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Input Impedance Spectrum for a Tenor Sax and a Bb Trumpet

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Input Impedance Spectrum for a Tenor Sax and a Bb Trumpet David Pignotti Department of Physics, University of Illinois, Urbana-Champaign Chris Van de Riet Department of Physics, University of Missouri—Rolla Abstract The input impedance spectrum of a reed-driven instrument gives information on which notes an instrument can produce and the relative difficulty a player encounters when playing a specific note. To measure the input impedance spectrum we used a piezoelectric transducer to excite the air within the instrument, the world’s smallest microphone to measure pressure, and a custom-built differential pressure microphone to measure air particle velocity. Using these same components we developed a method of measuring not only the input impedance, but the sound intensity of any vibrating air column as well as free air. I. Introduction When a player’s lips excite the air that is forced through a trumpet, the air There are two different types of wind resonates at the natural frequency of the instruments, those that exhibit low or tube in which the air flows. In the case high input impedance at the point at of the trumpet the tube appears to be which a player excites the air flowing open—It is open at the mouthpiece and through the instrument. The low- open at the bell. Thus, one would expect impedance group refers to instruments the trumpet to produce the fundamental such as the flute, organ pipe, or recorder. frequency associated with the ~147 cm We are interested in measuring the input tube length (f0 ≈ 117 Hz, C3) and all the impedance of the high impedance subsequent even and odd harmonics group—reed driven instruments (be they (C4, G4, C5, E5, G5, a flat Bb5 and C6) wooden reeds or lip reeds) such as the when no valves are pressed.2 This model clarinet, saxophone, or trumpet.1 For our agrees with the notes that can actually be experiment we will focus specifically on played when all 3 of the valves of the the input impedance of a trumpet and a trumpet are up. However, even though tenor saxophone, but our method can the tubing of a trumpet is open, there is easily be applied to any musical such a large impedance mismatch at the instrument, as well as for any arbitrary bell end of the trumpet that the tubing is sound field. effectively closed at the bell end. As a result, a trumpet will produce a 1 fundamental frequency that is a factor of II. Background two times lower than the open-tube model (f0 ≈ 58.5 Hz) and only the odd Acoustical input impedance is harmonics will be present.2 How does defined as a measure of resistance to the trumpet overcome this problem? putting a pressure wave through a tube.3 The quotient of the complex pressure By measuring the acoustical input wave inside the instrument and the impedance spectrum we can see how the complex air particle velocity gives the frequencies that should be produced by a value of the complex impedance: trumpet are shifted to the frequencies Pr ( ) that are observed. Zr ()= Ur () The saxophone is a woodwind instrument, and therefore operates in a The SI units of impedance are Pa·s/m somewhat different manner than the (= kg/m2·s) or simply “acoustical ohms.” trumpet. The vibrations of the air When a trumpet is played, the player’s column are created by a cane reed, lips vibrate, injecting bursts of air and which is clamped to the mouthpiece at pressure into the trumpet tubing. The one end with a metal ligature. The reed pressure wave travels through the receives feedback from the reflected trumpet and is reflected back towards the pressure waves regarding the proper mouthpiece when it encounters a sharp vibrational frequency needed to excite a change of the radius of the tube at the resonance of the instrument. Unlike the bell end of the trumpet. The reflected trumpet, where one valve combination is pressure wave gives the player’s lips used to play many notes, the saxophone “information” on the frequency at which has a different key combination for each to vibrate. Therefore, the higher the note - the player closes keys which impedance, the greater amount of become subsequently closer to the bell pressure waves reflected, and thus the of the instrument in order to play easier it is to play that note.3 This subsequently lower-frequency notes. phenomenon allows the trumpet to sound a desired note with ease. The saxophone family consists of various members which vary mainly by Backus investigated the change in the their physical size (and consequently, input impedance spectrum throughout their available frequency ranges); the the various stages of the construction of middle-sized tenor saxophone, pitched in a trumpet. The lowest mode is actually Bb, is examined in this experiment. the second harmonic since the fundamental frequency for brass Even though the saxophone has a instruments cannot be used. Due to the conical bore, it can be roughly construction of the mouthpiece which approximated as an open-open caters to the higher harmonics, the cylindrical tube, and therefore excites fundamental frequency is only available both even and odd harmonics, with the to extremely experienced players. lowest available frequency for the tenor Nevertheless, the trumpet is driven by saxophone (given an overall length of the player’s lips at the fundamental 1.4 m) being concert-pitch Ab2 at 103.83 frequency, thus higher harmonics appear Hz. at integer multiples of the fundamental. 