We will consider only members of the first two groups above in this section. These instruments will typically not produce harmonic overtones, though a few partials can often be ``tuned'' to approximate integer ratios of a fundamental.

**Vibrations of Bars:**

- Vibrations in bars are primarily one-dimensional.
- Longitudinal vibrations are harmonic with frequencies for , where is the speed of sound in the rod, is Young's elastic modulus, is the density, and is the length of the rod.
- Longitudinal vibrations are much higher in frequency than transverse vibrations.
- Transverse vibration frequencies for a bar free at both ends are , where is the radius of gyration (= thickness/3.46 for a flat bar) and . This results in modal frequency ratios of 1.0 : 2.76 : 5.40 : 8.90, etc.

**Rectangular Bars (The Glockenspiel):**

- The typical range of the glockenspiel is from (f = 784 Hz) to (f = 4186 Hz).
- All but the lowest transverse mode die out quickly when the bar is struck. Thus, we hear a clear ring at the frequency of the fundamental transverse mode.

**The Marimba, Xylophone, and Vibes:**

- The bars of the marimba, xylophone, and vibraphone are cut with an arch on their underside. This produces lower pitches for a given length (compared to the uniform bar) and also allows tuning of the partials.
- These instruments are outfitted with tubular resonators under each bar, tuned to the fundamental.
- The second partial of the marimba is tuned to about a two-octave interval (a ratio of 4.0). The third partial has a frequency around 9.2 times the fundamental (about three octaves plus a minor third above).
- The bars of the xylophone are less undercut and the first overtone is tuned to a twelfth (a ratio of 3.0).
- Vibraphones use deeply undercut aluminum bars with the first overtone tuned to a two-octave interval. The metal bars tend to have much longer decay times than the wood or synthetic bars of marimbas and xylophones.
- The most distinctive feature of vibraphones is the vibrato introduced by motor-driven discs at the top of the resonators.

**Chimes:**

- Tubular bells are fabricated from lengths of brass tubing. The upper end of each tube is partially or completely closed by a brass plug with a protruding rim.
- The strike tone one hears is based on modes 4, 5, and 6. The frequencies associated with these modes form rough ratios of 2:3:4, and thus cause our auditory system to perceive a fundamental at one octave below the 4th mode (though no such mode exists).

**Vibrations of Plates:**

- The restoring force in plates, like bars, results from the stiffness of the solid material.
- The overtones tend to be substantially higher than the fundamental.

**Cymbals:**

- Coupling between vibrational modes is very strong in cymbals, so that a large number of partials quickly appear in the spectrum when it is excited.
- Low-frequency energy is converted via a passive nonlinearity to high-frequency energy over several seconds after striking.

**Gongs and Tamtams:**

- Gongs are tuned to a definite pitch. When struck near the center with a soft mallet, their sound builds up slowly and continues for quite a while.
- Tamtams do not have the dome of the gong and have much less of a definite pitch.

**Bells and Carillons:**

- The strike tone is determined by partials with ratios 2:3:4 (the octave, twelfth, and double-octave), as in the chimes. Unlike chimes, however, the carillon bells have a prime mode at or near the strike tone (though the pitch of the strike tone is apparently more dependent on the upper partials).

**Vibrations of Membranes:**

- The vibrations of a membrane are primarily two-dimensional.
- Analytic solutions for various geometries (square, circular, elliptical) are possible and involve the solution of the two-dimensional wave equation.
- For a circular membrane, the modal frequencies are given by , where is the radius of the membrane, is the tension, is the area density, and is the th zero value of the th-order Bessel function.

**Timpani:**

- The mass of the air contained inside the kettle lowers the frequencies of the non-circularly concentric modes [(1,1), (2,1), (3,1), etc.].
- The stiffness of the air contained inside the kettle raises the frequencies of the circularly concentric modes [(0,1), (0,2), (0,3), etc.].
- The circularly concentric modes damp out rather quickly and do not contribute greatly to the timbre of the drum.
- The other modes produce frequencies nearly in the ratios 1 : 1.5 : 2 : 2.5, though a ``missing fundamental'' is not perceived.

**Bass Drum and Snare Drum:**

- These drums do not convey a definite pitch.
- The bass drum has many nearly harmonic modes in its low-frequency range, as in the timpani. However, the abundance of louder sounding non-harmonic modes above 200 Hz mask out any sense of pitch.

©1999 CCRMA, Stanford University. All Rights Reserved. Maintained by Gary P. Scavone, gary@ccrma.stanford.edu. |