Macintosh and MIDI
By Robbie Rhodes
Scientists use these definitions of resonant systems:
Fundamental: the lowest natural resonant frequency
Overtones: the remaining frequencies above the fundamental
Harmonics: Overtones which are an integral multiple of the fundamental frequency (2,3,4, etc).
I think in the piano trade "Partials" is used like "Overtones".
An air column resonator, as in an organ pipe, has overtones which are truly harmonic, due to the coupling between the modes. Wurlitzer "violin" pipes featured the "Gavioli Frein", a blade of brass intercepting the wind sheet near the mouth of the pipe. ("Frein" is French for "brake".) When the little frein is properly adjusted the pipe will always produce the fundamental tone, and increased pressure just enriches the harmonics. Without the frein the pipe would cease oscillating at the fundamental. Another open pipe design intentionally suppresses the fundamental because of a tiny hole half-way up the resonator. (David Wasson, can you elaborate on these statements?)
The vibrating string in the piano has non-harmonic overtones. When the hammer strikes the string all of the possible modes are excited. There is inadequate coupling to synchronize the different frequencies, and so they are "out-of-tune" with each other. Moreover, the sound amplitude of each overtone can decay differently from all the others.
A common experiment in the college physics lab uses a thin string of music wire stretched across a heavy cast-iron base, as in a piano. The students measure the frequency of each of the resonance modes, aided by an oscilloscope. The instructor waits patiently for the question, "Why aren't the overtones harmonic?" Then he explains that the cast-iron frame, seemingly stationary, is also vibrating with the string, especially at the higher frequencies. The result of this undesirable motion is that the overtones resonate at frequencies a little less than an integral harmonic. In radio antennas this is called the "end effect".
Scientists "build a model" in order to validate a theory. The mathematical model of the vibration of a struck piano string is quite complex -- but the more complex it is, the better it approximates the real-world action of the piano. By studying the model (and perhaps using it as the basis of a computer simulation) scientists have been able to achieve better pianos, and faster than if only "cut and try" experiments were employed.
I suspect that the strides in piano technology made by Steinway & Sons a century ago were due to both analytical studies and empirical trials. _Scientific American_ magazine frequently publishes interesting articles, adapted from the scientific tomes, about musical instrument science and technology.
Observed natural phenomena may be explained using a mathematical model, but there is surely nothing wrong in expressing the conclusions in the everyday words of the musical world. Thus the "warm tone" of a given piano or violin is explained quantitatively by the mathematical model, which shows that certain overtones dominate, or whatever. ...
Aren't pianos wonderful? The sound of the violin may tug at your heart, but the hammered piano hits you in the belly!
-- Robbie Rhodes <rrhodes@foxtail.com> |
(Message sent Sun 5 May 1996, 01:05:58 GMT, from time zone GMT-0700.) |
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