In the phase space formulation of quantum mechanics, solutions to the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form.

For instance, for a single bosonic quantum harmonic oscillator we have that the thermal expectation value of the number operator is simply the Bose-Einstein distribution

Of course, since many musical instruments produce an amount of even order harmonic distortion and so do all practical loudspeakers, reversing the phase of the recorded signal would change the harmonic distribution between odd and even harmonics.

Various experimental techniques were used for this study, such as Haynes-Shockley experiment on the transit times of injected carriers, time-of-flight ( TOF ) experiment for the determination of carrier mobility, pressure-wave propagation experiment for probing electric-field distribution in insulators, organic monolayer experiment for probing orientational dipolar changes, optical time-resolved second harmonic generation ( TRM-SHG ), etc.

The form of this distribution is similar to the solution of the classical equation of motion for a damped harmonic oscillator driven by a sinusoidal external force.

On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i. e. with a preferred orientation.

Each tonewheel assembly creates audio with low harmonic content, close to a sine wave.

The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

Mathematically, the most basic wave is the ( spatially ) one-dimensional sine wave ( or harmonic wave or sinusoid ) with an amplitude described by the equation:

When the input is a pure sine wave, the measurement is most commonly the ratio of the sum of the powers of all higher harmonic frequencies to the power at the first harmonic, or fundamental, frequency:

It can generate a constant 15 Hz sine wave tone at 140 dB for 24 hours a day, seven days a week with extremely low harmonic distortion.

A pure sine wave inverter produces a nearly perfect sine wave output ( less than 3 % total harmonic distortion ) that is essentially the same as utility-supplied grid power.

Square waveform with fundamental sine wave component, 3rd harmonic and 5th harmonic

Typically, an amplifier's power specifications are calculated by measuring its RMS output voltage, with a continuous sine wave signal, at the onset of clipping — defined arbitrarily as a stated percentage of total harmonic distortion ( THD )— into specified load resistances.

The method of EOF is similar in spirit to harmonic analysis, but harmonic analysis typically uses predetermined orthogonal functions, for example, sine and cosine functions at fixed frequencies.

The Fourier theorem states that any periodic waveform can be approximated as closely as desired as the sum of a series of sine waves with frequencies in a harmonic series and at specific phase relationships to each other.

YIG multipliers often want to select an arbitrary harmonic, so they use a stateful distortion circuit that converts the input sine wave into an approximate impulse train.

When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion.

In Fourier analysis, harmonic sine and cosine waves are multiplied into the signal, and the resultant integration provides indication of a signal present at that frequency ( i. e. the signal's energy at a point in the frequency domain ).

A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i. e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f,.

An overtone is a partial ( a " partial wave " or " constituent frequency ") that can be either a harmonic partial ( a harmonic ) other than the fundamental, or an inharmonic partial.

The distortion of a waveform relative to a pure sinewave can be measured either by using a wave analyser to analyse the output wave into its constituent harmonics and noting the amplitude of each relative to the fundamental ; or by cancelling out the fundamental with a notch filter and measuring the remaining signal, which will be total aggregate harmonic distortion plus noise.

Equipment distortion can be measured using a very-low-distortion audio generator as the signal source, with appropriate equipment to measure output distortion harmonic-by-harmonic with a wave analyser, or simply total harmonic distortion.

However, the higher harmonics roll off much faster than in a square wave ( proportional to the inverse square of the harmonic number as opposed to just the inverse ).

It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every ( 4n − 1 ) th harmonic by − 1 ( or changing its phase by π ), and rolling off the harmonics by the inverse square of their relative frequency to the fundamental.

( harmonic oscillator, linear wave equation ), statistical or quantum-mechanical systems of

In an elastic medium with rigidity, a harmonic pressure wave oscillation has the form,

Doing this modifies the harmonic content of the output wave in real time, producing sounds that can imitate acoustic instruments or be totally abstract, which is where this method of sound creation excels.

The square wave output has a high harmonic content, not suitable for certain AC loads such as motors or transformers.

With Carlton M. Caves, Thorne invented the back-action-evasion approach to quantum nondemolition measurements of the quadrature amplitudes of harmonic oscillators — a technique applicable both in gravitational wave detection and quantum optics.

The frequency spectrum of the modulated color signal overlaps that of the baseband signal, and separation relies on the fact that frequency components of the baseband signal tend to be near harmonics of the horizontal scanning rate, while the color carrier is selected to be an odd multiple of half the horizontal scanning rate ; this produces a modulated color signal that consists mainly of harmonic frequencies that fall between the harmonics in the baseband luma signal, rather than both being in separate continuous frequency bands alongside each other in the frequency domain.

In 1956, the station moved to 102. 5, because when channel 11 ( now WXIA-TV ) came on the air in Atlanta, it created a second harmonic at 199. 0, on top of the video carrier at 199. 25.

Each channel is centred on a specific frequency carrier so that there is no interference with adjacent or harmonic channels.

Beside the harmonic filters, equipment is also provided to eliminate spurious signals in the frequency range of power-line carrier equipment in the range of 30 kHz to 500 kHz.

