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Saxophone mouthpiece study

Discussion in 'General Acoustics Discussion' started by jbtsax, Apr 21, 2014.

  1. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    First I should apologize for dissing jazz players. I should have said "good enough for government work."

    I'm a little bit unclear on what you aim to accomplish by all these tedious measurements of conic volumes, John. The theory is very clear, and I don't think you are going to disprove it empirically. I'm assuming you expect to find some practical use for your measurements, but what it is escapes me.
  2. jbtsax

    jbtsax Distinguished Member Distinguished Member

    I should have time to finish writing down my findings in a day or two. I believe all of this will become more clear when I post that information. A brief way to say what all of my "tedious measurements" are meant to accomplish is to show what taper measurements don't work in trying to calculate the missing cone length and volume. Besides, the journey is an insightful learning experience whether it arrives at the desired destination or not. :smile:
  3. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    Just in case one of the reasons has to do with intonation, we have to remember that both temperature and air composition affect intonation independent of actual volume. These are not trivial factors. Temperature we all know about, but one researcher found that pitch drops 20 cents at the end of a long-held note on bassoon due to CO2 buildup in the lungs.

    SOTSDO Old King Log Staff Member CE/Moderator

    Hell, I learned about that over thirty years ago without ever having to hold a note on the bassoon.

    Back when I used to work a ten hour day downtown, pick up my vocalist/performer girlfriend and a fast food meal, and then speed out to work the evening shift at a theme park, I would set everything up before popping the burger and fries into the microwave to get them warmed back up. Then, I would wolf down the meal while the first show was going down.

    However, if you eat in a hurry, you tend to take in a lot of air as you swallow, plus there was the carbonation from the then-huge (but now just "large") cup of Coke that washed it all down.

    The predictable result was the occasional burp while playing, often at a time when there wasn't time to get the mouthpiece clear of the mouth. The "air" from the burp would go down the bore, and then when you started playing, the CO2-enriched air column in the horn would instantly drop your pitch by as much as a half step.

    If nothing else, I learned not to gulp the dinner down so quickly...
  5. jbtsax

    jbtsax Distinguished Member Distinguished Member

    We know from Benade that the frequency = speed of sound divided by the wavelength for which the formula is f = C/W where f is the frequency C is the speed of sound in meters per second and W represents the wavelength of the sound in meters. On the saxophone W = 2L where L is the physical length of the sounding tube.

    The natural resonant frequency of an alto saxophone in good repair can be found by fingering low Bb on the instrument and neck without its mouthpiece, and by "popping" one of the right hand keys. One can hear a pitch clearly by putting the end of the neck up to one's ear like a stethoscope. This pitch can also be measured by placing the opening of the neck against the speaker/microphone of a tuner.

    On a Selmer SBA alto the natural resonant frequency was measured at E3 Concert 45 cents sharp.

    At C = 345m/sec the frequency of E3 45c sharp is 169.22hz.

    Using the formula W = C/f the wave length for the 45c sharp E3 is 2.0388m or 203.88cm.

    For the saxophone W = 2L, so the "acoustic" length of the saxophone "tube" is 203.88cm/2 or 101.94cm.
    [This is in close agreement with the measured physical dimensions of 102.1cm given in the attachment above.]

    The lowest played note on an alto saxophone is written Bb or Db3 concert with a frequency of 138.59hz and wavelength of 248.93cm
    Since W = 2L, the "acoustic length" of a saxophone playing its lowest note is 248.93cm/2 or 124.47cm.

    • 124.47cm Db3 concert
    • 101.94cm E3 concert 45c sharp
    • 22.53cm difference

    If we assume 22.53cm to be the length of the missing cone, then it is possible to calculate the slope that would produce that length given the small opening of the neck. Using Ferron's* formula slope = 28.65(d - 0)/l where 28.65 is the multiplier to convert the slope to degrees, d is the diameter of the small opening of the neck, 0 is the diameter of the point of the cone and l is the length. The calculation would be as follows.

    slope = 28.65(12.6mm - 0)/225.3mm = 1.60°

    Adding the measured length of the neck including the tenon 19.7cm to the calculated length of the missing cone 22.53cm gives 42.23cm
    Since the wavelength of the pitch produced is 2L, the result is 84.46cm. The frequency of a note with this wavelength is 408.48hz.

    • 415.3hz Ab3
    • 408.5hz pitch of neck + missing cone
    • 6.8hz difference 1hz = 4.3c 6.8hz = 29c so the theoretical pitch of the neck plus missing cone is Ab3 29 cents flat.

