Search for the missing cone

Discussion in 'Books, Literature, and Websites' started by jbtsax, Mar 7, 2010.

  1. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    I have undertaken a study to investigate ways to calculate the so called "missing cone" of the saxophone that the volume of the mouthpiece is supposed to substitute in order for the sax to play properly.

    The findings of the study are at this link: http://jbtsaxmusic.homestead.com/Missing_Cone_Volume_Comparison_Study.pdf

    It is important to note that this is just one saxophone neck and one mouthpiece. Further tests are needed before any general conclusions can be drawn that apply to all sizes, makes, and models of saxophone and mouthpieces.

    Two related studies are: http://jbtsaxmusic.homestead.com/Detailed_SBA_neck_measurements.pdf and http://jbtsaxmusic.homestead.com/mouthpiece_equivalent_volume_study.pdf

    John
     
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  2. MartinMods

    MartinMods

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    Hi John,

    I have some questions about the figured mouthpiece equivalent volume.

    In your MEV study the MEV added 2.78cc or 28% to the physical mouthpiece chamber volume used.

    In your MCV study, MEV added only 1.28cc or 18.63% to the physical mouthpiece chamber volume used.

    The same mouthpiece was used for both tests.

    Benade's conclusions to his similar MEV study showed that the MEV was quite constant over a considerable portion of the instrument's range. Understandable since, also being termed "reed compliance" one might see it as a property of the reed and the mouthpiece facing, rather than the changing mouthpiece chamber volume (used) and tube air column length.

    Can you explain how you were able to get from 28% in the first study to 18.63% in the second study?
     
  3. pete

    pete Brassica Oleracea Staff Member Administrator

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    I have a couple questions, too:

    * First, wouldn't a straight soprano with a fixed neck be easier to work with, like, a Mark VI? You wouldn't have to worry about the neck as a measurement or any bends. (Of course, if you had the measurements from another source, just disregard this comment.)

    * Second, don't the tone holes do something to the overall volume of the cone or is that included in the measurements?

    * Third, do you really want a "full" cone? If you taper off into infinity (on the mouthpiece end, that is), wouldn't that mean that you would have an unplayable instrument? In other words, doesn't the cone HAVE to have a part that's missing?

    * Fourth, do you have any conclusions?

    As always, I wish to point out that I have no knowledge of any acoustics, but I find discussions on the topic intriguing.
     
  4. MartinMods

    MartinMods

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    Intriguing and important......

    I get lots of emails from players asking if the Benade Pip will fix their sharp D2-F2 problems. Preliminary screening tests frequently indicate that mouthpiece volume issues are the main cause of the problem. Though the saxophone is a very flexible instrument in this regard, there are limits, and the refined player will reap noticeable benefit from matching his mouthpiece to his instrument optimally.

    Just an observation about John's Missing Cone Volume Study results. The debate is still on as to which taper - the neck taper or the body taper, or a length proportional average of the two - most accurately reflects the instrument's volume requirement. John claims that his results, based upon the more acute neck taper angle, match his Classical mouthpiece to within 1%, and that to match the volume requirement based upon the body tube taper angle, the mouthpiece would have to be pulled 3 mm (if I recall correctly ) off the neck cork. I don't wish to debate John's number results here. I only wish to point out, that one does not balance mouthpiece volume requirements by pulling the mouthpiece out or pushing it in. Those are pitch adjustments exclusively. Though they do change the volume, that fact is secondary. Were in fact, more mouthpiece volume required, one would simply use a different mouthpiece design, a wider, fatter chamber combined with throat shape and dimensions which would balance the volume with the Frs pitch requirements, and the mouthpiece would stay firmly on the cork.
     
  5. pete

    pete Brassica Oleracea Staff Member Administrator

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    Mmmm. I don't understand this. Volume is volume is volume. I can grasp that if you're talking about a sax has to be in a "sax shape" for it to sound like a sax. In other words, if I just stuck a straight length of pipe on an alto that has the same volume as a sax neck, it's not really going to sound all that sax-like. But I think that's confusing the difference (or maybe relationship) of "taper" and "volume".

    I also don't get this. if you're talking about just adjusting one mouthpiece dimension (bigger chamber, whatever), you're not necessarily changing the volume of the mouthpiece. Am I missing something, there?

    BTB, what's an "Frs pitch requirement"?
     
