In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.

Most of this article is about real-valued functions of one real variable. A discussion of the multivariable case is presented towards the end.

Differentiability classes

Consider an open set on the real line and a function ''f'' defined on that set with real values. Let ''k'' be a non-negative integer. The function ''f'' is said to be of class ''C^k'' if the derivatives ''f'', ''f'''', ..., ''f^(k)'' exist and are continuous (the continuity is automatic for all the derivatives except for ''f^(k)''). The function ''f'' is said to be of class ''C^∞'', or smooth, if it has derivatives of all orders. ''f'' is said to be of class ''C^ω'', or analytic, if ''f'' is smooth and if it equals its Taylor series expansion around any point in its domain.

To put it differently, the class ''C''⁰ consists of all continuous functions. The class ''C''¹ consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a ''C''¹ function is exactly a function whose derivative exists and is of class ''C''⁰. In general, the classes ''C^k'' can be defined recursively by declaring ''C''⁰ to be the set of all continuous functions and declaring ''C^k'' for any positive integer ''k'' to be the set of all differentiable functions whose derivative is in ''C''^''k''−1. In particular, ''C^k'' is contained in ''C''^''k''−1 for every ''k'', and there are examples to show that this containment is strict. ''C''^∞ is the intersection of the sets ''C^k'' as ''k'' varies over the non-negative integers. ''C''^ω is strictly contained in ''C''^∞; for an example of this, see bump function or also below.

Examples

The function

:

is continuous, but not differentiable at $\backslash scriptstyle\; x=0$, so it is of class ''C⁰'' but not of class ''C¹''.

The function : is differentiable, with derivative : Because cos(1/''x'') oscillates as ''x'' approaches zero, ''f'' ’(''x'') is not continuous at zero. Therefore, this function is differentiable but not of class ''C¹''. Moreover, if one takes ''f''(''x'') = ''x^3/2'' sin(1/''x'') (''x'' ≠ 0) in this example, it can be used to show that the derivative function of a differentiable function can be unbounded on a compact set and, therefore, that a differentiable function on a compact set may not be locally Lipschitz continuous.

The functions

: $f(x)=|x|^\{k+1\}$

where ''k'' is even, are continuous and ''k'' times differentiable at all ''x''. But at $\backslash scriptstyle\; x=0$ they are not (''k''+1) times differentiable, so they are of class ''C ^k'' but not of class ''C ^j'' where j>k.

The exponential function is analytic, so, of class ''C^ω''. The trigonometric functions are also analytic wherever they are defined.

The function :

is smooth, so of class ''C^∞'', but it is not analytic at $\backslash scriptstyle\; x=\backslash pm\; 1$, so it is not of class ''C''^ω. The function ''f'' is an example of a smooth function with compact support.

Multivariate differentiability classes

Let ''n'' and ''m'' be some positive integers. If ''f'' is a function from an open subset of R^''n'' with values in R^''m'', then ''f'' has component functions ''f''₁, ..., ''f''_''m''. Each of these may or may not have partial derivatives. We say that ''f'' is of class ''C^k'' if all of the partial derivatives $\backslash frac\{\backslash partial^l\; f\_i\}\{\backslash partial\; x\_\{i\_1\}^\{l\_1\}\backslash partial\; x\_\{i\_2\}^\{l\_2\}\backslash cdots\backslash partial\; x\_\{i\_n\}^\{l\_n\}\}$ exist and are continuous, where each of $i,\; i\_1,\; i\_2,\; \backslash ldots,\; i\_k$ is an integer between 1 and ''n'', each of $l,\; l\_1,\; l\_2,\; \backslash ldots\; l\_n$ is an integer between 0 and ''k'', $l\_1+l\_2+\backslash cdots\; +l\_n=l$. The classes ''C^∞'' and ''C^ω'' are defined as before.

These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space is called a ''C''^''k'' manifold.

If one wishes to start with a coordinate-independent definition of the class ''C^k'', one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is an affine map which approximates it at that point. The derivative of the map assigns to the point ''x'' the linear part of the affine approximation to the map at x. Since the space of linear maps from one Banach space to another is again a Banach space, we may continue this procedure to define higher order derivatives. A map ''f'' is of class ''C^k'' if it has continuous derivatives up to order ''k'', as before.