* Wave Sequence wavetables often have sudden changes in harmonic content and are typically modulated with one or more LFO to provide rhythmic or loop-like sound effects.

Two oscillators, whose frequencies were harmonically related and ring modulated against each other, produce sounds that still adhere to the harmonic partials of the notes, but contain a very different spectral make up.

Stringing a plant's leaf or a piece of paper over one side of the comb and humming with cropped lips on the opposite side dramatically increases the high-frequency harmonic content of the hum produced by the human voice box, and the resulting spread sound spectrum can be modulated by changing the resonating frequency of the oral cavity.

It should not be confused with harmonic distortion ( which does have widespread use in audio effects processing ), nor with intentional modulation ( such as a frequency mixer in superheterodyne receivers ) where signals to be modulated are presented to an intentional nonlinear element ( multiplied ) ( see non-linear mixers such as mixer diodes and even single-transistor oscillator-mixer circuits ).

Another reviewer wrote, of a concert in Kassel, " He entranced all listeners with a slow, harmonic, and studiously modulated prelude and chorale.

In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic.

Similar to the view of Leopold Kronecker that " God made the integers ; all else is the work of man ," musicians drawn to the alphorn and other instruments that sound the natural harmonics, such as the natural horn, consider the notes of the natural harmonic series — particularly the 7th and 11th harmonics — to be God's Notes, the remainder of the chromatic scale enabled by keys, valves, slides and other methods of changing the qualities of the simple open pipe being an artifact of mere mortals.

This simple melodic and harmonic progression had served as an inspiration for many baroque composers, and would be used by later romantic and post-romantic composers.

It is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies.

While the types of basslines performed by the bassist vary widely from one style of music to another, the bassist fulfills a similar role in most types of music: anchoring the harmonic framework and laying down the beat.

His melodic and harmonic sense was profoundly influenced by the folk music of Hungary, Romania, and other nations.

Also, slides ( or valves ) are used to change the length of the tubing, thus changing the harmonic series presented by the instrument to the player.

This lengthens the vibrating air column thus lowering the fundamental tone and associated harmonic series produced by the instrument.

This led to changes in the way music was performed, the most crucial of which was the move to standard instrumental groups and the reduction in the importance of the continuo — the harmonic fill beneath the music, often played by several instruments.

Without valves, the player could only produce a harmonic series of notes like those played by the bugle and other " natural " brass instruments.

Both natural harmonics and artificial harmonics, where the thumb stops the note and the octave or other harmonic is activated by lightly touching the string at the relative node point, extend the instrument's range considerably.

Notable jazz bassists from the 1940s to the 1950s included bassist Jimmy Blanton ( 1918 – 1942 ) whose short tenure in the Duke Ellington Swing band ( cut short by his death from tuberculosis ) introduced new melodic and harmonic solo ideas for the instrument ; bassist Ray Brown ( 1926 – 2002 ), known for backing Beboppers Dizzy Gillespie, Oscar Peterson, Art Tatum and Charlie Parker, and forming the Modern Jazz Quartet ; hard bop bassist Ron Carter ( born 1937 ), who has appeared on 3, 500 albums make him one of the most-recorded bassists in jazz history, including LPs by Thelonious Monk and Wes Montgomery and many Blue Note Records artists ; and Paul Chambers ( 1935 – 1969 ), a member of the Miles Davis Quintet ( including the landmark modal jazz recording Kind of Blue ) and many other 1950s and 1960s rhythm sections, was known for his virtuosic improvisations.

The acts also teamed for two yuletide favorites: a frantic, melodious, harmonic rendition of " A Merry Christmas at Grandmother's House ( Over the River and Through the Woods )", and another duet by Danny & Patty, " All I Want for Christmas Is My Two Front Teeth "

With a few exceptions related to high-energy photons ( such as fluorescence, harmonic generation, photochemical reactions, the photovoltaic effect for ionizing radiations at far ultraviolet, X-ray, and gamma radiation ), absorbed electromagnetic radiation simply deposits its energy by heating the material.

By varying the air pressure, a flute player can also change the pitch of a note by causing the air in the flute to resonate at a harmonic other than the fundamental frequency without opening or closing any holes.

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

Puccini succeeded in mastering the orchestra as no other Italian had done before him, creating new forms by manipulating structures inherited from the great Italian tradition, loading them with bold harmonic progressions which had little or nothing to do with what was happening then in Italy, though they were in step with the work of French, Austrian and German colleagues.

Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

The original Hammond organ used additive synthesis of waveforms from harmonic series made by mechanical tonewheels that rotate in front of electromagnetic pickups.

The original Hammond organ imitated the function of a pipe organ's ranks of pipes in multiple registers by using additive synthesis of waveforms from harmonic series to generate its sounds.

In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation or its frequency f =, the number of cycles per unit time.

: is a spherical harmonic function of degree ℓ and order m. Note that the generalized Laguerre polynomials are defined differently by different authors.

If a set of weights, ..., is associated to the dataset, ...,, the weighted harmonic mean is defined by