complete history of the bell together with much interesting information about it.

I am grateful to Stephen Ivin for his recording of this bell being swung. The analysis below is extracted from a talk I gave to the London branch of the Institute of Acoustics on 12th December 2001. The description goes into some detail on the perceived pitch and secondary strike of this bell. The issues are of importance to the sound of all big bells.

Bell Founder Tuning
Great Paul Taylors 1881 none since

Tuning of main partials

Nominal: 317.1 Hz

Measure Hum Prime Tierce Quint Nom'l S'quint O'nom.
Cents -2309 -1283 -875 -430 0 672 1206
Hertz 83.6 151.1 191.3 247.4 317.1 467.6 636.3

(The figures in the top row of this table are given in cents from the nominal. The Doppler shift due to the swinging of the bell means it is not possible to check for doublets.)

Tuning of Great Paul

At the time this bell was cast, all founders in the UK were casting 'old-style' bells. The chief characteristic of such bells is sharp hums and flat primes. Earlier founders both in the UK and abroad had been able by design or accident to produce bells without these deficiencies, but in the UK at least the practice had fallen into disuse. Ten years after this bell was cast, Taylors were experimenting with new bell shapes and methods of tuning which led to them being the first UK bellfounders to produce true-harmonic bells. Great Paul is not true-harmonic; its hum is a semitone sharp and its prime almost a semitone flat. Its octave nominal is however an almost perfect octave, something which modern founders do not usually achieve. For Taylors to produce a bell so much bigger than any other they had cast to date, with comparatively good partial tuning, suggests that they already had the design skills that would later lead to their development of true-harmonic profiles.

The pitch of Great Paul

The pitch of a bell is the note assigned to it by the casual listener. It can be measured by finding a sine tone which sounds the same note as the bell. In bells of typical size (say from 50kg to 2 tonnes) the pitch or strike note is about an octave below the nominal partial. This is because the nominal, superquint, octave nominal and higher partials form an approximate harmonic series with a 'missing fundamental'. In a harmonic series, the ratios of the partial frequencies to the lowest are integers. The partials listed are also those stimulated first by the impact of the clapper. Psycho-acoustical effects in the ear and brain supply this missing fundamental which is then heard as the pitch of the note. The portion of this site on pitch and timbre of bells goes into some detail on this. The pitcher program allows investigation of the effect with bell recordings.

In this bell, as in all big bells, the dominant pitch heard is not the octave below the nominal. The nominal of Great Paul is 317.1 Hz giving a half nominal of 158.6 Hz. Here is a sine-tone of that frequency, to be compared with the recording above. It will be immediately clear that the dominant pitch of this bell is not this, but a higher frequency. Experiments with pitcher show that this frequency is about 204 Hz. Here is a sine-tone of the higher frequency. To explain this, and confirm the theory of the origin of the strike-note, it is instructive to plot the spectrum of this bell, not against frequency or cents from the nominal as I would normally do, but against the ratio of the frequency to the pitch of the bell. First, here is the spectrum normalised to 158.6 Hz:

Great Paul lower strike

and here is the spectrum normalised to 204 Hz:

Great Paul higher strike

For a harmonic series to exist, we are looking for partials with a frequency related to the pitch frequency by an integer ratio, or something close to it. The first plot, giving the lower pitch, shows a harmonic series, with strong partials at about 2, 3 and 4 times the pitch. The strong partial at twice the pitch is the nominal. The higher partials move away from integral multiples, and are at about 5.2, 6.4, 7.7 and 8.5 times the pitch frequency.

In the second plot, normalised to the higher pitch, there is also clear evidence of a harmonic series, with partials at about 2, 3.1, 4, 5, 6, 7 and 8 times the pitch frequency. This harmonic series has rather more near-integral partials than that related to the nominal. This series is based not on the nominal but on a partial of frequency 408.6 Hz about 438 cents above the nominal - a somewhat sharp major third or very flat fourth above the nominal. This interval can be heard in the recording of the bell. The partial does not have a common name but is sometimes called the eleventh. In most bells including this one it is a set of three close-spaced partials; the pitch of this bell is based on the highest of the three.

The crucial question is: why should this higher pitch dominate in larger bells, including this one? The harmonic series for the higher pitch is clearly a good fit, but this would be true whatever the size of the bell. What sets big bells apart is the very low pitch of their partials. The ear's perception of the loudness of a tone decreases with frequency; low frequency sounds (below a few hundred hertz) have to have a considerably greater amplitude to sound equally loud as sounds with frequencies in the range above this up to several kilohertz. This means that it will be easier for the ear to hear the partials giving rise to the higher pitch.

Do not be misled by the strength of the partials either side of the eleventh. In bells of normal weight, the nominal can be much weaker than the prime and tierce, and yet the pitch of the bell is determined by the nominal and the harmonic series above it.

For all bells heavier than two to three tonnes, the secondary strike is evident. Bellfounders have tried both to explain and remove the phenomenon. The effect of the secondary strike is that heavy bells do not sound as deep as the weight of metal would suggest. The above explanation shows that the secondary strike is implicit in the design and partial structure of bells. The only way to avoid it would be to dramatically reduce the intensity of the high partials through a complete redesign of the bell shape. The spectrum in the next section, plotted against cents from the nominal, shows this bell to have a quite typical design despite its weight.

Intensity profile

Here to allow comparison with other bells analysed on this website, is the spectrum of Great Paul plotted against cents from the nominal:

Great Paul spectrum

This spectrum is very typical of bells a fraction of the weight. However, the very low frequencies may be below the sensitivity threshold of the recorder: not too much should be read into the hum and prime intensities. The fairly close tuning of the low five partials can be seen, together with the modest intensity of the higher partials. The eleventh, which sets the pitch of the bell, can be seen just above 400 cents. Its significance to the bell's sound is not apparent from this plot!

Conclusion

I hope the above discussion has thrown some light on the pitch effects heard in big bells. Great Paul is a good specimen of the founder's art, and it is a great pity it is not easier to hear from the ground when it is rung.


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Last updated December 19, 2001. Site created by Bill Hibbert, Great Bookham, Surrey