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Bijection

A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is mapped to from exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

A function is bijective if and only if it is invertible; that is, a function $$f:X\to Y$$ is bijective if and only if there is a function $$g:Y\to X,$$ the inverse of $f$, such that each of the two ways for composing the two functions produces an identity function: $$g(f(x)) = x$$ for each $$x$$ in $$X$$ and $$f(g(y)) = y$$ for each $$y$$ in $$Y.$$

For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.

A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped to from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped to from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective.

The elementary operation of counting establishes a bijection from some finite set to the first natural numbers $(1, 2, 3, ...)$, up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.

A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group.

Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature.

Definition
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold:
 * 1) each element of X must be paired with at least one element of Y,
 * 2) no element of X may be paired with more than one element of Y,
 * 3) each element of Y must be paired with at least one element of X, and
 * 4) no element of Y may be paired with more than one element of X.

Satisfying properties (1) and (2) means that a pairing is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".

Batting line-up of a baseball or cricket team
Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.

Seats and students of a classroom
In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that: The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
 * 1) Every student was in a seat (there was no one standing),
 * 2) No student was in more than one seat,
 * 3) Every seat had someone sitting there (there were no empty seats), and
 * 4) No seat had more than one student in it.

More mathematical examples

 * For any set X, the identity function 1X: X → X, 1X(x) = x is bijective.
 * The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. More generally, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (or paired with) the real number x = (y − b)/a.
 * The function f: R → (−π/2, π/2), given by f(x) = arctan(x) is bijective, since each real number x is paired with exactly one angle y in the interval (−π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.
 * The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers $$\R^+ \equiv \left(0, \infty\right)$$, then g would be bijective; its inverse (see below) is the natural logarithm function ln.
 * The function h: R → R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to $$\R^+_0 \equiv \left[0, \infty\right)$$, then h would be bijective; its inverse is the positive square root function.
 * By Schröder–Bernstein theorem, given any two sets X and Y, and two injective functions f: X → Y and g: Y → X, there exists a bijective function h: X → Y.

Inverses
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.

Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition
 * for every y in Y there is a unique x in X with y = f(x).

Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.

Composition

The composition $$g \,\circ\, f$$ of two bijections f: X → Y and g: Y → Z is a bijection, whose inverse is given by $$g \,\circ\, f$$ is $$(g \,\circ\, f)^{-1} \;=\; (f^{-1}) \,\circ\, (g^{-1})$$.

Conversely, if the composition $$g \, \circ\, f$$ of two functions is bijective, it only follows that f is injective and g is surjective.

Cardinality
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.

Properties

 * A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.
 * If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).
 * Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities:
 * f(A)| = |A| and |f−1(B)| = |B|.
 * If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:
 * f is a bijection.
 * f is a surjection.
 * f is an injection.
 * For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.

Category theory
Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.

Generalization to partial functions
The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup.

Another way of defining the same notion is to say that a partial bijection from A to B is any relation R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.

When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.

title: Bijection


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Binary function

In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.

Precisely stated, a function $$f$$ is binary if there exists sets $$X, Y, Z$$ such that
 * $$\,f \colon X \times Y \rightarrow Z$$

where $$X \times Y$$ is the Cartesian product of $$X$$ and $$Y.$$

Alternative definitions
Set-theoretically, a binary function can be represented as a subset of the Cartesian product $$X \times Y \times Z$$, where $$(x,y,z)$$ belongs to the subset if and only if $$f(x,y) = z$$. Conversely, a subset $$R$$ defines a binary function if and only if for any $$x \in X$$ and $$y \in Y$$, there exists a unique $$z \in Z$$ such that $$(x,y,z)$$ belongs to $$R$$. $$f(x,y)$$ is then defined to be this $$z$$.

Alternatively, a binary function may be interpreted as simply a function from $$X \times Y$$ to $$Z$$. Even when thought of this way, however, one generally writes $$f(x,y)$$ instead of $$f((x,y))$$. (That is, the same pair of parentheses is used to indicate both function application and the formation of an ordered pair.)

Examples
Division of whole numbers can be thought of as a function. If $$\Z$$ is the set of integers, $$\N^+$$ is the set of natural numbers (except for zero), and $$\Q$$ is the set of rational numbers, then division is a binary function $$f:\Z \times \N^+ \to \Q$$.

In a vector space V over a field F, scalar multiplication is a binary function. A scalar a ∈ F is combined with a vector v ∈ V to produce a new vector av ∈ V.

Another example is that of inner products, or more generally functions of the form $$(x,y)\mapsto x^\mathrm{T}My$$, where $x$, $y$ are real-valued vectors of appropriate size and $M$ is a matrix. If $M$ is a positive definite matrix, this yields an inner product.

Functions of two real variables
Functions whose domain is a subset of $$\mathbb{R}^2$$ are often also called functions of two variables even if their domain does not form a rectangle and thus the cartesian product of two sets.

Restrictions to ordinary functions
In turn, one can also derive ordinary functions of one variable from a binary function. Given any element $$x \in X$$, there is a function $$f^x$$, or $$f(x,\cdot)$$, from $$Y$$ to $$Z$$, given by $$f^x(y) = f(x,y)$$. Similarly, given any element $$y \in Y$$, there is a function $$f_y$$, or $$f(\cdot,y)$$, from $$X$$ to $$Z$$, given by $$f_y(x) = f(x,y)$$. In computer science, this identification between a function from $$X \times Y$$ to $$Z$$ and a function from $$X$$ to $$Z^Y$$, where $$Z^Y$$ is the set of all functions from $$Y$$ to $$Z$$, is called currying.

Generalisations
The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number. This example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables simultaneously, because (for example) f (2,4) = f (1,2).

One can also consider partial binary functions, which may be defined only for certain values of the inputs. For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero. But this function is undefined when the second input is zero.

A binary operation is a binary function where the sets X, Y, and Z are all equal; binary operations are often used to define algebraic structures.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f x and fy are all linear transformations. A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear. However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product $$X \otimes Y$$ to Z.

Generalisations to ternary and other functions
The concept of binary function generalises to ternary (or 3-ary) function, quaternary (or 4-ary) function, or more generally to n-ary function for any natural number n. A 0-ary function to Z is simply given by an element of Z. One can also define an A-ary function where A is any set; there is one input for each element of A.

Category theory
In category theory, n-ary functions generalise to n-ary morphisms in a multicategory. The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Binary function


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Blue Velvet (film)

Blue Velvet is a 1986 American neo-noir mystery thriller film written and directed by David Lynch. Blending psychological horror with film noir, the film stars Kyle MacLachlan, Isabella Rossellini, Dennis Hopper, and Laura Dern, and is named after the 1951 song of the same name. The film concerns a young college student who, returning home to visit his ill father, discovers a severed human ear in a field. The ear then leads him to uncover a vast criminal conspiracy and enter into a romantic relationship with a troubled lounge singer.

The screenplay of Blue Velvet had been passed around multiple times in the late 1970s and early 1980s, with several major studios declining it due to its strong sexual and violent content. After the failure of his 1984 film Dune, Lynch made attempts at developing a more "personal story", somewhat characteristic of the surrealist style displayed in his first film Eraserhead (1977). The independent studio De Laurentiis Entertainment Group, owned at the time by Italian film producer Dino De Laurentiis, agreed to finance and produce the film.

