Why are spherical instruments not more common?

[Alimuir]

When it comes to waves, spherical harmonics are, like, da bomb. I'm no expert - probably obvious from the question - but it seem to me that an instrument which maximises the utilisation of harmonics/resonances would be spherical.

And yet, I can think of no spherical instruments - the most ubiquitous is the "figure-of-8" shape of most stringed instruments, and the quasi-cylindrical shape of woodwind/pipe instruments. Both shapes clearly have strong harmonic modes, but why is the sphere not equally prolific a shape upon which to model an instrument?

 

[robphy]

Seems to me, the issue would have to do with how you would set the boundary conditions.
I can see two 0-dimensional ends of a 1-dimensional string
and a 1-dimensional boundary of a 2-dimensional drumhead.

 

[ZapperZ]

When it comes to waves, spherical harmonics are, like, da bomb. I'm no expert - probably obvious from the question - but it seem to me that an instrument which maximises the utilisation of harmonics/resonances would be spherical.

Why would you think that? What makes spherical geometry more special than others? Cylindrical harmonics, etc. are equally useful. That's why we have the flute, recorder, etc. The cavity in an acoustic guitar is definitely not spherical.

As has been said, the geometry and the boundary conditions dictate the waveform. There's nothing special about a sphere that makes it any better than other geometry.

Zz.

 

[AlephNumbers]

Ocarinas are somewhat spherical. They aren't particularly loud, though.

 

[Alimuir]

Why would you think that? What makes spherical geometry more special than others? Cylindrical harmonics, etc. are equally useful. That's why we have the flute, recorder, etc. The cavity in an acoustic guitar is definitely not spherical.

As has been said, the geometry and the boundary conditions dictate the waveform. There's nothing special about a sphere that makes it any better than other geometry.

Zz.

Infinite symmetry?

 

[ZapperZ]

Infinite symmetry?

Yeah, and why would that be special and of any use for a musical instrument? Does that mean that someone can make that instrument while standing on his head or lying on his side?

Zz.

 

[Alimuir]

Yeah, and why would that be special and of any use for a musical instrument? Does that mean that someone can make that instrument while standing on his head or lying on his side?

Zz.

Surely predictable/manipulatable harmonics depend upon symmetry - hence why pretty much all resonant instruments exhibit it. It just seems to me that a perfectly-symmetrical shape is the perfect candidate for an instrument dependent upon harmonic properties. As for boundary conditions, I'm sure you'll agree a sphere has a pretty well-defined surface. Why is the 2-D boundary of a sphere inferior to the 1-D boundary of a drum?

I'm not trying to start an argument - there's clearly a reason why spheres aren't deployed in the construction of instruments (with the exception of the ocarina, thanks AlephNumbers), I'd just like to know what it is.

 

[ZapperZ]

Surely predictable/manipulatable harmonics depend upon symmetry - hence why pretty much all resonant instruments exhibit it. It just seems to me that a perfectly-symmetrical shape is the perfect candidate for an instrument dependent upon harmonic properties. As for boundary conditions, I'm sure you'll agree a sphere has a pretty well-defined surface. Why is the 2-D boundary of a sphere inferior to the 1-D boundary of a drum?

I don't know. No one has claimed that a boundary of a sphere is inferior to boundary of a drum. I haven't seen any such claim other than from you.

I'm not trying to start an argument - there's clearly a reason why spheres aren't deployed in the construction of instruments (with the exception of the ocarina, thanks AlephNumbers), I'd just like to know what it is.

But this whole thing started on a premise that you built, that somehow having all these symmetries must be good. I can show you many instances that higher symmetries are NOT necessarily a good thing to have! So it all comes down to what it is being used for and they are made. The consideration for something to be "better" must account for those as well, not just on ONE criteria!

Zz.

 

[Alimuir]

I don't know. No one has claimed that a boundary of a sphere is inferior to boundary of a drum. I haven't seen any such claim other than from you. Zz.

Robphy said mentioned the boundary conditions of a string and a drum, citing the boundary conditions of these things as a reason why spherical instruments aren't as common. You subsequently endorsed his comment. If boundary conditions are the important factor, why is a sphere any less useful?

But this whole thing started on a premise that you built, that somehow having all these symmetries must be good. I can show you many instances that higher symmetries are NOT necessarily a good thing to have! So it all comes down to what it is being used for and they are made. The consideration for something to be "better" must account for those as well, not just on ONE criteria!
Zz.

I didn't mean to imply that spherical instruments are definitely better than any other - I just don't understand why not. I'm aware spherical harmonics appears everywhere in physics, from GR to QM, and so I find it odd that they aren't so present in music.

And yes, please do show me why higher symmetries aren't a good thing - that's exactly what I want to know!

 

[robphy]

As for boundary conditions, I'm sure you'll agree a sphere has a pretty well-defined surface. Why is the 2-D boundary of a sphere inferior to the 1-D boundary of a drum?

For strings and drumheads, the boundary is fixed in space (zero displacement there) and the oscillations are in a direction perpendicular to the boundaries.
For something analogous and spherical, would it be some elastic material between two rigid concentric spheres? We might be out of spatial dimensions for some other perpendicular directions.

I guess one could relax the boundaries and have something like "Basketballs as spherical acoustic cavities"
http://www.acs.psu.edu/drussell/Publications/Basketball.pdf
With the NBA playoffs going on, the bouncing of a basketball could be music to some ears.
(Sorry, off-topic: Fancy pre-playoff dribbling by Stephen Curry.)
Maybe some professional "deflating" methods could produce a variety of interesting tones.

 

[nasu]

Spherical harmonics are frequently used in physics because systems with spherical symmetry are usually between the few cases with analytic solutions.
But this does not give them some extra power.:) If we were to use only objects whose behavior is described by analytic solutions we won't have to much around us.
A shape that gives nice, harmonic modes may not give the most "pleasant" sounds.

Spherical harmonics and the other "special" functions "rule" in the physics and math books but not in the real world. Not to say they are not useful.

 

[TeethWhitener]

If you consider a 1D string with fixed endpoints, you see intuitively that the only waveforms that satisfy the boundary conditions are exactly the ones that divide the string into halves, thirds, fourths, etc. (Mathematicians give this the very musically suggestive name "harmonic series.") This gives you a very simple eigenspectrum: fundamental frequency f, and overtones 2f, 3f, 4f, etc. Likewise, for a cylindrical column of air (like you would find in a flute or a clarinet), the resonant frequencies are those that divide the cylinder into halves, thirds, fourths, etc. (NB--in a string instrument, the string generates the sound; the shape of the body simply enhances/damps various overtones of the excited strings)

For a circular drumhead, the eigenvalues are related to the Bessel function zeroes, which gives an eigenspectrum that is not nearly as straightforward as the case of the 1D string. This is why a drum sounds like a drum. For whatever reason, our ears seem to find the former eigenspectrum more "musical" and the latter more "rhythmic" (very loosely speaking). For comparison, a square drumhead has eigenvalues that are proportional to f, √2f, √3f, etc., and this spectrum doesn't sound particularly musical either.

For a ball of air within a rigid sphere, the eigenspectrum is more "Bessel" than "harmonic," so it probably won't sound so musical. For the vibration of the spherical surface itself, the frequency is proportional to √f(f+1), so that probably wouldn't be too musical either. So it'd probably be a percussion instrument. But there's also the difficulty of playing a spherical instrument. Do you bounce it? Do you hit it with a stick and then run after it as it rolls away?