Physics 1251 The Science and Technology of Musical Sound

1. Physics 1251The Science and Technology of Musical Sound Unit 3 Session 35 Pipes, Voice and Percussion Review

2. Physics 1251 Unit 3 Session 35 Pipes, Voice and Percussion Foolscap Quiz: What are the first three harmonics that “speak” in the harmonic series of a Tympani tuned to A3 (110 Hz)? Answer: approximately 220 Hz, 330 Hz, 440 Hz. The fundamental 110 Hz is missing.

3. Physics 1251 Unit 3 Session 35 Pipes, Voice and Percussion 1′ Lecture: • The pitch of pipes and the voice is determined by a harmonic series. • Percussion most often have no pitch because they lack a harmonic series. • Vibration recipes of pipes, the voice and percussion arise from the modes of vibration the air column, of the vocal folds and of the membrane, plate, or block, respectively.

4. Physics 1251 Unit 3 Session 35 Pipes, Voice and Percussion The key to understanding the Sound of Music! 6 2 4 1 3 7 5 Harmonic Series produce a sense of musical intonation.

5. Physics 1251 Unit 3 Session 35 Pipes, Voice and Percussion 80/20The timbre of an instrument’s sounds depends on its vibration recipe. fn = n f1 Pitched Amplitude f1 2f1 3f1 4f1 fn m = xn m f1 Unpitched Amplitude f01 Frequency

6. Physics 1251 Unit 3 Session 26 Sound in Pipes 1′ Lecture: • Sound in pipes can produce standing waves in the air column. • Standing waves in air columns produce pressure nodes and displacement nodes (and antinodes) at different places. • A change in the acoustic impedance of the air column produces a reflection. • Organ pipes and the flute are examples of open or unstopped pipes.

7. Physics 1251 Unit 3 Session 26 Sound in Pipes Standing Waves in a Cylindrical Pipe: • A Closed or Stopped Pipe – the pressure wave reflects without inversion, but the displacement wave inverts upon reflection. • Thus, a pressureanti-node will occur at the wall; but, on the other hand, a displacementnode will occur at the same place.

8. Physics 1251 Unit 3 Session 26 Sound in Pipes Comparison of Pressure and Displacement Standing Wave in a Double Stopped Pipe λ/4 λ/4 Pressure Wave Displacement Wave

9. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20Acoustic Impedance: Z = p/U Acoustic Impedance is the ratio of the pressure p of a sound wave to the flow U (= u S) that results. For a plane wave in a tube of cross section S (m2) in air the acoustic impedance is: Z = ρv/S = 415/ S rayl

10. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20For Stopped Pipe: Nna = odd number = 2n-1, n=1,2,3,4 … λn = 4L/ Nna = 4L / (2n-1) fstopped = f2n-1 = v/ λ2n-1 = (2n-1) v/ 4L 80/20Only odd harmonics of fstopped 1 = v/4L.

11. Physics 1251 Unit 3 Session 26 Sound in Pipes 80/20For Open Pipe: Nna = even number = 2n, n=1,2,3,4… λn = 4L / Nna = 4L/(2n) = 2L/ n fopen = fn = v/ λn = n ‧ v/2L 80/20All harmonics of fopen1 = v/2L [= 2 fstopped 1 ]

12. Physics 1251 Unit 3 Session 26 Sound in Pipes End Correction for Open Pipe without Flange δ ≈ 0.6 a for a ≪λ ; δ ≈ 0 a for a > λ / 4 a Radius L + δ δ

13. Physics 1251 Unit 3 Session 26 Sound in Pipes Transverse Flute 80/20The transverse flute is a cylindrical open pipe. Mouthpiece is open

14. Physics 1251 Unit 3 Session 26 Sound in Pipes Summary: • fopen = fn = n ‧ v/2L • fstopped = f2n-1 = (2n-1) v/ 4L • Stopped and open cylindrical pipes have different timbres. • Impedance: Z = p/U • An abrupt change in Z is responsible for the reflections that lead to standing waves in pipes.

15. Physics 1251 Unit 3 Session 27 Flutes et cetera 1′ Lecture: • Flutes and flue pipes are driven by fluid flow instabilities at their mouth. • Standing waves in open air columns of flutes determine the pitch. • Open holes in the flute tube change the effective length of the air column.

