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On the previous page, the results from Perrin et al's detailed investigation into the 'Perrin and Charnley' bell are given alongside actual measurements from a recording. This page explains the effects observed.
Given the extensive investigation by Perrin et al, it is not credible that modes of vibration exist in the bell which were not noted by them. However, frequencies of 1278.2 Hz and 2342.8 Hz are present in the recording and were not seen by Perrin et al.
More thorough analysis of the recording shows that frequencies of about 2342, 3515, 4686, 5858, 7029, 8197 and 9372 can be found in the early part of the waveform. These are all exact integer multiples of the (very strong) nominal frequency. As it is implausible that real untuned partials should be so exactly aligned, it has been assumed that these frequencies are harmonic distortion introduced at some point in the recording cycle - the video camera was rather close to the bell when it was stuck. This effect has not been noted with other recordings.
No such simple explanation attaches to the 1278.2 Hz partial. However, the bell was housed in a complex mechanical structure - a relatively flimsy steel supporting frame, and an electric chiming mechanism inside the bell. It has been assumed that this frequency arises from some part of this structure. Frequencies arising from steel or iron rather than bronze have been noted in other recordings.
All these frequencies have been ignored in the analysis.
A quick scan of the list shows that all the RIR partials are present, and are generally stronger than the others. This has a very simple mechanical explanation. When the clapper first hits the soundbow, it distorts at the point of clapper impact. It is likely therefore that the rim driven partials, i.e. those in which the soundbow is in motion, will be prominent. This point sounds obvious - it was actually arrived at by the author via trial and error in attempts to accurately reproduce the sound of a bell.
The next point worthy of note is that no RA modes are audible. In these modes of vibration, the bell twists axially, and the surface of the bell has no appreciable movement along the line between bell and listener. In consequence, the energy transmitted by the bell's motion towards the listener is likely to be minimal. (In fact, some energy can be detected at these frequencies with very careful analysis).
The remaining audible partials are all shell driven - the rim of the bell is stationary and the waist vibrates. A list of the audible modes is as follows:
Apart from the missing R=2 2m=0 and additional R=5 2m=4, these are the 'simplest' modes, i.e. those with the least number of radial and axial nodes.
A further effect - not particularly noticeable with this bell but common enough to be of interest - is that modes of vibration which have a nodal circle near the point at which the clapper strikes are often low intensity. This is because a ring of metal at the nodal circle is stationary, and the mode of vibration cannot be stimulated directly by the clapper's blow, though it can begin to vibrate via transfer of energy from other modes. The most notable partial with this quality is the quint, which is often inaudible in a bell whose clapper hits the correct spot on the soundbow.
It is notable that modern true-harmonic bells such as this one have very few audible partials above the octave nominal. Other bells are much more partial-rich.
The first graph below shows the variation over time of all the partials listed on the previous page:
The most prominent partials; 292.77 Hz (hum), 586.13 Hz (prime), 693.02 (tierce) and 1171.4 (nominal) are labelled. The hum starts at a low intensity, grows and decays again. The nominal appears to be the largest near the beginning and decays sharply. The tierce and prime start moderately loud, drop, grow and drop again. The oscillation is due to doublets - it is usually possible to estimate doublet separation from a graph such as this by determining the beat frequency.
The next graph shows the same information over a shorter timebase - the first half second:
Again, the prominent partials are labelled. It is clear again that the nominal dominates from early on. However, the strength of nominal, prime and tierce obscures the behaviour of the higher partials. Therefore, the last graph shows the same time period but with the nominal omitted:
It is is clear that much of interest occurs in the first 50 or 100 ms. The partials can be divided into two groups: those which start at higher intensity and fall (in the first 100 ms) and those which start at lower intensity and rise. In order of decreasing amplitude at the start, the frequencies are: 2441, 1764, 1526 and 292. All but one are RIR partials. It is actually very difficult to determine the amplitude of the partials in the first few milliseconds, because of a fundamental physical limit. If a transform is performed using a sound sample of length t, the best frequency discrimination achievable is 1 / t. Therefore, over 25 ms (the shortest sample used to produce the above graph) the minimum discrimination is 40 Hz - insufficient to distinguish between the partials at 1764 and 1776 Hz. We can never decide using analysis alone which partials are the first to sound - to discover this we have to proceed by inference.
When the clapper first hits the bell, only certain of the partials sound - mostly the RIR ones. As time progresses, energy swaps between the partials in a complex way. The initial burst of partials is very short - about 100ms for this sound recording. The importance of this short time period, as explained earlier in the paper, is that (in change ringing) this is the time during which each bell can be heard before the next strikes. Based on the observations on this recording, the most important partials in decreasing order of amplitude during the first 100ms are: 1171.4 (nominal), 693.02 (tierce), 586.13 (prime), 2441.22 (octave nominal), 1764.16 (superquint), 1526.9, and 292.77 (hum). This is not the order of importance that true-harmonic tuning would assign.
If the theory advanced earlier about origin of the strike note is correct, it is plausible that the sequence of RIM partials from the nominal upwards (i.e. 1171.39, 1764.16, 2441.22, 3184.60 and 3982.30) together form the series that creates the perception of the strike note, causing the ear to fill in the 'missing fundamental' an octave below the nominal. One would suppose that these frequencies will in some way determine the quality or timbre of the strike - and yet only one of them, the nominal, is conventionally tuned.
Last updated January 14, 2002. Site created by Bill Hibbert, Great Bookham, Surrey