William B wondered

My questions come in here.  I've become reasonably convinced that 'pop 
tone' is an important consideration in the performance of any given 
mouthpiece design.  What is 'proper' pop tone?  What influence does mass 
of the mouthpiece have on the effect of pop tone?  What factor does 
annealing play in the role of pop tone effect?
***********
By pop tone, I assume you mean the sound you hear when
(for example) you hit the rim of the mouthpiece against
the palm of your hand.  When the mouthpiece volume is larger, the 
pop tone is lower in frequency.  The most important effect of
the cup of the mouthpiece is to amplify harmonics near
the popping frequency.  If the popping frequency is high,
the mouthpiece will result in a brighter sound than if the
popping frequency is low.  Another effect is that a high
popping frequency makes it a little easier to hit high notes.
Yet another effect is that the acoustical length of a mouthpiece,
which differs from its actual length, is affected by mouthpiece
volume.  So you may need to retune your horn when you switch 
from a shallow to a deep mouthpiece, even when they have the 
same actual length.

The pop tone that is "proper" depends on what results you want.

I would guess that the mass and the annealing of the mouthpiece
are far less important than the volume.  Most mouthpieces
are pretty thick (compared to the bell and much of the cylindrical
tubing of the horn) and small in diameter; this makes it difficult
for the walls of the mp to vibrate very much.  It is true
that the pressure variations in the sound going through the 
narrowest part of the mp are greater than in any other
part of the horn.  Even so, I remain dubious that the
effect would be very significant.  I am unaware of any
scientific studies that have been made on this, however.
I think it is likely that a player would sense that one mp
vibrates more than another; but this does not necessarily
translate into a different in playing quality, IMO.

Bill also wrote

At a temperature, just short of the softening point, the atomic 
structure can move just enough that all the stresses are equalized. 
During a slow, uniform, drop in temperature, the crystal structure 
remains homogeneous, and at room temperature would be considered 'dead 
soft annealed'.  Some times this process is repeated, but this time the 
metal is cooled very quickly once it reaches a predetermined 
temperature.  This locks the entire crystal structure into the 
characteristic form for that temperature, and sets the metal to a 
'harder' 'temper', that generally favors higher frequency vibrations.

************
You might want to took at a book about copper and its alloys
to learn more about the effects of heating and cooling the metal.  
Steel can be hardened by tempering; but brass does not show this
effect.  If you took some "soft annealed" brass and raised it to
some temperature below the annealing temperature, you would
detect no change in the softness of the metal.  You stiffen brass
by work-hardening.  This may be why so many local thugs carry
around horn mouthpieces instead of brass knuckles.  

Gotta go,
Cabbage  
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