Blue Velvet initially received a divided critical response, with many stating that its explicit content served little artistic purpose. Nevertheless, the film earned Lynch his second nomination for the Academy Award for Best Director, and received the year's Best Film and Best Director prizes from the National Society of Film Critics. It came to achieve cult status. As an example of a director casting against the norm, it was credited for revitalizing Hopper's career and for providing Rossellini with a dramatic outlet beyond her previous work as a fashion model and a cosmetics spokeswoman. In the years since, the film has been re-evaluated, and it is now widely regarded as one of Lynch's major works and one of the greatest films of the 1980s. Publications including Sight & Sound, Time, Entertainment Weekly and BBC Magazine have ranked it among the greatest American films ever. In 2008, it was chosen by the American Film Institute as one of the ten greatest American mystery films.

Plot
College student Jeffrey Beaumont returns to his hometown of Lumberton, North Carolina, after his father, Tom, has a near-fatal attack from a medical condition. Walking home from the hospital, Jeffrey cuts through a vacant lot and discovers a severed human ear, which he takes to police detective John Williams. Williams' daughter Sandy tells Jeffrey that the ear somehow relates to a lounge singer named Dorothy Vallens. Intrigued, Jeffrey enters her apartment by posing as an exterminator. While there, he steals a spare key while she is distracted by a man in a distinctive yellow sport coat, whom Jeffrey nicknames the "Yellow Man".

Jeffrey and Sandy attend Dorothy's nightclub act, in which she sings "Blue Velvet", and leave early so Jeffrey can infiltrate her apartment. Dorothy returns home and undresses; she finds Jeffrey hiding in a closet and forces him to strip at knifepoint, but he retreats to the closet when Frank Booth, a psychopathic gangster and drug lord, arrives and interrupts their encounter. Frank beats and rapes Dorothy while inhaling gas from a canister, alternating between fits of sobbing and violent rage. After Frank leaves, Jeffrey sneaks away and seeks comfort from Sandy.

Surmising that Frank has abducted Dorothy's husband Don, and son Donnie, to force her into sex slavery, Jeffrey suspects that Frank cut off Don's ear to intimidate her into submission. While continuing to see Sandy, Jeffrey enters into a sadomasochistic relationship with Dorothy, in which she encourages him to hit her. Jeffrey sees Frank attending Dorothy's show and later observes him selling drugs and meeting with the Yellow Man. Jeffrey then sees the Yellow Man meeting with a "well-dressed man".

When Frank catches Jeffrey leaving Dorothy's apartment, he abducts them and takes them to the lair of Ben, a criminal associate holding Don and Donnie hostage. Frank permits Dorothy to see her family and forces Jeffrey to watch Ben perform an impromptu lip-sync of Roy Orbison's "In Dreams", which moves Frank to tears. Afterwards, he and his gang take Jeffrey and Dorothy on a high-speed joyride to a sawmill yard, where he again attempts to sexually abuse Dorothy. When Jeffrey intervenes and punches him in the face, an enraged Frank and his gang pull him out of the car. Replaying the tape of "In Dreams", Frank smears lipstick on his face and violently kisses Jeffrey. Frank then has Jeffrey restrained and beats him unconscious, while Dorothy pleads for Frank to stop. Jeffrey awakens the next morning, bruised and bloodied.

While visiting the police station, Jeffrey discovers that the Yellow Man is Detective Williams's partner Tom Gordon, who has been murdering Frank's rival drug dealers and stealing confiscated narcotics from the evidence room for Frank to sell. After Jeffrey and Sandy declare their love for each other at a party, they are pursued by a car which they assume belongs to Frank; as they arrive at Jeffrey's home, Sandy realizes the driver is her ex-boyfriend, Mike. After Mike threatens to beat Jeffrey for stealing his girlfriend, Dorothy appears on Jeffrey's porch naked, beaten, and confused. Mike backs down as Jeffrey and Sandy whisk Dorothy to Sandy's house to summon medical attention.

When Dorothy calls Jeffrey "my secret lover", a distraught Sandy slaps him for cheating on her. Jeffrey asks Sandy to tell her father everything, and Detective Williams then leads a police raid on Frank's headquarters, killing Frank's men. Jeffrey returns alone to Dorothy's apartment, where he discovers Don dead and Gordon mortally wounded. As Jeffrey leaves the apartment, the "Well-Dressed Man" arrives, sees Jeffrey in the stairs, and chases him back inside. Realizing that the "Well-Dressed Man" is Frank in disguise, Jeffrey uses Gordon's walkie-talkie to say he is in the bedroom (to distract Frank, who has a police radio) before hiding in the closet. When Frank arrives, he starts shooting around the apartment, in the process killing Gordon. Frank deduces where Jeffrey is hiding, and Jeffrey kills Frank with Gordon's gun upon Frank opening the door. Moments later, Sandy and Detective Williams arrive.

Some time later, Jeffrey and Sandy have continued their relationship, Tom Beaumont has recovered, and Dorothy has been reunited with her son.

Origin
The film's story originated from three ideas that crystallized in the filmmaker's mind over a period of time starting as early as 1973.

The first idea was only "a feeling" and the title, as Lynch told Cineaste in 1987.

The second idea was an image of a severed, human ear lying in a field. "I don't know why it had to be an ear. Except it needed to be an opening of a part of the body, a hole into something else ... The ear sits on the head and goes right into the mind so it felt perfect," Lynch remarked in a 1986 interview to The New York Times.

The third idea was Bobby Vinton's rendition of "Blue Velvet" and "the mood that came with that song a mood, a time, and things that were of that time."

The scene in which Dorothy appears naked outside was inspired by a real-life experience Lynch had during childhood when he and his brother saw a naked woman walking down a neighborhood street at night. The experience was so traumatic to the young Lynch that it made him cry, and he had never forgotten it.

After completing The Elephant Man (1980), Lynch met producer Richard Roth over coffee. Roth had read and enjoyed Lynch's Ronnie Rocket script, but did not think it was something he wanted to produce. He asked Lynch if the filmmaker had any other scripts, but the director only had ideas. "I told him I had always wanted to sneak into a girl's room to watch her into the night and that, maybe, at one point or another, I would see something that would be the clue to a murder mystery. Roth loved the idea and asked me to write a treatment. I went home and thought of the ear in the field." Production was announced in August 1984. Lynch wrote two more drafts before he was satisfied with the script of the film. The problem with them, Lynch has said, was that "there was maybe all the unpleasantness in the film but nothing else. A lot was not there. And so it went away for a while." Conditions at this point were ideal for Lynch's film: he had made a deal with Dino De Laurentiis that gave him complete artistic freedom and final cut privileges, with the stipulation that the filmmaker take a cut in his salary and work with a budget of only $6 million. This deal meant that Blue Velvet was the smallest film on De Laurentiis's slate. Consequently, Lynch would be left mostly unsupervised during production. "After Dune I was down so far that anything was up! So it was just a euphoria. And when you work with that kind of feeling, you can take chances. You can experiment."

Casting
The cast of Blue Velvet included several then-relatively unknown actors.

Lynch met Isabella Rossellini at a restaurant, and offered her the role of Dorothy Vallens. Helen Mirren had been Lynch's first choice for the role. Rossellini had gained some exposure before the film for her Lancôme ads in the early 1980s and for being the daughter of actress Ingrid Bergman and director Roberto Rossellini. After completion of the film, during test screenings, ICM Partners—the agency representing Rossellini—immediately dropped her as a client. Furthermore, the nuns at the school in Rome that Rossellini attended in her youth called to say they were praying for her.

