The Sound of Bells

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The musical pitch of bells

Bells are musical instruments, and are assigned a note on a musical scale which identifies them, and which is used to judge whether a bell is 'in tune' with others in a peal or chime. When asked what (single) note a church bell is sounding, most people pick a note about an octave below the bell's nominal partial. This often does not co-incide with any partial in the bell, though by coincidence it is the same note as the prime if the prime is tuned to an octave below the nominal. This effect was first noted in a scientific context by Rayleigh in his 1890 paper but has only recently been explained.

It's worth at this point introducing a formal definition of pitch, to aid discussion and measurement. The pitch of a sound can be defined as the frequency of a pure tone which sounds neither higher nor lower than the sound. Pitch is an artifact of the human hearing system and pitch perception has unexpected complexities. Even with a pure tone, pitch depends slightly on the loudness of the sound being listened to. Sounds with multiple partial frequencies, especially if they are not harmonically related, can be heard in two ways by different listeners, and by the same listener under different circumstances. If the pitch heard by the listener corresponds closely to a partial frequency, it is called a spectral pitch. This type of hearing is common if the sound has one dominant partial, if the listener is a trained musician, or if the listener concentrates on the partial structure of the sound (so-called analytical hearing). However, the pitch heard is very frequently not that of one constituent partial, but rather is a synthesis of a number of partials. This effect, called virtual pitch, is very important - it is the most common mode of pitch perception.

Where the partials in a sound are harmonically related, but with the first member of the series missing (for example, a sound with partials at 500Hz, 750Hz, 1000Hz, 1250Hz etc.) a virtual pitch can be heard at 250Hz - the missing fundamental. Where the partials are not exactly harmonic, a virtual pitch is still heard at about the same place, but the exact frequency turns out to be determined in quite a complicated way by the frequencies of the individual partials. No comprehensive rule for determining virtual pitch is yet known. Virtual pitch has been extensively researched, and has been proved not to be due to a simple explanation such as difference tones, but rather is a side effect of the human hearing mechanism. There is no doubt that the strike note of a bell is a virtual pitch, as will be explained below. Virtual pitch effects often dominate spectral pitches - for example, in bells the strike note is about an octave below the nominal even if the tierce, only a minor third away, is very strong.

Frequencies of partials present in sounds can be measured with scientific instruments, spectrum analysers or Wavanal. Pitches cannot be measured with instruments, they exist only in our perception of a sound. Only a human listener can tell us the pitch of a sound - and different listeners will not all agree.

Research into virtual pitch and strike notes

A good overview of virtual pitch effects and some recent research is given in a paper by Julyan Cartwright. The papers of Meyer and Klaes, Arts, and Shouten and t'Hart apply virtual pitch principles to bells. However, the first conclusive proof that the strike note is a virtual pitch effect was given by some experiments by Terhardt and Seewann in the early 80s. In their paper, translated here for the first time, they compare the results of listening tests on 137 bells with an algorithm to determine pitch based on virtual pitch theory. The correspondence is good, though Terhardt and Seewan conclude that accurate (better than 15 cents) measurements of pitch are not possible by their methods. Prof. Terhardt has published several related articles on the web. Lehr's paper Partial groups in the bell sound, in section IV - Musical Consequences - also gives a good overview and several references to other work including a further description of Terhardt's paper.

In a paper published shortly after the work of Terhardt, Eggen and Houtsma reported on some careful measurements of the effect on the pitch of a bell of the tuning of various partials. Their results show that the pitch, or strike note, of a bell is determined primarily by the nominal, superquint and octave nominal, and only to a very small extent by the other low partials (hum, prime, tierce and quint). The nominal, superquint and octave nominal are the second, third and fourth members of an approximate harmonic series based on the (missing) strike note. Their results show that, for a given nominal, as the higher partials get sharper, the pitch sharpens, and conversely as the higher partials get flatter the pitch flattens. (This result only holds good for bells of middling size. For small bells, the pitch is determined by the hum. In large bells, the pitch is determined by partials higher than the nominal, the so-called 'secondary strike'.)

The assumption that the strike note of a bell is an octave below the nominal is universal in bell-founding and tuning. To admit that the strike note can be affected by other partials is counter-intuitive. To prove that this is the case, in a recent article on strike notes, I give two examples taken from peals of bells in the UK where the departure of the strike note from the half-nominal can clearly be heard by anyone with a musical ear. The effect is quite common once one learns to listen for it. The recordings of bells of different provenance given earlier all have identical nominals but show differences in pitch.

