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Discussion Starter #1
In a previous thread there were several members who wrote that they were interested in saxophone acoustics. I'd like to start a discussion on finding the length and volume of the "missing cone" for each member's particular set up. None of these exercises requires more than high school algebra to accomplish. Then I would like to follow up with a separate thread showing a method to determine the "effective volume" of a mouthpiece as it is being played. Comments, questions, criticisms of the methods being used are most welcome. I am certainly not an expert or authority on any of this, but I believe a healthy debate can get us all closer to the answers we are looking for.


 

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Your formula is too complicated: you don't need the intermediate angle S . In fact, the expression of S given by Ferron is only approximate; but both formulae are exact when you replace the angle S by its tangent (and, of course, get rid of the factor 28.65 which is there to get the measure of the angle in degrees).
A more straightforward formula for l is:
l=dL/(D-d).
(indeed, this formula is a simple corollary of Thales's theorem!).

Also, M. Postma
http://sax.mpostma.nl
>measurements>bore profiles
uses the average taper of the whole saxophone to estimate the length of the missing cone. The neck is not a truncated cone (its taper varies along the neck) and the taper of the whole body is a better defined quantity. What is in fact the relevant quantity?
 

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Discussion Starter #3
Your formula is too complicated: you don't need the intermediate angle S . In fact, the expression of S given by Ferron is only approximate; but both formulae are exact when you replace the angle S by its tangent (and, of course, get rid of the factor 28.65 which is there to get the measure of the angle in degrees).
A more straightforward formula for l is:
l=dL/(D-d).

Also, M. Postma
http://sax.mpostma.nl
>measurements>bore profiles
uses the average taper of the whole saxophone to estimate the length of the missing cone. The neck is not a truncated cone (its taper varies along the neck) and the taper of the whole body is a better defined quantity. What is in fact the relevant quantity?
Thank you for that simplification. It took a few minutes for me to figure out how that equation was derived as I don't have a background in mathematics. Acousticians tend to refer to the taper (slope) of a conical instrument in degrees. I suppose that is why Ferron used that formula. Using that equation the value for l was 217.41. In response to your comments about how the taper varies along the length of the neck, I have included an illustration of some measurements taken from the neck used in this study. Your question "what is the relevant quantity" is the essential question at the core of this entire discussion. In other words, which taper gives the most accurate length and therefore the most accurate volume of the "missing cone"? I do not know the answer to that and I have not found an answer in all of the published literature I have read to date. That said, I have also included some measurements of the body and the calculated tapers of the rest of the saxophone to add to the discussion.

View attachment 189714 View attachment 189722
 

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The neck is not a truncated cone (its taper varies along the neck) and the taper of the whole body is a better defined quantity.
Might this varying taper be a parabolic approximation? Do all necks have a varying taper?

I can further appreciate how local variations in the neck taper might affect timbre.

Whew...

Yes, John, I used to chase some of this stuff as well, but at the end of the day, determined that playing is more important to me. It's not like we can access and review the tapers of every horn. At the end of the day, the players just have to find what works best for them - regardless of the numbers. I used to get pretty numbers intensive about guitars, bicycles, and canoe hulls too... And then we mess up the best intentions of the designers by adding a human to the mix!

Best o' luck with your quest. I'll continue watching and reading.
 

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Discussion Starter #5
Might this varying taper be a parabolic approximation? Do all necks have a varying taper?
The almost have to in order to accommodate a more cylindrical area for the cork.

I can further appreciate how local variations in the neck taper might affect timbre.
My study reveals that the greater effect of different tapers is upon the harmonics produced, which in turn can have an effect upon the timbre.

Best o' luck with your quest. I'll continue watching and reading.
Your learned input is always welcome George. We need you to keep us on our tones with terms like "parabolic approximation".
 

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Might this varying taper be a parabolic approximation? Do all necks have a varying taper?

I can further appreciate how local variations in the neck taper might affect timbre.

Whew...

Yes, John, I used to chase some of this stuff as well, but at the end of the day, determined that playing is more important to me. It's not like we can access and review the tapers of every horn. At the end of the day, the players just have to find what works best for them - regardless of the numbers. I used to get pretty numbers intensive about guitars, bicycles, and canoe hulls too... And then we mess up the best intentions of the designers by adding a human to the mix!

Best o' luck with your quest. I'll continue watching and reading.
No, I don't think it can be considered as a parabolic approximation.
The website I quoted in my previous post has graphics of the bore profile: using his measurements, the author plots points on a graduated paper ("papier millimétré", I don't know the English word for it...). The vertical scale (for the diameter) is bigger than the horizontal one (position along the horn) and you see clearly that the neck is not (in fact, is almost never...) a straight part of a cone -and how it deviates from a truncated cone. With rare exceptions, the body (if you except the bow and bell) is much closer to a truncated cone.
Interestingly, the most obvious differences are not between manufacturers, but between saxophone voices: you clearly have separate traditions for the sopranos, altos, tenors, baritones.
You should really open the pages of M. Postma with his bore profiles (around 20 different instruments for each voice!).