Note that R^''n'' is a Banach space for any value of ''n'', so the coordinate-free approach is applicable in this instance. It can be shown that the definition in terms of partial derivatives and the coordinate-free approach are equivalent; that is, a function ''f'' is of class ''C^k'' by one definition iff it is so by the other definition.

The space of ''C''^''k'' functions

Let ''D'' be an open subset of the real line. The set of all ''C^k'' functions defined on $\backslash scriptstyle\; D$ and taking real values is a Fréchet space with the countable family of seminorms

: $p\_\{K,\; m\}=\backslash sup\_\{x\backslash in\; K\}\backslash left|f^\{(m)\}(x)\backslash right|$

where ''K'' varies over an increasing sequence of compact sets whose union is ''D'', and ''m'' = 0, 1, …, ''k''.

The set of ''C^∞'' functions over $\backslash scriptstyle\; D$ also forms a Fréchet space. One uses the same seminorms as above, except that $\backslash scriptstyle\; m$ is allowed to range over all non-negative integer values.

The above spaces occur naturally in applications where functions having derivatives of certain orders are necessary; however, particularly in the study of partial differential equations, it can sometimes be more fruitful to work instead with the Sobolev spaces.

Parametric continuity

Parametric continuity is a concept applied to parametric curves describing the smoothness of the parameter's value with distance along the curve.

Definition

A curve can be said to have ''C''^''n'' continuity if

:$\backslash frac\{d^n\; s\}\{dt^n\}$

is continuous of value throughout the curve.

As an example of a practical application of this concept, a curve describing the motion of an object with a parameter of time, must have ''C''¹ continuity for the object to have finite acceleration. For smoother motion, such as that of a camera's path while making a film, higher levels of parametric continuity are required.

Order of continuity

The various order of parametric continuity can be described as follows:

''C''⁻¹: curves include discontinuities

''C''⁰: curves are joined

''C''¹: first derivatives are equal

''C''²: first and second derivatives are equal

''Cⁿ'': first through ''n''^th derivatives are equal

The term ''parametric continuity'' was introduced to distinguish it from ''geometric continuity'' (''Gⁿ'') which removes restrictions on the speed with which the parameter traces out the curve.

Geometric continuity

The concept of geometrical or geometric continuity was primarily applied to the conic sections and related shapes by mathematicians such as Leibniz, Kepler, and Poncelet. The concept was an early attempt at describing, through geometry rather than algebra, the concept of continuity as expressed through a parametric function.

The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An ellipse tends to a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines. Thus, there was ''continuity'' between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle of infinite radius. For such to be the case, one would have to make the line closed by allowing the point ''x'' = ∞ to be a point on the circle, and for ''x'' = +∞ and ''x'' = −∞ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of ∞.

Smoothness of curves and surfaces

A curve or surface can be described as having ''Gⁿ'' continuity, ''n'' being the increasing measure of smoothness. Consider the segments either side of a point on a curve:

''G''⁰: The curves touch at the join point.

''G''¹: The curves also share a common tangent direction at the join point.

''G''²: The curves also share a common center of curvature at the join point.

In general, ''Gⁿ'' continuity exists if the curves can be reparameterized to have ''C''^''n'' (parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions $\backslash scriptstyle\; f(s)$ and $\backslash scriptstyle\; g(t)$ have ''Gⁿ'' continuity if $\backslash scriptstyle\; f^\{(n)\}(t)\; \backslash ,\backslash neq\backslash ,\; 0$ and $\backslash scriptstyle\; f^\{(n)\}(t)\; \backslash ,=\backslash ,\; kg^\{(n)\}(t)$, for a scalar $\backslash scriptstyle\; k\; \backslash ,>\backslash ,\; 0$ (i.e., if the direction, but not necessarily the magnitude, of the two vectors is equal).

While it may be obvious that a curve would require ''G''¹ continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has ''G''² continuity.