  6. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    You are right. A soprano would have been easier. An alto sax was used in the above study to compare the results of using the taper of the neck to define the missing cone with using the different taper of the body to define the missing cone.
    That is correct. The effect of the closed toneholes is to make the "acoustic" geometry of the body wider and longer than its physical measurements. This is why I believe that using the physical measurements of the body to define the missing come gives an inaccurate representation.
    First of all the "taper" can't go to infinity like parallel lines can. Because they are at an angle they will eventually meet to form the "apex" of the cone. If the saxophone were a complete cone to an apex, there would be nowhere to blow into to make the sound. That is why it is cut off or "truncated". The acoustics of the saxophone are such that the instrument will behave like a completed cone ie. play overtones that are whole number multiples of the fundamental and overblow octaves that are in tune ONLY if the body of the saxophone can be fooled into thinking there is an cone all the way to an apex at the top. The way the saxophone is fooled into thinking the top of its cone isn't missing is when the effective volume* of the saxophone mouthpiece matches that of the "ideal missing cone". The question we have been wrestling with is how to determine the volume and length of the "missing cone" from the known dimensions of the saxophone and its neck.
    My preliminary work points to the conclusion that the missing cone extrapolated from the main body tube is far too large to be a workable choice. Although I will be the first to admit that further study is needed using different size saxophones, and different style mouthpieces. I do believe that a productive approach is to measure the effective mouthpiece volume that works on a given instrument and "reverse engineer" that measurement to try to discover a way to calculate a missing cone volume of similar value. My thinking is that knowing the effective mouthpiece volume that allows the saxophone to play in tune and in tune with itself, by definition, is that volume of the missing cone we are looking for.
    I wish to point out that I am not an acoustic scientist, I just play one on woodwind discussion forums. :)

    John

    * The effective or equivalent volume of the mouthpiece is its geometric volume plus the volume added by the movement of the reed and the effects of the player's embouchure.
     
  7. MartinMods

    MartinMods

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    The saxophone is basically a cone with a bit of the end chopped off. In order for it to work correctly, whatever we stick on the end to make noise with, should have the same internal volume as the part that we chopped off - the mouthpiece (from the end of the neck) + the area that the reed occupies in it's motion + to some extent, the player's vocal tract. This makes the saxophone respond and tune well in the lower frequency resonances - 1st 2 octaves.

    Frs: Volume is the first requirement of the substitution. The second is Frs, or (I'm guessing) "frequency of the substitution". The resonance frequency of the mouthpiece + the neck, when played alone, should equal the theoretical frequency of the the theoretical cone having the same length as the part we cut off + the neck. This serves to insure alignment of the higher frequency resonances.

    The ratio of the mouthpieces chamber width to it's length, must be within certain parameters for it to be in tune. Not all volume is equal, depending on it's shape and location. You can't take a small chambered mouthpiece and turn it into a large chambered mouthpiece by pulling it off the cork some, as the increase in length will affect the pitch excessively.
     
  8. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    Frs is frequency of the (oboe) reed on its staple. F - frequency, r - reed, s - staple.

    Benade equates this acoustically to the bassoon reed on its bocal, and the saxophone mouthpiece on its neck. He also sometimes refers to the staple, bocal, and neck as the "constriction".
     
  9. pete

    pete Brassica Oleracea Staff Member Administrator

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    First, thank you, gentlemen for answering me/indulging me. As I've said a couple times, I'm fairly interested in the conversation, but I have little to no background in acoustics. It gives me the opportunity to question folks that have more of a clue than me.

    Second, thanks to JBT for recognizing that I was thinking of "zero" when I said "infinity". Some of the more interesting hallucinations I'm havening on my current medications are mathematical and I see formulae from my calculus classes I took 20+ years ago with + and - infinity. For some reason I was thinking "infinite zero". Told ya: too many drugs.

    The way I'm looking at it is that you do have to account for the mouth cavity of the person playing, to some extent. I also still hold out some hope with the theory that there has to be a "missing cone" for the sax to be a sax.

    Again, I'm not getting this. The formula for volume is always the same. However, I get the fact that if you have a mouthpiece that is, say, 12" long and has a chamber of 1", it's not going to play properly. What I'm trying to say is that you can't just completely divorce the volume measurement from the design considerations. Or maybe I'm completely wrong and you could have a 12" x 1" mouthpiece and it's work juuuust fine. Are we talking about the same thing?
     