Edit: Saxoclese, I'm not really convinced by your explanation for the cylindrical portion near the cork end, because the cork can (should!) be sanded to be cylindrical. I believe the explanation has more to do with acoustics than convenience.
 

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Discussion Starter #7
Edit: Saxoclese, I'm not really convinced by your explanation for the cylindrical portion near the cork end, because the cork can (should!) be sanded to be cylindrical. I believe the explanation has more to do with acoustics than convenience.
You may be right that there are acoustic reasons as well. In my SBA neck measurements above, the area that typically is covered with cork has less taper than the rest of the neck. I too sand neck corks to be cylindrical and accept the mouthpiece all the way to the end of the cork. You can call me by my name which is John.

My hypothesis is that once we begin to calculate the length of the "missing cone" using different tapers, we will find the most accurate one is the one that gives the proper wavelength which coincides with the played frequency of the mouthpiece + neck off the saxophone which combined can also represent the "missing cone to its apex" with the truncation at the neck receiver. If this is not clear, it will become more so as the discussion progresses. I hope.
 

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John, I think we should use the measurements by M. Postma (he even is a member here, though he doesn't participate a lot).
Also, "les études acoustiques de B.B. Ninob" quoted by Dirk (Vries 1) are a fantastic resource (with, for example, an estimation of the correction to the mouthpiece volume to obtain its effective volume).

With your notations, the formula for the volume of the missing cone is
V=∏ l d^2/12
(l and d in cm, if you want the volume unit to be cm^3).
For the SBA alto you measured:
V=9 cm^3
(9.04 indeed, but the extra precision is illusory, see below).
For the SBA alto, the cross section at the cork end is (very) close to 1 cm^2, so pushing/pulling the neck by 1mm causes a variation of 0.1 cm^3 in the volume. I think it would be completely unrealistic to expect results with a higher precision. Even this precision would not be very realistic, because with a same sax, mouthpiece and reed, from player to player (and also when the outer temperature varies) the variations of the tuning position are quite bigger than 1mm.
 

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The Postma data looks really good. He appears to be very careful about taking measurements.

A few years ago at the Navy Sax Symposium, Walter Martin gave a presentation on his efforts to take measurements and compare them to calculations. He had pretty good agreement for SAT saxes but was way off for Bari sax. His data set was from participants at some kind of sax retreat. You need to load the Prezi viewer app to see his presentation.

Search for Completing the Cone Walter Martin Prezi

https://prezi.com/user/o5bgtxczpg1b/

On my iPad the viewer has a right arrow you can press to go through the slides. Sometimes it zooms 3X before bringing up the next slide. Keep trying.
 

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A few years ago at the Navy Sax Symposium, Walter Martin gave a presentation on his efforts to take measurements and compare them to calculations. He had pretty good agreement for SAT saxes but was way off for Bari sax.
Might the deviations from ideal for bari sax be due to the compensation/correction for the more tortuous bends in the neck? If one is measuring diameter as a function of bore position (distance from a fiducial), should that be inner (shorter) path or outer?
 

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George, I would not be surprised ...

Alain, in English we say "quadrille paper", or more simply "graph paper".

The several jumps in bore size (reed tip, MP to neck, then the straight section at the tenon) concern me more than the variable taper of the neck itself.

Have fun with this. I am going to practice.

dsm
 

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Discussion Starter #12
Might the deviations from ideal for bari sax be due to the compensation/correction for the more tortuous bends in the neck? If one is measuring diameter as a function of bore position (distance from a fiducial), should that be inner (shorter) path or outer?
In reading Nederveen (acoustical aspects of woodwind instruments) the curve of a saxophone neck has an insignificant effect upon the soundwave. Benade (Fundamentals of Musical Acoustics) states that at the "U" bend of the bow, the soundwave "sees" a slightly shorter tube with a slightly larger diameter. One of my "pet" theories is that sometimes instrument makers do not take this into consideration and make the diameter of the bow proportionate to the taper of the body and bell therefore making the volume too large. This is why adding something inside the bell fixes the warble creating inharmonicity by lessening the volume.
 

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In reading Nederveen (acoustical aspects of woodwind instruments) the curve of a saxophone neck has an insignificant effect upon the soundwave. Benade (Fundamentals of Musical Acoustics) states that at the "U" bend of the bow, the soundwave "sees" a slightly shorter tube with a slightly larger diameter. One of my "pet" theories is that sometimes instrument makers do not take this into consideration and make the diameter of the bow proportionate to the taper of the body and bell therefore making the volume too large. This is why adding something inside the bell fixes the warble creating inharmonicity by lessening the volume.
I can accept that the curve has little effect on the soundwave itself - the challenge would come when translating theoretical to reality. Is the ideal characterization performed by determining the bore diameter as a function of length relative to the center of the bore?

