Why does the clarinet sound different than the soprano saxophone? Part 1

Why does the clarinet sound different than the soprano saxophone?  This is an interesting comparison and has implications for hearing, speech acoustics and, oh yes, musical instruments.

It is also one of the most misunderstood comparisons in acoustics.

Both the soprano saxophone and the clarinet have identical lengths; both are “closed” at the mouth piece end, and “open” at the bottom flared end.  So, both should be viewed as quarter wavelength resonators that generate resonances according to the formula we all learned in high school: 

F = (2k-1)v/4L

In this familiar formula, v is the speed of sound (close to 34,000 cm/sec or 340 m/sec), L is the length of the resonating tube (in cm), and the (2k-1) part is a multiplier that gives odd numbered multiples of v/4L.

In 2k-1, if k=1, then 2k-1 is 1; if k=2, then 2k-1 is 3; if k=3, then 2k-1 is 5.  This gives odd numbered multiples of 1, 3, 5, and so on.  All quarter wavelength resonators (“closed” at one end and “open” at the other end) have odd numbered multiples of the first resonance. 

In speech acoustics the mid, unstressed vowel shwa, has odd numbered multiples of its formant structure with formants at F1=500 Hz, F2=1500 Hz, F3=2500 Hz.  And the resonances of a clarinet are also at odd numbered multiples of the first mode (where the length L depends on the fingering).

In hearing aid acoustics, a conventional behind the ear hearing aid is “closed” at the receiver end and “open” at the end of the earmold in the earcanal.  For adult-sized hard of hearing people, we know to expect a primary resonance at 1000 Hz, with subsequent ones at 3000 Hz and 5000 Hz (and 7000 Hz).  The intermediary resonances are receiver related and depend on the mechanical characteristics of the hearing aid receivers themselves.

Every audiologist has learned about this in their speech sciences training and in their very first hearing aid course.  And every linguistic student (assuming that they have taken my course on acoustic phonetics) knows this as well.

The soprano saxophone is like this as well: closed at one end and open at the other end, and given the same length as the clarinet, we would expect it to be identical sounding.  But that is not the case.

While it is true that the soprano saxophone looks a lot like the clarinet (except that it is usually a pretty gold color), the saxophone (like its oboe and bassoon cousins) has an octave key.  The clarinet has a “register” key.  When an octave key is pressed in on the saxophone, the note becomes twice as high in frequency.  When the clarinet register key is pressed, the note becomes three times as high in frequency.

So, what is going on?

Well, it’s not the gold color!  The saxophone is a “conical” instrument whereas the clarinet is merely a “cylindrical” one.

This seems to be a rather trivial difference but for those who really like math (and physics) the pressure variations (or as we audiologists like to say, the volume velocity variations) vary as a function of sin(x)/x where “x” is the distance along the cone.  For a cylinder like a clarinet or behind the ear hearing aid tubing, there is no change in diameter of the resonating tube so all “x” perturbations are the same.  Unfortunately this is also the case for those who do not like math (and physics)!

This seemingly trivial difference is the basis for the difference in the generation of sound:  “open” and “closed” cylinders function as quarter wavelength resonators with register keys; “open” and “closed” cones function as half wavelength resonators with octave keys.

So, is the human ear canal a cylinder or a cone?  Stay tuned for part 2.