Overtone in the context of "Sine wave"

Distortion and overdrive are forms of audio signal processing used to alter the sound of amplified electric musical instruments, usually by increasing their gain, producing a "fuzzy", "growling", or "gritty" tone. Distortion is most commonly used with the electric guitar, but may be used with other instruments, such as electric bass, electric piano, synthesizer, and Hammond organ. Guitarists playing electric blues originally obtained an overdriven sound by turning up their vacuum tube-powered guitar amplifiers to high volumes, which caused the signal to distort. Other ways to produce distortion have been developed since the 1960s, such as distortion effect pedals. The growling tone of a distorted electric guitar is a key part of many genres, including blues and many rock music genres, notably hard rock, punk rock, hardcore punk, acid rock, grunge and heavy metal music, while the use of distorted bass has been essential in a genre of hip hop music and alternative hip hop known as "SoundCloud rap".

The effects alter the instrument sound by clipping the signal (pushing it past its maximum, which shears off the peaks and troughs of the signal waves), adding sustain and harmonic and inharmonic overtones and leading to a compressed sound that is often described as "warm" and "dirty", depending on the type and intensity of distortion used. The terms distortion and overdrive are often used interchangeably; where a distinction is made, distortion is a more extreme version of the effect than overdrive. Fuzz is a particular form of extreme distortion originally created by guitarists using faulty equipment (such as a misaligned valve (tube); see below), which has been emulated since the 1960s by a number of "fuzzbox" effects pedals.

A vibration in a string is a wave. Initial disturbance (such as plucking or striking) causes a vibrating string to produce a sound with constant frequency, i.e., constant pitch. The nature of this frequency selection process occurs for a stretched string with a finite length, which means that only particular frequencies can survive on this string. If the length, tension, and linear density (e.g., the thickness or material choices) of the string are correctly specified, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. For a homogeneous string, the motion is given by the wave equation.

An acoustic guitar is a musical instrument in the string family. When a string is plucked, its vibration is transmitted from the bridge, resonating throughout the top of the guitar. It is also transmitted to the side and back of the instrument, resonating through the air in the body, and producing sound from the sound hole. While the original, general term for this stringed instrument is guitar, the retronym 'acoustic guitar' – often used to indicate the steel stringed model – distinguishes it from an electric guitar, which relies on electronic amplification. Typically, a guitar's body is a sound box, of which the top side serves as a sound board that enhances the vibration sounds of the strings. In standard tuning the guitar's six strings are tuned (low to high) E₂ A₂ D₃ G₃ B₃ E₄.

Guitar strings may be plucked individually with a pick (plectrum) or fingertip, or strummed to play chords. Plucking a string causes it to vibrate at a fundamental pitch determined by the string's length, mass, and tension. (Overtones are also present, closely related to harmonics of the fundamental pitch.) The string causes the soundboard and the air enclosed by the sound box to vibrate. As these have their own resonances, they amplify some overtones more strongly than others, affecting the timbre of the resulting sound.

In mathematics, a Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space, to infinite dimensions. The inner product, which is the analog of the dot product from vector calculus, allows lengths and angles to be defined. Furthermore, completeness means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.

Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input, and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs (tines) formed from a U-shaped bar of elastic metal (usually steel). It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone once the high overtones fade out. A tuning fork's pitch depends on the length and mass of the two prongs. They are traditional sources of standard pitch for tuning musical instruments.

The tuning fork was invented in 1711 by British musician John Shore, sergeant trumpeter and lutenist to the royal court.

In music, inharmonicity is the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency (harmonic series).

Acoustically, a note perceived to have a single distinct pitch in fact contains a variety of additional overtones. Many percussion instruments, such as cymbals, tam-tams, and chimes, create complex and inharmonic sounds.