    In the next installment this "calculated taper" to find the missing cone will be compared to the measured tapers on the instrument and there will be a demonstration of how a measurement being off by just .1mm can have a dramatic effect upon the missing cone calculations using Ferron's* methodology.

    *The Saxophone is my Voice" by Ernest Ferron.
  6. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    Color me skeptical, but the resonant frequency of the the horn without the mpc is that of a conic frustum, which is open at both ends, not of a complete cone closed at one end. Do you really think that the resonant frequency of a truncated cone is equivalent to that of a closed cone of the same length?

    Where does Ferron's formula come from? He is not known in the acoustics community for his accuracy, you know. Add to that that Nederveen states that cone angles of saxes are 3-4 degrees and I think you'd better rethink your methodology. Here BTW is a cone calculator:


    Here is another:


    Plugging in your diameter and length gives an angle of 3.2 degrees, which seems better.
    Last edited: Jun 7, 2014
  7. jbtsax

    jbtsax Distinguished Member Distinguished Member

    I just wanted to drop in and let you know I haven't gone away to sulk. ;) In fact I have the answers to all of the concerns posted above. I know that Ferron's formula which uses a trigonometry "short cut" to be accurate, I am just trying to bone up on enough trig to explain why and how it works if I can. I will get back with my response in a day or two. Thanks Toby for your "peer review" of my efforts. I am grateful to have another individual who can ask questions, force me to think through my work again, and to hone in on what is and isn't true.
  8. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    Just remember: garbage in, garbage out. If your sax is in tune then somehow the volume past the truncation is correct, excluding all those pesky variables like bore geometry, mechanical reed effects, air temperature composition and temperature and oral cavity resonance, then the effective volume is correct by definition. That is a physical law. If it doesn't look right it's you who made the mistake, not physics ;)
  9. jbtsax

    jbtsax Distinguished Member Distinguished Member

    Interestingly enough I do. The quote below is from Wikipedia

    * x is the length of the truncated cone to its apex.

    Point taken, but his formula does check out with flying colors. See the attachment entitled: Proof of Ferron's formula. The fact that he uses the diameter measurements instead of the radius measurements tends to conceal what he is doing. I rediscovered grey cells I haven't used in over 50 years trying to figure it out. As usual, the difficult problem has an amazingly simple answer once you find it.

    Important to this discussion is the fact that the taper or slope of a conical woodwind is its"half-angle".

    I'm not sure where you found that information, but here are a couple of sources that agree with my finding of 1.6°. The first is from Fletcher/Rossing p.464

    The second is from http://www.phys.unsw.edu.au/jw/reprints/ChenetalAA.pdf

    It is not hard to extrapolate that the alto sax is in the vicinity of 1.60° from these figures.

    Sorry, I feel as if Ferron and I set you up on this one. If you put the radius into the calculators instead of the diameter, the result agrees with Ferron's formula.

    The bottom line is that I am encouraged by the fact that skipping trying to measure the neck, body, bow, and bell and using acoustic measurements of wavelengths gives the most accurate figure for both the slope and length of the missing cone. However, using Ferron's method of measuring the neck came closer in my calculations than any other method.

    Another discovery I made in doing this work is how much a measurement which is off by 1/10 of 1 millimeter (.1mm) affects the outcome. It appears Lance was right about that all along. The last attachment entitled "effects of variations. . ." shows how small changes in the neck measurements can produce significant differences in the taper and the calculated length of the missing cone.

    After a bit of a rest, the next installment will try to tie all this together with the topic of this thread which is the study of saxophone mouthpiece shapes and volumes.

    Attached Files:

  10. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    So since it is my job to nitpick, let me point out that you should close the top of the sax so that it is a closed conic frustum when measuring the frequency.

    I am actually amazed that Ferron got something right. Yes, half angle makes sense. But we hit a snag with the fact that the cone is not constant. Lance and I had a long argument about this. I don't think there is an easy solution, because it depends on how much the cone varies and where and for what length. But it's totally obvious that if the very end of the neck is slightly enlarged (to make it cylindrical, as many are), that local variation is not going to greatly affect the overall conic geometry, but if you use the end of the neck for calculation (id of end of neck as compared to id of end of sax) you are going to get a very erroneous picture of the effective conic angle (or half-angle if you will).

    It seems likely to me (but I don't have numbers) that one has to take the average conic angle, but variations in the cone angle are going to create different tuning effects in different places. Particularly, if the angle increases, generally speaking (have a look at Nederveen), correct tuning of the registers becomes impossible, but not if the angle contracts. And of course an increase at the end of the sax, in the bell, is only going to affect the lowest notes. A change at the top of the neck will only materially affect the intonation of the highest notes.

