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Discussion Starter #1
From the moment I first saw Ernest Ferron's diagram of a saxophone with its "missing cone" in The Saxophone is My Voice I have been fascinated by that concept. What I still don't completely understand is how notes generated at the front part of the mouthpiece by the vibrations of the reed can have wavelengths that go beyond the tip of the mouthpiece. The illustration below is simplified by leaving out the "end correction" and using the first tone hole on horn's without high F# and representing the frequency without using the octave key to avoid the complicated effects of closed toneholes.

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I also don't completely understand what's going on inside. But when I see diagrams of saxophone acoustics I'm always skeptical. They always seem too simplistic. The missing cone theory seems logical, but I'm not certain it translates as accurately as is often shown. We would like to use the "tip" of the mouthpiece as the exact distance measurement and the volume of the mouthpiece (with the reed completely sealing the window) as the exact volume measurement. I think it is just as likely that distance and volume are "flexible" to a degree because both are effected by a "flexible" reed. The jumble of distance and volume changes produced in the mouthpiece, even though minute, might be a large part of what we hear and identify as the sound of a reed instrument, with the conical bore causing what we identify as a saxophone.

I'm not sure if this adds anything to your query, it's just that the "pulse" sent down the body tube is probably a lot more complex than most diagrams can show. Here are some ideas on how messy what is going on inside a sax mouthpiece might be. The distance to tone holes and tube volume sorts this mess into notes.

Mark
 

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Discussion Starter #3
It is accurate to state that the volume inside the mouthpiece past the end of the neck is not the same as it's measured volume. It is somewhat greater due to the motion of the and other factors. Benade calls this the "effective volume" and provides a method of measuring it in his book. I followed the same procedure using my Rousseau 4R classical mouthpiece and found the "effective volume" to be 33% greater than its measured volume.
 

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What I still don't completely understand is how notes generated at the front part of the mouthpiece by the vibrations of the reed can have wavelengths that go beyond the tip of the mouthpiece.
It excites the air in the whole tube. Or am I misunderstanding the question?

FWIW Ferron is too wrong about some things to be a useful starting point for understanding maybe anything.

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All that it means is that there is neither a pressure node nor a displacement node at the tip of the mouthpiece, right?
 

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Ferron wet my appetite for acoustics. But a lot of what he has in his book is wrong.

A player can bend the pitch a lot without changing the length of the system. I’m not sure this can be explained by what I have read other than some researchers have said that the role of the player as part of the system needs to be studied more.
 

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All that it means is that there is neither a pressure node nor a displacement node at the tip of the mouthpiece, right?
I think there has to be a pressure node at the tip.
Indeed, in a conical instrument the relevant quantity is the product rp of the distance r from the apex of the cone and the pressure p. In mathematical terms, the reason is the expression of the Laplace operator in spherical coordinates (r, theta, phi). The apex of the cone is a node of the quantity rp (r=0, so rp is constant!). In a real instrument, the tip will not be a node of rp (nor of p) and neither will it be an antinode of p, like it is on a clarinet.
 

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Discussion Starter #9
Ferron wet my appetite for acoustics. But a lot of what he has in his book is wrong.

A player can bend the pitch a lot without changing the length of the system. I’m not sure this can be explained by what I have read other than some researchers have said that the role of the player as part of the system needs to be studied more.
That is often repeated about Ferron's work. I believe saxophone players interested in the acoustics of their instrument can still learn a great deal from the information that he got right reading and studying his book. It would be unfortunate if saxophonists dismiss his work out of hand entirely because others have said he gets some things wrong.

As far as player's being able to bend the pitch my understanding is that although the natural resonant frequency of the length of the tube of each note determines the frequencies the reed vibrates at, the players embouchure and air pressure still has some degree of control over the exact pitch. This allows the player to "dial in" the desired pitch and tonal center using the embouchure pressure and air. Also on notes high A and above on the saxophone, the shorter "tube" does not have as strong a natural resonant frequency which allows the players oral cavity to also take over control of the pitch. Ultimately using altissimo fingerings which give the body tube even weaker resonances, the oral cavity can take over completely.
 

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That is often repeated about Ferron's work. I believe saxophone players interested in the acoustics of their instrument can still learn a great deal from the information that he got right reading and studying his book. It would be unfortunate if saxophonists dismiss his work out of hand entirely because others have said he gets some things wrong.
The problem is, how do you parse what's correct from what's wrong? The guidance can only come from more reliable sources.
As far as player's being able to bend the pitch my understanding is that although the natural resonant frequency of the length of the tube of each note determines the frequencies the reed vibrates at, the players embouchure and air pressure still has some degree of control over the exact pitch. This allows the player to "dial in" the desired pitch and tonal center using the embouchure pressure and air. Also on notes high A and above on the saxophone, the shorter "tube" does not have as strong a natural resonant frequency which allows the players oral cavity to also take over control of the pitch. Ultimately using altissimo fingerings which give the body tube even weaker resonances, the oral cavity can take over completely.
As you stated, the vibration of the reed (+ interaction with the player's oral cavity) accounts for a part of the "missing volume" (your estimation is 33%, cf. post #3). The player has some control on this variable.
 

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Thanks for weighing in here. So in a real instrument, where is it?

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I don't know exactly. Ideally, at the vertex of the cone (so, this node of the product rp is an idealization, it is not physically realized).
 

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Why assume the saxophone to be a perfect cone? It’s not.

