Physics 1251 The Science and Technology of Musical

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

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.

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.

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

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

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

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

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

Physics 1251 Unit 3 Session 33 Percussion Orchestral Percussion Tympani

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

Physics 1251 Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of an (Ideal) Clamped Membrane Surface density σ Mode: (0, 1) f 0 1 = 0. 7655/ d ‧ √(S/ σ) Surface Tension S Mode: (1, 1) Mode: (2, 1) f 1 1 = 1. 594 f 0 1 f 2 1 = 2. 136 f 0 1

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

Physics 1251 Unit 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

Physics 1251 Unit 3 Session 34 Percussion with Pitch The Modes of Oscillation of Tympani Strike point Mode: (0, 1) (1, 1)1. (2, 1)2. (0, 2)2. (3, 1)2. fn m/f 01 : 1 594 136 296 653 (1, 2)2. (4, 1)3. (2, 2)3. (0, 3)3. (5, 1)3. 918 156 501 600 652

Physics 1251 Unit 3 Session 34 Percussion with Pitch 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 80/20 Tympani f 0 (0, 1) 2 f 0 3 f 0 (1, 1) 4 f 0 5 f 0 6 f 0 (2, 1) (3, 1) (4, 1) (0, 3) (0, 2) (1, 2) (2, 2) (0, 2) (1, 2) Frequency (3, 2) (5, 1)

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

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

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

Physics 1251 Unit 3 Session 33 Percussion Bending Wave in a Bar vbend h: thickness ρ: density E: Young’s Modulus • Density ρ= mass/volume • Young’s Modulus stress/elongation =stiffness • v. L = √E/ ρ E= Longitudinal Wave Velocity fnm = ynm h v. L /L 2

Physics 1251 Unit 3 Session 34 Percussion with Pitch Bending Modes in Bars: End Clamped f 1= 0. 1782 fo f 2= 1. 116 fo f 3=3. 125 fo

Physics 1251 Unit 3 Session 34 Percussion with Pitch Bending Modes in Bars: Free Ends f 1= 1. 133 fo. 224 L f 2= 3. 125 fo f 3=6. 125 fo

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

Physics 1251 Unit 3 Session 34 Percussion with Pitch Orchestral Chimes Free Ends End Plug f 1= 1. 133 fo f 2= 3. 125 fo f 3=6. 125 fo

Physics 1251 Unit 3 Session 34 Percussion with Pitch Marimba

Physics 1251 Unit 3 Session 34 Percussion with Pitch Mode Frequencies in Undercut Bar: Undercut Bar in Xylophone, Marimba and Vibraphone Xylophone f 1/f 1 = 1. 00 f 2 /f 1 = 3. 00 f 3 /f 1 =6. 1 λ/4 Marimba/Vibes f 1 /f 1 = 1. 00 f 2 /f 1 = 4. 00 f 3 /f 1 =6. 5 Vibraphone

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 3 x the fundamental, for a marimba and “vibe” it is 4 x. The xylophone sounds “brighter” and the marimba more “mellow. ” Vibes have a tremolo mechanism.

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.