Problem Sets
Interlude 1: Brass instruments
I1.1
There was a very popular movie made in 1958 called
The Vikings, which claimed to have taken great pains to
achieve historical accuracy. At one point in the movie, as the Viking
ship sails back up to their village up the fiord, a look-out sights
the ship and wakes the village by blowing on a great horn, which
looks like this.
This is roughly what you hear on the
soundtrack:
Leaving aside the question of what animal
that horn might have come from, could that melody have been played on
that instrument?
I1.2
Imagine a fictitious instrument, rather like a
trumpet. It consists of a perfectly cylindrical tube, about 140 cm
long, bent into a double U shape to make it more manageable. There is
no bell or mouthpiece and the player puts his/her lips to one end and
blows (so that it sounds like an organ pipe closed at one
end).
(a) Exactly how long (to the nearest mm)
should the tube be if the fundamental note of this instrument is to
be a Bb (i.e. B flat)?
Be careful that you are in the right octave. You should find that
the fundamental frequency of an open/closed organ pipe of length 140
cm is around 60 Hz, hence the Bb you are looking for must be Bb1
(frequency 58.3 Hz).
(b) This instrument is now fitted with a
valve which, when opened, adds a short length of tubing to the air
column, increasing its length.
If this valve is designed to lower the pitch by a semitone when it is
open, how long must this extra length of tubing be?
(c) The instrument is now fitted with a
second valve, designed so that, when this valve is open but the first
one is closed, the pitch is lowered by a whole tone . How long must
be the extra length of tubing connected to this valve?
(d) Now when both valves are opened, what
is the fundamental note of the instrument? Is this exactly in tune
with what you would want? (If there is a discrepancy, express it in
cents).
(e) Comment on the significance of this
calculation for brass players.
(f) A real Bb trumpet is very close to 140
cm long, and the pitch of its "fundamental" is Bb 2. But that note is
essentially impossible to sound. The lowest note it plays, the "open"
note is the first harmonic, Bb3.
Even so, your calculation seems to be way out. In part (a) you
calculated the fundamental was Bb1, so the "open" note ought to be be
Bb2. Can you offer any explanation why the
simple calculation you did should be wrong by so much?
I1.3
(a) The trumpet, trombone and tuba are very
different in size but are constructed from metal tubes which are, in
many ways, similar in shape (forgetting the fact that one uses a
slide and the others use valves. It might be expected therefore that
the frequencies of the notes they play might be in proportion to
their lengths.
Draw up a table which compares the lengths
of a trumpet, a trombone and a tuba, with the frequencies of
their "open" notes (which are Bb3, Bb2 and Bb1 respectively). Are
they in proportion?
(b) Add the French horn to that table. Is
it in proportion? If not, can you think of a reason why it
isn't?