So what good is it to surmise where the volume of the missing conic is made up?
The acoustical models of the real instrument take into account the fact that it's not a perfect cone, but they consider the saxophone as a perturbation of a perfect cone. So, the question makes sense.
 

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Discussion Starter #15
Why assume the saxophone to be a perfect cone? It’s not.

So what good is it to surmise where the volume of the missing conic is made up?
Whether the body and neck of a saxophone have a uniform taper from top to bottom or not, the saxophone still performs acoustically as a truncated cone, albeit an imperfect one. As such it is still possible to estimate the length and volume of an extension of that truncation to its imaginary apex in order to make an educated guess as to what "effective volume" the mouthpiece needs to have in order for the saxophone to "see" an object at its upper end whose acoustical behavior is quire similar to that of the missing apical cone". [Benade's words not mine] Benade also writes "For a conical woodwind instrument to work properly the equivalent volume of the reed cavity (mouthpiece) added to the mechanical volume of the neck must closely match the volume of the missing part of the cone." This language interprets the "truncation" as beginning at the neck receiver at the top of the saxophone, and the rest of the cone to it's apex as the neck plus the "effective volume" of the mouthpiece.
 

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The acoustical models of the real instrument take into account the fact that it's not a perfect cone, but they consider the saxophone as a perturbation of a perfect cone. So, the question makes sense.
Yeah, but to what end? The theoretical model does not take into consideration the amount of physical fudge factor that goes into the production of a real saxophone whose taper deviates considerably from a true cone, especially at the neck. Thinking that the mystical “missing cone” volume could somehow steer you towards that special mouthpiece that slots intonation in is a fool’s errand even if you did possess the insight and theoretical understanding of the physics of acoustics.
 

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Billy Baggins has a point. Am I wrong to think that the interior of any given sax isn't really a perfect cone, given the inevitable irregularities in the surfaces of pads? Surely those perturbations, no matter how small, would introduce turbulences that complicate the model.
 

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Discussion Starter #18
Yeah, but to what end? The theoretical model does not take into consideration the amount of physical fudge factor that goes into the production of a real saxophone whose taper deviates considerably from a true cone, especially at the neck. Thinking that the mystical “missing cone” volume could somehow steer you towards that special mouthpiece that slots intonation in is a fool’s errand even if you did possess the insight and theoretical understanding of the physics of acoustics.
Each closed tonehole that the soundwave passes is also a "perturbation" inside the conical instrument making the physics of what takes place incredibly complex. Yet instrument makers and designers work with this to create instruments that demonstrate the near perfect "harmonicity" that would characterize a perfect cone. The most interesting part of all of this to me is Benade's assertion that the "correct playing frequency" (Frs) of a mouthpiece on its neck can be calculated from a knowledge of the length Xo of the missing part of the cone by the formula Frs = v/2Xo where v is the wave velocity of sound. All of this can be found on p.70 of Benade's Fundamentals of Musical Acoustics.

This discussion however, has strayed from the original topic of this thread which was to show the phenomenon that the wavelength of any note on a saxophone from the first open tonehole and back goes beyond the tip of the mouthpiece where the tone is generated. I for one find it absolutely amazing how that can happen since in reality that "missing cone to its apex" does not actually exist.
 

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Yeah, but to what end? The theoretical model does not take into consideration the amount of physical fudge factor that goes into the production of a real saxophone whose taper deviates considerably from a true cone, especially at the neck. Thinking that the mystical “missing cone” volume could somehow steer you towards that special mouthpiece that slots intonation in is a fool’s errand even if you did possess the insight and theoretical understanding of the physics of acoustics.
The steeper taper of the neck is precisely there to compensate other perturbations, making the whole instrument closer to an ideal cone.
The R&D departements of big manufacturers have detailed models allowing to compute those perturbations.

Why is the cone an ideal ? Because you don't have a lot of tube shapes that produce harmonic partials -and the "Bessel horn" shape, which is possible on a brasswind, would not be possible on a woodwind because on a woodwind you want to keep the harmonicity for all the fingerings. It leaves only two shapes: cylindrical or conical.

Concerning the missing cone volume, it won't "steer you towards that special mouthpiece that slots intonation", but if the volume in your mouthpiece is wrong, you will have intonation problems (and the sound of the sax will lack harmonics). Nothing "mystical" there. If you have doubts about the practical interest of this concept, see the end of this thread:
https://forum.saxontheweb.net/showt...-fingering-confusion/page3&highlight=cornetto

Before the era of scientific acoustics (which was already developing at the time of Adolphe Sax), this "missing cone volume" was not understood. The only conical instruments were the oboe and the bassoon (and folkloric variants of them), which have a quite small missing volume (by chance, exactly compensated by the double reed). If you want to build a single reed instrument with a conical bore, you need to understand this constraint (essays and errors can not lead you very far when you don't know the relevant variable you have to vary).
 

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(...)
This discussion however, has strayed from the original topic of this thread which was to show the phenomenon that the wavelength of any note on a saxophone from the first open tonehole and back goes beyond the tip of the mouthpiece where the tone is generated. I for one find it absolutely amazing how that can happen since in reality that "missing cone to its apex" does not actually exist.
The fact that the apex of the cone is a node of the product rp is a simple consequence of the spherical symmetry (in a conical wind instrument, you have spherical waves centered on the apex whereas in a cylindrical instrument you have plane waves). The image you are using (of a reflected wave propagating beyond the tip) makes no sense to me and I'm afraid it is misleading.
 
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