Kyle MacLachlan had played the central role in Lynch's critical and commercial failure Dune (1984), a science fiction epic based on the novel of the same name. MacLachlan later became a recurring collaborator with Lynch, who remarked: "Kyle plays innocents who are interested in the mysteries of life. He's the person you trust enough to go into a strange world with."

Dennis Hopper was the best-known actor in the film, having directed and starred in Easy Rider (1969). Hopper—said to be Lynch's third choice (Michael Ironside has stated that Frank was written with him in mind) —accepted the role, reportedly having exclaimed, "I've got to play Frank! I am Frank!". Harry Dean Stanton and Steven Berkoff both turned down the role of Frank because of the violent content in the film.

Laura Dern, then 18 years old, was cast as Sandy after several already-successful actresses turned the role down; among these was Molly Ringwald.

Shooting
Principal photography of Blue Velvet began in August 1985 and completed in November. The film was shot at EUE/Screen Gems studio in Wilmington, North Carolina, which also provided the exterior scenes of Lumberton. The scene with a raped and battered Dorothy proved to be particularly challenging. Several townspeople arrived to watch the filming with picnic baskets and rugs, against the wishes of Rossellini and Lynch. However, they continued filming as normal, and when Lynch yelled cut, the townspeople had left. As a result, police told Lynch they were no longer permitted to shoot in any public areas of Wilmington.

The Carolina Apartments in downtown Wilmington served as Dorothy's apartment building, with the adjacent Kenan fountain featured prominently in many shots. The building is also the birth place and death place of noted artist Claude Howell. The apartment building stands today, and the Kenan fountain was refurbished in 2020 after sustaining heavy damage during Hurricane Florence.

Editing
Lynch's original rough cut ran for approximately four hours. He was contractually obligated to deliver a two-hour movie by De Laurentiis and cut many small subplots and character scenes. He also made cuts at the request of the MPAA. For example, when Frank slaps Dorothy after the first rape scene, the audience was supposed to see Frank actually hitting her. Instead, the film cuts away to Jeffrey in the closet, wincing at what he has just seen. This cut was made to satisfy the MPAA's concerns about violence, though Lynch thought that the change made the scene more disturbing.

In 2011, Lynch announced that footage from the deleted scenes, long thought lost, had been discovered. The material was subsequently included on the Blu-ray Disc release of the film. Among the deleted footage was Megan Mullally as Jeffrey's college sweetheart Louise Wertham, whose entire role was cut from the theatrical release. The final cut of the film runs at just over two hours.

Distribution
Because the material was completely different from anything that would be considered mainstream at the time, De Laurentiis Entertainment Group's marketing employees were unsure of how to promote the film, or even if it would be promoted at all; it wasn't until the positive reception the film received at various film festivals that they began to promote it.

Interpretations
Despite Blue Velvet's initial appearance as a mystery, the film operates on a number of thematic levels. The film owes a large debt to 1950s film noir, containing and exploring such conventions as the femme fatale (Dorothy Vallens), a seemingly unstoppable villain (Frank Booth), and the questionable moral outlook of the hero (Jeffrey Beaumont), as well as its unusual use of shadowy, sometimes dark cinematography. Blue Velvet establishes Lynch's famous "askew vision" and introduces several common elements of his work, some of which would later become his trademarks, including distorted characters, a polarized world, and debilitating damage to the skull or brain. Perhaps the most significant Lynchian trademark in the film is the unearthing of a dark underbelly in a seemingly idealized small town; Jeffrey even proclaims in the film that he is "seeing something that was always hidden". Lynch's characterization of films, symbols, and motifs have become well known, and his particular style, characterised largely in Blue Velvet for the first time, has been written about extensively using descriptions like "dreamlike", "ultraweird", "dark", and "oddball". Red curtains also appear in key scenes, specifically in Dorothy's apartment, which have since become a Lynch trademark. The film has been compared to Alfred Hitchcock's Psycho (1960) because of its stark treatment of evil and mental illness. The premise of both films is curiosity, leading to an investigation that draws the lead characters into a hidden, voyeuristic underworld of crime.

The film's thematic framework harks back to Edgar Allan Poe, Henry James, and early gothic fiction, as well as films such as Shadow of a Doubt (1943) and The Night of the Hunter (1955) and the entire notion of film noir. Lynch has called it a "film about things that are hidden—within a small city and within people."

Feminist psychoanalytic film theorist Laura Mulvey argues that Blue Velvet establishes a metaphorical Oedipal family—"the child", Jeffrey Beaumont, and his "parents", Frank Booth and Dorothy Vallens—through deliberate references to film noir. Michael Atkinson claims that the resulting violence in the film can be read as symbolic of domestic violence within real families. He reads Jeffrey as an innocent youth who is both horrified by the violence inflicted by Frank and tempted by it as the means of possessing Dorothy for himself. Atkinson takes a Freudian approach to the film, considering it to be an expression of the traumatised innocence which characterises Lynch's work. He states, "Dorothy represents the sexual force of the mother [figure] because she is forbidden and because she becomes the object of the unhealthy, infantile impulses at work in Jeffrey's subconscious."

Symbolism
Symbolism is used heavily in Blue Velvet. The most consistent symbolism in the film is an insect motif introduced at the end of the first scene, when the camera zooms in on a well-kept suburban lawn until it unearths a swarming underground nest of bugs. This is generally recognized as a metaphor for the seedy underworld that Jeffrey will soon discover under the surface of his own suburban paradise. The severed ear he finds is being overrun by black ants. The bug motif is recurrent throughout the film, most notably in the bug-like gas mask that Frank wears and Jeffrey's exterminator disguise. One of Frank's accomplices is also consistently identified through the yellow jacket he wears, possibly referencing the name of a type of wasp. Finally, a robin eating a bug on a fence becomes a topic of discussion in the last scene of the film.

The severed ear that Jeffrey discovers is also a key symbolic element, leading Jeffrey into danger. Indeed, just as Jeffrey's troubles begin, the audience is treated to a nightmarish sequence in which the camera zooms into the canal of the severed, decomposing ear.

Soundtrack
The Blue Velvet soundtrack was supervised by Angelo Badalamenti (who makes a brief cameo appearance as the pianist at the Slow Club where Dorothy performs). The soundtrack makes heavy usage of vintage pop songs, such as Bobby Vinton's "Blue Velvet" and Roy Orbison's "In Dreams", juxtaposed with an orchestral score inspired by Shostakovich. During filming, Lynch placed speakers on set and in streets and played Shostakovich to set the mood he wanted to convey. The score alludes to Shostakovich's 15th Symphony, which Lynch had been listening to regularly while writing the screenplay. Lynch had originally opted to use "Song to the Siren" by This Mortal Coil during the scene in which Sandy and Jeffrey share a dance; however, he could not obtain the rights for the song at the time. He would go on to use this song in Lost Highway eleven years later.

Entertainment Weekly ranked Blue Velvet soundtrack on its list of the 100 Greatest Film Soundtracks, at the 100th position. Critic John Alexander wrote, "the haunting soundtrack accompanies the title credits, then weaves through the narrative, accentuating the noir mood of the film." Lynch worked with music composer Angelo Badalamenti for the first time in this film and asked him to write a score that had to be "like Shostakovich, be very Russian, but make it the most beautiful thing but make it dark and a little bit scary." Badalamenti's success with Blue Velvet would lead him to contribute to all of Lynch's future full-length films until Inland Empire as well as the cult television program Twin Peaks. Also included in the sound team was long-time Lynch collaborator Alan Splet, a sound editor and designer who had won an Academy Award for his work on The Black Stallion (1979) and been nominated for Never Cry Wolf (1983).

