Math for tapering of conical music instrument like Horn, Sax, Oboe ...

[science4ever]

To get the inner dimensions right one have to see to it that overblowing

produces the overtones in tune and that require one get the tapering right.

 

Could some Math person help me set up the 1732 tapering.

 

suppose the music instrument is 600mm long and the inner tube

start as small as 6mm diameter and in the end it is about 30 mm.

 

How is the 1732 related to all the other numbers how does one set it up?

 

I ask because I want to build such an instrument say a Horn in wood or a Sax

without mechanics just fingerholes or a oboe without mechanics just fingerholes

so would love to get how one set up the equation. I am an old retired person

and I have never understood Math so this should be easy for those fluent in it?

[studiot]

I would be interested to learn the meaning of 1732 in relation to musical instruments.

 

Acoustic horn maths I do understand, but have never heard of this.

[science4ever]

Oh I am sorry. I trusted that I had proof read it.

I mean 1/32 the European key board has 7 below the /

while it is on another place on other key boards.

Sorry for the confusion. Here is the important part

 

suppose the music instrument is 600mm long

and the inner tube start as small as 6mm diameter

and in the end the inner tube it is about 30 mm diameter.

 

then the tapering how fast it incline or rise to wider and wider

has the relation 1/32 so how does one set that up as an equation with x?

[Daedalus]

I've been researching the problem and I have found this at Wikipedia:

Cylindrical bore[edit]

The diameter of a cylindrical bore remains constant along its length. The acoustic behavior depends on whether the instrument is stopped (closed at one end and open at the other), or open (at both ends). For an open pipe, the wavelength produced by the first normal mode (the fundamental note) is approximately twice the length of the pipe. The wavelength produced by the second normal mode is half that, that is, the length of the pipe, so its pitch is an octave higher; thus an open cylindrical bore instrument overblows at the octave. This corresponds to the second harmonic, and generally the harmonic spectrum of an open cylindrical bore instrument is strong in both even and odd harmonics. For a stopped pipe, the wavelength produced by the first normal mode is approximately four times the length of the pipe. The wavelength produced by the second normal mode is one third that, i.e. the 4/3 length of the pipe, so its pitch is a twelfth higher; a stopped cylindrical bore instrument overblows at the twelfth. This corresponds to the third harmonic; generally the harmonic spectrum of a stopped cylindrical bore instrument, particularly in its bottom register, is strong in the odd harmonics only.
Instruments having a cylindrical, or mostly cylindrical, bore include:

• Clarinet (stopped)
• Flute (Boehm system — open)
• Sudrophone
• Baritone Horn

Conical bore[edit]

The diameter of a conical bore varies linearly with distance from the end of the instrument. A complete conical bore would begin at zero diameter—the cone's vertex. However, actual instrument bores approximate a frustum of a cone. The wavelength produced by the first normal mode is approximately twice the length of the cone measured from the vertex. The wavelength produced by the second normal mode is half that, that is, the length of the cone, so its pitch is an octave higher. Therefore, a conical bore instrument, like one with an open cylindrical bore, overblows at the octave and generally has a harmonic spectrum strong in both even and odd harmonics.
Instruments having a conical, or approximately conical, bore include:

• Bassoon
• Cornet
• Euphonium
• Flugelhorn
• Flute (pre-Boehm)
• Oboe
• Saxophone
• Tuba
• Uilleann pipes

Most sites refer to the following: "The conical bore in musical acoustics," by R. D. Ayers, L. J. Eliason, and D. Mahgerefteh, American Journal of Physics, Vol 53, No. 6, pgs 528-537, (1985).

I have never designed a musical instrument. So take caution with my interpretation in applying the math. However, if 1/32 defines the slope and you start with a 6 mm opening and have 600 mm length of tubing, then your equation would be:

Wikipedia "The diameter of a conical bore varies linearly with distance from the end of the instrument"

[math]y=\frac{1}{32}x+6[/math]

Such that at the smallest end ( [math]x=0\,\text{mm}[/math] ) you will get 6 mm, and at the other end ( [math]x=600\,\text{mm}[/math] ) you will get 24.75 mm, which is pretty close to the numbers you have posted.

Acoustic horn maths I do understand, but have never heard of this.

Perhaps studiot will verify the math, but I believe it to be correct. Of course, this does not include the bell, which would vary non-linearly with distance.

The bell of a wind instrument is the round, flared opening opposite the mouthpiece. It is found on horns, trumpets and many other kinds of instruments. On brass instruments, the acoustical coupling from the bore to the outside air occurs at the bell for all notes, and the shape of the bell optimizes this coupling. On woodwinds, most notes vent at the uppermost open tone holes; only the lowest notes of each register vent fully or partly at the bell, and the bell's function in this case is to improve the consistency in tone between these notes and the others.

Edited July 2, 2013 by Daedalus

[science4ever]

Daedalus that was exactly what I needed the way to set it up.

and the text thatexplain it you provided

 

 

Such that at the smallest end ( ) you will get 6 mm, and

at the other end ( ) you will get 24.75 mm,

which is pretty close to the numbers you have posted.