A ''rounded rectangle'' (with ninety degree circular arcs at the four corners) has ''G''¹ continuity, but does not have ''G''² continuity. The same is true for a ''rounded cube'', with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with ''G''² continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.

Smoothness

Relation to analyticity

While all analytic functions are smooth on the set on which they are analytic, the above example shows that the converse is not true for functions on the reals: there exist smooth real functions which are not analytic. For example, the Fabius function is smooth but not analytic at any point. Although it might seem that such functions are the exception rather than the rule, it turns out that the analytic functions are scattered very thinly among the smooth ones; more rigorously, the analytic functions form a meagre subset of the smooth functions. Furthermore, for every open subset A of the real line, there exist smooth functions which are analytic on A and nowhere else.

It is useful to compare the situation to that of the ubiquity of transcendental numbers on the real line. Both on the real line and the set of smooth functions, the examples we come up with at first thought (algebraic/rational numbers and analytic functions) are far better behaved than the majority of cases: the transcendental numbers and nowhere analytic functions have full measure (their complements are meagre).

The situation thus described is in marked contrast to complex differentiable functions. If a complex function is differentiable just once on an open set it is both infinitely differentiable and analytic on that set.

Smooth partitions of unity

Smooth functions with given closed support are used in the construction of smooth partitions of unity (see ''partition of unity'' and topology glossary); these are essential in the study of smooth manifolds, for example to show that Riemannian metrics can be defined globally starting from their local existence. A simple case is that of a bump function on the real line, that is, a smooth function ''f'' that takes the value 0 outside an interval [''a'',''b''] and such that

:

Given a number of overlapping intervals on the line, bump functions can be constructed on each of them, and on semi-infinite intervals (-∞, ''c''] and [''d'',+∞) to cover the whole line, such that the sum of the functions is always 1.

From what has just been said, partitions of unity don't apply to holomorphic functions; their different behavior relative to existence and analytic continuation is one of the roots of sheaf theory. In contrast, sheaves of smooth functions tend not to carry much topological information.

Smooth functions between manifolds

Smooth maps between smooth manifolds may be defined by means of charts, since the idea of smoothness of function is independent of the particular chart used. If ''F'' is a map from an ''m''-manifold ''M'' to an ''n''-manifold ''N'', then ''F'' is smooth if, for every $\backslash scriptstyle\; p\; \backslash in\; M$, there is a chart $\backslash scriptstyle\; (U,\; \backslash varphi)$ in ''M'' containing ''p'' and a chart $\backslash scriptstyle\; (V,\; \backslash psi)$ in ''N'' containing ''F''(''p'') with $\backslash scriptstyle\; F(U)\; \backslash subset\; V$, such that $\backslash scriptstyle\backslash psi\backslash circ\; F\; \backslash circ\; \backslash varphi^\{-1\}$ is smooth from $\backslash scriptstyle\backslash varphi(U)$ to $\backslash scriptstyle\backslash psi(V)$ as a function from $\backslash scriptstyle\backslash mathbb\{R\}^m$ to $\backslash scriptstyle\backslash mathbb\{R\}^n$.

Such a map has a first derivative defined on tangent vectors; it gives a fibre-wise linear mapping on the level of tangent bundles.

Smooth functions between subsets of manifolds

There is a corresponding notion of smooth map for arbitrary subsets of manifolds. If $\backslash scriptstyle\; f\; :\; X\; \backslash to\; Y$ is a function whose domain and range are subsets of manifolds $\backslash scriptstyle\; X\; \backslash subset\; M$ and $\backslash scriptstyle\; Y\; \backslash subset\; N$ respectively. $\backslash scriptstyle\; f$ is said to be smooth if for all $\backslash scriptstyle\; x\; \backslash in\; X$ there is an open set $\backslash scriptstyle\; U\; \backslash subset\; M$ with $\backslash scriptstyle\; x\; \backslash in\; U$ and a smooth function $\backslash scriptstyle\; F\; :\; U\; \backslash to\; N$ such that $\backslash scriptstyle\; F(p)=f(p)$ for all $\backslash scriptstyle\; p\; \backslash in\; U\; \backslash cap\; X$.