Box office
Blue Velvet premiered in competition at the Montréal World Film Festival in August 1986, and at the Toronto Festival of Festivals on September 12, 1986, and a few days later in the United States. It debuted commercially in both countries on September 19, 1986, in 98 theatres across the United States. In its opening weekend, the film grossed a total of $789,409. It eventually expanded to another 15 theatres, and in the US and Canada grossed a total of $8,551,228. Blue Velvet was met with uproar during its audience reception, with lines formed around city blocks in New York City and Los Angeles. There were reports of mass walkouts and refund demands during its opening week. At a Chicago screening, a man fainted and had to have his pacemaker checked. Upon completion, he returned to the cinema to see the ending. At a Los Angeles cinema, two strangers became engaged in a heated disagreement, but decided to resolve the disagreement to return to the theatre.

Critical reception
Blue Velvet was released to a very polarized reception in the United States. The critics who did praise the film were often vociferous. The New York Times critic Janet Maslin directed much praise toward the performances of Hopper and Rossellini: "Mr. Hopper and Miss Rossellini are so far outside the bounds of ordinary acting here that their performances are best understood in terms of sheer lack of inhibition; both give themselves entirely over to the material, which seems to be exactly what's called for." She called it "an instant cult classic", concluding that Blue Velvet "is as fascinating as it is freakish" and "confirms Mr. Lynch's stature as an innovator, a superb technician, and someone best not encountered in a dark alley."

Sheila Benson of the Los Angeles Times called the film "the most brilliantly disturbing film ever to have its roots in small-town American life," describing it as "shocking, visionary, rapturously controlled". Film critic Gene Siskel included Blue Velvet on his list of the best films of 1986, at the fifth spot. Peter Travers, film critic for Rolling Stone, named it the best film of the 1980s and referred to it as an "American masterpiece". Upon its initial release, Woody Allen and Martin Scorsese called Blue Velvet the best film of the year.

On the other hand, Paul Attanasio of The Washington Post said "the film showcases a visual stylist utterly in command of his talents" and that Angelo Badalamenti "contributes an extraordinary score, slipping seamlessly from slinky jazz to violin figures to the romantic sweep of a classic Hollywood score," but stated that Lynch "isn't interested in communicating, he's interested in parading his personality. The movie doesn't progress or deepen, it just gets weirder, and to no good end." A general criticism from US critics was Blue Velvet's approach to sexuality and violence. They asserted that this detracted from the film's seriousness as a work of art, and some condemned the film as pornographic. One of its detractors, Roger Ebert, stated that the large amount of "jokey small-town satire" in the film made it impossible to take its themes seriously. Ebert praised Rossellini's performance as "convincing and courageous" but criticized how she was depicted in the film, even accusing David Lynch of misogyny: "degraded, slapped around, humiliated and undressed in front of the camera. And when you ask an actress to endure those experiences, you should keep your side of the bargain by putting her in an important film." While Ebert in later years came to consider Lynch a great filmmaker, his negative view of Blue Velvet remained unchanged after he revisited it in the 21st century.

The film is now widely considered a masterpiece and has a score of 95% on Rotten Tomatoes based on 80 reviews with an average rating of 8.8/10. The website's critical consensus states: "If audiences walk away from this subversive, surreal shocker not fully understanding the story, they might also walk away with a deeper perception of the potential of film storytelling." The film also has a score of 76 out of 100 on Metacritic based on 15 critics, indicating "generally favorable reviews". Looking back in his Guardian/Observer review, critic Philip French wrote, "The film is wearing well and has attained a classic status without becoming respectable or losing its sense of danger."

Mark Kermode walked out on the film and gave the film a poor review upon its release, but revised his view of the film over time. In 2016, he remarked, "as a film critic, it taught me that when a film really gets under your skin and really provokes a visceral reaction, you have to be very careful about assessing it ... I didn't walk out on Blue Velvet because it was a bad film. I walked out on it because it was a really good film. The point was at the time I wasn't good enough for it."

Accolades
David Lynch was nominated for the Academy Award for Best Director and the Golden Globe Award for Best Screenplay for his work on the film. Dennis Hopper was nominated for the Golden Globe Award for Best Supporting Actor – Motion Picture for his performance, while Isabella Rossellini won the Independent Spirit Award for Best Female Lead for her performance. Lynch won Best Director and Hopper won Best Supporting Actor at the Los Angeles Film Critics Association awards in 1987. That same year, the film received four National Society of Film Critics awards: Best Film, Best Director (Lynch), Best Cinematography (Frederick Elmes), and Best Supporting Actor (Hopper).

Home media
Blue Velvet was released on VHS by Karl-Lorimar Home Video in 1987 and re-issued by Warner Home Video in 1992. After that, it was DVD in 1999 and 2002 by MGM Home Entertainment. The film made its Blu-ray debut on November 8, 2011, with a special 25th-anniversary edition featuring never-before-seen deleted scenes. On May 28, 2019, the film was re-released on Blu-ray by the Criterion Collection, featuring a 4K digital restoration, the original stereo soundtrack and other special features, including a behind-the-scenes documentary titled Blue Velvet Revisited.

Legacy
Although it initially gained a relatively small theatrical audience in North America and was met with controversy over its artistic merit, Blue Velvet soon became the center of a "national firestorm" in 1986, and over time became regarded as an American classic. In the late 1980s, and early 1990s, after its release on videotape, the film became a widely recognized cult film, for its dark depiction of a suburban America. With its many VHS, LaserDisc and DVD releases, the film reached broader American audiences. It marked David Lynch's entry into the Hollywood mainstream and Dennis Hopper's comeback. Hopper's performance as Frank Booth has itself left an imprint on popular culture, with countless tributes, cultural references and parodies. The film's success also helped Hollywood address previously censored issues, as Psycho (1960) had. Blue Velvet has been frequently compared to that ground-breaking film. It has become one of the most significant, well-recognized films of its era, spawning countless imitations and parodies in media. The film's dark, stylish and erotic production design has served as a benchmark for a number of films, parodies and even Lynch's own later work, notably Twin Peaks (1990–91), and Mulholland Drive (2001). Peter Travers of Rolling Stone magazine cited it as one of the most "influential American films", as did Michael Atkinson, who dedicated a book to the film's themes and motifs.

Blue Velvet now frequently appears in various critical assessments of all-time great films, also ranked as one of the greatest films of the 1980s, one of the best examples of American surrealism and one of the finest examples of David Lynch's work. In a poll of 54 American critics ranking the "most outstanding films of the decade", Blue Velvet was placed fourth, behind Raging Bull (1980), E.T. the Extra-Terrestrial (1982) and the German film Wings of Desire (1987). An Entertainment Weekly book special released in 1999 ranked Blue Velvet 37th of the greatest films of all time. The film was ranked by The Guardian in its list of the 100 Greatest Films. Film Four ranked it on their list of 100 Greatest Films. In a 2007 poll of the online film community held by Variety, Blue Velvet came in at the 95th-greatest film of all time. Total Film ranked Blue Velvet as one of the all-time best films in both a critics' list and a public poll, in 2006 and 2007, respectively. In December 2002, a UK film critics' poll in Sight & Sound ranked the film fifth on their list of the 10 Best Films of the Last 25 Years. In a special Entertainment Weekly issue, 100 new film classics were chosen from 1983 to 2008: Blue Velvet was ranked at fourth.

