The Stiff String

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Physical Audio Signal Processing

Digital Waveguide Models

Frequency-Dependent Damping

Stiff String Synthesis Models

A sinusoidal model for sound approximates each tonal component of the sound as a sum of slowly varying sinusoids. For tonal sounds such as from vibrating strings or wind instruments (including voiced speech), a sinusoidal model can provide a compact, high-fidelity representation. In addition to providing an intuitive, malleable representation for sound, sinusoidal models are also used in advanced audio compression. — Click for https://ccrma.stanford.edu/~jos/sasp/Sinusoidal_Modeling_Sound.html

Every periodic signal has a fundamental frequency given by the inverse of the period. If the period is in units of seconds, then the fundamental frequency is in units of Hertz (cycles per second). According to Fourier theory, every periodic signal can be expressed as a sum of sinusoids at frequencies given by integer multiples of the fundamental frequency. For integers greater than 1, these frequencies are called harmonic frequencies, and the sinusoids at harmonic frequencies are typically called harmonics. — Click for https://www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics

An overtone is a sinusoidal component of a waveform of greater frequency than its fundamental frequency. — Click for http://en.wikipedia.org/wiki/Overtone

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Pressing a key on the piano causes a felt-tipped hammer to strike a vibrating string. — Click for http://www.speech.kth.se/music/5_lectures/

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Traveling waves propagate (i.e. travel) through space. — Click for http://ccrma.stanford.edu/realsimple/travelingwaves

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In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

The wave equation refers to the partial differential equation governing wave phenomena in elastic media such as air and isotropic solids. — Click for https://en.wikipedia.org/wiki/Wave_equation

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

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Physical Audio Signal Processing

Digital Waveguide Models

Frequency-Dependent Damping

Stiff String Synthesis Models

The Stiff String

Stiffness in a vibrating string introduces a restoring force proportional to the bending angle of the string. As discussed further in §C.6, the usual stiffness term added to the wave equation for the ideal string yields

$\displaystyle \epsilon {\ddot y}= Ky''- \kappa y''''.$

When this wave equation is solved in terms of traveling waves (§C.6), it emerges that high-frequency wave components travel faster than low-frequency components. In other words, wave propagation in stiff strings is dispersive. (See §C.6 for details.)

Stiff-string models are commonly used in piano synthesis. In §9.4, further details of string models used in piano synthesis are described (§9.4.1).

Experiments with modified recordings of acoustic classical guitars indicate that overtone inharmonicity due to string-stiffness is generally not audible in nylon-string guitars, although just-noticeable-differences are possible for the 6th (lowest) string [226]. Such experiments may be carried out by retuning the partial overtones in a recorded sound sample so that they become exact harmonics. Such retuning is straightforward using sinusoidal modeling techniques [362,459].

Subsections

Stiff String Synthesis Models

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

The Stiff String

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University