Helmholtz

Human beings distinguish sounds on the basis of their frequency. Actually what really matters is the ratio between their frequencies. The natural scale is attributed to the Grecian philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies. After fixing the frequency of the first note — the C of the scale — the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the fol ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

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The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator relating and at a fixed frequency (in linear media): is usually a Hermitian operator under the inner product for vector fields and . (Technically, this unconjugated form is not a true inner product because e.g. it is not positive-definite for complex-valued fields, but that is not a problem here. In this sense, the operator is not truly Hermitian but is rather complex-symmetric.) This is true whenever the dielectric fu ...

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Reciprocity electromagnetism, Reciprocity electromagnetism - Lorentz reciprocity, Reciprocity electromagnetism - Reciprocity for electrical networks, Reciprocity electromagnetism - Conditions for reciprocity, Reciprocity electromagnetism - Surface-term cancellation, Reciprocity electromagnetism - Reciprocity and the Green's function, Reciprocity electromagnetism - Lossless and magneto-optic materials, Reciprocity electromagnetism - Generalization to non-symmetric materials, Reciprocity electromagnetism - Exceptions to reciprocity, Reciprocity electromagnetism - Feld-Tai reciprocity

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Gibbs' scientific career can be divided into four phases: Until 1879: theoretical thermodynamics. 1880-1884: vector analysis. 1882 to 1889: optics and the electromagnetic theory of light. After 1889: statistical mechanics, laying a foundation and "providing a mathematical framework for quantum theory and for Maxwell's theories" [1] He also wrote classic textbooks on these subjects. See also:

Josiah Willard Gibbs, Josiah Willard Gibbs - Biography, Josiah Willard Gibbs - Early years, Josiah Willard Gibbs - Middle years, Josiah Willard Gibbs - Later years, Josiah Willard Gibbs - Scientific recognition, Josiah Willard Gibbs - Quotations, Josiah Willard Gibbs - External articles and references, Josiah Willard Gibbs - Cited material, Josiah Willard Gibbs - General

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Germany - Legal system. Main article: Judiciary of Germany Germany has a civil or statute law system based ultimately on Roman law. Legislative power is divided between the Federation and the individual federated states. While criminal law and private law have seen codifications on the national level (in the Strafgesetzbuch and the Bürgerliches Gesetzbuch respectively), no such unifying codification exists in administrative law where a lot of the fundamental matters remai ...

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Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Länder, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

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Germany - Federal States Länder. Main article: States of Germany Germany is divided into sixteen federal states (in German called Länder, singular Land; commonly Bundesländer, singular Bundesland). It is further subdivided into 439 districts (Kreise) and cities (kreisfreie Städte) (2004). Germany - Territory. Since reunification Germany has resumed its role as a major centre bet ...

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Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Länder, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

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Germany is the largest European economy and the fifth largest economy in the world measured by gross domestic product purchasing power parity, placed behind the United States, China, Japan and India. According to the World Trade Organization, Germany is also the world's top exporter, ahead of the United States and China. Its major trading partners include France, the United States, the United Kingdom, Italy and the Netherlands. Germany is the largest trading partner of most European countries. A major issue of concern remains the persistentl ...

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Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Länder, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

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The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics partials and overtones

When a periodic wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), the summed wave is half-wave symmetric; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (0f, 2f, 4f, 6f, ...), it will be asymmetrical; the top half will not be a mirror image of the bottom. The opposite is also true. A system which changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics (harmonic distortion). This is called a non-linear system< ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics and non-linearities

Germany - Demographics. Main article: Demographics of Germany Due to the country's federal and decentralized structure Germany has a number of larger cities. The most populous cities of Germany are Berlin, Hamburg, Munich, Cologne, Frankfurt and Dortmund. By far the largest conurbation is the Rhine-Ruhr region, including the Düsseldorf-Cologne district and the cities of Dortmund, Duisburg and Bochum. The federal structure keeps the population oriente ...

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Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Länder, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

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The timbre of a sound is also greatly effected by the following factors: attack or Interonset interval, decay, sustain, release, and transients. Attack, decay, sustain, and release are thus all common controls on samplers. For instance, if one takes the attack off of the sound of a piano or trumpet, one much less readily identifies the sound correctly, since the sound of the hammer hitting the strings or the first blat of the players lips are highly c ...