  10. MartinMods

    MartinMods

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    Yes, but not as far as pitch is concerned. The volume distribution (length, width, tip, throat, must always take into account the varied effect that it will have on the pitch of the mouthpiece.
     
  11. pete

    pete Brassica Oleracea Staff Member Administrator

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    Then I think we're talkin' about the same thing: the mouthpiece must be sax-mouthpiece shaped, but can have some variations and should have a specific volume.

    At least, I think so :).
     
  12. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    I don't agree with everything that has been said about mouthpiece volume. Here is why.

    Of course a small chamber mouthpiece will still have the volume and tonal characteristics of a small chamber mouthpiece even when it is pulled farther off the cork and given a larger effective volume. The same is true of a large chamber mouthpiece put farther on to the cork having some of its volume displaced by the neck. The fact is in both cases each retains its tonal characteristics, but the pitch and intonation produced by each changes with the volume added and taken away when the mouthpiece is moved on or off the cork.

    I am convinced that it is the volume inside the mouthpiece that determines the pitch and not the length of the instrument measured to the tip of the mouthpiece. Two mouthpiece insert studies were done that show a definite pitch change when the volume inside the mouthpiece is added to or taken away from while the mouthpiece remains in the same position. In the first one Proposition B was found to be the most accurate description.
    In the second, actual pitch measurements were taken and recorded.

    http://jbtsaxmusic.homestead.com/Neck_insert_study.pdf

    http://jbtsaxmusic.homestead.com/Neck_insert_study_text_2.pdf

    Of course there are other factors that influence the volume inside the mouthpiece and the pitch as well, such as the hardness (compliance) of the reed, and the player's embouchure.
     
  13. pete

    pete Brassica Oleracea Staff Member Administrator

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    Why can't it be both? Additionally, what would happen with a mouthpiece that has a more conical bore rather than cylindrical?
     
  14. MartinMods

    MartinMods

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    I respectfully disagree with your assessment of the insert, based upon the most essential description of a conical wind instrument, clearly illustrated and explained by A. Benade, on page 21, here:

    https://ccrma.stanford.edu/marl/Benade/documents/Benade-Physics323-1977.pdf

    A conical wind instrument is a truncated cone and the surrogate for the missing section, consists of 2 parts, a cavity and a constriction of the bore. This can be clearly observed, if we examine the cross section of the most essentially simple, real world saxophone bore - a one piece, straight walled soprano, with an Otto Link STM mouthpiece.

    From the mouthpiece end, we clearly see a defined chamber cavity. Our constriction occurs at the neck opening obviously, however, as the bore immediately begins to expand, this real constriction, is effectively lengthless. The truncated conical body section begins where the bore begins to expand. Easy.

    Inserting a snug-fitting, 2mm long cylindrical tube, with a neck opening bore diameter, into the throat of the Otto Link, so it snugs up against the neck opening, we reexamine our cross-section.

    Once again we have a clearly defined chamber cavity, and a clearly defined constriction - the opening of the insert. As the bore of the insert remains cylindrical, the impedance in the air column in that 2mm section does not change, until we reach the opening of the neck, where the bore begins to expand. Thus the insert can only be seen as a 2mm long constriction, who's length and volume belong to the missing cone surrogate, both physically and acoustically.

    Were the bore of the insert conical however, the same slope and diameter as that of the neck or close, then, it would be both a physical and an acoustical extension of the neck.

    Pushing the extension up in the throat, towards the mouthpiece tip, causes a double, chamber, double constriction, which Benade claims, does not have much going for it.
     
  15. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    Quoting Benade Acoustical Evolution of Wind Instruments page 21. next to last paragraph.

    CAVITY: it is within the oboe or bassoon reed or sax mouthpiece
    CONSTRICTION: it is the oboe reed staple, bassoon bocal, sax neck

    It couldn't be stated any more clearly than this..
     
  16. MartinMods

    MartinMods

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    Then please explain my one-piece, straight sided soprano example.

    Further, if you accept Benade's statement here, unqualified, then your entire Missing Cone Study is invalid, as the entire neck belongs to the substitution, and the body is truncated at the tenon.
     
  17. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    It is very simple. The saxophone, like the oboe and bassoon is a woodwind with a conical bore. That conical bore must have the end cut off in order to attach an apparatus to create the soundwave. That apparatus is the saxophone mouthpiece, the oboe reed, or bassoon reed.

