Overtone Tracking in Music and Performance Systems
Overtone Tracking in Music and Performance Systems
Demonstration 1: Harmonic Vibration with Time-Varying Overtone Amplitudes
In an NRI patent-pending technology (US Patent Application 10/676,926, published April 15, 2004 as Pub. No. 2004/0069128), individual real-time amplitude measurements of the fundamental and portions of the overtone series of a pitched audio-frequency electrical signal are used to create control signals. These signals may be used by associated internal subsystems, may be put in the forms 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\blah-blah-blah.
This NRI demonstration page illustrates an overtone series of a vibrating string or air column wherein the amplitude of each sinusoidal frequency component individually varies over time. The fundamental and first four overtones are included, and the composite sum of these five amplitude-varying sinusoidal frequency components is also shown. Although for the sake of illustration the vibration speed is greatly slowed, the harmonic ratios of frequencies are accurately depicted and the rates of amplitude decay are typical. At the bottom of the animation, a bar graph depicts the values of control signals that are extracted and made available for applications and/or further processing. A subsequent page in this demonstration series (see links below) illustrates the technology applied to an inharmonically-vibrating drumhead, and a number of subsequent pages illustrate a few exemplary applications for these control signals.
Harmonic Overtone Example: Vibrating Strings and Air Columns |
|
 |
|
Fund |
OT 1 |
OT 2 |
OT 3 |
OT 4 |
Harm 1 |
Harm 2 |
Harm 3 |
Harm 4 |
Harm 5 |
Such a harmonic overtone series may be created by a vibrating string under tension, a vibrating reed, a vibrating air column (as in an organ pipe, flute, or recorder), etc. In the discussion below, a plucked vibrating string will be considered in detail. The behavior of a bowed-string is not dissimilar, and other harmonically-vibrating examples (reeds, arir columns, etc.) differ in some aspects but in many ways exhibit closely related behavior.
As an example, consider the case where a string is plucked (by extension and release) at a point one-fourth of its length. The drawing and analysis leading to the immediately following Fourier coefficents is adapted from Swenson, Principles of Modern Acoustics [2].
Swenson applies the standard separation of variables analysis to the differential equation for the string and gives the vibrational solution to the string equation with this initial excitation as:
which, as Swenson points out, does not include losses due to radiating sound energy. This simple model also does not include the effects of resonances, effects of nonlinearities in the string mechanics, the sympathetic acoustic effects of electronic amplification, stimulation by a bow, mechanical transducer or acoustic-frequency magenetic field, etc. These important effects will be considered en masse shortly.
For an initial study, it is helpful to reorganize the terms so as to apply some useful interpretations. The following figure shows an example reorganize of terms and an associated interpretation.
Adding to the simple model of Swenson, time-varying decay can be introduced. If the system is assumed to be linear, the theory of linear differential equations (with, as one has here, no repeated eigenvalues, i.e., no non-degenerate solutions), the time-varying decay occurs in the form of exponential damping. Admitting that each frequency likely has unique details in its energy loss mechanism, a separate exponential damping can be provided for each term.
One can lump together the pluck-dependent scalling
and the variational part of the roll-off term into a harmonic-dependent scaling
term 
; one can also lump together
the pluck amplitude, exponential decay, and constant terms to form a time-varying
harmonic-dependent amplitude term 
.
These associated groupings are shown below.
In this representation, a number of sins of earlier ommision ( resonances, effects of nonlinearities in the string mechanics, the sympathetic acoustic effects of electronic amplification, stimulation by a bow, mechanical transducer or acoustic-frequency magenetic field, etc. ) can be abstractly folded into the time-varying harmonic-dependent amplitude term 
(and, in some cases, to the harmonic-dependent scaling term 
as well)
. Acoustic resonances can change the value of the decay constants 
. More profoundly, the sympathetic acoustic effects of electronic amplification, stimulation by a bow, mechanical transducer or acoustic-frequency magenetic field, and certain types of nonlinearities can introduce a wide range of additional behaviours into the time-varying harmonic-dependent amplitude term 
(as well as slightly shift the frequencies of the overtone series), including increasing trends and non-exponential decays. Further, one can expect the harmonic-dependent scaling term 
, even in the initial simple model, to change form and values with variation in where the string is plucked or bowed as is well know and used by musicians for basic timbre means of musical expression. Specifically, generalizing the pluck-location aspect of the simplified model of Swenson, a more general pluck location scenario is depicted below.
To simplify the math, the string is assumed to be one unit in length (rather than L units as it was earlier), so that 
and 
. The equation for this piecewise-constant curve shown in the above drawing is:
for 
for 
The Fourier coefficients for this piecewise-constant curve in the interval [0,1] is:
Since 
for non-negative integer
n
, this simplifies to
Thus the terms in this more general Fourier expansion may be interpretted as show below.
The overtone detection technology effectively in U.S. Patent Application 10/676,926 [1] detects the time-varying harmonic dependent product 
for selected overtones and produces an initial signal corresponding to the value of this time-varying harmonic dependent product for a number of selected components of the overtone series (i.e., for a selected number of values of
n
). These can be subsequently processed in a wide variety of ways as described in US Patent Application 10/676,926 [1]. The processing can include individual normalization, sorting into odd-harmonic and even-harmonic energy measurments, detection of the introduction of artificial nodes suppressing specific harmonic families in the overtone series, etc.
The next page in this demonstration series (see link below) illustrates the technology applied to an inharmonically-vibrating drumhead. Then a number of 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] Swenson, George W., Jr., Principles of Modern Acoustics, Boston Technical Publishers, Inc., Cambridge, 1965, LCN 53-6199.