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Timbre, Timbre - Terms, Timbre - American Standards Association definition, Timbre - Attributes, Timbre - Spectra, Timbre - Envelope, Timbre - In music, Timbre - Sources

Read more here: » Timbre: Encyclopedia II - Timbre - Envelope

Specifically, suppose that one has a current density that produces an electric field and a magnetic field , where all three are periodic functions of time with angular frequency ω, and in particular they have time-dependence exp( − iωt). Suppose that we similarly have a second current at the same frequency ω which (by itself) produces fields and . The Lorentz reciprocity theorem then states, under certain simple conditions on the materials of the medium described below, that for an arbitrar ...

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Reciprocity electromagnetism, Reciprocity electromagnetism - Lorentz reciprocity, Reciprocity electromagnetism - Reciprocity for electrical networks, Reciprocity electromagnetism - Conditions for reciprocity, Reciprocity electromagnetism - Surface-term cancellation, Reciprocity electromagnetism - Reciprocity and the Green's function, Reciprocity electromagnetism - Lossless and magneto-optic materials, Reciprocity electromagnetism - Generalization to non-symmetric materials, Reciprocity electromagnetism - Exceptions to reciprocity, Reciprocity electromagnetism - Feld-Tai reciprocity

Read more here: » Reciprocity electromagnetism: Encyclopedia II - Reciprocity electromagnetism - Lorentz reciprocity

Above, Lorentz reciprocity was phrased in terms of an externally applied current source and the resulting field. Often, especially for electrical networks, one instead prefers to think of an externally applied voltage and the resulting currents. The Lorentz reciprocity theorem describes this case as well, assuming ohmic materials (i.e. currents that respond linearly to the applied field) with a 3×3 conductivity matrix σ that is required to be symmetric, which is implied by the other conditions below. In order to properly describe this situ ...

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Reciprocity electromagnetism, Reciprocity electromagnetism - Lorentz reciprocity, Reciprocity electromagnetism - Reciprocity for electrical networks, Reciprocity electromagnetism - Conditions for reciprocity, Reciprocity electromagnetism - Surface-term cancellation, Reciprocity electromagnetism - Reciprocity and the Green's function, Reciprocity electromagnetism - Lossless and magneto-optic materials, Reciprocity electromagnetism - Generalization to non-symmetric materials, Reciprocity electromagnetism - Exceptions to reciprocity, Reciprocity electromagnetism - Feld-Tai reciprocity

Read more here: » Reciprocity electromagnetism: Encyclopedia II - Reciprocity electromagnetism - Reciprocity for electrical networks

In the natural scale the ratio of the frequencies of two notes which differ for one tone is not always the same. Consequently a certain melody cannot be played starting from a random note of the scale. For instance, a melody starting with the two notes C and D (ratio 9/8) cannot be transposed one tone higher, since the ratio of the frequencies of E and of D is very near ((5/4)/(9/8) = 10/9), but not equal to 9/8. To obviate this inconveniency, we today use the so-called Equal Temperament, which constitutes the compromise adopted in moder ...

See also:

Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - The equal tempered scale

Germany - Legal system. Main articles: Judiciary of Germany, and [[]], and [[]], and [[]], and [[]]See also:

Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Bundesländer, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

Read more here: » Germany: Encyclopedia II - Germany - Politics

If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave will still be periodic with a short period, and the combination will sound consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave will repeat thrice and the 200 Hz wave will repeat twice. Note that the total wave repeats at 100 Hz, but there is not ac ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmony

Timbre is often cited as one of the fundamental aspects of music. Formally, timbre and other factors are usually secondary to pitch. "To a marked degree the music of Debussy elevates timbre to an unprecedented structural status; already in L'Apres-midi d'un Faune the color of flute and harp functions referentially," according to Jim Samson (1977). Surpassing Debussy is Klangfarbenmelodie and surpassing that the use of sound masses. Erickson (ibid, p.6) gives a table of subjective experiences and related physical phenomena b ...

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Timbre, Timbre - Terms, Timbre - American Standards Association definition, Timbre - Attributes, Timbre - Spectra, Timbre - Envelope, Timbre - In music, Timbre - Sources

Read more here: » Timbre: Encyclopedia II - Timbre - In music

Germany is the largest European economy and the fifth largest economy in the world measured by gross domestic product purchasing power parity, placed behind the United States, China, Japan and India. According to the World Trade Organization, Germany is also the world's top exporter, ahead of the United States and China. Its major trading partners include France, the United States, the United Kingdom, Italy and the Netherlands. Germany is the largest trading partner of most European countries. A major issue of concern remains the persistentl ...