    There is also a part of the cone that is not missing, but is detachable from the rest of the body of the instrument for practical reasons. That is the staple of the oboe reed, the bocal of the bassoon, and the neck of the saxophone. A notable exception is the older style one piece soprano saxophone that could fit comfortably in a case without a detachable neck. It is this part that Benade also calls the "constriction".

    It is impractical if not impossible to analyze the frequency produced by the oboe reed without its staple, the bassoon reed without its bocal or the saxophone mouthpiece without its neck. It is for this reason and this reason only that Benade discusses this combination as an alternate way to view the truncation.

    He gives two requirements for a conical woodwind to perform its best acoustically:

    1. That the surrogate for the "missing cone" (the oboe reed, the bassoon reed, and the saxophone mouthpiece past the end of the neck) must have an effective volume equal to that of the actual missing cone.

    2. That the frequency produced by playing the oboe reed on its staple (Frs), the bassoon reed on its bocal, or the saxophone mouthpiece on its neck match the natural resonant frequency that would be produced by a cone that is the calculated length of the "missing cone" plus the measured length its attachment (staple, bocal, neck) combined. That frequency is found by using the formula:

    Frs = V/2 Xo

    Frs - Frequency of the oboe reed on its staple, bassoon reed on its bocal, sax mouthpiece on its neck.
    V - speed of sound (345 m/s at 74 degrees F)
    Xo - calculated length of the missing cone + the length of its attachment.

    It is no more complicated than this. The true missing cone is the part beyond the end of the saxophone neck (if it has a neck), the part beyond the end of the staple of the oboe reed, and the part beyond the end of the bassoon's bocal.

    The alternate way to describe the missing cone as the reed on its staple, the reed on its bocal, and the mouthpiece on its neck is only to elucidate the above mentioned frequency requirement to make the conical instrument work its best acoustically.
     
  18. pete

    pete Brassica Oleracea Staff Member Administrator

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    Excuse, please, another question: if you're talking about a "straight" conical bore, what happens with the extra bends at the neck and bow of a sax? For that matter, the bocal of a bassoon and the reed of an oboe aren't conical.

    As an observation, if the oboe is a true conical-bore instrument, it obviously has a truncated cone: just look at where you're putting the reed.
     
  19. jbtsax

    jbtsax Distinguished Member Distinguished Member

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    The saxophone neck and bell bow are bends in the conical tube. The acoustic effect as Benade puts it is that the sound wave "sees" a somewhat wider and longer tube as it goes around the bend. The manufacturers compensate (hopefully) by making the bore a bit narrower and shorter to compensate.

    The staple of the oboe reed is slightly conical in shape as is the bocal of the bassoon. The oboe is a good visual example of a truncated cone.
     
  20. MartinMods

    MartinMods

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    I agree, and that is not contradictory to what I posted. Benade's use of "mouthpiece on it's neck" and "....it (the constriction) is the sax neck." is vague, and not to be taken word-for-word. (Though you earlier seemed to imply that it was.) The sax neck is the constriction, but only in that it causes the reduction in bore diameter at it's opening, and nothing more, as my example demonstrates. It is a lengthless constriction, and adding cylindrical length to it, is length added to the substitution, not the neck.

    This is demonstrated by the Frs resonance of the mouthpiece/constriction, which exists in the saxophone air column at all times during normal playing. The difference in impedance between the neck opening (constriction) and the expanding neck bore, causes a partial reflection of the wave, back towards the mouthpiece, in the same manner that the wave is reflected back at an open tone hole.

    This independent resonance, is responsible for the reed re-opening after closure, and is the reason why the reed closed period = the ratio of the length of the truncation (to the apex) to the length of the truncated body, which means, the reed stays closed only about 25% of the time, compared to about 50%, for the clarinet.

    It causes a frequency hole or formant, in the tone produced by the saxophone's normal resonances, which is to a great extent responsible for the characteristic saxophone sound.

    It also affects the pitch of normal saxophone tone generating resonances (and these are higher resonances) in it's general area, so if the mouthpiece design considerations (volume, shape, etc) are not correct, and it is sharper or flatter than the theoretical resonance, based on the distance from the apex to it's location, the the surrounding higher resonances will be pulled out of alignment.

    There is a little more to it......
     

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