## Thursday, December 9, 2010

### Ring Them Bells, Part IV

We've been talking about bells this week in the blog. Today we get into a bit more math. We start with how do we display the sound of a ringing bell visually?

Here is a picture of the waveform generated by a ringing bell, from Bill Hibbert's doctoral dissertation website:

The splash is when the clapper strikes the bell, then the tail is the slowly decaying sound we hear afterward.

Now to tie this in with Excel Math, the chart below is the first line graph our students ever see, in fourth grade Lesson 80. Like most line graphs, it shows amplitude (change in temperature) on the vertical axis, and a period of time going from left to right on the horizontal axis. This way of displaying value vs time is used constantly, in all sorts of math problems.

Here we show the values as discrete points with straight lines connecting them, as opposed to the rounded curves in the chart above. Curves require many more individual points OR analog data (as opposed to digital or individual data points).

Here's another graph from Bill's dissertation, this time not showing a line plotted over time, but a series of peaks that represent a bell's loudness or amplitude at different frequencies (vibrations; cycles per second).

If I understand his theory, the ground-breaking contribution of this dissertation is concluding that bell sounds are assembled "in the ear of the hearer" (like beauty is "in the eye of the beholder"). Some of the "minor" vibrations combine to produce a set of tones that are not necessarily what we would expect.

One of his tests involved playing sounds from 25 bells for 30 test subjects who were strong and confident singers. After hearing the bell, each singer sang the sound he/she had heard. The research showed that people clearly heard significantly different tones from the same bells. Any set of bells that should have been "in tune" (according to the scientific equipment) was heard to be out of tune by the listeners.

The math behind these results overwhelmed me.