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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 ...
Wilson template for mapping higher limit systems A lattice showing Erv Wilson's Eikosany structure. This template can be used with any 6 ratios. Erv Wilson has made significant headway with developing lattices than can represent higher limit harmonics, meaning more than 2 dimensions, while displaying them in 2 dimensions. Here is a template he ...
The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2 1 is a 180° (twofold) rotation followed by a translation of 1 / 2 of the lattice vector. 3 1 is a 120° (threefold) rotation followed by a translation of 1 / 3 of ...
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
This is based on the fact that a reciprocal lattice vector (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave ...
The braid group B 3 is the universal central extension of the modular group, with these sitting as lattices inside the (topological) universal covering group SL 2 (R) → PSL 2 (R). Further, the modular group has a trivial center, and thus the modular group is isomorphic to the quotient group of B 3 modulo its center ; equivalently, to the ...
A Riesz space or a vector lattice is a preordered vector lattice whose preorder is a partial order. Equivalently, it is an ordered vector space for which the ordering is a lattice . Note that many authors required that a vector lattice be a partially ordered vector space (rather than merely a preordered vector space) while others only require ...