Music 170: Wind Instruments

Transcription

1 Music 170: Wind Instruments Tamara Smyth, Department of Music, University of California, San Diego (UCSD) December 4, 27 1

2 Review Question Question: A 440-Hz sinusoid is traveling in the x direction. Two microphones are placed at x = 0 and x = 1 foot, respectively. Assuming a propagation speed of 1000 ft/s, what is the phase difference, in radians, between the signals picked up by the two microphones? Music 170: Wind Instruments 2

3 Review Answer Question: A 440-Hz sinusoid is traveling in the x direction. Two microphones are placed at x = 0 and x = 1 foot, respectively. Assuming a propagation speed of 1000 ft/s, what is the phase difference, in radians, between the signals picked up by the two microphones? Answer: if the sinusoid has frequency f Hz, then the period is T = 1/f and the wavelength is λ = ct or c f ; the waveform completes one cycle (2π radians) over λ feet; the phase after 1 foot would then be x = 2π c/f = 2π ct. Music 170: Wind Instruments 3

4 Wind Instruments Woodwind and brass instrument, a family of instruments called aerophones, include: flute clarinet saxophone trumpet trombone Music 170: Wind Instruments 4

5 Wind Instrument Construction With the exception of the flute, most western wind instruments typically consists of some kind of mouthpiece, a bore, and a bell. Trumpet: Saxophone: Common to many aerophones (except the flute) is a reed a pressure controlled valve which introduces an oscillating (driving) pressure into the instrument s bore. Music 170: Wind Instruments 5

6 Waves in a cylindrical tube As with the string, waves in a cylindrical tube may be considered one-dimensional (in the first approximation) and are the sum of right and left going traveling waves; described by D Alembert s traveling wave solution to the wave equation: where y = f 1 (ct x)+f 2 (ct+x), f 1 (ct x) a wave traveling to the right f 2 (ct+x) a wave traveling to the left with velocity c. The sum of the right and left traveling waves creates standing wave patterns due to constructive and destructive interference. Music 170: Wind Instruments 6

7 Open Tube Displacement waves in a tube open at both ends: Figure 1: Standing displacement waves in an open tube. Though opposite-phase to displacement on a string, the harmonics follow the same relationship: for each harmonic number n, the wavelength is λ n = 2L n. the frequency is f n = nc 2L = nf 1. where c is the sound speed. Pressure waves are opposite phase to displacement waves (looking like a string fixed at both ends). Music 170: Wind Instruments 7

8 Closed Tube Displacement waves in tube closed at one end: Figure 2: Standing waves for pressure in an closed tube For the first harmonic, the wavelength is λ = 4L. The next possible standing wave pattern has a wavelength of λ = 4L 3, corresponding to frequency 3f 1 (3rd harmonic). Notice there are only odd harmonics. The closed tube harmonics n = 1,3,5,... have wavelength λ = 4L n and frequency f n = nc 4L. Music 170: Wind Instruments 8

9 Digital Waveguide As with the string, waves propagating in a cylindrical or conical tube may be modeled using delay lines (digital waveguide). INPUT delay T seconds HP OUTPUT open (flute) closed(clarinet) 1 +1 LP LP 1 open delay T seconds low- and high-pass filters account for reflection and transmission loss at the bell. for pressure, open-end reflection is inverting. mouthpiece end is either open (e.g. flute) or closed (e.g. clarinet). Music 170: Wind Instruments 9

10 Open-Closed vs Open-Open Notice a closed-open tube (e.g. clarinet) has a period that is twice as long as when its open-open (e.g. flute). closed open time open open time Try flute.pd, a very simple flute, more like blowing in a tube ). How does the sound change when you change from open-open to open-closed? Question: If modeling a clarinet, what would be the length of the delay line T (in seconds) for a fundamental frequency of 100 Hz? Music 170: Wind Instruments 10

11 Setting the Delay for a Tube (Closed) Question: If modeling a clarinet, what would be the value of T for a fundamental frequency of 100 Hz? Answer: the period of the waveform at 100 Hz is 1/100 seconds. since a pulse would have to travel 4 lengths of the bore (2 round trips from mouthpiece to bell and back again) to create one period, 4T = 1 100, 1 T = = Question: If modeling a flute (open-open), what would be the value of T for a fundamental frequency of 400? Music 170: Wind Instruments 11

12 Setting the Delay for a Tube (Open) Question: If modeling a flute (open-open), what would be the value of T for a fundamental frequency of 400? Answer: the period of the waveforem is 1/400 or seconds. since a pulse would have to travel 2 lengths of the bore (1 round trip from mouthpiece to bell and back again) to create one period, 2T = 1 440, 1 T = Music 170: Wind Instruments 12

