PHY-2464 Physical Basis of Music

Transcription

1 Physical Basis of Music Presentation 19 Characteristic Sound (Timbre) of Wind Instruments Adapted from Sam Matteson s Unit 3 Session 30 and Unit 1 Session 10 Sam Trickey Mar. 15, 2005 REMINDERS: Brass instruments are characterized by buzzing the lips in a mouth piece. KEY FACTS Pipe is closed at mouthpiece end to an excellent approximation. The buzzing is stabilized by feedback from the open pipe end which matches impedance peaks. There are valved and valveless brass instruments There are brasses made with little or no metal.

2 What is the key physical difference among a Cornet, Trumpet, and Flügelhorn gelhorn? Trumpet Cornet The fraction of the horn that is cone/cylinder/flare. Trumpet most cylindrical Cornet -- more conical Flugel Horn most conical Flügelhorn Transverse Flute open pipe instrument Driven by air flow against the edge of the embochure hole. A pressure node exists there. Pitch is controlled by holes in the tube. Air flow Embrochure hole

3 Single Reed instruments stopped pipe The reed opens and closes like a valve, pressurizing the pipe when open, closing due to the Bernoulli effect when the air flows. A pressure anti-node node exists at the reed. Pitch is controlled by holes in the pipe. Air flow Tonguing Reed Double Reed instruments stopped pipe As with single reeds, the reed opens and closes like a valve, pressurizing the pipe when open, closing due to the Bernoulli effect when the air flows. A pressure anti-node node exists at the reed. Pitch is controlled by holes in the pipe. Air flow Pressure Pulses Reed Tip

4 Brass instruments stopped pipe Brass instruments are played by the player s s lips that form a lip valve. The lip valve admits pressure pulses into the pipe. The frequency is determined by the breath air pressure, the lip tension and the resonances of the pipe. A pressure anti-node node exists at the player s s lips. Pitch is controlled by valves and extra tubing, and/or slides, and/or overblowing. Louis Armstrong trumpet ( ) 1971) Wind instrument comparison f 5f 1 4f 1 3f 1 2f 1 f 1 5f 1 3f 1 f 1 6f 1 5f 1 4f 1 3f 1 2f 1 f 1 6f O 5f O 4f O 3f O 2f O f O f 1 Pedal Tone L f 1 = v/2l f 1 = v/4l Flute = v/4l f 1 = v/2(l+c) Clarinet Other Woodwinds c f o = (1+ξ)v/4(L+c) Brass

5 More wind instrument comparison Open Cylinder Stopped Cylinder Stopped Cone Stopped Combination N p N p f n = nf 1 f 1 = v/2l A p N p f 2n-1 = (2n-1)f 1 f 1 = v/4l A p N p f n = nf 1 f 1 = v/2(l+c) A p N p f n = nf 0 f 0 = (1+ξ)v/4(L+c) L f 1 = v/2l f 1 = v/4l Flute = v/4l f 1 = v/2(l+c) Clarinet Other Woodwinds c f o = (1+ξ)v/4(L+c) Brass In the flute, feedback from the acoustic standing wave locks the frequency of the oscillation if the edge tone is near the fundamental frequency. Matches impedance minima. Displacement wave f edge = 0.2 v jet /b f n = n v/ 2L; f edge f n

6 In reed instruments, pressure standing wave feedback locks the frequency of the oscillation of the reed; matches impedance maxima. Pressure wave f 2n 2n-1 = (2n-1) v/ 4L L = L r Pressure inverts 0.6 r Brass Instruments are stopped pipes. The player s s lips produce a displace- ment node (pressure antinode) ) at the mouth- piece. A displacement anti-node node (pressure node) exists at the bell. Winton Marsalis Trumpet

7 Feedback from Resonances KEY FACT: Wind instrument pitch is determined by the influence on the jet/reed/lip-valve of feedback from the pressure/displacement standing waves in the pipe. The near matching of impedance maxima or minima determines the stable ( playable )) pitches. Interlude regarding basics of SPECTRA: The energy of an oscillator or of sound dissipates in an exponential decay. An oscillator can be caused to vibrate in sympathy when the driving frequency is close to that of a natural mode of oscillation. The timbre of an instrument is determined by its spectrum The spectrum of an instrument changes with time because of transients.

8 Exponential Decay (pressure as example) p envelope = p / 1/2 0, /2 t t env A Fourier Decomposition is a representation of all the components that comprise a waveform, amplitude versus frequency and phase versus frequency.

