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In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
Euler's Tonnetz. The Tonnetz originally appeared in Leonhard Euler's 1739 Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae.Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a ...
Riemann's "dualist" system for relating triads was adapted from earlier 19th-century harmonic theorists. The term "dualism" refers to the emphasis on the inversional relationship between major and minor, with minor triads being considered "upside down" versions of major triads; this "harmonic dualism" (harmonic polarity) is what produces the change-in-direction described above.
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Harmonic table note layout. The Harmonic Table note-layout , or tonal array, is a key layout for musical instruments that offers interesting advantages over the traditional keyboard layout. Its symmetrical, hexagonal pattern of interval sequences places the notes of the major and minor triads together.
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The aim of the annual FFT is to bring together researchers from academia, government, and industry, as a means to spur innovation and foster interaction in Harmonic Analysis and its Applications. The FFT lasts two days and consists of approximately 15 half-hour talks, a distinguished lecture, a colloquium, and a keynote lecture.