See also

Non-analytic smooth function

Quasi-analytic function

References

Guillemin, Pollack. Differential Topology. Prentice-Hall (1974).

Category:Mathematical analysis Category:Algebraic geometry Category:Differential calculus Category:Smooth functions Category:Differential structures Category:Types of functions

de:Glatte Funktion es:Función continuamente diferenciable fr:Classe de régularité ko:매끈한 함수 is:Þjált fall it:Funzione liscia he:פונקציה חלקה kk:Сыптығыр функция nl:Gladde functie ja:滑らかな関数 no:Glatt funksjon pl:Regularność funkcji pt:Função suave ru:Гладкая функция fi:Sileä funktio sv:Glatt funktion uk:Гладка функція zh:光滑函数

Coordinates	52°10′″N9°45′″N
name	Pete Rock
background	solo_singer
birth name	Peter Phillips
alias	Soul Brother #1The Chocolate Boy Wonder
born	June 21, 1970New York City, New York, United States
origin	Mount Vernon, New York, United States
instrument	Turntable, sampler, keyboard, bass guitar
genre	East Coast hip hop, jazz rap
occupation	Music producer, disc jockey, rapper, singer
years active	1987–present
label	Elektra, Loud/RCA/BMG, Rapster/BBE, Nature Sounds/Caroline/Virgin/EMI
associated acts	CL Smooth, Marley Marl, INI, Heavy D, The UN, YGz, DJ Premier, J Dilla, 9th Wonder, DJ Green Lantern, Rakim, Nas, Wu-Tang Clan, Pharoahe Monch, Kanye West
website	}}

Peter Phillips (born June 21, 1970), better known by his stage name Pete Rock, is an American record producer, DJ and rapper. He rose to prominence in the early 1990s as one half of the critically acclaimed group Pete Rock & CL Smooth. After the duo went their separate ways, Rock continued with a solo career that has garnered him worldwide respect, though little in the way of mainstream success. Along with groups such as Stetsasonic, A Tribe Called Quest, The Roots and Gang Starr, Rock played a major role in the merging of elements from jazz into hip hop music (also known as jazz rap). He is widely recognized as one of the greatest hip hop producers of all time, and is often mentioned alongside DJ Premier and RZA as one of the mainstays of 1990s East Coast hip hop production. Pete Rock is also the older brother and younger cousin, respectively, of rappers Grap Luva and Heavy D.

Biography

Early life

Pete Rock was born in The Bronx, New York, the sixth of seven children of Jamaican immigrant parents. His family moved to Mount Vernon, New York when he was seven years old. During high school, he met his future recording partner CL Smooth. According to Rock, his father was also a part-time DJ who had an impressive record collection. Rock would often accompany his father to a cricket club called Wembley in The Bronx and watch as he spun records for the guests. His first job was as a paperboy, in his neighborhood.

Early career

His first major exposure to the hip-hop audience was in 1987 with Marley Marl, as a DJ on New York's WBLS radio show "In Control With Marley Marl." The 17-year-old Phillips was recognized by the listening audience as the man "puttin' in work", as he would use double copies of each record to cut up every song he played, when most New York DJs would only use double copies on every third or fourth song. Propelled by the growth of his popularity, he began producing in the early 1990s. In 1991 he joined with CL Smooth to release the EP ''All Souled Out''.

The following year saw ''Mecca and the Soul Brother'', the first of two full-length albums, released to critical acclaim and hailed as a classic by many. During this period, Pete Rock began to produce songs for other acts, such as "Down With the King" for Run-DMC and "The World Is Yours" for Nas, as well remix singles for Jeru the Damaja ("You Can't Stop the Prophet"), Public Enemy ("Shut 'em Down"), and The Notorious B.I.G. ("Juicy" - the original of which was allegedly based on Rock's own demo). Pete and CL followed up ''Mecca'' in 1994 with ''The Main Ingredient''. Like its predecessor, ''The Main Ingredient'' also received wide critical acclaim. However, soon after the album's release, the duo split and went their separate ways, with Rock focusing on production work for other artists.

