# The Universe of Discourse

Thu, 16 Feb 2006

The rate of a vibrating string in 1666
In an earlier post, I discussed a report by Samuel Pepys, from 1666, that Robert Hooke claimed to be able to tell how often a fly beat its wings, by comparing the pitch of the sound with that of a known source such as a vibrating string. But I didn't know how he had determined the rate at which the string vibrated. I guessed a couple of ways of measuring this with 17th-century technology.

Clinton Pierce wrote to me with a clever and highly plausible suggestion: build a "buzzer". Take a spoked wheel, and spin it at a known rate, and hold some flexible object against the spokes. If the wheel spins, say, four times per second, and has 64 spokes, the buzzing will be a middle C. It should be possible to space the spokes evenly, and to spin the wheel uniformly at the correct speed.

My own idea involved attaching a bristle to the string, and letting it draw a sinusoid on a smoked glass plate that was drawn past it. The more I thought about the bristle, the more I was sure I had really read about it somewhere. This morning as I made breakfast, I asked myself "What is this 'bristle', anyway? Where do you get one?" and the answer came immediately "You pull it out of a hog." I'm sure I am recalling this and not just making it up. But I liked M. Pierce's suggestion much better than the bristle and smoked glass idea, because it seemed to me much more likely to work. What Hooke actually did, I still have no idea.

I did a little research today, and I have not yet been able to find out how Hooke measured the frequency of a vibrating string. But I have been able to get some information.

Hooke died in 1703. Many of his papers were given to Richard Waller, and in 1705 Waller published Posthumous Works of Dr. Robert Hooke. As an introduction, Waller wrote "The Life of Dr. Robert Hooke", an itemization of every important research or contribution made by Hooke. Two of the entries are as follows:

In July 1664. he produc'd an Experement to shew the number of Vibrations of an extended String, made in a determinate time, requisite to make a certain Tone or Note, by which it was found that a Wire making two hundred seventy two vibrations in one Second of Time, founded G Sol Re Ut in the Scale of all Musick.

(We would make it 384 vibrations; 272 is only C#. Whether this is due to a change in the definition of G, or experimental error, or both, I do not know. But it does resolve my question of whether Hooke did have a specific number for the rate at which a fly beat its wings, or whether he just identified it as low D and left it at that.)

[ Addendum 20060216: I realized this morning that it might also be caused by a change in the length of the second. ]

[ Addendum 20070917: I realized sometime later that this was a dumb suggestion. One second is 1/86400 of a day, then as it is now, and the length of the day has not changed significantly since 1664. ]

In July [1680] he shew'd a way of making Musical and other Sounds, by the striking of the Teeth of several Brass Wheels, proportionaly cut as to their numbers, and turned very fast round, in which it was observable, that the equal or proprtional stroaks of the Teeth, that is, 2 to 1, 4 to 3, &c. made the Musical Notes, but the unequal stroaks of the Teeth more answer'd the sound of the Voice in speaking.

There we have it: He didn't think of the buzzing rotating wheel until 1680, so he must have been doing something else in 1664. What, I do not yet know.