16. f1 f2 f3 f4 ♩ ♪ ♫ fn ~ ~ Physics 1251 Unit 3 Session 27 Flutes et cetera The Flute • The transverse flute is acoustically driven by the fluid flow instabilities whose frequency is controlled by the feedback of the resonances of the pipe. Standing wave frequencies Flow Instability Feedback

17. Physics 1251 Unit 3 Session 27 Flutes et cetera Transverse Flute 80/20The flute is driven by air flow against the edge of the embrochure. Air flow Embrochure

18. Physics 1251 Unit 3 Session 27 Flutes et cetera Edge Tone 80/20An air stream striking against an edge produces a fluctuating instability in flow. Air Stream Edge The flow alternates sides.

19. Physics 1251 Unit 3 Session 27 Flutes et cetera Why does the stream oscillate? Short answer: positive feedback. • When the stream bends to the left, the stream moves faster on the right side. • Bernoulli’s Principle tells us that the faster the flow, the lower the pressure. • Therefore, the left-flowing stream will bend back to the right

20. Physics 1251 Unit 3 Session 27 Flutes et cetera Bernoulli Effect • 80/20The pressure in a fluid decreases as the velocity increases. • Hold the foolscap by the edge and blow across the top. What do you observe?

21. Physics 1251 Unit 3 Session 27 Flutes et cetera fedge = 0.4 vjet / 2 b = 0.2 vjet /b Edge Tone u = 0.4 vjet b b vjet u

22. Physics 1251 Unit 3 Session 27 Flutes et cetera Feedback from the acoustic standing wave locks the frequency of the oscillation if the edge tone is near the fundamental frequency. fedge = 0.2 vjet /b fn = n v/ 2L; fedge≈ fn Displacement wave

23. Physics 1251 Unit 3 Session 27 Flutes et cetera The Problem with Flutes: • Only about 1% of the energy of the air stream produces sound. • Playing louder means more air flow. • More air flow means higher jet velocity • Edge tone goes sharp • Worse in Recorder than in Transverse Flute • Player must “lip” tone into tune

24. Leff Physics 1251 Unit 3 Session 27 Flutes et cetera How does one “play” the notes? • By effectively changing the length of the air column. • Opening holes introduces reflections that change the standing wave length. Displacement wave f n′ = n ‧ v/2Leff

25. ♩ ♩ ♯♩ ♩ ♯♩ ♩ Physics 1251 Unit 3 Session 27 Flutes et cetera Cross Fingering 80/20The position and size of the open holes modify the effective length of the air column and consequently the pitch.

26. Physics 1251 Unit 3 Session 27 Flutes et cetera Why does the size of the hole matter? Z =p/U Impedance = pressure/flow Displacement →Flow U: Z Z ′

27. Physics 1251 Unit 3 Session 27 Flutes et cetera Summary: • Flutes and flue pipes are open columns of air, with fn = n v/2L, n = 1,2,3,4…. • Flue pipes are excited by flow instabilities of the air steam in the embrochure or fipple. • The frequency range is selected by the edge tone. • The pitch is determined by the effective length of the pipe. • Open holes determine the effective length of the pipe.

28. Physics 1251 Unit 3 Session 28 Clarinets et cetera 1′ Lecture: • Reed instruments are stopped pipes. • The clarinet has a cylindrical bore and is a stopped pipe; consequently, only odd harmonics are significant. • Conical pipes exhibit all harmonics, even in stopped pipes. • The saxophone, oboe and bassoon‒all have conical bores.

29. Physics 1251 Unit 3 Session 28 Clarinets et cetera Comparison of Flute and Clarinet Registers • Overblown flutes jump from a fundamental f1= v/2L to an octave f2 = 2f1 in the second register; an octave (2x) and a perfect fifth (3/2) f3 = 3 f1 =3 (v/2L) in the third register. • Overblown clarinets jump from a fundamental f1 = v/4L to an octave (2x) and a fifth (3/2)‒“the twelfth‒” in the second register, because only odd harmonics produce standing waves in a stopped cylindrical pipe.

30. f1 f2 f3 f4 ♩ ♪ ♫ fn ~ ~ Physics 1251 Unit 3 Session 28 Clarinets et cetera Reed Instruments • The reed produces a pulsation in the pressure admitted to the pipe; the pressure standing wave feeds back to control the oscillations of the reed. Standing wave frequencies Reed pulsations Feedback

31. Air flow Tonguing Physics 1251 Unit 3 Session 28 Clarinets et cetera The Single Reed 80/20The reed opens and closes like a valve, pressurizing the pipe when open, closing due to the Bernoulli effect when the air flows. Reed

32. Physics 1251 Unit 3 Session 28 Clarinets et cetera Hard and Soft Reeds 80/20A hard reed is one for which the frequency is determined by its stiffness and dimensions. A soft reed flexes easily and vibrates at the frequency of external pressure fluctuations. Soft Reeds Hard Reed: Harmonica Clarinet Oboe

