Physics 1251 The Science and Technology of Musical Sound

1. Physics 1251The Science and Technology of Musical Sound Unit 3 Session 33 MWF Percussion Instruments

2. Physics 1251 Unit 3 Session 33 Percussion In the video (Amadeus), why did the soprano smile so broadly when she sang the high notes? She was deliberately raising the pitch of the formant in order to match the pitch of the notes she was singing by changing the shape of her vocal tract.

3. Physics 1251 Unit 3 Session 33 Percussion Q&A: • Throat Singers? • Dogs barking? • Smoking? • Whistling? • Hyoid bones?

4. Physics 1251 Unit 3 Session 33 Percussion 1′ Lecture: • Percussion instruments are instruments that are struck. • The timbre of their sound is determined by their vibration recipe. • Their vibration recipe is determined by the modes of oscillation that are excited. • Often percussion instruments do not have pitch.

5. Physics 1251 Unit 3 Session 33 Percussion The Rain Stick By Seamus Heaney Upend the rain stick and what happens next Is a music that you never would have known To listen for. In a cactus stalk Downpour, sluice-rush, spillage and backwash Come flowing through. You stand there like a pipe Being played by water, you shake it again lightly And diminuendo runs through all its scales Like a gutter stopping trickling. And now here comes A sprinkle of drops out of the freshened leaves,

6. Physics 1251 Unit 3 Session 33 Percussion The Rain Stick By Seamus Heaney Then subtle little wets off grass and daises; Then glitter-drizzle, almost-breaths of air. Upend the stick again. What happens next Is undiminished for having happened once, Twice, ten, a thousand times before. Who cares if all the music that transpires Is fall of grit or dry seeds through cactus? You are like a rich man entering heaven Through the ear of a raindrop. Listen now again. From The Spirit Level (New York: Noonday Press, 1996) p3.

7. Physics 1251 Unit 3 Session 33 Percussion The Percussion Instruments Strings Membranes Drums Piano Hammer dulcimer Percussion – striking Blocks, bells, shells Bars Plates Cymbals, Gongs, Pans Others Xylophones, chimes

8. Physics 1251 Unit 3 Session 33 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

9. Surface density σ Surface Tension S Physics 1251 Unit 3 Session 33 Percussion The Oscillation of a Clamped Membrane Mode: (0,1) d f0 1 = v/λ; v = √(S/ σ) f0 1 = x0 1 /(π d) ‧ √(S/ σ) x0 1 = 2.405 Surface density σ= mass/area σ= density ‧ thickness Surface Tension S= force/length

10. Surface density σ Surface Tension S Physics 1251 Unit 3 Session 33 Percussion Tension T Clamped Membrane vs String L d Linear density μ • Linear density μ= mass/length • Surface density σ= mass/area • Tension T= force • Surface Tension S= force/length fn = n /(2 L) ‧ √(T/ μ) n= 1, 2, 3, 4, 5, 6, 7…. fn m = xn m /(π d) ‧ √(S/ σ) x0 1 = 2.405

11. Surface density σ Surface Tension S Physics 1251 Unit 3 Session 33 Percussion Mode: (0,1) The Oscillation of a Clamped Membrane f0 1 = v/λ; v = √(S/ σ) f0 1 = x0 1 /(π d) ‧ √(S/ σ) x0 1 = 2.405 d Example: d = 0.30 m; m = 58 gm; T = 474 N• C = π d = .94 m • Area= 0.73 m2• σ= 0.058 kg/0.073 m2=0.8 kg/m2 • S = T/C = 503. N/m f0 1 = x0 1 /(π d) ‧ √(S/ σ) = 2.405/(0.94)√(503/0.8)= 64 Hz

12. Surface density σ Surface Tension S Physics 1251 Unit 3 Session 33 Percussion The Modes of Oscillation of an (Ideal) Clamped Membrane Mode: (0,1) f0 1 = x0 1 /(π d) ‧ √(S/ σ) x0 1 = 2.405 Mode: (1,1) Mode: (2,1) f1 1 = (x1 1 / x0 1) f0 1 x1 1 / x0 1 = 1.594 f2 1 = (x2 1 / x0 1) f0 1 x2 1 / x0 1 = 2.136

