The harmonic distribution of a sine wave carrier modulated by such a sinusoidal signal can be represented with Bessel functions ; this provides the basis for a mathematical understanding of frequency modulation in the frequency domain.

Simple harmonic motion provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis.

In the first decade of the development of the theory, G. N. Clements developed a number of influential aspects of the theory involving harmonic processes, especially vowel harmony and nasal harmony, and John McCarthy generalized the theory to deal with the conjugational system of classical Arabic, on the basis of an autosegmental account of vowel and consonant slots on a central timing tier ( see also nonconcatenative morphology ).

In tubular bells, modes 4, 5, and 6 appear to determine the strike tone and have frequencies in the ratios 9 < sup > 2 </ sup >: 11 < sup > 2 </ sup >: 13 < sup > 2 </ sup >, or 81: 121: 169, " which are close enough to the ratios 2: 3: 4 for the ear to consider them nearly harmonic and to use them as a basis for establishing a virtual pitch ," presumably on 2.

The Italian giga is rarer than the gigue, and is faster with running passages over a harmonic basis.

The first tenet of Schenkerian harmony is that nature, through the harmonic series, gives us the triad as the ultimate ( and only possible ) basis for musical composition.

The Serenade, an orchestral work, evinces madrigalist harmony re-worked by Moeran into an astringent style in which acerbic tonal and harmonic patterns are grafted onto the madrigalist basis to produce music of outstanding freshness and originality that surely places Moeran into the genre of inventive twentieth-century music, rather than into the " English Pastoral School ", which, in itself, is arguably a misnomer.

* The Pythagorean theory of harmonic ratios is basis of studies on music theory in Islamic world, for example the al-Farabi's Kitab al-Musiqa al-kabir.

Now consider a single quantum harmonic oscillator in a superposition of basis states, evolving for a time under a Hamiltonian.

Their archaic melodies float above a modally oriented harmonic basis.

The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the " number basis ".

In the Modern period, sonata form became detached from its traditional harmonic basis.

Consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic basis

While appearing simpler in style than his instrumental music, these pieces are often related to the larger works, both technically ( for example ' At the Fountain ', the last of the Four American Choruses, has the same melodic and harmonic basis as a passage from Alhambra Fantasy ) and aesthetically ( the American poet Emily Dickinson is a recurring presence, as are themes of non-denominational spirituality or a secularised Christianity ).

The one downside to this method is that computing cube maps to properly represent a skylight is very complex ; one recent process is computing the spherical harmonic basis that best represents the low frequency diffuse illumination from the cube map.

Partch chose the 11 limit ( i. e. all rational numbers with odd factors of numerator and denominator not exceeding 11 ) as the basis of his music, because the 11th harmonic is the first that is utterly foreign to Western ears.

Also, it should be noted that the polytonal freedom of his melodies and harmonic modulations, the brilliant orchestrations, the adroitness for evading the heaviness of figured bass, the skill in florid counterpoint were not lost in his mature output, even in the spectacular historical dramas of the stage and cinema, where a large, dramatic canvas of sound was required.

The acoustic equations for the conservation of momentum and the conservation of mass are often expressed in time harmonic form ( at fixed frequency ).

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.

Moreover, Mozart also had a taste for more chromatic chords ( and greater contrasts in harmonic language generally ), a greater love for creating a welter of melodies in a single work, and a more Italianate sensibility in music as a whole.

Molecular mechanics simulations, for example, use a single classical expression for the energy of a compound, for instance the harmonic oscillator.

Software has also been written to assist with the identification of musical keys for use in harmonic mixing, which analyze computer media files for musical content and automatically determine the musical key.

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.

Of course, if two stations transmit on the same frequency, it is practically impossible for the receiver to separate them ; so instead of all stations transmitting at the same frequency, each chain was allocated a nominal frequency, 1f, and each station in the chain transmitted at a harmonic of this base frequency, as follows:

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.

Also, many audiophiles believe that class A gives the best sound quality ( for their absence of crossover distortion and reduced odd-harmonic and high-order harmonic distortion ) which provides a good market for expensive high fidelity class-A amps.

The euphonium ( like the baritone ; see below for differences ) is pitched in concert B, meaning that when no valves are in use the instrument will produce partials of the B harmonic series.

However, the opera has been acclaimed for its incorporation of advanced harmonic language and rhythmic complexity into the Italian operatic form.

The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations.

Other writers, such as F. G. Edwards in 1900 and Robert Buckley in 1905, have held that the theme is a " countermelody to some other unheard tune ": it would fit when played simultaneously, but does not necessarily contain any of its characteristics other than the most general harmonic or structural outline.

Comparison of many of Johnson's recording's of a given tune over the years does indeed demonstrate a good degree of variation from one performance to another, characterised by respect for the melody, and reliance upon a well worked out set of melodic, rhythmic, and harmonic devices, such as repeated chords, serial thirds ( hence his admiration for Bach ), and interpolated scales, on which the improvisations were based.

Because of the problem of inharmonicity, electronic piano tuning devices used by some piano technicians are not designed to tune according to a simple harmonic series.

These traits included the assignment of the melody to the tenors, harmonic structure emphasizing fourths and fifth, and the distinction between the ordinary four-part hymn (" plain tune "), the anthem, and the fuging tune.

In the fuging tune all the parts start together and proceed in rhythmic and harmonic unity usually for the space of four measures or one musical sentence.

Many of these composers were amateurs, and many were singers: they developed new forms of sacred music, such as the fuguing tune, suitable for performance by amateurs, and often using harmonic methods which would have been considered bizarre by contemporary European standards.

" In an extremely unusual example of melodic setting prior to the 20th century, portions of the tune with strong tonic harmonic leanings are harmonized with the dominant, and vice versa ; the harmonic clash is harsh, and many listeners have found this portion of the quartet to be quite amusing, especially as contrasted with the prosaic, almost " exercise-book " counterpoint which precedes it ( another example of Beethoven parodying a student counterpoint exercise can be found in the scherzo of the Quartet No. 10, opus 74 ).

The tune, in the Dorian mode, is wistful, the lyric is a happy one about true love, and Grainger's setting is atmospheric and uses a creative harmonic treatment.

Put simply, there is no tune and little definition of any harmonic base, creating a world alluring, fragile and magical.

This is reinforced by his extensive use of the pentatonic scale ( C-D-E-G-A ) in the context of the A minor tonality and a slow-moving harmonic progression that suggests a “ blues .” Because of these factors, this tune has been a favorite of jazz performers for decades and can be done in a variety of tempos and styles.

The songwriters combined memorable melodies — which could be anything from pentatonic, as in a Gershwin tune like " I Got Rhythm ", to sinuously chromatic, as in many of Cole Porter's tunes — and great harmonic subtlety, a good example being Kern's " All the Things You Are ", with its winding modulations.

The melody of the song is a memorable tune made easy to learn because of its combination of harmonic repetition and contrast — elements that make music easy to remember while avoiding monotony.

Although theorists like Hindemith and Schoenberg have suggested that the 11th harmonic is implied by, e. g. F # in the key of C, the fact is that it is simply too far out of tune, and " if the ear does not realize an implication, it does