Some time ago Keith and I communicated about mouthpiece volume and he
sent me a spreadsheet of some measurements he had done.  Being
interested in the subject, I have tried to recreate the technique
described by Benade (FMA p. 466) in his study of mouthpiece acoustics
with Jim Gebler to determine the "equivalent volume" of a mouthpiece
under playing conditions.

By using some simple techniques and some easy math, I found that the
effective or equivalent volume" of a Rousseau classical alto sax
mouthpiece was approximately 28% larger that the physical volume past
the end of the tube it was inserted upon.  This was in good agreement
with the chart in Fig. 22.1 on p. 466 showing the equivalent volume of a
soprano sax mouthpiece at various pitches.

The steps of the study can be seen at this web page:  
<http://jbtsaxmusic.homestead.com/mouthpiece_equivalent_volume_study.pdf\
> Equivalent volume of a saxophone mouthpiece
<http://jbtsaxmusic.homestead.com/mouthpiece_effective_volume_study.pdf>

There seems to be a bit of a dicey complication which nobody mentions, but which you and I know from our debate on SOTW on this topic. Benade has it hidden talking about oboes, and that is that both volume and length play a role in determining tuning, although volume is by far the more important
 of the two. Still if you read Benade's 1977 paper carefully, you will see that adjusting length will move all the modes up or down about the same amount, whereas varying volume changes the pitch, but does so unequally across the modes. The problem is that with the sax, with the mpc fitting over
 the neck, we change both length and volume at the same time, which makes sax intonational issues rather complex.

Toby

John <jtalcott47@...> wrote:                                           Some time ago Keith and I communicated about mouthpiece volume and he sent me a spreadsheet of some measurements he had done.  Being interested in the subject, I have tried to recreate the technique described by Benade (FMA
 p. 466) in his study of mouthpiece acoustics with Jim Gebler to determine the "equivalent volume" of a mouthpiece under playing conditions.

By using some simple techniques and some easy math, I found that the effective or equivalent volume" of a Rousseau classical alto sax mouthpiece was approximately 28% larger that the physical volume past the end of the tube it was inserted upon.  This was in good agreement with the chart in Fig.
 22.1 on p. 466 showing the equivalent volume of a soprano sax mouthpiece at various pitches.

The steps of the study can be seen at this web page:  Equivalent volume of a saxophone mouthpiece  
 
 
      
                 
                 
 

Forgive me for the long post.  To save Toby the trouble and to provide
information for others to join the discussion, I have extracted
information from Benade's "Acoustical Evolution of Wind Instruments"
that is on point to the discussion of the "equivalent volume" of a sax
mouthpiece and its effect upon both pitch and intonation---which are two
separate and distinct elements.

I will not debate Toby on this issue upon which we still have our
disagreements until he addresses which of Benade's statements he draws
his conclusions from.  If there are portions of the text that pertain to
the discussion that I have somehow missed, I'm sure Toby will point that
out as well.   [;)]

Benade:   Acoustical Evolution of Wind Instruments
A course taught by A.H. Benade in the fall of 1977
Case Western University

Some properties of air columns

Most important:

Enlarging toward the open end [of an incomplete cone], or contracting
toward the closed end:

(a) shrinks f n +1/ f n  ratios,
(b) raises f  1  most, higher  f n �s less.

Contracting toward the open end, etc. has the opposite effect.

Basic Acoustical Resources for Adjustment of Cone-Plus-Reed Combination

(1) At very low frequencies (i.e., near fundamental component of lowest
note on the instrument)
the air column is fooled if the total volume (cavity plus constriction)
is pretty much equal to the volume of the missing cone.

Note: The effective [equivalent] volume [of the mouthpiece] includes
effects produced by springiness of the reed cavity wall plus some
[effects] associated with the dynamics of the oscillation process.

Cavity:    It is within the oboe or bassoon reed or sax mouthpiece.
Constriction:    It is the oboe reed staple, bassoon bocal, sax neck

(2) At frequencies that are fairly high (e.g., 2nd-mode frequency of
shortest used tube), there is a resonant frequency of the cavity plus
constriction [the playing frequency of this object using standard
embouchure].  Call this  f r s.   Now  f r s.  must pretty well match
the natural frequency of the missing cone segment.

(3) The general behavior of the air column in response to changes in the
effective volume [of the mouthpiece plus constriction] and the  f r s. :

a) Enlarging the total effective volume lowers mode 1 some, mode 2 more,
and mode 3 a lot---it NARROWS FREQUENCY RATIOS---this is in cents or
percent, not in Hz!

b) Lowering  f r s lowers the frequencies of the air column modes that
lie in its neighborhood.

c) Pinching with lips [playing too high on the mouthpiece pitch?] or
blowing harder collapses cavity, which reduces V eff  and raises f r s. 
NOT BIG EFFECT ON SAX.

Benade's references to oboe reed adjustment for intonation

(12) A little about tinkering [with] the total volume and  f r s for a
reed:

Elongating, widening, or thinning the cane enlarges V total  and lowers 
f r s.
Enlarging or shortening the staple enlarges   V total   and raises  f r
s.

Provided the instrument pretty well sings with the reed you have and
provided the scale runs a trifle short [flat as you go higher] . . .the
cure is to use a lightly SHORTER staple and/or reed a narrower one of
each will stretch the scale rather than raise it en masse.  THIS SEEMS
OBVIOUS ---IT IS NOT,  HOWEVER.  It is a special quirk of a cone of
variable length provided at the top with a cavity somewhat elongated and
a constriction.

As little as 3 mm shortening will raise the overall playing pitch of
essentially the whole scale by 15 to 20 cents on an oboe if the
cooperations are in reasonably good order.

EVERYTHING IN THESE NOTES TRANSLATES WITHOUT ESSENTIAL CHANGE FOR THE
BASSOON; WITH ONLY A LITTLE CHANGE IT ADAPTS THINGS TO THE SAXOPHONE;
AND IT IS NO GOOD AT ALL FOR FLUTE OR CLARINET.

Note: Capitalizations for emphasis are Benade's [italics in brackets]
are my clarifications.

The entire text can be downloaded at
<https://ccrma.stanford.edu/marl/Benade/documents/Benade-Physics323-1977\
.pdf>   MARL <https://ccrma.stanford.edu/marl/>     Click Benade
Archive, Writings, 70's