In all this we overlook the transient conditions of a reed, starting and stopping contributes much to
how we identify a sound with or human ear but this is more difficult to document because FFT is not mad for this.
This subject is Really of a lot of importance. Much of what we perceive as "tone' in an instrument is really in the transient section when a reed starts. The influence of the surroundings on the reed here can be great, Some circumstances can cause a rapid build up of vibration in the reed with a sharp crisp start and impression of brightness that is stronger than the actual harmonic content of the steady tone would imply. Other circumstances make a slow starting reed that sounds smooth and soft in the attack transient. Each reed has it's own attack transient pattern. This can be compressed at higher starting pressures, or extended at lower ones, but holds for that reed. There is always a slight delay from a reed being open to the air pressure / flow and it's ramping up to full volume for that pressure. What it does during that time I find nearly as important a part of how the instrument sounds as the overall timbre.
In some experiments I did a number of years ago, I watched the way different reeds ramped up in volume. At first, only the slightest but quite sinusoidal vibration was present, then as soon as any harmonic content became noticeable, the amplitude went up very rapidly. Some surroundings damped the harmonics more and left the reed slow to start, while others allowed the harmonic content to build and the reeds started very fast. ( same reed used in different material surroundings and also tested with different set gaps. It seemed fairly clear that the reed needed to be able to build it's lower harmonic content to gain any power. There is sort of a chicken and egg problem though, since as reeds gained volume the harmonic content tends to follow as a part of that very hard to seperate cause and effect.. Still, reeds that were in surroundings that supported richer low harmonics ( harmonic numbers 1-6) started much faster at the same playing pressures and final sound levels. Reeds that are set too low never can build up much volume, but I find instruments that have fairly low damping tolerate both lower and higher sets of the reed gap and still provide fast starting. Often now when I find an instrument that has some slow starting reeds, the first place I look is at the reed pan support. Lack of firm contact can greatly increase the damping.