In addition to Blue Velvet various "all-time greatest films" rankings, the American Film Institute has awarded the film three honors in its lists: 96th on 100 Years ... 100 Thrills in 2001, selecting cinema's most thrilling moments and ranked Frank Booth 36th of the 50 greatest villains in 100 Years ... 100 Heroes and Villains in 2003. In June 2008, the AFI revealed its "ten Top Ten"—the best ten films in ten "classic" American film genres—after polling over 1,500 people from the creative community. Blue Velvet was acknowledged as the eighth best film in the mystery genre. Premiere magazine listed Frank Booth, played by Dennis Hopper, as the 54th on its list of 'The 100 Greatest Movie Characters of All Time', calling him one of "the most monstrously funny creations in cinema history". The film was ranked 84th on Bravo Television's four-hour program 100 Scariest Movie Moments (2004). It is frequently sampled musically and an array of bands and solo artists have taken their names and inspiration from the film. In August 2012, Sight & Sound unveiled their latest list of the 250 greatest films of all time, with Blue Velvet ranking at 69th.

Blue Velvet was also nominated for the following AFI lists:
 * AFI's 100 Years...100 Movies
 * AFI's 100 Years...100 Heroes & Villains
 * Frank Booth – ranked 36th-greatest film villain
 * AFI's 100 Years...100 Songs:
 * "In Dreams" - nominated
 * AFI's 100 Years...100 Movies (10th Anniversary Edition)

Inspired by the film, pop singer Lana Del Rey recorded a cover version of "Blue Velvet" in 2012. Used to endorse clothing line H&M, a music video accompanied the track and aired as a television commercial. Set in post-war America, the video drew influence from Lynch and Blue Velvet. In the video, Del Rey plays the role of Dorothy Vallens, performing a private concert similar to the scene where Ben (Dean Stockwell) pantomimes "In Dreams" for Frank Booth. Del Rey's version, however, has her lip-syncing "Blue Velvet" when a little person dressed as Frank Sinatra approaches and unplugs a hidden Victrola, revealing Del Rey as a fraud. When Lynch heard of the music video, he praised it, telling Artinfo: "Lana Del Rey, she's got some fantastic charisma and—this is a very interesting thing—it's like she's born out of another time. She's got something that's very appealing to people. And I didn't know she was influenced by me!"

"Now It's Dark", a song by American heavy metal band Anthrax on their 1988 album State of Euphoria, was directly inspired by the film, and specifically the character of Frank Booth. The same phrase appeared in the liner notes of Rush's album Roll the Bones, and drummer Neil Peart later explained: "The phrase occurs in David Lynch's comedy classic Blue Velvet."

The sludge metal band Acid Bath sampled several of Frank Booth's lines on the song "Cassie Eats Cockroaches" from their 1994 debut album When the Kite String Pops, and industrial metal band Ministry sampled the movie in their song "Jesus Built My Hotrod". The experimental rock band Mr. Bungle also sampled lines from Blue Velvet on the songs "Squeeze Me Macaroni", "Stubb (A Dub)", and "My Ass Is On Fire" from their debut self-titled album.

title: Blue Velvet (film)


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Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.

More specifically, a binary operation on a set is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.

An operation of arity two that involves several sets is sometimes also called a binary operation. For example, scalar multiplication of vector spaces takes a scalar and a vector to produce a vector, and scalar product takes two vectors to produce a scalar. Such binary operations may also be called binary functions.

Binary operations are the keystone of most structures that are studied in algebra, in particular in semigroups, monoids, groups, rings, fields, and vector spaces.

Terminology
More precisely, a binary operation on a set $$S$$ is a mapping of the elements of the Cartesian product $$S \times S$$ to $$S$$:
 * $$\,f \colon S \times S \rightarrow S.$$

The closure property of a binary operation expresses the existence of a result for the operation given any pair of operands.

If $$f$$ is not a function but a partial function, then $$f$$ is called a partial binary operation. For instance, division of real numbers is a partial binary operation, because one can not divide by zero: $$\frac{a}{0}$$ is undefined for every real number $$a$$. In both model theory and classical universal algebra, binary operations are required to be defined on all elements of $$S \times S$$. However, partial algebras generalize universal algebras to allow partial operations.

Sometimes, especially in computer science, the term binary operation is used for any binary function.

Properties and examples
Typical examples of binary operations are the addition ($$+$$) and multiplication ($$\times$$) of numbers and matrices as well as composition of functions on a single set. For instance,
 * On the set of real numbers $$\mathbb R$$, $$f(a,b)=a+b$$ is a binary operation since the sum of two real numbers is a real number.
 * On the set of natural numbers $$\mathbb N$$, $$f(a,b)=a+b$$ is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
 * On the set $$M(2,\mathbb R)$$ of $$2 \times 2$$ matrices with real entries, $$f(A,B)=A+B$$ is a binary operation since the sum of two such matrices is a $$2 \times 2$$ matrix.
 * On the set $$M(2,\mathbb R)$$ of $$2 \times 2$$ matrices with real entries, $$f(A,B)=AB$$ is a binary operation since the product of two such matrices is a $$2 \times 2$$ matrix.
 * For a given set $$C$$, let $$S$$ be the set of all functions $$h \colon C \rightarrow C$$. Define $$f \colon S \times S \rightarrow S$$ by $$f(h_1,h_2)(c)=(h_1 \circ h_2)(c)=h_1(h_2(c))$$ for all $$c \in C$$, the composition of the two functions $$h_1$$ and $$h_2$$ in $$S$$. Then $$f$$ is a binary operation since the composition of the two functions is again a function on the set $$C$$ (that is, a member of $$S$$).

Many binary operations of interest in both algebra and formal logic are commutative, satisfying $$f(a,b)=f(b,a)$$ for all elements $$a$$ and $$b$$ in $$S$$, or associative, satisfying $$f(f(a,b),c)=f(a,f(b,c))$$ for all $$a$$, $$b$$, and $$c$$ in $$S$$. Many also have identity elements and inverse elements.

The first three examples above are commutative and all of the above examples are associative.

On the set of real numbers $$\mathbb R$$, subtraction, that is, $$f(a,b)=a-b$$, is a binary operation which is not commutative since, in general, $$a-b \neq b-a$$. It is also not associative, since, in general, $$a-(b-c) \neq (a-b)-c$$; for instance, $$1-(2-3)=2$$ but $$(1-2)-3=-4$$.

On the set of natural numbers $$\mathbb N$$, the binary operation exponentiation, $$f(a,b)=a^b$$, is not commutative since, $$a^b \neq b^a$$ (cf. Equation xy = yx), and is also not associative since $$f(f(a,b),c) \neq f(a,f(b,c))$$. For instance, with $$a=2$$, $$b=3$$, and $$c=2$$, $$f(2^3,2)=f(8,2)=8^2=64$$, but $$f(2,3^2)=f(2,9)=2^9=512$$. By changing the set $$\mathbb N$$ to the set of integers $$\mathbb Z$$, this binary operation becomes a partial binary operation since it is now undefined when $$a=0$$ and $$b$$ is any negative integer. For either set, this operation has a right identity (which is $$1$$) since $$f(a,1)=a$$ for all $$a$$ in the set, which is not an identity (two sided identity) since $$f(1,b) \neq b$$ in general.

