2. Physics 1251 The Science and Technology of Musical Sound Unit 3 Session 34 MWF Percussion with Pitch Unit 3 Session 34 MWF Percussion with Pitch

3. Physics 1251Unit 3 Session 34 Percussion with Pitch A percussionist has two nearly identical cymbals. They have identical fundamental frequencies, but one is 15 inches in diameter while the other is 14 inches in diameter. What must be true about the two? The larger cymbal must be about 15% thicker than the smaller one, since the frequency is proportional to the thickness and inversely proportional to the square of the diameter.

4. Physics 1251Unit 3 Session 34 Percussion with Pitch 1′ Lecture: Piano strings exhibit inharmonicity because of the stiffness of the wire. Piano strings exhibit inharmonicity because of the stiffness of the wire. Some percussion instruments have pitch. Some percussion instruments have pitch. Pitch results from a harmonic series of overtones. Pitch results from a harmonic series of overtones. Tympani and Tabla are pitched drums. Tympani and Tabla are pitched drums. Orchestra Chimes, Glockenspiel, Xylophone, Marimba and Vibraphone have intonation. Orchestra Chimes, Glockenspiel, Xylophone, Marimba and Vibraphone have intonation.

5. Physics 1251Unit 3 Session 34 Percussion with Pitch The Percussion Instruments Percussion – striking Piano Hammer dulcimer Cymbals, Gongs, Pans Xylophones, chimes Others Drums Membranes Blocks, bells, shells Plates Bars Strings

6. Physics 1251Unit 3 Session 34 Percussion with Pitch 80/20 The task of producing pitch in a percussion instrument is an exercise in manipulating the overtones into a harmonic series. Frequency Amplitude f 01 f n m = x n m f 10 Unpitched Amplitude f1f1f1f1 2f 1 3f 1 4f 1 f n = n f 1 Pitched

7. Physics 1251Unit 3 Session 33 Percussion The Modes of vibration of an ideal string are harmonic. 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…. Tension T Linear density μ L The stiffness of the wire increases the frequency of the higher frequency harmonics. ₧ = 3986¢ Log(nf 1 /440) + I (₧) I (₧) = Inharmonicity

8. Physics 1251Unit 3 Session 34 Percussion with Pitch Inharmonicity of Piano Pitch (¢) Inharmonicity Because of the inharmonicity of strings the octaves are “stretched” in a piano. 20¢ 40¢ -20¢

9. Physics 1251Unit 3 Session 34 Percussion with Pitch Tympani and Tabla

10. Physics 1251Unit 3 Session 33 Percussion Orchestral Percussion Tympani

11. Physics 1251Unit 3 Session 33 Percussion Tympani are tuned by adjusting the tension on the head. Tension device Tension pedal

12. Physics 1251Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of an (Ideal) Clamped Membrane Mode: (0,1) f 0 1 = 0.7655/ d ‧ √(S/ σ) Mode: (1,1) f 1 1 = 1.594 f 0 1 Mode: (2,1) f 2 1 = 2.136 f 0 1 Surface Tension S Surface density σ

13. Physics 1251Unit 3 Session 34 Percussion with Pitch Air Loading of a Clamped Membrane Surface Tension S Surface density σ Air mass The mass of air moved by the membrane adds to the effective surface density, lowering the frequency.

14. Physics 1251Unit 3 Session 34 Percussion with Pitch 80/20 The kettle of Tympani modifies the membrane frequencies by the interaction of the air resonances with the surface modes. Modes of air vibration

15. Physics 1251Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of Tympani Mode: (0,1) f n m /f 01 : 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 Strike point

16. Physics 1251Unit 3 Session 34 Percussion with Pitch 80/20 Tympani achieve pitch by (1) suppression of “radial” modes; (2) modification of other mode frequencies by air loading and the effect of the kettle ; (3) attenuation of the lowest mode. Frequency Amplitude(0,1)(1,1)(2,1) (0,2) (3,1) (1,2) (4,1) (2,2) (0,3) (5,1) (3,2)(0,1)(1,1)(2,1) (0,2) (3,1) (1,2) (4,1) (2,2) (0,3) (5,1) (3,2) 5f 0 2f 0 3f 0 4f 0 6f 0 f0f0f0f0

