The sound of bells, and how it is perceived, is of high significance for the understanding of auditory perception, in particular, pitch perception. As was pointed out in the topic definition of pitch, pitch is a multiple auditory attribute, and the sound of a bell makes this drastically apparent. A considerable number of spectral pitches can be "heard out" which correspond to the frequencies of the bell's eigen-oscillations. Besides these "analytic" auditory aspects of the sound, there is a "holistic" one, namely, the auditory percept of a musical pitch that is more or less pronounced and fairly unambiguous - depending on the quality of the bell. That holistic pitch, which by the ear is associated with the sound immediately after the strike, has traditionally been termed the strike note. Explanation of the strike not is not trivial, as will become apparent from the following.
The sound of a bell is like certain pictures by Pablo Picasso: Many of the objects included are familiar, but they are not found quite at those places where they were supposed to be. The objects included in the sound of a bell are its part tones, and the intervals between them. The part tones of a typical contemporary church bell are by convention intended to come as close as possible to the frequency ratios 1:2:2.4:3:4:5:6:8 (followed by some higher, less well defined part tones). Even if those frequencies were exactly in the above ratios, the part-tone pattern would not quite correspond to what the auditory system "supposes". While some features of the pattern are familiar, as one can find two harmonic series (1:2:3:4:5:6:8, and 2:4:6:8), other features are not quite as they were "supposed to be": The minor third (2.4, with respect to 2) does neither fit into the harmonic series (1:2:3:4:5:6:8) nor into the series (2:4:6:8). And the fifth (3, with respect to 2) does not fit into the series (2:4:6:8). Even if instead of the minor third a bell is cast with a major third (2.5 instead of 2.4), the discrepancy remains. In reality, the part tone frequencies may depart considerably (e.g., by a few percent) from the above ratios - which provides for additional unexpected features of the sound. The part tones of church bells cast in previous Centuries may not even nearly follow the above pattern.
So it is apparent that both the auditory system and an acoustic theorist should find it difficult to assign a pitch to such type of sound. The theorist's problem is that neither from the time signal nor from the part-tone spectrum one can "read" a period or a fundamental frequency, respectively, that would tell the pitch. By contrast, the auditory system does not seem to have much difficulty in assigning a strike tone to the sound, at least for bells whose part tones largely follow the above pattern. This is why through decades the strike note of bells has been regarded as a kind of acoustic paradox.
The explanation is that the strike note indeed is a pitch, namely, the most prominent one of the multiple pitches that are elicited by the bell sound. The seeming paradox that one can assign one pitch to a sound that actually has several pitches is resolved by the notion that perception is hierarchically organized. An object that at the bottom of the hierarchy appears just as a collection of multiple elements (spectral pitches) may on a higher level of the hierarchy be represented by just one perceptual object, i.e., a kind of Gestalt (virtual pitch). The theory of virtual pitch a priori is based on this hierarchical concept, and this is why that theory explains the strike note of bells.
Grossly, the explanation is as follows. Any collection of simultaneous part tones elicits a number of pitches which are both of the spectral and virtual type. The relative prominence of all these pitches depends in a fairly complicated manner on the frequencies and amplitudes of the part tones. The most prominent pitch then is apprehended as the strike note. Whether that pitch is of the virtual or spectral type, depends on the frequencies (not just frequency ratios) and amplitudes of the part tones.
This suggests that the strike note - in the sense of a musical pitch - is by far not as pronounced and well defined as that of the conventional tones of music, i.e., harmonic complex tones. That this indeed is true becomes immediately apparent when one listens to music played on a carillon (Glockenspiel). It is one of the main appeals of carillons that recognition of a familiar melody requires some extra effort of auditory analysis, and that in polyphonic music played on a carillon there occur strange (dis)harmonies that can not - or not easily - be created by conventional music instruments.
A most important implication of the aforementioned explanation of the strike note is, that the theory accounts for the influence of the bell's size, i.e., of the absolute part-tone frequencies. Assume, e.g., that the part-tone pattern is of the aforementioned "ideal" type, and that the bell is fairly big, such that the first part tone's frequency is 100 Hz. Then the first four part tones are below the so-called dominant frequency region of the ear, which means that they hardly contribute to the holistic pitch percept. The next part tones (400, 500, 600, 800 Hz), however, are well in the dominant region and will therefore be employed by the auditory inference system of virtual pitch. The auditory inference system will "conclude" that these part tones either are the 4th, 5th, 6th, and 8th harmonics of 100 Hz or that they are a mixture of the 2nd, 3rd, and 4th harmonics of 200 Hz with an extra 500-Hz tone that does not count. As a result, both a virtual pitch corresponding to 100 Hz and another, corresponding to 200 Hz will be signaled. As these two pitches are octave equivalent, the musical pitch category is well defined and it is safe to predict that the strike note will correspond to about the note G. The octave region of the pertinent pitch is ambiguous, i.e., the strike note may be heard both as G2 and G3, although it turns out that often G3 is somewhat more pronounced. Note that to get this result, it is not required that the part tones at and below 300 Hz are present or that they have the correct frequencies. In this example, the strike-note pitch is of the virtual-pitch type.
Now consider the case of a medium-sized bell of which the first part tone of the series has the frequency of 500 Hz. In this case essentially the first four part tones (500, 1000, 1200, 1500 Hz) must be considered, because the higher ones are above the dominant frequency region. If the 500-Hz component is quite intense, it may alone suffice to determine the pitch, such that the latter is a spectral pitch corresponding to 500 Hz. If it the first part tone is not that strong, a virtual pitch will dominate that is "deduced" from the 500, 1000, and 1500 Hz components and which, of course, corresponds to 500 Hz, as well.
For a small-sized bell, e.g., with the first part tone at 1000 Hz or higher, it is the first part tone alone that determines the pitch of the bell, i.e., the strike note, because the higher part tones are above the ear's dominant frequency region. Thus for small, i.e. high-pitched bells the strike note throughout corresponds to the spectral pitch of the first part tone.
From these considerations it is apparent that automatic, quantitative prediction of strike notes is a challenge to any theory of pitch perception. In a study of 17 and 137 historical church bells, respectively, we have tested how well the predictions of the virtual-pitch theory agreed with auditory pitch evaluation by listeners. The results were indeed satisfying [46] , [65] , [104] p. 219-220.