In 1995, he formed the group INI, with Grap Luva, Rob-O, and I Love H.I.M. and released the classic 12" "Fakin' Jax". The trio recorded an album, ''Center of Attention''(originally named ''The Life I Live''), which was heavily bootlegged and remained unreleased until 2003, when BBE Records picked it up for distribution. The release of this album was coupled with another previously unreleased mid-'90s Pete Rock-produced album, DeDa's ''The Original Baby Pa''.

In 1996, he appeared on the Red Hot Organization's compilation CD, America is Dying Slowly, alongside Wu-Tang Clan, Coolio, and Fat Joe, among others. The CD, meant to raise awareness of the AIDS epidemic among African American men, was heralded as a masterpiece by The Source magazine.

Solo career

While working as a DJ with Marley Marl on Hot 97's Future Flavas show, Rock was able to forge a relationship with Loud Records, allowing him to release his solo album ''Soul Survivor'' in the summer of 1998. After being dropped from the Loud imprint, Rock signed with Rapster/BBE under his own label Soul Brother Recordings. His first release on the imprint was ''PeteStrumentals'', which was mostly made up of previously unreleased Rock beats; all recorded between 1990 and 1995 but remixed and mastered for the 2001 BBE Project. The album did however feature two additional 2001 songs that featured the vocals from The UN. An expanded and revisited version of the album was released less than one year later; three instrumentals had been replaced with three new songs performed by C.L. Smooth, Freddie Foxxx and Nature separately.

Since then he has continued releasing solo albums including 2004's ''Soul Survivor II''. In that same year he also produced the bulk of Edo G's ''My Own Worst Enemy'', as well as a track for the all-girl rap group Northern State.

Pete Rock returned to greater visibility through a closer relationship with the Wu-Tang Clan. Their collaborations began with ''Soul Survivor'', which featured Inspectah Deck, Method Man, Ghostface Killah and Raekwon on various tracks as well as producing a track each for Inspectah Deck's ("Trouble Man") and Raekwon's ("Sneakers") respective 1999 albums. Rock continued to work with the Wu on ''Soul Survivor II'' which featured both GZA and RZA. In 2006 he also produced three tracks for Ghostface's release ''Fishscale'', including the first single, and Nature Sounds labelmate Masta Killa's second album, ''Made in Brooklyn''.

Pete Rock has cultivated a relationship with Brooklyn-based Nature Sounds Records. He recorded the song "The PJs", which also features Wu-Tang Clan's Raekwon and Masta Killa, released on the Nature Sounds compilation ''Natural Selection''. He released an album entitled ''NY's Finest'' in February 2008, featuring Raekwon, Masta Killa, D-Block, Redman, Papoose, Slum Village and Jim Jones, among others, also on Nature Sounds. The lead single from the album is entitled "914" and features rappers Sheek Louch and Styles P. He is scheduled to appear on albums by Bishop Lamont, Cannibal Ox, Termanology, Freddie Foxxx, Royal Flush, Verbal Threat, LL Cool J, JoJo Pellegrino, La The Darkman, Cormega, R.A. The Rugged Man, and others. He is also working with DJ/Rapper Doo Wop under the name Tango & Cash (taken from the film of the same name).

Planet Asia has recently claimed that later on in 2009 he will release a collaboration fully produced album with Pete Rock called "Planet Rock". On December 2009 Rock denied that, but talked about collabo projects with Tha Dogg Pound, Camp Lo, Smif-N-Wessun.

