You have all experienced that it is more complicated to play the lowest notes of the saxophone softly than those of the higher register. Do you know why?
The input impedance is our friend
NB : If you don't know what the input impedance is, I invite you to first read my article on acoustic input impedance.
To answer this question, let’s start by looking at the input impedance of a tenor saxophone in C fingering (heard note Bb2, 117Hz) for which all holes, except the two last ones, are closed:
As you can see, the first peak, at about 120Hz, which is related to the fundamental of this fingering, has a lower amplitude than peaks 2 and 3 (248Hz and 382Hz). Peaks 2 and 3 correspond to the second and the third harmonics (NB: this measurement was made at the crook of the instrument, without an adaptation device to take the mouthpiece into account. Therefore the instrument is smaller than in reality, which explain why the measured frequencies are a bit higher than those produced when playing). The amplitude difference is about 1,7dB between peaks 1 and 2, and about 2,3dB between peaks 1 and 3. As the first peak is lower than the two next, the musician needs to blow harder in the instrument to play the corresponding note. This explains why it is harder to play the notes of the low register softly, in comparison with the higher register, which corresponds to the second peak.
Why is the first peak lower?
We may now try to figure out why the first peak is lower than peaks 2 and 3. To try to find a reason for this, let’s have a look at the input impedance of an oboe for its lowest note, the Bb3 (234Hz), for which all holes are closed:
As for the saxophone, the first peak (234 Hz) is lower in amplitude than the two next peaks (465 Hz and 708 Hz), with differences of 3,7 dB between peaks 1 and 2 and between peaks 1 and 3.
Finally, let’s have a look at the input impedance of a simplified clarinet (a cylindrical tube equipped with a clarinet mouthpiece), equivalent to a clarinet with all holes closed, designed to produced the note C#2:
The behavior of the peaks is different than before. Here, the first peak is higher than the next peaks (about 7 to 8dB higher than peaks 2 and 3).
What is the fundamental difference between these three instruments? The saxophone and the oboe have a conical bore, while the clarinet has a cylindrical bore. Therefore, it seems that the difficulties in playing the fundamentals of the lowest notes of the saxophone softly are due to its shape.
Cone VS Cylinder
What is the acoustical difference between a cone and a cylinder? Look at the curve below. It represents the input impedance of a cylinder and of several cones of the same length but with different opening angles.
We can see that when the opening angle of the cone is higher, the amplitude of the first impedance peak decreases in comparison with the next peaks.
Plitnik & Strong 1979 : Numerical method for calculating input impedances of the oboe, J. Acoustc. Soc. Am. 65, 1979