 

That will help me much becase the wood planks that I ahve at home don't allow the 28mm or higher

so I will most likely start with 6mm and test what happens if I only use 25mm instead of 28mm.

 

so the wood is 15mm and gluing two togther then makes 30mm thick and then if bore is 25mm

then I will have at least 2.5 mm thick wall and that is a bit thin but hopefully works.

 

 

Much appreciated you took time to explain it to me.

 

So typical of me not being clever enough to know how to set up that one.

Looks so easy now when one see it

Edited July 2, 2013 by science4ever

[Daedalus]

I'm glad I was able to help

Daedalus that was exactly what I needed the way to set it up.
and the text thatexplain it you provided


That will help me much becase the wood planks that I ahve at home don't allow the 28mm or higher
so I will most likely start with 6mm and test what happens if I only use 25mm instead of 28mm.
or even go down to 5 mm or smaller and see if it fail to get the lower register okay or if that is too thin
diameter 6mm seems to be among the smallest in actual usage and the tool I will buy
are 6mm and widest 24mm so that is cool.

Much appreciated you took time to explain it to me.



So typical of me not being clever enough to know how to set up that one.
Looks so easy now when one see it

Just remember, the 6 in the equation is the 6 mm measurement. So if you start with 5 mm, then the equation would be:

[math]y=\frac{1}{32}x+5[/math]

Then you plug in the length of tubing (600 mm) into [math]x[/math] to get the other end's diameter.

Edited July 2, 2013 by Daedalus

[science4ever]

Yes theorethically I know but fail to set it up but will try out until I get it.

 

What I know from practice is that the thinner inner dimension the more diffult

to get the lower register strong and stable and the opposite is that if one make

the inner much thicker like 16mm then the upper register second overblowing

most likely fail and one get an instrument with strong lower register and it has

trouble with the highest notes getting unstable and hard to blow. One need to compromize.

[studiot]

It is considered good practice in the design of horn loudspeakers to size so that the perimeter of the wide end is at least one wavelength at the cutoff frequency of the horn.

The cutoff frequency is the lowest frequency the horn will transmit.

[science4ever]

studiot yes that explains why a sax or Tuba has such wide horns.

And why the instrument that I will try is named a "muted" horn?

Itlack that loudspeaker that one are so used to see on brass instruments .

[studiot]

I think that brass instruments do not have a linear taper like you are attempting, but an exponential one.

 

There is also the resonator approach as in the Cor Anglais.

Incidentally I commend Daedalus on his research and clear presentation.

Edited July 2, 2013 by studiot

[science4ever]

 

There is also the resonator approach as in the Cor Anglais.

 

 

The name “English Horn” is a translation of the French cor anglais which is probably a corruption of cor anglé, meaning “angled horn”, referring to an early form of the instrument which was bent in the middle at an angle. from here http://www.organstops.org/e/englishhorn.html

 

Yes the exponential seems to have been known when they did

the Cornettos/Zink too the tapering is not linear but

I have to start with something that is at least close to ideal

 

Some say that the first bore to the thumbhole is almost cylindrical

and then it is conical. Others say one need to do two such with different tapering

after the other so they should have different climb for to get the overbklown right.

 

I will be happy if I get an instrument that at least sound each note even if a bit false

it is just a proof of concept that I make one for to later maybe buy a real one?

Edited July 2, 2013 by science4ever

[studiot]

I am not an expert in musical instrument names, so I may be wrong in the name but I thought the Cor Anglais was the alto member of the oboe family that had the bulb shaped resonator at the end. A smaller version is the Oboe D'Amore. The point of the bulb is that the exit orifice is smaller in diameter than the bulb itself.

I have never seen mention of a bent one, although I know that happens with some larger recorders.

 

(Source the Oxford Companion to Music)

Edited July 2, 2013 by studiot

[imatfaal]

I seem to remember that the cor anglais has the split reed in a metal tube that is angled (perhaps thats where they mean) - whereas the oboe has the split reed plugged directly into the wooden body of the instrument. BTW I have seen bonkers clarinets, oboes and cor anglais with angles between the upper and lower hand; but I thought this was a modern and silly affectation - I am not sure it makes any difference apart from making some of the long keys amazingly intricate

[science4ever]

Cornetto Zink is a lip vibrating instrument blown almost identical to Trumpet

but with a conical instead of cylindrical bore for to allow one to have both the lower

the overblown higher register in tune and to use just plain fingerholes with no mechanics.

 

So it is seen as a very primitive instrument now not easily integrated in modern music.

 

To me it would be a huge improvement over the Cowhorn and Goathorn used

on Folkmusic which is my main interest. these instrument play only in lowest register

and don't make use of overblowing.

Edited July 2, 2013 by science4ever

[science4ever]

Doesn's seem that we have that wood plancks here locally.

They are double size so I got curious on what if I make it

oval/eliptic diameter instead of round circle so it is say 22mm

in height but 35 or 40 in width? that way one could get same

air volume area and that way have strong walls and same

aucoustic properties?

 

Would that maybe work out well?

Edited July 5, 2013 by science4ever