17. Physics 1251Unit 3 Session 34 Percussion with Pitch Metalophones: Glockenspiels, Xylophones, Marimbas and Vibes Xylo : wood Phone : sound

18. Physics 1251Unit 3 Session 34 Percussion with Pitch Metalophones: Glockenspiels, Xylophones and Marimbas h thickness L Length Density ρ = mass/volume Young’s Modulus E= Force/elongation Bar w width

19. Physics 1251Unit 3 Session 34 Percussion with Pitch Metalophones: Glockenspiels, Xylophones and Marimbas v L = √E/ ρ Longitudinal Wave Velocity Density ρ = mass/volume Young’s Modulus E= Stress/Elongation Longitudinal Waves in a Bar f n = n/2L√E/ ρ like an open pipe node Anti-node Anti-node

20. Physics 1251Unit 3 Session 33 Percussion Bending Wave in a Bar 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 Longitudinal Wave Velocity h: thickness v bend ρ: density E: Young’s Modulus f nm = y nm h v L /L 2

21. Physics 1251Unit 3 Session 34 Percussion with Pitch Bending Modes in Bars: f 1 = 0.1782 f o f 2 = 1.116 f o f 3 =3.125 f o End Clamped

22. Physics 1251Unit 3 Session 34 Percussion with Pitch Bending Modes in Bars: f 1 = 1.133 f o f 2 = 3.125 f o f 3 =6.125 f o.224 L Free Ends

23. Physics 1251Unit 3 Session 34 Percussion with Pitch Glockenspiel, Orchestra Bells:

24. Physics 1251Unit 3 Session 34 Percussion with Pitch Orchestral Chimes f 1 = 1.133 f o f 2 = 3.125 f o f 3 =6.125 f o Free Ends End Plug

25. Physics 1251Unit 3 Session 34 Percussion with Pitch Marimba

26. Mode Frequencies in Undercut Bar: Xylophone f 1 /f 1 = 1.00 f 2 /f 1 = 3.00 f 3 /f 1 =6.1 Undercut Bar in Xylophone, Marimba and Vibraphone Marimba/Vibes f 1 /f 1 = 1.00 f 2 /f 1 = 4.00 f 3 /f 1 =6.5 λ/4 Vibraphone

27. Physics 1251Unit 3 Session 34 Percussion with Pitch What is the different between a Xylophone, a Marimba and a Vibraphone? The depth of the undercut: a marimba is undercut more than a xylophone. The depth of the undercut: a marimba is undercut more than a xylophone. The first harmonic of a xylophone is 3x the fundamental, for a marimba and “vibe” it is 4x. The first harmonic of a xylophone is 3x the fundamental, for a marimba and “vibe” it is 4x. The xylophone sounds “brighter” and the marimba more “mellow.” The xylophone sounds “brighter” and the marimba more “mellow.” Vibes have a tremolo mechanism. Vibes have a tremolo mechanism.

28. Physics 1251Unit 3 Session 34 Percussion with Pitch Summary: Piano strings exhibit inharmonicity because of the stiffness of the wire. Piano strings exhibit inharmonicity because of the stiffness of the wire. Some percussion instruments have pitch. Some percussion instruments have pitch. Pitch results from a harmonic series of overtones. Pitch results from a harmonic series of overtones. Tympani and Tabla are pitched drums. Tympani and Tabla are pitched drums. Orchestra Chimes, Glockenspiel, Xylophone, Marimba and Vibraphone have intonation. Orchestra Chimes, Glockenspiel, Xylophone, Marimba and Vibraphone have intonation. Marimba are undercut more than xylophones. Marimba are undercut more than xylophones.