Pete Rock oversaw the production of ''Jay Stay Paid'', a posthumous album by the producer J Dilla, released June 2, 2009, on Nature Sounds. Lately he has been in Hawaii, working with Kanye West on the latter's fifth album, ''My Beautiful Dark Twisted Fantasy''. In addition, he and DJ Premier have announced that they are working on a joint album together, although further details are unknown. Recently in London he confirmed that Big Pooh & C.L. Smooth will be on his half of the VS album and he plans on dropping 5 albums in 2011 including reuniting with C.L. Smooth for a third album & drop his 4th album on Nature Sounds. His next few collaborative albums are both due for a summer release with Monumental first then with Camp Lo's "80 Blocks From Tiffanys" LP. In an April 2011 interview on Conspiracy Worldwide Radio, Pete Rock discussed his new solo work including his album with DJ Premier, as well as exploring the fact that he has had numerous beats rejected by Eminem over the years Pete Rock Uncensored Radio Interview

Affiliates

Proteges

Through the years, Rock has helped to jump-start the careers of several artists. His first project outside of Pete Rock & CL Smooth was the hardcore duo YG'z, who released an EP called ''Street Nigga'' in 1993, with four out of the six tracks produced by Rock; however, they were quickly dropped from their deal with Reprise Records. His next venture, INI, was a group featuring Rock, his younger brother Grap Luva, Rass and rapper Rob-O. They released a single, "Fakin' Jax", through Elektra Records in 1995, before their debut album, ''Center of Attention'', was shelved by the label. The other two members continue to record solo material, albeit only sporadically. In an interview Rock elaborated on the situation:

Another mid-1990s artist, DeDa, also met the same fate with his album ''The Original Baby Pa'', although both this and INI's album were eventually released as a double album package in 2003. Other associated artists include Meccalicious, who recorded a few songs under Pete Rock's guidance (sometime around 1997), before disappearing from the music scene altogether.

Rock has had some success, however, overseeing and jump-starting the career of hardcore underground favorites The UN, a group featuring four MCs, including former Flipmode Squad member Rock Marciano. Rock premiered the group on the single "Nothin' Lesser" from his ''PeteStrumentals'' album, and they went on to release the fairly successful album ''UN or U Out'' in July 2004, featuring production by Rock, Large Professor, and several others.

Relationship with CL Smooth

Since their split in 1995, Pete Rock's relationship with CL Smooth has been highly unpredictable. Although the pair briefly united for the reflective "Da Two" from Rock's ''Soul Survivor'' album in 1998, they avoided entertaining requests for a reunion album until 2001, when they once again teamed up for "Back on Da Block" from Rock's ''PeteStrumentals''. In their interviews during this period, it appeared as though a new album was underway. As Rock would explain:

The pair went on a short international tour culminating in their well-received show at London's Jazz Cafe; however, soon after this they declined to comment any further on the new album, which never materialized (although Smooth did make three separate appearances on ''Soul Survivor II''). Eventually, Smooth would confirm rumors of a rift in an interview with AllHipHop.com, in which he appeared angry and frustrated with his former partner, saying "I didn’t ask him to be a superhero" and "I’m not the problem." In an interview taken in December 2006, Rock ruled out any further collaborations with Smooth but stated that he holds no grudges against his former partner. He recently confirmed that he will be recording a third album with C.L. Smooth.

Musical style

Production

Pete Rock builds his beats from samples, the majority of which are taken from obscure R&B;, funk, and jazz records. Early on in his career he would also sample drum breaks such as Black Heat's "Zimba Ku" for Heavy D & The Boyz's "Letter To The Future". Pete Rock heavily used the E-mu SP-1200 as well as the AKAI [S950]—later moving onto using the MPC—for his productions. Pete Rock tends to use the samples as palettes for his beats, chopping (cutting the sample into smaller parts), filtering (altering the frequencies of the sample), and layering several samples, often within the same song. While this technique was applied long before Rock (on De La Soul's ''Three Feet High and Rising'' or the work of The Bomb Squad for example), Rock's work is distinctive for the way in which he uses samples to achieve a hazy, droning effect. He is also noted for his resonant basslines, horn samples, and gritty sounding drums. His beats often sound as though they were being played from an old vinyl record; he samples many of his sounds straight off these records.

Another trait of his, more so in the earlier part of his career, is the way he uses horn samples to supplement his grooves. With perhaps the most famous example being "They Reminisce Over You (T.R.O.Y.)" (on which he uses a horn sample from Tom Scott's "Today"), Rock has also used horns on several other productions such as "Straighten It Out", Public Enemy's "Shut 'Em Down", Rah Digga's "What They Call Me", and A.D.O.R.'s "Let It All Hang Out".