    My point here is that it thus becomes most difficult to try to pin down the variables enough to make any sense of trying to measure mpc volume. You have noted that your empirical measurements of mpc volume (also a slippery slope that) do not seem to correspond with that of the imaginary truncated conic apex, and is off by 30% or whatever by your calculations. And for this you propose your very own cosmological constant? A correction corresponding to some idea of the upstream resonator somehow sharing responsibility on the wrong side of the reed for effective volume?

    John, you have done yeoman's work, but I feel duty-bound to point out to you again: garbage in, garbage out. I don't think you have adequately covered all the bases here, hard as you have tried to quantify this. The fact remains that the effective volume of the mpc IS equal to that of the missing apex when the intonation is correct in the first register and a half or so. That is the constant, from which you can move in either direction if you are able to quantify the other direction adequately. If and when the numbers don't match up, then is the time to start looking at your measurements and methodology, not proposing some fantasy condition to make inadequate methodology seem valid.
  11. jbtsax

    jbtsax Distinguished Member Distinguished Member

    These discussions have a way of breaking into an assortment of subtopics so that the focus becomes unclear. Let me take one statement at a time. It is a fact that the saxophone + neck in the study has a natural resonant frequency of E3 concert 45 cents sharp. It is an open conic frustum which whose frequency and wavelength are determined by the formula:

    f = V/l
    where V is the speed of sound and l is the wavelength. Since on the saxophone l = 2L, the formula becomes f = V/2L

    My reasoning is that when the mouthpiece is added and the instrument is played, that portion of the saxophone is not a closed conic frustum
    . The system as a whole behaves as a closed conic frustum only when the mouthpiece substituting for the apex of the cone is added to the system and even then as far as the fundamental frequency is concerned it still behaves like a cylinder open at both ends using the formula given above.

    Had I used the frequency generated by closing the end of the neck (D3 concert) and its wavelength to represent the saxophone + neck, the calculations would have produced a missing cone length of 15.54 cm. and an alto saxophone half angle of 2.32° instead of a half angle of 1.6° which corresponds exactly with Fletcher's figure for an alto saxophone.
    Last edited by a moderator: Jun 12, 2014
  12. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    This gets interesting. The sax with mpc acts as a cone closed at one end, which has the same harmonic series as a cylinder open at both ends, but it works on impedance maxima instead of impedance minima like the flute. They are not analogous.
    Do me a favor: do a pop test on a tenor neck only, both open and closed. Are the frequencies the same? If not, you are going to have to rethink this, unless the pop frequency of a sax plus mpc is the same as the played frequency of the sax.
    I'm not trying to break your cojones, but I am deeply skeptical of some of your assumptions. By the way a frustum is a conic section. So a sax plus mpc acts as a cone, not as a frustum. The question at hand is what relationship a cone open at both ends (a frustum) has to a complete cone, as far as resonant frequency goes, and what relationship a frustum closed at one end has to both of those.
  13. jbtsax

    jbtsax Distinguished Member Distinguished Member

    I think I understand your concerns. What I believe that you are missing is that the body of the saxophone with its neck attached [open frustum] only contributes its length as far as determining the fundamental frequency is concerned. If we were dealing with the harmonics produced, then the maxima and minima of the impedance would come into the equation. After Benade, here is my version of his numbered statements that are true.

    1. The resonant frequency of an open frustum is the same as that of an open cylinder.
    2. The frequency of a "popped" open frustum can be measured and used to determine its wavelength with this formula w = v/f
    3. The length (L) of an open frustum, open cylinder, or closed cone to its apex is 2 times its wavelength or L = 2w.
    4. The frequency (f) of the missing cone = v/2xo where v is the velocity of sound, and xo is the length of the missing cone. [FMA p.470]
    5. The wave length of the entire cone (low Bb) played with the mouthpiece = the wave length of the body and neck + the wavelength of the missing cone.
    6. The length of the entire cone (low Bb) played with the mouthpiece = the length of the body and neck + the length of the missing cone therefore:
    7. The length of the entire cone (low Bb) played with the mouthpiece - the length of the body and neck = the length of the missing cone.
  14. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    The unanswered question is whether the resonant frequency of an open frustum is the same as that of a complete cone of the same length. Or did I just miss that?
  15. jbtsax

    jbtsax Distinguished Member Distinguished Member

    That was answered in the quote from Wikipedia in post #69.

