Computing the Frequency (FR S )* of the Alto Saxophone Mouthpiece and Neck
It is noted in the book "The Saxophone Is My Voice" by Ernest Ferron that the volume inside the mouthpiece must closely match the volume of the missing part of the "cone" if the neck were extended to its apex. This claim is supported by such leaders in musical instrument acoustics as Benade, Fletcher, and Nederveen in their written works. Benade (Fundamentals of Music Acoustics p. 470) writes that the (acoustically) correct playing frequency of a saxophone reed on its mouthpiece and neck can be calculated with this formula: F = v/2Xo Where F is the frequency, v is the velocity of sound (347 m/sec for the warm damp air at the upper end of a woodwind) and Xo is the length of the "missing part of the cone". Note that the missing part of the cone is now the length of the neck + its imaginary extension to an apex (point of the cone) because the formula is estimating the pitch of the mouthpiece and the neck together. Ferron (The Sax Is My Voice p. 101) provides the formula to calculate the length of the missing (imaginary) cone to its apex. It is possible with a flexible ruler to measure the length of a sax neck in millimeters. My Selmer alto neck measures .197 m to the end of the tenon and the length of the missing cone is calculated at .213 m (longer than the neck itself)! The total length then = .410 m. Putting this figure into Benade's equation: F = 347/(2)(.410) or F = 423 vps somewhere between Ab = 415.3 and A = 440. If the "end correction" http://www.phys.unsw.edu.au/jw/musFAQ.html of the resonant frequency of the neck is taken into consideration the "effective length" of the missing cone is .417 m. Putting this into the equation gives: F = 347/(2)(.417 m) or F = 416 vps (Ab = 415.3) I was taught that the ancillary concept to the mouthpiece pitch (for classical playing) = A 880 is the neck + the mouthpiece = Ab Concert (Ab = 415.3). When I play the neck and mouthpiece with the same embouchure that produces A = 880, an Ab concert is produced, provided the mouthpiece is set on the cork where the sax normally is tuned. If this is true, then why do lower mouthpiece pitches for jazz playing also work on the saxophone? I believe that when a jazz player plays G concert or lower on the alto mouthpiece, that player typically tunes the saxophone by pushing the mouthpiece farther on the neck to compensate for the lower mouthpiece pitch. This removes a portion of the volume inside the mouthpiece needed to match that of the missing cone. This discrepancy is resolved by the by the player adding back to the “effective volume” of the mouthpiece by relaxing the embouchure and opening the oral cavity to produce the characteristic jazz sound
* Benade in the “Fundamentals of Musical Acoustics” calls this the frequency of the reed on its staple or “F RS”. He compares the oboe reed on its staple and the bassoon reed on its bocal, to the saxophone mouthpiece on its neck. He states that the frequency of the reed plus its “constriction” must match that of the length of the missing cone to its apex as determined by its wavelength. The other requirement is that the “effective volume” of the double reed, or the single reed on its mouthpiece must match that of the missing cone as well in order for the instrument to play properly.