33. Air flow Physics 1251 Unit 3 Session 28 Clarinets et cetera The Double Reed 80/20The reed opens and closes like a valve, pressurizing the pipe when open, closing due to the Bernoulli effect when the air flows. Pressure Pulses Reed Tip

34. Physics 1251 Unit 3 Session 28 Clarinets et cetera Bernoulli Effect • 80/20The pressure in a fluid decreases as the velocity increases. • Thus, as the air flows past the reed, it is forced closed. Bernoulli Effect

35. Pressure inverts Physics 1251 Unit 3 Session 28 Clarinets et cetera 80/20Feedback from the pressure standing wave locks the frequency of the oscillation of the reed. f2n-1 = (2n-1) v/ 4L′ Pressure wave L′ = L + 0.3 d 0.3 d

36. Physics 1251 Unit 3 Session 28 Clarinets et cetera 80/20For a stopped conical pipe: fn≈ nv / 2(L′ + c) if c << λ L′ = L + 0.3 d L′ d c 0.3 d

37. Physics 1251 Unit 3 Session 28 Clarinets et cetera Summary: • Reed Instruments are stopped pipes. • L′ = L + 0.3 d • f2n-1 = (2n-1) v/4L′ for stopped cylindrical pipes such as the clarinet. • fn = n v/ 2(L′+c) for stopped conical pipes such as the saxophone, oboe, bassoon, etc. • Soft reeds act as pressure valves that respond to the frequency fed back from the standing waves of the pipe.

38. f1 f2 f3 f4 ♩ ♪ ♫ fn ~ ~ Physics 1251 Unit 3 Session 29 Brass Instruments Brass Instruments • The lips produce a pulsation in the pressure admitted to the pipe; the pressure standing wave feeds back to control the oscillations of the plays lips. Lip-valve pulsations Standing wave frequencies Feedback

39. Physics 1251 Unit 3 Session 29 Brass Instruments The Lip Valve 80/20Brass instruments are played by the player’s lips. Breath pressure, muscle tension and pressure feedback from the pipe determine the frequency of the opening and closing of the lips. Louis Armstrong – trumpet (1901-1971)

40. Physics 1251 Unit 3 Session 29 Brass Instruments Lip Valve • The lips of the player act as a valve that admits pressure pulses into the pipe. • The frequency is determined by the breath air pressure, the lip tension and the resonances of the pipe.

41. Physics 1251 Unit 3 Session 29 Brass Instruments 80/20Brass Instruments are stopped pipes. • The player’s lips produce a displacement node (pressure antinode) at the mouthpiece. • A displacement anti-node (pressure node) exists at the bell. Winton Marsalis Trumpet

42. Physics 1251 Unit 3 Session 29 Brass Instruments The Mouthpiece Cup Volume 80/20The Cup Volume and the diameter of the constriction leading to the back bore are more important than the shape of the cavity. Diameter

43. Physics 1251 Unit 3 Session 29 Brass Instruments 80/20The Brass mouthpiece lowers the high frequency resonances. Resonance for Combination Pipes f Cone with mouthpiece Cone

44. Physics 1251 Unit 3 Session 29 Brass Instruments The pitch is changed by pipe length and excitation of resonances. By means of slides and valves the length is changed.

45. Pedal Tone Physics 1251 Unit 3 Session 29 Brass Instruments Resonance for Combination Pipes f Cone/ Cylinder % 0/100% 25/75% 40/60% 50/50% 20/80% 100/0%

46. Physics 1251 Unit 3 Session 29 Brass Instruments Resonances for Combination Bores in Brass Instruments 80/20A 50% cylindrical ‒ 50% conical bore has a nearly harmonic series.

47. x Physics 1251 Unit 3 Session 29 Brass Instruments Exponential Horn The Bell a = ao exp(m x)+ b 80/20m is called the “flare constant.” Larger m means more rapid flare.

48. a x Physics 1251 Unit 3 Session 29 Brass Instruments Bessel Horns The Bell a = ao e-(εx) +b 80/20Called “Bessel Horns” because the standing wave follows a Bessel Function.

49. Physics 1251 Unit 3 Session 29 Brass Instruments Summary: • Brass Instruments are stopped pipes. • The pipe bore is designed to give resonances that are harmonic. • The pedal tone (the lowest note) is not harmonic. • The player’s lips are a soft reed. • The pitch is changed by changing the length and exciting resonances.

50. Physics 1251 Unit 3 Session 30 The Timbre of Wind Instruments 1′ Lecture: • 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.