Physics 1251 The Science and Technology of Musical Sound

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

2. Physics 1251 Unit 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.

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

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

5. fn = n f1 Pitched Amplitude f1 2f1 3f1 4f1 fn m = xn m f10 Unpitched Amplitude f01 Physics 1251 Unit 3 Session 34 Percussion with Pitch 80/20The task of producing pitch in a percussion instrument is an exercise in manipulating the overtones into a harmonic series. Frequency

6. Tension T L Linear density μ Physics 1251 Unit 3 Session 33 Percussion The Modes of vibration of an ideal string are harmonic. • Linear density μ= mass/length The stiffness of the wire increases the frequency of the higher frequency harmonics. • Tension T= force fn = n /(2 L) ‧ √(T/ μ) n= 1, 2, 3, 4, 5, 6, 7…. ₧ = 3986¢ Log(nf1 /440) + I(₧) I(₧) = Inharmonicity

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

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

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

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

11. Surface density σ Surface Tension S Physics 1251 Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of an (Ideal) Clamped Membrane Mode: (0,1) f0 1 = 0.7655/ d ‧ √(S/ σ) Mode: (1,1) Mode: (2,1) f1 1 = 1.594 f0 1 f2 1 = 2.136 f0 1

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

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

14. Strike point Physics 1251 Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of Tympani Mode: (0,1) fn m/f01 : 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. 2f0 3f0 4f0 5f0 6f0 (0,1) f0 (1,1) (2,1) (3,1) (4,1) (0,3) (3,2) (2,2) (5,1) (0,2) (1,2) (0,1) (1,1) (2,1) (3,1) (4,1) (0,3) (3,2) (5,1) (0,2) (1,2) (2,2) Physics 1251 Unit 3 Session 34 Percussion with Pitch 80/20Tympani 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. Amplitude Frequency

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

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

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

19. Physics 1251 Unit 3 Session 33 Percussion vbend h: thickness Bending Wave in a Bar ρ: density E: Young’s Modulus • Density ρ= mass/volume • vL = √E/ ρ Longitudinal Wave Velocity • Young’s ModulusE= stress/elongation =stiffness fnm = ynm h vL/L2

20. End Clamped f1= 0.1782 fo f2= 1.116 fo f3=3.125 fo Physics 1251 Unit 3 Session 34 Percussion with Pitch Bending Modes in Bars:

21. Physics 1251 Unit 3 Session 34 Percussion with Pitch Free Ends Bending Modes in Bars: f1= 1.133 fo f2= 3.125 fo f3=6.125 fo .224 L

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

23. Physics 1251 Unit 3 Session 34 Percussion with Pitch Free Ends Orchestral Chimes End Plug f1= 1.133 fo f2= 3.125 fo f3=6.125 fo

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

25. λ/4 Vibraphone Physics 1251 Unit 3 Session 34 Percussion with Pitch Mode Frequencies in Undercut Bar: Undercut Bar in Xylophone, Marimba and Vibraphone Xylophone f1/f1 = 1.00f2 /f1 = 3.00f3 /f1 =6.1 Marimba/Vibes f1 /f1 = 1.00f2 /f1 = 4.00f3 /f1 =6.5

26. Physics 1251 Unit 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 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.” • Vibes have a tremolo mechanism.

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