Overtone Tracking in Music and Performance Systems
Overtone Tracking in Music and Performance Systems
Demonstration 2: Inharmonic Vibration with Time-Varying Overtone Amplitudes
Continuing, the NRI patent-pending technology (US Patent Application 10/676,926, published April 15, 2004 as Pub. No. 2004/0069128) additionally provides for individual real-time amplitude measurements of the fundamental and portions of the inharmonic overtone series of a pitched audio-frequency electrical signal and the use of these measurements to create real-time control signals. These real-time control signals may be used by associated internal subsystems, may be put in the form of outgoing control signals (such as MIDI, USB, serial, analog), or both.
These control signals may be used to control signal processing equipment, audio synthesis equipment, stage lighting systems, instrument lighting, or combinations of these in response to changes of timbre and amplitude of an audio or musical sound. The control signals can also be interpretted in other ways: for example, they may be used for the acoustic monitoring of moving machinery wherein changes of timbre signify component wear, lubrication problems, excessive loading, failing mounting, etc. An NRI whitepaper on the technology and example applications is available at www.nri-licensing.com\XXXXXX.
This NRI demonstration page illustrates an overtone series of inharmonically-related frequencies of a vibrating drumhead wherein the amplitude of each sinusoidal frequency component individually varies over time.
The vibration of circular membrane (drumhead) rigidly supended alone its edge and freely vibrating (i.e., not sealing an enclosed chamber of air) is given by (adapted from Morse [2]):
where c denotes the wave propagation constant of the membrane, t denotes time, r denotes radial distance from the center, 
denotes the circular angle with respect to a reference, and 
is the Bessel function of order
m
. If the membrane is fastened along a boundary circlar rim of radius
a
in a way such that it cannot vibrate, the allowed frequency values of 
are those that satisfy the boundary condition of there being no motion at the rim, i.e., the values of 
that satisfy
Since the Bessel functions of a given order have repeated zero crossings (as shown below),
for each value of 
there will be an entire associated set of values of 
that force the
m
th-order Bessel function to zero at radius
a
. These are the allowed frequencies of vibration. Using the notation of Morse [2] one can index these allowed values of the frequency as (x)
, where
, 
, 
, ..., are the sequence of solutions of
;
, 
, 
, ..., are the sequence of solutions of
,
etc. The lowest audible frequency 
is taken as the fundamental frequency of vibration.
In terms of this fundamental frequency, the results are
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None of these frequency ratios are integers or ratios of integers. As a result, unlike the vibrating string or air column, the overtones are inharmonic.
The animation following illustrates an overtone series of inharmonically-related frequencies of a vibrating drumhead wherein the amplitude of each sinusoidal frequency component individually varies over time. Such behavior represents an examplary impulse excitation of the drum head (for example, being struck by a hand, stick, mallet, etc. The fundamental and first eight overtones are shown. Although for the sake of illustration the vibration speed is greatly slowed, the ratios of frequencies are accurately depicted and the rates of amplitude decay are typical.
At the bottom of the animation, a three-dimensional bar graph array depicts the values of control signals that are extracted and made available for applications and/or further processing. An earlier page in this demonstration series (see links below) illustrates the technology applied to a harmonically-vibrating string or air column, and subsequent pages illustrate a few exemplary applications for these control signals.
REFERENCES
[1] U.S. Patent Application 10/676,926, published April 15, 2004 as Pub. No.: 2004/0069128.
[2] Morse, Philip M., Vibration and Sound, McGraw-Hill Book Company, Inc., New York, 1948.