Properties of an "ideal" metal tube

I'm messing around with ideas for tuning up steel tubes. William
Sethares in "Tuning, Timbre, Spectrum, Scale" gives the modes of
vibration and spectrum for an 'ideal' metal bar but I can't find such
information for the 'ideal' metal tube. Does anyone have any references
to literature, sites, past postings on this? Thanks in anticipation.

Best Wishes.

In-Reply-To: <3BAF6DAF.BF996681@which.net>
Alison wrote:

> I'm messing around with ideas for tuning up steel tubes. William
> Sethares in "Tuning, Timbre, Spectrum, Scale" gives the modes of
> vibration and spectrum for an 'ideal' metal bar but I can't find such
> information for the 'ideal' metal tube. Does anyone have any references
> to literature, sites, past postings on this? Thanks in anticipation.

I found it! It's in
<http://www.mills.edu/LIFE/CCM/ftp/tuning/list/archive/jul945>, towards
the bottom. The formula is sqrt[(s^2)[(s^2 - 1)^2]/(s^2 + 1)], and you
can search on that to get the rest. You might also find it in McLaren's
Xenharmonikon 13 article.

Graham