The practice of change ringing gives special emphasis to virtual pitch effects. Each bell sounds for only a short period of time (200mS to 250mS) before the next. This favours impressionistic rather than analytical hearing and hence emphasises virtual pitches over spectral pitches. The pitch of bells can sound different in changes from when they are rung alone.

One question that has not yet been explicitly answered (though the Terhardt pitch algorithm predicts it by implication) is why it should be the nominal, superquint and octave nominal, rather than any other partials, which determine the strike note. The answer, also covered briefly in my article on strike notes, has two components. First, the three partials approximate to a harmonic series in bells of normal shape. This is not chance, but rather informs the process of evolution of the western bell shape to give bells with defined pitch. However, other sets of partials in a bell also approximate to a harmonic series. The explanation for the nominal's importance lies in the fact that the ear does not respond equally to partials of every frequency. Frequencies in a range roughly from 500Hz to 1500Hz are accentuated by the physical construction of the ear (an effect first quantified by Fletcher and Munson) so that partials lying in this range are given preference for pitch determination. Spectral plots of bells with the intensity at each frequency weighted by Fletcher-Munson curves make it immediately obvious which partials contribute most to virtual pitch. A sample plot is included in my paper.

Practical implications of virtual pitch effects

If all bells had nominal, superquint, octave nominal etc. spaced in the same ratios, virtual pitch effects would only be of academic interest. The pitch of all bells would be shifted equally; bells would be 'in tune with each other' provided their nominals were in tune. The only occasion when the virtual pitch effect would be noticed would be when sounding bells together with other musical instruments; a very rare occurrence.

However, this assumption of equal spacing of upper partials is far from the case. Bells from different founders, from the same founder at different dates, and particularly, from a single founder at different places in a peal, can have different intervals between the upper partials. In peals of eight or more, it is very common for the smaller bells to be cast to different, heavier profiles for mechanical reasons. This closes up the upper partials and makes the smaller bells sound proportionately flatter. I first encountered virtual pitch effects, long before I could explain them, when I noticed that older peals of bells which had been retuned to have 'correct' nominals often had trebles that sounded flat. In fact, peals of eight from the nineteenth century or earlier commonly have the treble nominals sharp to offset the flattening of the virtual pitch; this is called stretch tuning.

Once the principle of true-harmonic tuning had been established in UK bellfounding, the practice of stretch tuning stopped. However, in the early 1950s the Taylor bellfoundry began to stretch the trebles on higher numbers again. The Whitechapel bellfoundry also produced stretched peals in the middle of the 20th century. The first such Taylor peal was probably Evesham, closely followed by Bengeo in Hertfordshire. Chris Povey has recounted discussions between Paul Taylor of Loughborough and Geoffrey Hemming of Evesham which suggest that Paul was aware of the flattened pitch of smaller, proportionately heavier bells and tuned them up to compensate. If so, he recognised the effect well in advance of academic researchers, though he did not attempt to explain it scientifically. The shift in bell pitch due to these effects should not be underestimated; shifts of 20 or 30 cents are common.

Banding of strike notes

The most striking example of virtual pitch is the secondary strike of large bells. The much lower frequency of the various partials brings a different set into the zone of emphasis, which as explained before is broadly 500Hz to 1500Hz. This allows a different and higher series of partials to create a virtual pitch, often a major third or a fourth above the strike note derived from the nominal. UK bellfounding practice and bell profiles seems to produce a secondary strike a fourth above the primary strike derived from the nominal. Some continental founders produce bells with a secondary strike a major third higher, which sounds much more pleasant. As a practical example, Big Ben has a nominal of about 335Hz (it is doubletted) which in musical notation is E + 27 cents. Actually, the dominant note we hear with this bell is a secondary strike of about 440Hz, i.e. an A, based on a partial above the octave nominal with a frequency of 883Hz.

As a further example, listen to this recording of Great Bede at Downside Abbey (Taylor, 1900). Here are the two strike notes a fourth apart: Lower strike Upper strike. There is nothing at all unusual about the partial structure of this bell. However, the lower strike (based on the nominal) is almost outside the band suggested above, and the ear provides a second 'missing fundamental' based on a strong partial a fourth above the nominal. The full list of lower partial frequencies of Great Bede in Hz is 102.5 (hum), 205.5 (prime), 246 (tierce), 306 (quint), 349.5, 409.5 (nominal), 539, 609, 672.5, and 836 (octave nominal), and the two strike notes are 204.8 (half the nominal) and 269.5 (half of 539). For a detailed analysis of a bell with a secondary strike see the description of Great Paul.