Division ($$\div$$), a partial binary operation on the set of real or rational numbers, is not commutative or associative. Tetration ($$\uparrow\uparrow$$), as a binary operation on the natural numbers, is not commutative or associative and has no identity element.

Notation
Binary operations are often written using infix notation such as $$a \ast b$$, $$a+b$$, $$a \cdot b$$ or (by juxtaposition with no symbol) $$ab$$ rather than by functional notation of the form $$f(a, b)$$. Powers are usually also written without operator, but with the second argument as superscript.

Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses. They are also called, respectively, Polish notation $$\ast a b$$ and reverse Polish notation $$a b \ast$$.

Binary operations as ternary relations
A binary operation $$f$$ on a set $$S$$ may be viewed as a ternary relation on $$S$$, that is, the set of triples $$(a, b, f(a,b))$$ in $$S \times S \times S$$ for all $$a$$ and $$b$$ in $$S$$.

Other binary operations
For example, scalar multiplication in linear algebra. Here $$K$$ is a field and $$S$$ is a vector space over that field.

Also the dot product of two vectors maps $$S \times S$$ to $$K$$, where $$K$$ is a field and $$S$$ is a vector space over $$K$$. It depends on authors whether it is considered as a binary operation.

title: Binary operation


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Bagpipes

Bagpipes are a woodwind instrument using enclosed reeds fed from a constant reservoir of air in the form of a bag. The Great Highland bagpipes are well known, but people have played bagpipes for centuries throughout large parts of Europe, Northern Africa, Western Asia, around the Persian Gulf and northern parts of South Asia.

The term bagpipe is equally correct in the singular or the plural, though pipers usually refer to the bagpipes as "the pipes", "a set of pipes" or "a stand of pipes".

Construction
A set of bagpipes minimally consists of an air supply, a bag, a chanter, and usually at least one drone. Many bagpipes have more than one drone (and, sometimes, more than one chanter) in various combinations, held in place in stocks—sockets that fasten the various pipes to the bag.

Air supply
The most common method of supplying air to the bag is through blowing into a blowpipe or blowstick. In some pipes the player must cover the tip of the blowpipe with the tongue while inhaling, in order to prevent unwanted deflation of the bag, but most blowpipes have a non-return valve that eliminates this need. In recent times, there are many instruments that assist in creating a clean air flow to the pipes and assist the collection of condensation.

The use of a bellows to supply air is an innovation dating from the 16th or 17th century. In these pipes, sometimes called "cauld wind pipes", air is not heated or moistened by the player's breathing, so bellows-driven bagpipes can use more refined or delicate reeds. Such pipes include the Irish uilleann pipes; the border or Lowland pipes, Scottish smallpipes, Northumbrian smallpipes and pastoral pipes in Britain; the musette de cour, the musette bechonnet and the cabrette in France; and the Dudy, koziol bialy, and koziol czarny in Poland.

Bag
The bag is an airtight reservoir that holds air and regulates its flow via arm pressure, allowing the player to maintain continuous, even sound. The player keeps the bag inflated by blowing air into it through a blowpipe or by pumping air into it with a bellows. Materials used for bags vary widely, but the most common are the skins of local animals such as goats, dogs, sheep, and cows. More recently, bags made of synthetic materials including Gore-Tex have become much more common. Some synthetic bags have zips that allow the player to fit a more effective moisture trap to the inside of the bag. However, synthetic bags still carry a risk of colonisation by fungal spores, and the associated danger of lung infection if they are not kept clean, even if they otherwise require less cleaning than do bags made from natural substances.

Bags cut from larger materials are usually saddle-stitched with an extra strip folded over the seam and stitched (for skin bags) or glued (for synthetic bags) to reduce leaks. Holes are then cut to accommodate the stocks. In the case of bags made from largely intact animal skins, the stocks are typically tied into the points where the limbs and the head joined the body of the whole animal, a construction technique common in Central Europe. Different regions have different ways of treating the hide. The simplest methods involve just the use of salt, while more complex treatments involve milk, flour, and the removal of fur. The hide is normally turned inside out so that the fur is on the inside of the bag, as this helps to reduce the effect of moisture buildup within the bag.

Chanter
The chanter is the melody pipe, played with two hands. All bagpipes have at least one chanter; some pipes have two chanters, particularly those in North Africa, in the Balkans, and in Southwest Asia. A chanter can be bored internally so that the inside walls are parallel (or "cylindrical") for its full length, or it can be bored in a conical shape. Popular woods include boxwood, cornel, plum or other fruit wood.

The chanter is usually open-ended, so there is no easy way for the player to stop the pipe from sounding. Thus most bagpipes share a constant legato sound with no rests in the music. Primarily because of this inability to stop playing, technical movements are made to break up notes and to create the illusion of articulation and accents. Because of their importance, these embellishments (or "ornaments") are often highly technical systems specific to each bagpipe, and take many years of study to master. A few bagpipes (such as the musette de cour, the uilleann pipes, the Northumbrian smallpipes, the piva and the left chanter of the surdelina) have closed ends or stop the end on the player's leg, so that when the player "closes" (covers all the holes), the chanter becomes silent.

A practice chanter is a chanter without bag or drones and has a much quieter reed, allowing a player to practice the instrument quietly and with no variables other than playing the chanter.

The term chanter is derived from the Latin cantare, or "to sing", much like the modern French verb meaning "to sing", chanter.

A distinctive feature of the gaida's chanter (which it shares with a number of other Eastern European bagpipes) is the "flea-hole" (also known as a mumbler or voicer, marmorka) which is covered by the index finger of the left hand. The flea-hole is smaller than the rest and usually consists of a small tube that is made out of metal or a chicken or duck feather. Uncovering the flea-hole raises any note played by a half step, and it is used in creating the musical ornamentation that gives Balkan music its unique character.

Some types of gaida can have a double bored chanter, such as the Serbian three-voiced gajde. It has eight fingerholes: the top four are covered by the thumb and the first three fingers of the left hand, then the four fingers of the right hand cover the remaining four holes.

Chanter reed
The note from the chanter is produced by a reed installed at its top. The reed may be a single (a reed with one vibrating tongue) or double reed (of two pieces that vibrate against each other). Double reeds are used with both conical- and parallel-bored chanters while single reeds are generally (although not exclusively) limited to parallel-bored chanters. In general, double-reed chanters are found in pipes of Western Europe while single-reed chanters appear in most other regions.

They are made from reed (arundo donax or Phragmites), bamboo, or elder. A more modern variant for the reed is a combination of a cotton phenolic (Hgw2082) material from which the body of the reed is made and a clarinet reed cut to size in order to fit the body. These type of reeds produce a louder sound and are not so sensitive to humidity and temperature changes.

Drone
Most bagpipes have at least one drone, a pipe that generally is not fingered but rather produces a constant harmonizing note throughout play (usually the tonic note of the chanter). Exceptions are generally those pipes that have a double-chanter instead. A drone is most commonly a cylindrically bored tube with a single reed, although drones with double reeds exist. The drone is generally designed in two or more parts with a sliding joint so that the pitch of the drone can be adjusted.

