An understanding of the acoustical physics of the clarinet is aided by mentally disassembling it into three parts: the reed, the bore and the side holes. Vibrations in the column of air in the bore (standing pressure waves) are set up by air blown into the clarinet through the reed and mouthpiece. It is the vibrating column of air that produces the clarinet's sound. The frequency at which the air vibrates is determined mainly by the bore dimensions which can be changed by different open and closed combinations of the side holes.
The clarinet is the only modern wind instrument that functions acoustically like a stopped cylindrical pipe. This means that the wavelength corresponding to a given frequency is (with minor corrections) four times the length of the instrument from mouthpiece to the first open hole. This contrasts with the flute and oboe in which the wavelength formed is only twice the length of the corresponding length of tubing. Knowing that wavelengths double in length for every octave lower in frequency and that the wavelength formed in the clarinet is four times the length of tube used versus only two times the length of tube used in the flute or oboe explains why though all three woodwinds are similar in length the clarinet can play almost a full octave lower than the flute or oboe.
The clarinet is further distinguished from other wind instruments by the relatively low intensity of the even numbered partials (this is more prominent in modern German system clarinets than modern French system clarinets) in the composite sound of a given note. This is most pronounced in the chalumeau register. In higher registers both odd and even partials are more equally present. Therefore the clarinet loses its distinctive tone quality the higher in pitch it plays.
To produce the resonating air column in the clarinet, energy is admitted to the clarinet from the player's higher-pressured mouth cavity. The reed and lip unit acts like a valve admitting energy at the right time into the resonating air column whose resonant frequency is determined by the open and closed holes on the clarinet. In very soft playing the tip of the reed moves almost sinusoidally without contacting the tip of the mouthpiece at all and in loud playing the reed moves far enough to completely close the end of the clarinet tube during part of each cycle of vibration. In this type of coupled system energy must be fed into the resonating tube at the right time during each cycle for the vibration in the tube to continue. This only works if the frequency of vibration of the reed/lip system is higher than the frequency of resonance inside the clarinet. Therefore even if the correct fingering is used for an upper register note it will not sound unless the vibrating frequency of the reed is high enough. This is accomplished primarily by controlling the length of the reed that is actually vibrating. Through embouchure control the player can adjust where on the lay of the mouthpiece the reed makes contact and hence the reed's vibrating length. If the reed is too soft it may not be possible to find a position on the reed relative to the lay of the mouthpiece that will allow production of high altissimo or altissimo notes. The strength of the reed also affects how quickly the reed restores its shape upon deflection by the air stream.
There is a discrepancy between the length of the resonating air column in the clarinet for a given frequency and the value that is one quarter of the corresponding free-air wavelength at room temperature. The factors that account for this difference are: that the air in the clarinet is moist and is warmed above room temperature, the bore of the clarinet is marked by irregularities of tone holes and absorbent pads, and the mouthpiece is not completely closed and cylindrical. Clarinet makers have individually by trial and error made alterations of a few thousandths of an inch in the cross section of the bore at different places to compensate for unwanted effects produced by the complex behavior of the reed and holes.
With any fundamental frequency, the third and fifth harmonics of the many generated by the reed are also strongly present resonances which help to define the clarinet's tone color. The ability of the clarinet to play in upper registers is dependent on the presence of these harmonics. When the very small register hole is opened it is very difficult to make the air column resonate at the frequency associated with the tube to this length. The register hole does not vent the tube well enough. In this situation the next possible primary resonance to form in the clarinet is for the second resonance (third harmonic or the interval of a 12th higher) which is present for the length of tube established ignoring the register hole.
Another factor greatly affecting the tone color of the clarinet is that there is a large drop off in the energy of resonances above the cut-off frequency of ca. 1500 cycles per second in the modern clarinet. This value is determined by the array of open holes (or the design of the bell when all the holes are closed) that radiate the sound for a given note. Early clarinets with a very different array of tone holes will therefore have different cut-off frequencies and different sound characteristics.
The exact relationship between resonances of different harmonics affects both tone and intonation. If the second resonance is not an exact 12th above the fundamental, a duller sound will result because the relevant partial in the reed motion will not adequately excite this harmonic. There will not be as much "ring" in the sound. The intonation between 12ths will also be out of tune if the second resonance is not an exact 12th above the fundamental.
Another factor affecting the perceived tone of the clarinet is that for a given note the lower frequencies of the composite pitch are emitted relatively weakly from the first one or two open holes omnidirectionally while the higher components are radiated from all the open holes but in a highly directional manner.