13 Flute and Clarinet Timbres Why are the timbres of clarinets and flutes different? flute solo clarinet solo In the lower note clarinet (from exam2) you can see there is more emphasis on odd harmonics frequency (Hz) frequency (Hz) Music 170: Wind Instruments 13

14 Flute vs. Clarinet at Higher Pitch Listen to notes at the same pitch. Flute.mf.B4.wav, Flute (no attack), Clarinet.mf.B4.wav, Clarinet (no attack), What do you notice in the spectrum? The theoretical odd harmonics only applies to lower frequnencies. 0 Flute frequency (Hz) Clarinet frequency (Hz) The instrument itself has a cutoff frequency after which it has less influence over the resulting sound. Music 170: Wind Instruments 14

15 Flute The flute is perhaps one of the best examples of an instrument with a purely cylindrical bore. Of course, there are toneholes... There is a reflection and transmission in the propagating wave at the position of each hole. Figure 3: 9000 year old chinese neolithic flutes. Music 170: Wind Instruments 15

16 Flute toneholes and Sound examples Figure 4: Measurement of toneholes. Opening a tone hole has the effect of effectively shortening the bore length; it also introduces radiation characteristics that change the sound spectrum. Play Chris Chafe s Oxygen Flutes: oxyflutes-cc.wav. Music 170: Wind Instruments 16

17 Open Cylinder and Cylinder+Cone Music 170: Wind Instruments 17

18 Trombone Without a mouthpiece, the trombone is well modeled as a cylinder (or a piecewise connection of cylinders). Figure 5: The trombone in fully retracted and extended positions One consideration, however, is finding a filter to model the acoustic characterisitcs of the bell. Music 170: Wind Instruments 18

19 Characteristics of Bell Closed-open cylindrical bore produces only odd harmonics: 1. resonant frequencies are too far apart 2. produced sound is not loud enough. Closed-open conical pipes have resonances that are higher and more closely spaced. A flared bell raises frequencies, particularly low frequencies: longer wavelengths: less able to follow curve of the bell are reflected back, effectively see a shorter pipe; shorter waveglengths go into the bell and radiate more efficiently out the bell as sound. (see Brass Instruments by UNSW). The bell serves to change frequencies and amplitudes of resonant peaks make radiation pattern more directional at high frequencies Music 170: Wind Instruments 19

20 make radiation more efficient by matching the high pressure inside the horn with the outside lower pressure. Music 170: Wind Instruments 20

21 Piecewise Bell Model Since no 1-D wave solution exists for the propagation of waves in the bell (as it does for the cylindrical bore), an approximation of some kind must be made. Figure 6: Bell profile. bell radius meters bell length meters Figure 7: Modeling the bell profile. Or... the bell can be measured to obtain its acoustic characteristics. Music 170: Wind Instruments 21

22 Pressure Controlled Valves Many sounds are produced by coupling the mechanical vibrations of a source to the resonance of an acoustic tube: in vocal systems, air pressure from the lungs controls the oscillation of a membrane (or vocal fold), creating a variable constriction through which air flows. blowing into the mouthpiece of a clarinet will cause the reed to vibrate, narrowing and widening the airflow aperture to the bore. blowing into the mouthpiece of a brass instrument will create a pressure difference accross the lips, also causing them to vibrate. Sound sources of this kind are called pressure-controlled valves and they may simulated in various ways to create models of woodwind and brass instruments as well as vocal systems. Music 170: Wind Instruments 22

23 Classifying Pressure-Controlled Valves Pressure-controlled valves (PCVs) have two sides: one that sees 1. the upstream (blowing) pressure, 2. the downstream (bore) pressure. The PCV may be characterized by whether an additional pressure on either side causes the valve to open or close further. This characteristic may be described using the couplet (σ 1,σ 2 ), where subscript indicates up and downstream sides, respectively, +1 indicates an opening of the valve 1 indicates a closing of the value in response to a pressure increase. Music 170: Wind Instruments 23

24 Three simple configurations of PCVs 1) (,+) p 1 p 2 U 2) (+, ) p 1 p 2 U 3) (+,+) p 1 p 2 U Figure 8: Simplified models of three common configurations of pressure-controlled valves. 1. (,+): the valve is blown closed (as in woodwind instruments or reed-pipes of the pipe organ). 2. (+, ): the valve is blown open (as in the simple lip-reed models for brass instruments, the human larynx, harmonicas and harmoniums). 3. (+, +): the transverse (symmetric) model where the Bernoulli pressure causes the valve to close perpendicular to the direction of airflow. Music 170: Wind Instruments 24