TonalityTonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is called the tonic. The root of the tonic chord forms the name given to the key, so in the key of C major, the note C can be both the tonic of the scale and the root of the tonic chord (when it is C–E–G). The tonic can be a different note in the same scale, when the work is said to be in one of the modes of the scale.
HarmonizationIn music, harmonization is the chordal accompaniment to a line or melody: "Using chords and melodies together, making harmony by stacking scale tones as triads". A harmonized scale can be created by using each note of a musical scale as a root note for a chord and then by taking other tones within the scale building the rest of a chord.
HarmonyIn music, harmony is the process by which individual sounds are joined or composed into whole units or compositions. More concretely, harmony often refers to the effects created by distinct musical pitches or tones coinciding with one another. These effects are variously identified, defined, and categorized as harmonic objects like chords, textures and tonalities. Harmony is broadly understood to involve both a "vertical" dimension (frequency-space) and a "horizontal" dimension (time-space), and often overlaps with related musical concepts such as melody, timbre, and form.
Compact operatorIn functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of (subsets with compact closure in ). Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces. Any bounded operator that has finite rank is a compact operator; indeed, the class of compact operators is a natural generalization of the class of finite-rank operators in an infinite-dimensional setting.
Bounded operatorIn functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces (a special type of TVS), then is bounded if and only if there exists some such that for all The smallest such is called the operator norm of and denoted by A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.
Unbounded operatorIn mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.
Multispectral imagingMultispectral imaging captures image data within specific wavelength ranges across the electromagnetic spectrum. The wavelengths may be separated by filters or detected with the use of instruments that are sensitive to particular wavelengths, including light from frequencies beyond the visible light range, i.e. infrared and ultra-violet. It can allow extraction of additional information the human eye fails to capture with its visible receptors for red, green and blue.
ImageAn image is a visual representation of something. An image can be a two-dimensional (2D) representation, such as a drawing, painting, or photograph, or a three-dimensional (3D) object, such as a carving or sculpture. An image may be displayed through other media, including projection on a surface, activation of electronic signals, or digital displays. Two-dimensional images can be still or animated. Still images can usually be reproduced through mechanical means, such as photography, printmaking or photocopying.
Fredholm operatorIn mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel and finite-dimensional (algebraic) cokernel , and with closed range . The last condition is actually redundant. The index of a Fredholm operator is the integer or in other words, Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored.
Differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
