Mike Hoff asked me to review for him what I had been covering the past few weeks. I thought that these rough notes might also be useful to the class.

I apologize for the roughness of these notes, as I have not had time to polish them (or even get rid of all typos).

> Have you got notes up on the web, though?  I don't know anything about
> what you've been teaching these last couple of weeks.

The main thing I have been teaching is musical instruments-- woodwinds,
trumpet, violin and voice.

Three things are important. 
a) The amplifier feedback sound procduction. -- the key idea is that the
feedback amplifier must be designed to increase whatever the effect in the
instrument is.  In the ordinary reeds, it is because in a certain range the
reed closes as the pressure inide drops. This means that less air goes in
when the pressure inside is low and more when it is high. This amplifies.
What gets amplified is the mode in the instrument .

For the lip reed ( trumpet family) it is the time delayin the reed. The
trumpet reed opens as less pressure is inside. This would damp out any
mode. However if you tune the reed ( your lips) to have their resonance
just under the resonance of the mode, then the lips will have a greater
than 90 degree lag wrt the driving force inside the instrument. Ie, they
will open when the pressure in the instrument is high, and close when it is
low. Again the air flow in is high when the pressure inside is high, and
low when the pressure is low, which again amplifies.

The flute (and beer bottle) , air reed, the operation is different. The
oscillation of the air in the mode causes air to flow into and out of the
instrument. the player directs a stream of air across the opening so that
if there were no air flow in the instrument, about half would go in, and
half out. Now, the mode air flow deflects the air stream from the mouth of
the player. but it takes time for that deflection to affect how much of the
air actually enters the instrument-- that air stream has to travel across
the hole and to the other side. If this delay ( which is roughly the side
of the hole divided by the velocity of the stream) is a quarter period,
then the deflection, which occurs when the velocity of the mode at the
mouth is max or min, will enter the instrument when the pressure is max
 or min. Again, a large air flow in when the pressure is high, will
 heighten the pressure, and a flow out when the pressure is low will
 lower the pressure more. Ie, again amplification.

For a bow, the friction curve of the rosin on the bow plays the same role.
The friction coefficient is high when the velocity is low and low when the
velocity is high. Again this is a direct amplifier. If the velocity of the
string wrt the bow (remember that the average velocity of the string wrt
the bow is just the velocity of the bow) is lower than average, the
friction force is higher, slowing it down even more. If the velocity is
greater than the average, the friction force drops, allowing it to move
even faster. The bow on the string enters this stick-slip regime, where the
bow spends time sticking to the string and then breaking free. The velocity
is a square wave with the duty cycle ( time spent sticking over time spent
at high velocity) equal to the distance from the bow to the bridge vs the
distance to the finger holding down the string.

In the case of the pitch, in all the air instruments the pitch corresponds
to the frequency of the particular mode that is being amplified. In a
clarinet, the modes are the usual "closed tube" modes, with frequencies f1=
c/4L (c=vel of sound, L=lengthof instrument) , f2=3f1, f3=5f1, ...
Thus the sound coming out tends to be only the odd harmonics esp for the
lower notes on the clarinet. The pitches are changed by a) shortening the
length of the air column, (finger holes), and b) playing a higheri (second) mode (
going up in register) by opening the tiny register hole to damp out the
lowest mode.

In an oboe ( and bassoon, saxaphone), it is a conical bore instrument. Here the modes have
frequencies of 
f1=c/2L, (ie an octave higher than for a clarinet ofthe same length),
f2=2f1, f3=3f1,... The reed operates in the same way as for 
a clarinet. The modes however are different and thus the tuning of the
second register is at only 2 the freq of the lower register (second mode is
2f1). Again the length is altered by finger holes.

In a  flute, the modes again have frequencies
f1= c/2L, f2=2f1, f3=3f1 ( tube open at both ends).