Along with Gang Starr, The Roots and A Tribe Called Quest, Pete Rock played a large role in the fusing of jazz and funk music into Hip hop. The aforementioned "Reminisce..." withstanding, Rock used many jazz samples on his album ''Mecca and The Soul Brother'', such as Cannonball Adderley's "Country Preacher", for the song "Return of the Mecca", or "Capricorn" for the song "In the House" from ''The Main Ingredient''. Pete Rock's heavy use of intro and outro beats has also been widely influential. To introduce feature songs, he often plays a short instrumental excerpt, completely different from the rest of the song. Aside from their role as transitions, these are widely regarded as a way of displaying his large collection and as a challenge to other hip-hop producers to identify the records that the breaks come from. ''Mecca & the Soul Brother'' and ''The Main Ingredient'' use intro/outro beats on nearly every track to great effect, and the tradition continues to the present on Rock's recent releases.

Remixes

"Another Pete Rock Remix" is Pete Rock's trademark catchphrase, heard on countless singles which he has remixed. In addition to Hip-hop artists he has done remix work for artists from other genres such as his 1995 remix of "Before You Walk Out Of My Life" for R&B; singer Monica. In 1992 he collaborated with Mary J. Blige on the ''What's the 411?'' single "Reminisce", which utililized the same sample from his own single "T.R.O.Y. (They Reminisce Over You)". Rock claims to have done several high profile remixes which remain unreleased including one of Madonna's "Secret". He also claims to have produced the original beat for The Notorious B.I.G.'s "Juicy", and that it was recreated by P. Diddy and Poke (of Tone & Poke fame), without consent. However, he was invited to produce the remix, which utilizes the same sample as the original - Mtume's "Juicyfruit". Although he received no official producer credit, he made the original demo beat for A Tribe Called Quest's "Jazz (We've Got)", which was then recreated by the group for the album ''The Low End Theory''. He remixed Public Enemy's "Shut 'em Down" and "Nighttrain", in the same day, starting at 12pm and finishing at 12am.

Up until 2003, he created all of his productions on the E-mu SP1200, thereafter using the AKAI MPC2000XL. He also has a collection of about 90,000 records and looks for records at least once a week. Pete Rock was one of 9 artists who participated in thetruth.com’s Remix Project, where he remixed the Sunny Side song “Magical Amount”.

Influence

Pete Rock has had a considerable impact on a number of record producers who have emerged in the hip hop scene since the late 1990s. Many critics have compared newcomers to him, generally favorably. Perhaps most notable among these comparisons are the late Detroit producer J Dilla, and North Carolina's 9th Wonder, both of whom have worked with Rock during their recording careers. Several of the comparisons stem from the fact that these producers have created the bulk of their productions out of samples, as well as the warm, mellow, and exuberant undertones apparent in their work. Pete Rock himself has added validation to the comparisons with J Dilla by stating "he's the only producer in this game that was just as serious [as me]."

Many other producers, including Kev Brown and Kanye West, have also found themselves compared to Pete Rock, with the latter glowingly referring to himself as "the new version of Pete Rock" on Slum Village's "Selfish" from the group's 2004 album ''Detroit Deli (A Taste of Detroit)''. Pete Rock has acknowledged his relevance to these artists, releasing an instrumental record with 9th Wonder (''Class Is in Session''), as well as recording his own remix of West's "Heard 'Em Say". Rock himself has named his main musical influences as being influential hip hop producer and close friend Marley Marl and legendary soul musician James Brown.