Depending on the type of pipes, the drones may lie over the shoulder, across the arm opposite the bag, or may run parallel to the chanter. Some drones have a tuning screw, which effectively alters the length of the drone by opening a hole, allowing the drone to be tuned to two or more distinct pitches. The tuning screw may also shut off the drone altogether. In most types of pipes with one drone, it is pitched two octaves below the tonic of the chanter. Additional drones often add the octave below and then a drone consonant with the fifth of the chanter.

Possible ancient origins
The evidence for bagpipes prior to the 13th century AD is still uncertain, but several textual and visual clues have been suggested. The Oxford History of Music posits that a sculpture of bagpipes has been found on a Hittite slab at Euyuk in Anatolia, dated to 1000 BC. Another interpretation of this sculpture suggests that it instead depicts a pan flute played along with a friction drum.

Several authors identify the ancient Greek (ἀσκός askos – wine-skin, αὐλός aulos – reed pipe) with the bagpipe. In the 2nd century AD, Suetonius described the Roman emperor Nero as a player of the tibia utricularis. Dio Chrysostom wrote in the 1st century of a contemporary sovereign (possibly Nero) who could play a pipe (tibia, Roman reedpipes similar to Greek and Etruscan instruments) with his mouth as well as by tucking a bladder beneath his armpit. Vereno suggests that such instruments, rather than being seen as an independent class, were understood as variants on mouth-blown instruments that used a bag as an alternative blowing aid and that it was not until drones were added in the European Medieval era that bagpipes were seen as a distinct class.

Spread and development in Europe

In the early part of the second millennium, representation of bagpipes began to appear with frequency in Western European art and iconography. The Cantigas de Santa Maria, written in Galician-Portuguese and compiled in Castile in the mid-13th century, depicts several types of bagpipes. Several illustrations of bagpipes also appear in the Chronique dite de Baudoin d’Avesnes, a 13th-century manuscript of northern French origin. Although evidence of bagpipes in the British Isles prior to the 14th century is contested, they are explicitly mentioned in The Canterbury Tales (written around 1380): "A baggepype wel coude he blowe and sowne, /And ther-with-al he broghte us out of towne."

Bagpipes were also frequent subjects for carvers of wooden choir stalls in the late 15th and early 16th century throughout Europe, sometimes with animal musicians.

Actual specimens of bagpipes from before the 18th century are extremely rare; however, a substantial number of paintings, carvings, engravings, and manuscript illuminations survive. These artefacts are clear evidence that bagpipes varied widely throughout Europe, and even within individual regions. Many examples of early folk bagpipes in continental Europe can be found in the paintings of Brueghel, Teniers, Jordaens, and Durer.

The earliest known artefact identified as a part of a bagpipe is a chanter found in 1985 at Rostock, Germany, that has been dated to the late 14th century or the first quarter of the 15th century.

The first clear reference to the use of the Scottish Highland bagpipes is from a French history that mentions their use at the Battle of Pinkie in 1547. George Buchanan (1506–82) claimed that bagpipes had replaced the trumpet on the battlefield. This period saw the creation of the ceòl mór (great music) of the bagpipe, which reflected its martial origins, with battle tunes, marches, gatherings, salutes and laments. The Highlands of the early 17th century saw the development of piping families including the MacCrimmonds, MacArthurs, MacGregors, and the Mackays of Gairloch.

The earliest Irish mention of the bagpipe is in 1206, approximately thirty years after the Anglo-Norman invasion; another mention attributes their use to Irish troops in Henry VIII's siege of Boulogne. Illustrations in the 1581 book The Image of Irelande by John Derricke clearly depict a bagpiper. Derricke's illustrations are considered to be reasonably faithful depictions of the attire and equipment of the English and Irish population of the 16th century.

The "Battell" sequence from My Ladye Nevells Booke (1591) by William Byrd, which probably alludes to the Irish wars of 1578, contains a piece entitled The bagpipe: & the drone. In 1760, the first serious study of the Scottish Highland bagpipe and its music was attempted in Joseph MacDonald's Compleat Theory. A manuscript from the 1730s by a William Dixon of Northumberland contains music that fits the border pipes, a nine-note bellows-blown bagpipe with a chanter similar to that of the modern Great Highland bagpipe. However, the music in Dixon's manuscript varied greatly from modern Highland bagpipe tunes, consisting mostly of extended variation sets of common dance tunes. Some of the tunes in the Dixon manuscript correspond to those found in the early 19th century manuscript sources of Northumbrian smallpipe tunes, notably the rare book of 50 tunes, many with variations, by John Peacock.

As Western classical music developed, both in terms of musical sophistication and instrumental technology, bagpipes in many regions fell out of favour because of their limited range and function. This triggered a long, slow decline that continued, in most cases, into the 20th century.

Extensive and documented collections of traditional bagpipes may be found at the Metropolitan Museum of Art in New York City, the International Bagpipe Museum in Gijón, Spain, the Pitt Rivers Museum in Oxford, England and the Morpeth Chantry Bagpipe Museum in Northumberland, and the Musical Instrument Museum in Phoenix, Arizona.

The International Bagpipe Festival is held every two years in Strakonice, Czech Republic.

Recent history
During the expansion of the British Empire, spearheaded by British military forces that included Highland regiments, the Scottish Great Highland bagpipe became well known worldwide. This surge in popularity was boosted by large numbers of pipers trained for military service in World War I and World War II. This coincided with a decline in the popularity of many traditional forms of bagpipe throughout Europe, which began to be displaced by instruments from the classical tradition and later by gramophone and radio.

As pipers were easily identifiable, combat losses were high, estimated at one thousand in World War I. A front line role was prohibited following high losses in the Second Battle of El Alamein in 1943, though a few later instances occurred.

In the United Kingdom and Commonwealth Nations such as Canada, New Zealand and Australia, the Great Highland bagpipe is commonly used in the military and is often played during formal ceremonies. Foreign militaries patterned after the British army have also adopted the Highland bagpipe, including those of Uganda, Sudan, India, Pakistan, Sri Lanka, Jordan, and Oman. Many police and fire services in Scotland, Canada, Australia, New Zealand, Hong Kong, and the United States have also adopted the tradition of fielding pipe bands.

In recent years, often driven by revivals of native folk music and dance, many types of bagpipes have enjoyed a resurgence in popularity and, in many cases, instruments that had fallen into obscurity have become extremely popular. In Brittany, the Great Highland bagpipe and concept of the pipe band were appropriated to create a Breton interpretation known as the bagad. The pipe-band idiom has also been adopted and applied to the Galician gaita as well. Bagpipes have often been used in various films depicting moments from Scottish and Irish history; the film Braveheart and the theatrical show Riverdance have served to make the uilleann pipes more commonly known.

Bagpipes are sometimes played at formal events at Commonwealth universities, particularly in Canada. Because of Scottish influences on the sport of curling, bagpipes are also the official instrument of the World Curling Federation and are commonly played during a ceremonial procession of teams before major curling championships.

Bagpipe making was once a craft that produced instruments in many distinctive, local and traditional styles. Today, the world's largest producer of the instrument is Pakistan, where the industry was worth $6.8 million in 2010. In the late 20th century, various models of electronic bagpipes were invented. The first custom-built MIDI bagpipes were developed by the Asturian piper known as Hevia (José Ángel Hevia Velasco).

Astronaut Kjell N. Lindgren is thought to be the first person to play the bagpipes in outer space, having played "Amazing Grace" in tribute to late research scientist Victor Hurst aboard the International Space Station in November 2015.

