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

3. Physics 1251Unit 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.

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

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

6. Physics 1251Unit 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,

7. Physics 1251Unit 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.

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

9. Physics 1251Unit 3 Session 33 Percussion 80/20 The timbre of an instrument’s sounds depends on its vibration recipe. Frequency Amplitude Amplitude f1f1f1f1 2f 1 3f 1 4f 1 f 01 f n = n f 1 f n m = x n m f 1 Pitched Unpitched

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

11. Physics 1251Unit 3 Session 33 Percussion Clamped Membrane vs String f n m = x n m /(π d) ‧ √(S/ σ) x 0 1 = 2.405 Surface Tension S Surface density σ= mass/area Surface density σ= mass/area Surface Tension S= force/length Surface Tension S= force/length Tension T Linear density μ Linear density μ= mass/length Linear density μ= mass/length Tension T= force Tension T= force f n = n /(2 L) ‧ √(T/ μ) n = 1, 2, 3, 4, 5, 6, 7…. d Surface density σ L

12. Physics 1251Unit 3 Session 33 Percussion The Oscillation of a Clamped Membrane Mode: (0,1) f 0 1 = v/λ; v = √(S/ σ) f 0 1 = x 0 1 /(π d) ‧ √(S/ σ) x 0 1 = 2.405 Surface Tension S Surface density σ Example: d = 0.30 m; m = 58 gm; T = 474 N C = π d =.94 m Area= 0.73 m 2 σ= 0.058 kg/0.073 m 2 =0.8 kg/m 2 S = T/C = 503. N/m f 0 1 = x 0 1 /(π d) ‧ √(S/ σ) = 2.405/(0.94)√(503/0.8)= 64 Hz d

13. Physics 1251Unit 3 Session 33 Percussion The Modes of Oscillation of an (Ideal) Clamped Membrane Mode: (0,1) f 0 1 = x 0 1 /(π d) ‧ √(S/ σ) x 0 1 = 2.405 Mode: (1,1) f 1 1 = ( x 1 1 / x 0 1 ) f 0 1 x 1 1 / x 0 1 = 1.594 Mode: (2,1) f 2 1 = ( x 2 1 / x 0 1 ) f 0 1 x 2 1 / x 0 1 = 2.136 Surface Tension S Surface density σ

14. Physics 1251Unit 3 Session 33 Percussion The Modes of Oscillation of a Clamped Membrane Mode: (0,1) x n m / x 0 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

15. Physics 1251Unit 3 Session 33 Percussion 80/20 Membrane Acoustics: The overtones of a circular membrane clamped at the edge are not harmonic and, therefore, they have no pitch. The overtones of a circular membrane clamped at the edge are not harmonic and, therefore, they have no pitch. f n m = ( x n m / x 01 )f 01 The frequencies f nm 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. The frequencies f nm 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.

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

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

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

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

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

21. Physics 1251Unit 3 Session 33 Percussion Longitudinal Wave (Sound Wave) Density ρ= mass/volume Density ρ= mass/volume Young’s Modulus E= stress/elongation =stiffness Young’s Modulus E= stress/elongation =stiffness v L = √E/ ρ v L = √E/ ρ h: thickness vLvLvLvL ρ: density E: Young’s Modulus f n = n v L /(2L)

22. Physics 1251Unit 3 Session 33 Percussion Bending Wave in a Plate Density ρ= mass/volume Density ρ= mass/volume Young’s Modulus E= stress/elongation =stiffness Young’s Modulus E= stress/elongation =stiffness v L = √E/(.91 ρ) v L = √E/(.91 ρ) v bend = √[1.8 f h v L ] h: thickness v bend ρ: density E: Young’s Modulus f nm = 0.0459 h v L ( y nm /d) 2

23. Physics 1251Unit 3 Session 33 Percussion The Modes of Oscillation of a Flat Cymbal Mode: (2,0) f n m / f 0 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

24. Physics 1251Unit 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. The overtones of a circular plate clamped in the center are not harmonic and, therefore, have no pitch. f n m = ( y n m / y 20 ) 2 f 20 The frequencies f nm 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/d 2. The frequencies f nm 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/d 2.

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