FWIW, I have a horn with a patch of copper tape (approx 2 inches square) affixed to the inside of the bell. I am told that it fixed a warble, but I don't have firsthand experience of the warble before the patch was installed. What I do know is that it plays well with the patch, and I'm not going to mess with it.
 

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In reading Nederveen (acoustical aspects of woodwind instruments) the curve of a saxophone neck has an insignificant effect upon the soundwave. Benade (Fundamentals of Musical Acoustics) states that at the "U" bend of the bow, the soundwave "sees" a slightly shorter tube with a slightly larger diameter. One of my "pet" theories is that sometimes instrument makers do not take this into consideration and make the diameter of the bow proportionate to the taper of the body and bell therefore making the volume too large. This is why adding something inside the bell fixes the warble creating inharmonicity by lessening the volume.
The following study:
http://la.trompette.free.fr/Ninob/Coudes.pdf
(theoretical study+experiment) seems to support the assertion of Nederveen (at least for cylindrical tubes).
Also, Yamaha's YAS 62 and YTS 62 (instruments designed with a quite scientific approach) don't have radical perturbations in the bow to compensate the hypothetic effect of the bend.
 

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Might the deviations from ideal for bari sax be due to the compensation/correction for the more tortuous bends in the neck? If one is measuring diameter as a function of bore position (distance from a fiducial), should that be inner (shorter) path or outer?
George, the study I've just quoted
http://la.trompette.free.fr/Ninob/Coudes.pdf
addresses your question (at least for a tube with rectangular constant cross section, limited by circular walls of radius r_1 and r_2). The equivalent length is the length along the circle of radius
((r_1^2+r_1r_2+r_2^2)/3)^{1/2} (sic!)
(the author calls it "quadratic mean"). In practice (except maybe for a bassoon) you can replace it with the usual arithmetic mean (r_1+r_2)/2.
 

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George, the study I've just quoted
http://la.trompette.free.fr/Ninob/Coudes.pdf
addresses your question (at least for a tube with rectangular constant cross section, limited by circular walls of radius r_1 and r_2). The equivalent length is the length along the circle of radius
((r_1^2+r_1r_2+r_2^2)/3)^{1/2} (sic!)
(the author calls it "quadratic mean"). In practice (except maybe for a bassoon) you can replace it with the usual arithmetic mean (r_1+r_2)/2.
Arggh, all these years wasted when I should have been helping my daughter with her French language homework! :cry:

Does the author address a limiting ratio = r1/r2, beyond which one radius dominates the acoustic response? As long as r1/r2 is close to 1, all is well, but what happens when the threshold is exceeded?

This question is relevant to whether (or at what point) distortion from pulldown is an acoustic issue.
 

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Arggh, all these years wasted when I should have been helping my daughter with her French language homework! :cry:

Does the author address a limiting ratio = r1/r2, beyond which one radius dominates the acoustic response? As long as r1/r2 is close to 1, all is well, but what happens when the threshold is exceeded?

This question is relevant to whether (or at what point) distortion from pulldown is an acoustic issue.
One of the approximations made by the author in his theoretical study (namely, a(\rho) is constant) is valid only when r1/r2 is close to 1 -but in his experimental study, the smallest radius is 0 (!).
For the acoustic distortion due to pulldown, in the ovalized part the cross section of the tube decreases (the perimeter remains constant, so the area decreases: a disk realizes the maximal area for a given perimeter). So, at least, a pulldown causes a significant bore perturbation. I think this effect is the dominant effect.
 

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Thank you, Alain.
 

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Discussion Starter #19
John, I think we should use the measurements by M. Postma (he even is a member here, though he doesn't participate a lot).
Also, "les études acoustiques de B.B. Ninob" quoted by Dirk (Vries 1) are a fantastic resource (with, for example, an estimation of the correction to the mouthpiece volume to obtain its effective volume).
I have read through the entire website and I haven't been able to find the method M. Postma uses to determine the "average taper". Can you see what I am missing.
 

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I have read through the entire website and I haven't been able to find the method M. Postma uses to determine the "average taper". Can you see what I am missing.
In most cases, what he does is quite clear: the body is an almost perfect cone and he considers the taper of this cone (see the straight line on the bore profiles, which always matches the main cone... when there is a main cone. There are exceptions where I don't really understand his method:
-the Solotone and SML soprano
-the Goumas and Martin alto
In these exceptional cases, except for the Goumas alto, he seems to consider the upper part of the body.
Also, in the rubrique "A tribute to Sax", you see that he draws the line fitting the taper of the upper body up to the zero of the horizontal scale. For him, this seems to give the right value for the length of the missing cone.
If you need further explanations, you can write him an email. I've done it once; he is a friendly guy and was happy to have a conversation with a reader of his website.
 
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