13. Physics 1251 Unit 3 Session 33 Percussion The Modes of Oscillation of a Clamped Membrane Mode: (0,1) xn m / x0 1 : 1 (1,1)1.594 (2,1)2.136 (0,2)2.296 (3,1)2.653 (1,2)2.918 (4,1)3.156 (2,2)3.501 (0,3)3.600 (5,1)3.652

14. Physics 1251 Unit 3 Session 33 Percussion 80/20Membrane Acoustics: • The overtones of a circular membrane clamped at the edge are not harmonic and, therefore, they have no pitch. fn m = (xn m /x01)f01 • The frequencies fnm of a membrane are (1) proportional to the square root of the ratio of surface tension of the head to the surface density ∝√(S / σ) and (2) inversely proportional to its diameter ∝1/d.

15. Physics 1251 Unit 3 Session 33 Percussion Demonstration: Normal Modes of a Oscillation of a Clamped Membrane

16. Physics 1251 Unit 3 Session 33 Percussion Ideal vs Real Membranes: 80/20Real membranes have a lower frequencies than predicted for ideal membranes because of air loading; the lowest frequencies are lowered the most.

17. Physics 1251 Unit 3 Session 33 Percussion Mode Excitation: 80/20Only those frequencies for which the modes were excited will appear in the vibration recipe. 80/20The highest frequency that can be excited by a mallet that is in contact with the surface for a period of Tcontact is f max= 2/Tcontact

18. Physics 1251 Unit 3 Session 33 Percussion Mode Excitation: 80/20The highest frequency that can be excited by a mallet that is in contact with the surface for a period of Tcontact is f max= 2/Tcontact Tcontact = ½ Tperiod= 1/(2fmax )

19. Physics 1251 Unit 3 Session 33 Percussion Demonstration: Longitudinal Modes vs Transverse Modes for a Rod

20. Physics 1251 Unit 3 Session 33 Percussion vL h: thickness Longitudinal Wave (Sound Wave) ρ: density E: Young’s Modulus • Density ρ= mass/volume • vL = √E/ ρ • Young’s ModulusE= stress/elongation =stiffness fn= n vL/(2L)

21. Physics 1251 Unit 3 Session 33 Percussion vbend h: thickness Bending Wave in a Plate ρ: density E: Young’s Modulus • Density ρ= mass/volume • vL = √E/(.91 ρ) vbend = √[1.8 f h vL ] • Young’s ModulusE= stress/elongation =stiffness fnm = 0.0459 h vL( ynm /d)2

22. Physics 1251 Unit 3 Session 33 Percussion The Modes of Oscillation of a Flat Cymbal Mode: (2,0) fn m / f0 1 : 1 (0,1)1.730 (3,0)2.328 (1,1)3.910 (4,0)4.110 (5,0)6.30 (2,1)6.71 (0,2)3.600

23. Physics 1251 Unit 3 Session 33 Percussion 80/20 Plate Acoustics: • The overtones of a circular plate clamped in the center are not harmonic and, therefore, have no pitch. fn m = (yn m /y20)2 f20 • The frequencies fnm of a circular plate are (1) proportional to the thickness ∝h and (2) to the square root of the ratio of the stiffness and the density ∝√E/ρ and (3) inversely proportional to the square of the diameter ∝1/d2 .

24. Physics 1251 Unit 3 Session 33 Percussion Summary: • Percussion instruments are instruments that are struck. • Their vibration recipe is often not harmonic and, therefore, they do not have a definite pitch. • For ideal circular edge-clamped membranes: fnm ∝(xnm /d)√(S/σ). • For circular plates free at the edge: fnm ∝h ‧ (ynm /d) 2 √(E/ρ). • The maximum frequency excited by a mallet is f max= 2/Tcontact.