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Germany, Germany - History, Germany - Early history of the Germanic tribes 100 BC-300 AD, Germany - Migration Period and Franks 300-843, Germany - The Holy Roman Empire 843–1806, Germany - Restoration and revolution 1814–1871, Germany - German Empire 1871–1918, Germany - Weimar Republic 1919–1933, Germany - Third Reich 1933–1945, Germany - Division and reunification 1945–1990, Germany - Politics, Germany - Legal system, Germany - Foreign Relations, Germany - Armed Forces, Germany - Energy policy, Germany - Geography, Germany - Federal States Bundesländer, Germany - Territory, Germany - Climate, Germany - Economy, Germany - Exports, Germany - Imports, Germany - Agriculture, Germany - Industrial sector, Germany - Service sector, Germany - Natural resources, Germany - Society, Germany - Demographics, Germany - Religion, Germany - Education, Germany - Social issues, Germany - Culture, Germany - Miscellaneous topics

Read more here: » Germany: Encyclopedia II - Germany - Economy

The exponential function ex has  x ln x − x  as a Legendre transform since the respective first derivatives ex and ln x are inverse to each other. This example shows that the respective domains of a function and its Legendre transform need not agree. Similarly, the quadratic form with A a symmetric invertible n-by-n-matrix has< ...

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Legendre transformation, Legendre transformation - Applications, Legendre transformation - Examples, Legendre transformation - Legendre transformation in one dimension, Legendre transformation - Geometric interpretation, Legendre transformation - Legendre transformation in more than one dimension, Legendre transformation - Further properties, Legendre transformation - Scaling properties, Legendre transformation - Behavior under translation, Legendre transformation - Behavior under inversion, Legendre transformation - Behavior under linear transformations, Legendre transformation - Infimal convolution

Read more here: » Legendre transformation: Encyclopedia II - Legendre transformation - Examples

In the natural scale the ratio of the frequencies of two notes which differ for one tone is not always the same. Consequently a certain melody cannot be played starting from a random note of the scale. For instance, a melody starting with the two notes C and D (ratio 9/8) cannot be transposed one tone higher, since the ratio of the frequencies of E and of D is very near ((5/4)/(9/8) = 10/9), but not equal to 9/8. To obviate this inconveniency, we today use the so-called Equal Temperament, which constitutes the compromise adopted in moder ...

See also:

Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - The equal tempered scale

The spectral theorem depicts the importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions, a linear transformation of a vector can be expressed as the linear combination of the eigenvectors with coefficients equal to the eigenvalues times the scalar prod ...

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Eigenvalue eigenvector and eigenspace, Eigenvalue eigenvector and eigenspace - Definitions, Eigenvalue eigenvector and eigenspace - Examples, Eigenvalue eigenvector and eigenspace - Eigenvalue equation, Eigenvalue eigenvector and eigenspace - Spectral theorem, Eigenvalue eigenvector and eigenspace - Eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Computing eigenvalues and eigenvectors of matrices, Eigenvalue eigenvector and eigenspace - Properties, Eigenvalue eigenvector and eigenspace - Conjugate eigenvector, Eigenvalue eigenvector and eigenspace - Generalized eigenvalue problem, Eigenvalue eigenvector and eigenspace - Entries from a ring, Eigenvalue eigenvector and eigenspace - Infinite-dimensional spaces, Eigenvalue eigenvector and eigenspace - Applications, Eigenvalue eigenvector and eigenspace - Notes

Read more here: » Eigenvalue eigenvector and eigenspace: Encyclopedia II - Eigenvalue eigenvector and eigenspace - Spectral theorem

Seurat was born to a well-off family in Paris. His father, a legal official, was a solitary man, and so was his son. Seurat attended the École des Beaux-Arts in 1878 and 1879. After a year of military service at Brest military academy, he returned to Paris in 1880. He shared a small studio on the Left Bank with two student friends before moving to a studio of his own. For the next two years he devoted himself to mastering the art of black and white drawing. He spent 1883 on his first major painting — a ...

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Georges-Pierre Seurat, Georges-Pierre Seurat - Life, Georges-Pierre Seurat - Challenging the Impressionists, Georges-Pierre Seurat - Scientific background and influences, Georges-Pierre Seurat - Seurat's melding of science and emotion, Georges-Pierre Seurat - The crowning achievement, Georges-Pierre Seurat - Endnotes

Read more here: » Georges-Pierre Seurat: Encyclopedia II - Georges-Pierre Seurat - Life