9 Example: Synthesis of a Square wave- The Fourier Spectrum Fundamental 1 st Harmonic Amplitude 1 st Overtone 3 rd 2 nd Harmonic Overtone 5 th Harmonic f o 3f o 5f o Frequency 7f o Waveform Synthesis Harmonic n = Overtone f0 f1 f2 f3 f4 f5 f6 f7 f8 Amplitude = Amplitude (Pa) all modes =1/odd=2: Phase Frequency (Hz) = Component Amplitude (Pa) Fourier Spectrum Amplitude (Pa) Composite Waveform Frequency (Hz) Time (ms)

10 ADSR: Attack, Decay, Sustain, Release The envelope of the amplitude of all musical sound is described by ADSR. amplitude Attack Decay Sustain Release time Wind Instruments A jet produces a fluctuating air flow, while a reed or the lips produce pressure pulsations, the frequencies of which are controlled by impedance matching feedback from standing waves in the horn. Flow fluctuations or Pressure Feedback pulsations Standing waves in horn f 1 f 2 f 3 f 4 ~ ~ f n

11 Effect of Excitation The mode of excitation of the horn significantly influences the harmonic recipe (spectrum) of the air column. The relative strengths (amplitudes) of the harmonics are determined by the excitation of the jet/reed/lip-valve. The spectrum of the radiated sound determines the timbre. Driven Pipe Vibrational Spectrum ( Recipe( Recipe ) A Pipe Spectrum A A Mouthpiece Spectrum Driven Pipe Spectrum Frequency

12 Effect of the Pipe A pipe is three dimensional; therefore, 3-D 3 modes of oscillation are possible in the pipe. Only those modes with frequency above a cut- off frequency f c (low frequency cut-off) will exist in the pipe. f > f c for propagation. Transverse Modes of Vibration of an Air Column (0,0) D (1,0) (2,0) Cut-off Frequency f c = q n m v/d; for f < f c no propagation q 00 = 0; q 10 = 0.59; q 20 = 0.97

13 Effect of Transverse Modes on Spectrum More transverse modes implies more intensity. Most influential in high frequency harmonics. Shape and relative diameter of pipe influence transverse modes. Thus, a square, triangular, and circular cross- section organ pipes have different timbres. Reflections from the array of holes in a woodwind affect the relative strength of the high frequency harmonics in the pipe. Displacement wave Reflections from holes (closed and open)

14 Effect of Holes on Transmission Larger holes have greater effect. A high pass filter: Low frequencies tend to be reflected more and high frequencies transmitted more. Thus the holes make a brighter sounding instrument. Reflections from joints and imperfections affect the relative amplitude of the high frequency harmonics in the pipe. Reflections

15 Filtering of Wind Instrument Sound The specifics of transmission of the various frequency components in the pipe produce a filtering effect on the frequency spectrum of the sound. Transmission through horn f 1 f 2 f 3 f 4 ~ ~ f n Radiation of Sound from Wind Instruments The radiation characteristics of the bell alter ( shape )) the harmonic recipe, hence strongly influence the timbre of the instrument. Radiation Characteristics f 1 f 2 f 3 f 4 ~ ~ f n

16 The diameter of the mouth and the flare rate of the bell determine the radiation characteristics of brass instruments. Trumpet The larger the bore diameter, the more intense the low frequency harmonics. The more rapid the flare,, the more the low frequencies are reflected, and thus, the more high frequency harmonics are radiated. Cornet Flügelhorn The Bell Exponential Horn a = a o exp(m x)+ b m = flare constant. Larger m means more rapid flare accentuate high frequencies x

17 French Horn Bell Bessel Horn a a = a o e -(εx ) +b x Called Bessel Horns because the standing wave follows a Bessel Function. Mutes The French Horn player s s hand modifies the radiation characteristics of the horn, as well as the effective flare. Mutes reduce the effective area of the horn and, therefore, reduce the intensity. Mutes tend to reduce more the first and second harmonic of the pipe than higher frequency harmonics due to their internal modes of oscillation. Mutes make brass sound thin and reedy.

18 Summary - The pitch of a wind instrument is determined by the length and shape of its air column. The effective length of the air column is controlled with holes, valves and slides. Feedback from the resonances of the pipe select the frequency of oscillation of the jet, reed or lip-valve. The excitation, transmission and emittance of the sound in the horn determine the timbre of the instrument.