2 Figure 1 shows the input impedance somewhat better matched to the desired peaks for the construction of a trumpet. frequencies. When the bell is added to The cylindrical tubing alone shows a this 3rd system, the input impedance steady decrease in the magnitude of the peaks shift to almost exactly the input impedance peaks as frequency frequencies required for a trumpet.4 increases, but the peaks do not line up Figure 2 shows the discrepancies with the desired note frequencies for a th th trumpet. When the bell is added, the between the n resonance and the n input impedance peaks still steadily harmonic throughout the stages of decrease but the higher modes decrease construction of a trumpet. Before the more rapidly than they did without the instrument is put together, the deviation from the resonances and the harmonics bell. The bell also matches the input th impedance peaks with desired the for just the cylindrical tubing at the 5 mode is 1/4 an octave. When the bell is frequencies better than with just the th tubing. The tubing plus mouthpiece and added, the deviation at the 5 mode is leader pipe show a steady increase of the 1/6 an octave. When the mouthpiece is input impedance peaks of the modes as added, the deviation is lowered to 1/12 frequency increases and then displays a of an octave. Finally, when the sharp falloff after about the 8th harmonic. instrument is complete there is no The input impedance peaks are appreciable deviation from desired harmonics.4 3 Figure 1. Input impedance curves for the construction of a modern Bb trumpet. (a) a length of cylindrical tubing; (b) tubing plus trumpet bell; (c) tubing plus mouthpiece and leader pipe; and (d) tubing plus bell plus mouthpiece and leader pipe. Full scale is 1000 acoustic Ω.4 4 Figure 2. Plots of the discrepancies in cents between the nth resonance and the nth harmonic, with the second resonance coinciding with the second harmonic. (a) Cylindrical tubing; (b)tubing plus bell; (c) tubing plus leader pipe and mouthpiece; and (d) tubing plus bell plus leader pipe and mouthpiece.4 In order to measure the complex piezo-transducer. Benade used the input impedance of an instrument microphone to measure the pressure Benade developed a piezo-disk-driven response to the voltage-dependent impedance head apparatus.5 This oscillations provided by the transducer. technique involves attaching a piezo- He states that his microphone has a flat transducer onto the mouthpiece of the frequency response from 50 Hz to 5000 instrument. Benade states that if the Hz. The air particle velocity, Benade stiffness of the piezoelectric disk is large claims, is related to the amount of air compared to the highest possible input displaced by the piezo transducer. Using impedance of the air column under these two values of pressure and air study, then the acoustical pressure particle velocity the acoustical observed at the input end of the air impedance could easily be calculated column is an accurate measure of the air over a range of frequency.5 We can column’s own input impedance. He improve upon the piezo-transducer further states that this measurement of method by directly measuring the air the air column’s input impedance is particle velocity using a differential accurate as long as the frequency at pressure microphone. which the tube is excited is not a resonant frequency of the piezo- transducer. A microphone is then inserted into the mouthpiece near the 5 III. Experimental Apparatus world.6 This microphone was chosen because of its small size and its flat We measured the acoustic input frequency response up to 10,000 Hz impedance using a piezoelectric driver, (Figure 4). Keeping the components two 1/10˝ Knowles Acoustics FG-23329 inside the instrument’s mouthpiece as high performance omni-directional small as possible is important so as to microphones to measure pressure, and minimally disturb the air volume of the two modified RadioShack 270-090 mouthpiece to ensure negligible condenser microphones to measure disruption of airflow. The Knowles particle velocity. A pressure microphone Acoustics microphones were each and a particle velocity microphone were connected to custom built preamps secured inside the mouthpiece near the (Figure 5) to increase the voltage output piezo-transducer to determine the input from the microphone before being fed impedance.
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