Traditionally, one of the purposes of the bagpipe was to provide music for dancing. This has declined with the growth of dance bands, recordings, and the decline of traditional dance. In turn, this has led to many types of pipes developing a performance-led tradition, and indeed much modern music based on the dance music tradition played on bagpipes is suitable for use as dance music.

Types of bagpipes
Numerous types of bagpipes today are widely spread across Europe, the Middle East and North Africa as well as through much of the former British Empire. The name bagpipe has almost become synonymous with its best-known form, the Great Highland bagpipe, overshadowing the great number and variety of traditional forms of bagpipe. Despite the decline of these other types of pipes over the last few centuries, in recent years many of these pipes have seen a resurgence or revival as musicians have sought them out; for example, the Irish piping tradition, which by the mid 20th century had declined to a handful of master players is today alive, well, and flourishing, a situation similar to that of the Asturian gaita, the Galician gaita, the Portuguese gaita transmontana, the Aragonese gaita de boto, Northumbrian smallpipes, the Breton biniou, the Balkan gaida, the Romanian cimpoi, the Black Sea tulum, the Scottish smallpipes and pastoral pipes, as well as other varieties. Bulgaria has the Kaba gaida, a large bagpipe of the Rhodope mountains with a hexagonal and rounded drone, often described as a deep-sounding gaida and the Dzhura gaida with a straight conical drone and of a higher pitch. The Macedonian gaida is structurally between a kaba and dzhura gaida and described as a medium pitched gaida.

In Southeastern Europe and Eastern Europe bagpipes known as gaida include: the gajde, mishnica, bishnica, gaidã, гайда (gaida), γκάιντα (gáida) τσαμπούνα (tsaboúna) or ασκομαντουρα (askomandoura), гајда (gajda),, gayda also tulum and.

In Tunisia, it is known by the name "mezwed". It is used in the Tunisian pop music genre, also called mezwed, that is named after the instrument.

Usage in non-traditional music
Since the 1960s, bagpipes have also made appearances in other forms of music, including rock, metal, jazz, hip-hop, punk, and classical music, for example with Paul McCartney's "Mull of Kintyre", AC/DC's "It's a Long Way to the Top (If You Wanna Rock 'n' Roll)", and Peter Maxwell Davies's composition An Orkney Wedding, with Sunrise.

Periodicals
Periodicals covering specific types of bagpipes are addressed in the article for that bagpipe

Books

 * , 147 pp. with plates.
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title: Bagpipes


text:
Talk:Bagpipes

Roman dispute
I dispute "evidence for pre-Roman era bagpipes is still uncertain" -- I argue that evidence for pre-13th Century evidence is uncertain

The way it's phrased currently implies that evidence for Roman and post-Roman bagpipes may be certain. My understanding is that there is no absolutely ironclad evidence for bagpipes is in the Cantigas, from 13th century Spain. I would like to modify the article to be more explicit that even Roman pipes are still a contested issue. Any objections? MatthewVanitas (talk) 19:43, 25 April 2020 (UTC)
 * Contrarily, I have read in more than one place (in the 1980s to 1990s, in books I no longer have) that the bagpipe, originally of Middle Eastern origin, was spread throughout Europe by the Roman army. There's clearly more than one hypothesis, and we need to account for them, with sources.  — SMcCandlish ☏ ¢ 😼  21:08, 8 May 2023 (UTC)

wicklow pipes
no mention of this - whuich wer enmost liekly bagpuipes from 2000bc:

Bernice Molly at Greystones, Co. Wicklow. It consists of six carefully worked wooden pipe — Preceding unsigned comment added by 62.56.110.121 (talk) 10:13, 18 April 2024 (UTC)

 * That's an excellent lead, thank you! However, this research paper is very very careful to say that there is no indication what sort of instrument these pipes were part of .  https://www.ancientmusicireland.com/wp-content/uploads/2020/11/Holmes_Molloy_paper_Wicklow_Pipes.pdf  unless we have a source that definitively says they were 'most likely' bagpipes, we can't include it in this article. Jeff UK  10:21, 18 April 2024 (UTC)

What notes can it play?
It would be good if the article gave the notes that a bagpipe can play. I've seen sources that say that it goes from the G above Middle C to the A an octave higher, in Dorian Mode. I don't have a good reference. Bubba73 You talkin' to me? 04:21, 30 April 2022 (UTC)
 * This article is about a fairly disparate family of instruments, thus with a variety of ranges and notes available, dealt with in the specific articles. You're referring to the nominal range of the Great Highland Bagpipe, though they tend to be about 3/4s of tone sharp relative to concert pitch. Mutt Lunker (talk) 09:10, 30 April 2022 (UTC)

 * Thanks, and that is Mixolydian, not Dorian (as I thought). Bubba73 You talkin' to me? 01:42, 3 May 2022 (UTC)

 * My daughter plays chanter (I assume of this type) and she said that it can't play Happy Birthday to You in any key. But it looks it can play it in A.  Bubba73 You talkin' to me? 01:53, 3 May 2022 (UTC)

Merge from Gaida
Gaida seems to be just a word for bagpipes used in several European languages. I can't find anything to show this term is differentiated by scholars (what makes gaidas different from bagpipes except region of origin?). In other words, gaida seems to be a WP:POVFORK of dubious stand-alone WP:GNG. A merge and redirect seems warranted. PS. While arguably different typs of bagpipes, culturally connected to different regions, warrant subarticles (Kaba gaida, or Great Highland bagpipe), I can't find any sources that say that "gaidas" are a specific subtype of bagpipes. It's just just a word for "bagpipe" used in the Balkan region. PS. This merge proposal is inspired by the followng discussion on pl wiki, where several users expressed the sentiment I summarized above: pl:Wikipedia:Poczekalnia/artykuły/2022:07:14:Gajda_(instrument_muzyczny). Piotr Konieczny aka Prokonsul Piotrus| reply here 10:00, 20 July 2022 (UTC)
 * I'm uncertain about this. Greek WP has two separate articles for bagpipe and gaida (Άσκαυλος, Γκάιντα), as does Bulgarian (Волинка, Гайда), implying that these are considered two separate concepts in (at least those two) cultures where the gaida exists. Doremo (talk) 12:16, 5 August 2022 (UTC)
 * Support, Gaida does not meet the General notability guideline as it has very few reliable secondary sources, if any at all Quinnerwinner12 (talk) 12:47, 26 February 2023 (UTC)

 * ✅ Klbrain (talk) 09:00, 7 April 2023 (UTC)

title: Talk:Bagpipes


text:
Bedrock Records

Bedrock Records is an English record label for trance, progressive house and techno started by John Digweed. Its name comes from a long running and successful club night held in Hastings and also at Heaven nightclub, London - both also called Bedrock. Bedrock Records has released many singles from artists such as Astro & Glyde, Brancaccio & Aisher, Steve Lawler, Shmuel Flash, Steve Porter, Sahar Z, Guy J, Henry Saiz, Stelios Vassiloudis, Electric Rescue, The Japanese Popstars and Jerry Bonham. Bedrock is also the name that Digweed and Muir use as their production moniker.

Bedrock has had different imprints: Bedrock Breaks, B_Rock and Black (Bedrock). Currently it has Bedrock Digital and one called Lost & Found belonging to Guy J.

The first Bedrock album compiled and mixed by John Digweed was released in 1999, containing several tracks signed to the Bedrock label.

In 2018, Digweed marked the 20th anniversary of the label with the release of Bedrock XX.

title: Bedrock Records