The story starts with the human ear which receives the sound which we hear. As I understand things, there is a tube in an ear in which there are lines of innervated hairs, receptive to increasing but overlapping frequency bands of sounds, a more complicated version of what the three sorts of colour cones do in the eye. But the ear is not as clever as the eye and the signals from the ear are sent onto the brain in a more or less raw state, with the brain responsible for converting them into the sound that we hear.
This sound that we hear from a bell has something called pitch and most people are very good at telling one pitch from another. And most of those can go further saying that this pitch is higher, the same or lower than another. With the result that we can organise pitches into a well ordered sequence, in much the same way as we can put numbers, whole or otherwise, into a well ordered sequence.
Then, using modern, computerised audio equipment which can generate pure tones, tones dominated by just one frequency of sound, we are able to match the pitches that people hear with frequency. This pitch corresponds to that frequency.
Such equipment can also be used to analyse the tones coming from a bell, giving in the case of the bell at Ipswich above, sounds at a number of frequencies. But not much at the frequency of the pitch that we hear. We get all the overtones; two times the frequency, three times, four times and so on, but not the frequency itself, not the fundamental. The brain, on the basis of these more or less harmonic overtones, decides that what we really want to hear is that fundamental, which is half the frequency of the nominal in the spectrogram above. What you get is not what you hear. Sometimes called the virtual pitch, by analogy with the virtual world of computer games.
That said, people with good ears will sometimes hear the separate overtones and sometimes hear the pitch. There will also be variation arising from where and how the bell is hit by the clapper or hammer. Other complications arise because the various frequencies generated by one bell are not in tune with each other and because the frequencies generated by one bell are not in tune with those generated by another bell. A lot of old rings of bells are quite feeble in this respect, albeit noisily so. But for all that you must turn to reference 1.
While at reference 2, there is a mention of bells in connection with the threads of consciousness postulated there. So, it might be, that a concentrating campanologist would have several threads for the sound that a bell made, one for each principal partial, so that what he actually heard at any particular time would depend on the relative strength of the various threads.
No idea whether similar considerations apply to other western musical instruments – or to eastern temple bells, which tend to be cylindrical rather than bell shaped. In the latter case, I suspect not, it being the bell shape which gives us all the partials. Gives us the richness of sound that we know and love from a good bell.
It also occurs to me that, as the sounds of any particular frequency from a bell can be thought of as coming from the vibrations of some particular horizontal ring around the bell, the sound coming from the east (say) of the bell will be 180 degrees out of phase with that coming from the north and south. So, combining the sound from cunningly placed microphones might add up to zero. Something to be thought about in the pub.
With thanks to Bill Hibbert of Surrey, the owner of reference 1, from where I have taken much of the foregoing. With any errors being my own.
PS: with the answer to the question at the end of reference 3 being that the note that you hear is roughly half the frequency of the nominal, but being an octave lower there is no change to the note’s name.
Reference 1: http://www.hibberts.co.uk/.
Reference 2: http://psmv3.blogspot.co.uk/2017/01/layers-and-columns.html.
Reference 3: http://psmv3.blogspot.co.uk/2017/01/the-bells.html.
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