In a violin the frequency is determined by the mass of the string, the
tension in the string, and the length. The mass and tension are set
beforehand to tune the string to its standard frequencies ( G3 D4 A4 E5 on
a violin, lower for the others-- eg viola is C3 G3 D4 A4).
The pitch is changed either by choosing which string to bow, or by
shortening the string by putting your finger down hard on the string.

The next point is the transmission of sound out of the instrument-- how do
the vibrations in the instrument get out as ssound. In the case of the
wind instruments, it is fairly direct. The vibrations of the air in the
instrument cause the air in the openings (end of tube, finger holes, blow
hole of a flute) to vibrate back and forth. This acts like a speaker piston
which then forces the air outside to vibrate creating sound waves. This
translation ofthe piston motion to sound has the same efficiency problems
that a speaker has-- ie the sound that gets out is 6dB per octave below the
knee freq of the piston less than one would expect just due to the velocity
of the piston (air) at the opening.

The flaring of the trumpet say, does increase the diameter of the piston
for the higher frequencies, making the efficiency greater, but does little
for the lowest frequencies. In the woodwinds and flutes, since most of the
air vibration occurs at the finger holes or at the mouthpiece (for a flute)
the flaring plays very little role except for the lowest notes.

In the violin the strings move essentially no air. Thus the body of the
instrument. The bridge transmits the vibration of the string to the body by
a lever action ( one foot of the bridge is held relatively static by the
soundpost in the instrument. The bridge pivots around that foot, pushing up
and down on the top). Of course the more massive the bridge, the less it
moves, and the less the vibration is transmitted. The bridge also has its
own resonances ( the bridge as a whole, or the top of the bridge rocking in
opposite directions to the bottom) which tends to accentuate certain note
ranges. The main sound production  comes from the top and bottom of the
instrument vibrating, and from the air in the f holes vibrating. Thus the
resonances of the top plate (like a drum) and of the air inside the
instrument with the f holes as openings will accentuate certain
frequencies. The Helmholtz resonance is tuned to the D4, ( second lowest
string) and the first of the drum modes is usually tuned around A4. the
second drum mode is usually around E4 and then they start to crowd closer

The motion of the string against the bridge is very much of a saw-wave
pattern. But because of all of the resonances along the way, the sounds
that come out of the violin have a very complicated spectrum which changes
from note to note.

For the voice, my main emphasis was on the incredible flexibility of the
alterations of the vocal cavity ( throat mouth with tongue, teeth and lips)
which alter the resonances of this cavity. The vocal chords vibrate due to
a feedback mechanism (I hope to learn more in the colloquium this
afternoon about what the exact mechanism is) but that sound it altered by
the resonances of the vocal tract. Those resonances are called formants,
and largely determine the vowel sound that is produced ( also some of the
consonants-- the difference between thththt and sssss is the resonances
produced at the front of the mouth in the vocal cavity by changes in the
lips tongue and teeth.) I showed them the formant diagram showing how the
first two formants ( resonances) primarily determine which vowel is heard
and that the vowel sound is primarily determined by the location ofthose
first two formants.

I pointed out the singers problems-- When the voice sings at a certain
pitch only the harmonics of that pitch are produced by the vocal chords.
Thus the ear only has the intensity of the sound at those frequencies to
use to determine where the formants lie. For a soprano, singing high, those
harmonics are high in frequency and also widely spaced, making
determination of the formant difficult. Also the singer wants a loud sound
and so want to make sure that the vocal tract resonances lie on those
harmonics, as otherwise the sound production will be poor and quiet. Thus
sopranos tune the formants to the harmonics, rather than using the formants
to sing a particular vowel sound. Ie, they mess around with the vowels in
order to make a louder sound.

Tenors (operatic) develop their throats to produce a strong third formant up
at about 3KHz. This strengthens the sound at those frequencies, which is
where the orchestra/choir is getting quieter. This allows teh tenor voice
to "stick out" above the loudness of the orchestra/choir, even though on
average the orchestra is much louder than the tenor is. This tenor's
formant ( third resonance) is hard to develope, but is very effective.

Copyright W G Unruh