PHY-2464 Physical Basis of Music

Transcription

1 Physical Basis of Music Presentation 21 Percussion Instruments II Adapted from Sam Matteson s Unit 3 Session 34 Sam Trickey Mar. 27, 2005 Percussion = striking Percussion instruments divide nicely into those that lack a well-defined pitch (overtones not related to harmonics of the fundamental) - Snare drums, bass drums, wood blocks, castanets, cymbals those with well-defined pitch(es es) - Tympani, marimba, xylophone, carillon bells, steel drums

2 Example of percussion without 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. Something to recall Piano strings exhibit inharmonicity because of the stiffness of the wire. [Hall, Chapter 10, p. 197] 2 ( ) f 1 1 n nf 1 + n J J must be small! J = 0.02 will raise 2 nd mode frequency of a string by a semitone.

3 The Percussion Instruments (*already discussed) Strings* Membranes Drums Piano* Hammer dulcimer* Plates Bars Blocks, bells, shells Cymbals, Gongs, Pans Xylophones, chimes Others Fact: Production of pitch in a percussion instrument is an exercise in manipulating the struck-object overtones into a (approximately) harmonic series. Amplitude f 01 f n m = x n m f 01 Unpitched Amplitude f 1 2f 1 f n = n f 1 Pitched 3f 1 4f 1 Frequency

4 Recall: : The Modes of vibration of an ideal string are harmonic. Tension T L Linear density µ Linear density µ= = mass/length Tension T= = force f n = n /(2 2 L) x (T/ µ) n = 1, 2, 3, 4, 5, 6, 7. 7 The stiffness of the wire increases the frequency of the higher frequency harmonics. = 3986 Log(nf 1 /440) + I( ) I( ) ) = Inharmonicity Recall from Unit 8: Pitch interval for two frequencies f α, f β Interval in cents = I ( )( = log (f α /f β ) So = Log(nf 1 /440) + I( ) and I( ) = Inharmonicity & the value of n Give a determination of how far a given interval in an inharmonic spectrum is from harmonic

5 Inharmonicity of Piano Inharmonicity Pitch ( )( Because of the inharmonicity of strings, the octaves are stretched in a piano [Hall, pp ] Tympani and Tabla

6 Orchestral Percussion Tympani Tympani are tuned by adjusting the tension on the head. Tension device Tension pedal Tympanum: Greek drum

7 Reminder: Oscillation Modes of an (Ideal) Clamped Membrane Surface density σ Mode: (0,1) Surface Tension S f 0 1 = / d x (S/ σ) Mode: (1,1) f = f 0 1 Mode: (2,1) 0 1 f = f 0 1 F x,y,y:: x = # radial nodes, y = # circular nodes Reminder: : Oscillation Modes of Ideal Clamped Membrane F x,y,y:: x = # radial nodes, y = # circular nodes Mode: (0,1) x n m / x 0 1 : 1 (1,1) (2,1) (0,2) (3,1) (1,2) (4,1) (2,2) (0,3) (5,1) 3.652

8 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, thus lowers the working frequency. Air mass Key Fact: Tympani kettles modify the membrane frequencies by the interaction of the air resonances with the surface modes. Modes of air vibration

9 Modes of Oscillation of Tympani Strike point Mode: (0,1) n m /f 01 : 1 f n m /f 01 (1,1) (2,1) (0,2) (3,1) (1,2) (4,1) (2,2) (0,3) (5,1) Key Fact: Tympani achieve pitch by (1) suppression of radial modes; (2) modification of other mode frequencies by both air loading and the effect of the kettle ; (3) attenuation of the lowest mode. Amplitude f 0 2f 0 (0,1) 3f 0 4f 0 (1,1) (2,1) (3,1) (4,1) (0,3) (0,2) (1,2) (2,2) Frequency 5f 0 6f 0 (3,2) (5,1)

10 Steelpans (steel drums): Mid-20 th century metalophone innovation. Scribed areas on the end plate of a 55 gallon drum are tensioned to tuned notes by hammering & heating Note anti- clockwise circle of 5ths on outer two rings Figure from Rossing,, Moore, and Wheeler The Science of Sound 3 rd Ed. Steelpans (steel drums): Typical playable ranges Figure from Rossing,, Moore, and Wheeler The Science of Sound 3 rd Ed.

11 Steelpans (steel drums): Modes of vibration Figure from Rossing,, Moore, and Wheeler The Science of Sound 3 rd Ed. Steelpans (steel drums): Spectra G 3 (196 Hz), C # 4 (277 Hz) Figure from Rossing,, Moore, and Wheeler The Science of Sound 3 rd Ed.

12 Traditional metalophones: : Glockenspiels, Xylophones, Marimbas, & Vibraphones Xylo: : wood Phone: sound Metalophones: Glockenspiels, Xylophones and Marimbas Bar h = thickness L = Length w = width Density ρ = mass/volume Young s s Modulus Y = Force/elongation

13 Metalophones: Glockenspiels, Xylophones and Marimbas Longitudinal Waves in a Bar Anti-node node node Anti-node node v L = Y/ ρ Longitudinal Wave Velocity f n = n/2l Y/ ρ like an open pipe Density ρ = mass/volume Young s s Modulus Y= Stress/Elongation Reminder: Bending Wave in a Plate v bend h: thickness Y: Young s Modulus Density ρ= = mass/volume Young s s Modulus Y= = stress/elongation =stiffness v L = Y/(.91 ρ) ρ: density v bend = [1.8 f h v L ] f nm = h v L ( y nm /d) 2

14 Bending Modes in Bars: End Clamped f 1 = f o f 2 = f o f 3 =3.125 f o Bending Modes in Bars: Free Ends f 1 = f o f 2 = f o L f 3 =6.125 f o

15 Glockenspiel, Orchestra Bells: Orchestral Chimes Free Ends End Plug f 1 = f o f 2 = f o f 3 =6.125 f o

16 Marimba 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

17 What are the differences among a Xylophone, Marimba, and Vibraphone? The depth of the undercut: a marimba is undercut more than a xylophone. The first harmonic of a xylophone is 3x3 the fundamental; for a marimba and vibe 1 st harmonic is 4x 4 the fundamental. The xylophone sounds brighter and the marimba more mellow. mellow. Vibes have a tremolo mechanism. Carillons, Chimes, handbells Carillon by definition is 23 or more bells played from a keyboard (less than 23 is a Chime). Strike Note: determined by Octave, Twelth,, & Upper Octave (2:3:4 frequency ratio), not the Prime nor Hum Handbells: : pp Hall Figure from Rossing,, Moore, and Wheeler The Science of Sound 3 rd Ed.

18 Summary: Piano strings exhibit inharmonicity because of wire stiffness. Some percussion instruments have pitch. Pitch results from a harmonic series of overtones. Tympani, Tabla,, & Steelpans are pitched drums Orchestra Chimes, Glockenspiel, Xylophone, Marimba and Vibraphone have intonation. Marimba are undercut more than xylophones. Carillon, chimes, and handbells are tuned, deformed plates.