    The summation of my calculations are in the numbered statements 1 through 7. Please tell me which of those statements you believe to be untrue.

    The frequency of a frustum closed at one end that does not extend to its apex is irrelevant because,

    1) that geometric figure does not exist anywhere in the system,
    2) that has never been a part of my calculations.
  16. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    And anyway that is also answered in post #69, with the caveat that those numbers are first order approximations. So the next question is: is that first order approximation actually useful in figuring out the volume of the missing apex to a degree that is useful for your intended purpose, whatever that is--I'm still not clear if this is more than an obscure academic exercise...
  17. jbtsax

    jbtsax Distinguished Member Distinguished Member

    Let me try to explain my thinking.
    This means to me that the frequency of the mouthpiece + neck (Frs) is equal to the measured length of the neck (or 1/2 the wavelength of its "popped" frequency added to the calculated length of the missing cone. This gives us two ways to calculate the length of the missing cone.

    1. Play the mouthpiece + neck with a "normal" embouchure and record the frequency
    2. From this frequency, calculate the wavelength of that tone and divide by 2
    3. Subtract the length of the neck from that figure to get the length added by the "missing cone"

    1. Find the "popped" frequency of the saxophone + neck
    2. From this frequency calculate the wavelength of that tone and divide by 2
    3. Calculate the wavelength of a low Bb played on the instrument and divide by 2
    4. Subtract the length of the saxophone + neck from the length of the "complete cone" of low Bb to get the length added by the missing cone.

    Once the length of the missing cone is determined it is easy to determine its volume. Using Benade's method described on p. 466 where one can find the "equivalent" volume of a mouthpiece. These results can be verified by placing the mouthpiece on the cork the prescribed location to match the "equivalent" volume of the mouthpiece with the calculated volume of the missing cone. If the instrument plays well in tune at A=440 and the blown overtones are well matched, then all of this work can be verified.

    Where I want to go from that point is to displace the volume inside the mouthpiece 1cc at a time and record the pitch. Then pull the mouthpiece off the cork just enough to restore the volume displaced, and record the pitch again, this time blowing the overtones to see if the added physical length of the mouthpiece past the end of the neck has any effect upon the "harmonicity".

    That sums up my thinking. Whether or not this line of experimentation will produce any significant, repeatable date is still unclear. However, I am having a lot of fun testing my ideas and making my measurements and learning a great deal in the process.
  18. SOTSDO

    SOTSDO Old King Log Staff Member CE/Moderator

    And, this is where I add that I used to be able to "(In conics) I can floor peculiarities parabolous" (with apologies to W.S. Gilbert). These days, I consider myself fortunate if I can figure the sales tax on a retail purchase...

    For that matter, in a college production of HMS Pinafore (in addition to holding down Clarinet 1) I was the understudy for the guy playing Major General Stanley, since I a) fit the costume, and b) could handle the lyrics for his famous patter song. This included ripping through the encore verse, which is always done about 20% up tempo from the tune itself.)

    Just now, I've tried running through them again with the lyrics in front of me, and the tongue twisters are simply too much to handle. Part of it is the dysarthria from which I am suffering, but there's a lot of obscure stuff there, all delivered at about mm = 160. It's just too fast for my old body to handle.

    (Well, that and I persist in using "Chassepot rifle" in place of the more common "Mauser rifle" - call me an old fashioned traditionalist if you must.)

    However, dysarthria and all, I can still handle the delivery of the "When you say 'orphan'…" humor section just fine. Some talents you never lose…
  19. kymarto

    kymarto Content Expert/Moderator Staff Member CE/Moderator

    Why didn't you say it was all for fun in the beginning? I would have saved a lot of bandwidth. What you propose to do doesn't need all the elaborate head work. Simply tune the note by position of mpc on neck; displace the volume inside and retune...

    Do you know the story of Edison's question when hiring young engineers? He would give them the glass bulb from one of his light bulbs and ask them to find the volume. They would proceed to do careful measurements of the complex shape and then laboriously figure it all out using geometry.

    When they were done he'd say, "this is how I do it." He'd fill up the bulb with water, and then pour it into a graduated cylinder ;)

    Good luck!
  20. jbtsax

    jbtsax Distinguished Member Distinguished Member

    Thanks Toby. I am presently corresponding with Jer-Ming Chen at UNSW about how he and Joe Wolfe determined the half angle of the soprano and tenor in their study. If I don't wear him out with my questions, I would like to ask him if that taper is useful in determining the length of the "missing cone". I will keep you posted on what I learn from my discussion.
    Last edited by a moderator: Jun 14, 2014
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