At the other end of the scale, in very small bells such as are used in carillons, the ear picks out the hum (i.e. roughly two octaves below the nominal) as the strike note. Here is a small carillon bell: Carillon and here is its strike pitch. The full list of lower partial frequencies of this bell is 839.5 (hum), 1683.0 (prime), 2008.5 (tierce), 3376.5 (nominal), and 5029.5 (superquint) and the strike pitch is 839.5 Hz. See here for a description of a mini-ring giving more examples of this. The strike pitch dropping from the half-nominal to the hum in small bells causes some difficulties to designers of carillons in deciding where the break point will occur.

Strike notes with confused pitch

Non-true harmonic bells can confuse as to what the perceived pitch actually is. For example, this Whitechapel bell of 1921 has a very loud prime about an octave and a tone below the nominal. Is the pitch of the bell the half-nominal or the prime? (Here is more information on this bell.) Heavy and / or untuned bells produce a sound which is difficult to pitch. An example is the large (32 ton) bell from Rostov Cathedral, for which it is difficult (for me) to establish a note at all.

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Experiment on the strike note of bells

The previous version of the page, written before I undertook the research explained above, included a simple experiment showing how the strike note of a bell is generated by the presence of various partials. This now needs revision based on the latest work, I have included it here unchanged for the time being. The previous version of the page also included some speculative thoughts on the effect of prime tuning on bell pitch which have been deleted; they proved to be wrong.

I have constructed the demonstration below to show how the perception of the pitch of a bell arises. This demonstration gives you the opportunity to test for yourself the reality of the effect.

To summarise the results detailed in the demonstration:

1) The strike note of most bells is an octave below the nominal for reasons associated with the behaviour of the ear rather than the bell

2) It is necessary for the bell to have audible tierce and / or hum for the strike note to appear, but the frequency of these lower partials does not affect the pitch of the strike note.

In this experiment, a set of overtones from a non-true harmonic Mears bell is used to create a number of bell-like sounds. The difference between the sounds is that various overtones are omitted from the mix. The overtones exclude the prime - so that there is no partial at or near the expected pitch of the strike note - and yet the strike pitch can be clearly heard in some cases. A sound file containing only the pitch of the strike is also provided for comparison purposes. Details of the overtones and sound files are as follows. A * in a box indicates the partial is present in the sound file. The author's results (i.e. whether a strike note is apparent) are given in the last line of the table. Your perception may differ . . .

freq. partial file1 file2 file3 file4 file5 file6 file7 file8 file9 file10 file11 strike
176.6 hum

*

       

*

*

*

*

*

*  

390.2

tierce

*

*

 

 

 

 

*

*

*

*

   

640.9

nominal

*

*

*

 

 

 

 

*

*

*

*  

949.0

superquint

*

*

*

*

 

 

 

 

*

*

*  

1302.4

oct. nom.

*

*

*

*

*

 

 

 

 

*

*  

1690.9

 

*

*

*

*

*

 

        *  

2110.2

 

*

*

*

*

*

          *  

2604.6

 

*

*

*

*

*

          *  

3380.0

 

*

*

*

*

*

 

        *  

320.5

strike

                     

*

strike note? yes yes no no no no no yes yes yes maybe  

To conduct the experiment, listen to strike to familiarise your ear with the pitch to be detected, and then play the various sound files and decide for yourself which ones appear to sound at the pitch of strike. Another way to conduct the experiment is to listen to each file and quickly, without allowing yourself to think about individual partials, sing the pitch you have just heard, and then compare with strike. This second method works more objectively if you let some time pass between each trial to allow you to forget what pitch you are listening for.

Analysis of the sound files with Wavanal should convince you that none of them (except strike) contain a partial at 320.5 Hz.

The author's results in the above table indicate that, for a strike note at a frequency of half the nominal to be heard, the nominal must be present, and either the tierce or the hum, preferably both. (In fact the prime plays a part in strike note production also, but was omitted from the experiment to ensure that no physical partial exists at the relevant frequency.) The higher partials add brightness to the sound but do not affect the pitch heard. The nominal defines the pitch of the strike, appearing as the first harmonic of the 'missing fundamental'. However, lower partials (the hum and the tierce) are needed to 'lead the ear down' to expect a lower fundamental.


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Last updated February 10, 2004. Site created by Bill Hibbert, Great Bookham, Surrey