Discography

; Studio albums

''Soul Survivor'' (1998)

''PeteStrumentals'' (2001)

''Lost & Found: Hip Hop Underground Soul Classics'' (2003)

''Soul Survivor II'' (2004)

''The Surviving Elements: From Soul Survivor II Sessions'' (2005)

''NY's Finest'' (2008)

''PeteStrumentals: 10th Anniversary Expanded & Limited Edition'' (2011)

References

External links

Pete Rock — official website

Pete Rock discography at Discogs

Pete Rock at MySpace

Pete Rock Interview - February 2008

Pete Rock Interview Part 1

Pete Rock Interview Part 2

Riot Sound, Hip-Hop, Rap & Urban Culture - Pete Rock Interview

Interview with Pete Rock

Category:1970 births Category:Living people Category:African American rappers Category:Rappers from New York City Category:American hip hop record producers Category:American rappers of Jamaican descent Category:People from Westchester County, New York Category:Members of the Nation of Gods and Earths

da:Pete Rock de:Pete Rock es:Pete Rock fr:Pete Rock it:Pete Rock ja:ピート・ロック pl:Pete Rock pt:Pete Rock fi:Pete Rock sv:Pete Rock tr:Pete Rock

Coordinates	52°10′″N9°45′″N
name	C.L. Smooth
background	solo_singer
birth name	Corey Brent Penn
alias	Caramel KingThe Mecca DonC.L.El Presidente
birth date	October 08, 1968
origin	New Rochelle, New York, United States
genre	Rap, East Coast hip hop
occupation	Rapper, Producer
years active	1991–present
label	Elektra Records (1991–1995)Shaman Work/Koch (2006–present)
associated acts	Pete RockHeavy DAZPete Rock & C.L. SmoothNujabesDJ Krush }}

CL Smooth (born Corey Brent Penn, Sr., October 8, 1968, New Rochelle, New York) is an American rapper, poet, and is best known as the vocal half of the hip-hop duo Pete Rock & CL Smooth.

After the pair split in 1995 Pete Rock, whose work with CL had garnered him a reputation as one of the finest producers in hip-hop, went on to produce (and remix) tracks for dozens of marquee-name and/or well-respected acts, and to release a batch of solo and instrumental albums. C.L. Smooth remained musically inactive.

The pair managed to reconcile their differences only long enough to record five isolated tracks in the ten years since their split: "Da Two," "Back On Da Block," "Fly 'Til I Die," "Appreciate," and "It's A Love Thing." There were talks of a definitive reunion and a new album in 2004; however, the speculation was dismissed by both parties after another fallout.

During his period of hiatus, C.L. Smooth declined even to appear as a guest on other artists' albums, with the exception of "Only the Strong Survive" with DJ Krush in 1996. Recently, however, he has contributed featured guest vocals to songs by AZ ("Magic Hour"), Nujabes ("Sky is Falling") and Dipset affiliate J. R. Writer (the remix of "Mesmerize") or Supafuh ("Act of Faith").

He released his debut solo album, ''American Me'', in 2007. To promote the album, Shaman Work released a promotional mixtape compiled by DJ J Period called ''Man On Fire'' in late April 2006, featuring 36 minutes of freestyles and a remixed version of the track "Impossible". He was also been featured on the DJ Jazzy Jeff album ''The Return of the Magnificent'' on the track "All I Know", which was also featured in the video game NBA Live 08.

In October 2010 C.L. Smooth and Pete Rock headlined the "Clean Energy Tour," which had six shows all over California and focused on voter mobilization leading up to the November mid-term elections. He has developed an interest in climate change and become a vocal opponent of Proposition 23.

Discography

Solo albums

''American Me'' (2006)

''The Outsider'' (2007)

Featurings

Supafuh feat. CL Smooth ''Act Of Faith'' (2011)

Pete Rock & C.L. Smooth albums

! Album information
''Mecca and the Soul Brother''	*Released: June 9, 1992	*Billboard 200 chart position: #43	*R&B;/Hip-Hop chart position: #7	*Singles: "They Reminisce Over You (T.R.O.Y.)", "Straighten It Out" & "Lots of Lovin'"
	*Released: November 8, 1994	*Billboard 200 chart position: #51	*R&B;/Hip-Hop chart position: #9	*Singles: "I Got a Love", "CL Smooth at [[MySpace.com">Take You There (Pete Rock & CL Smooth song)

References

External links

CL Smooth at [[MySpace.com

Category:1968 births Category:Living people Category:Members of the Nation of Gods and Earths Category:African American rappers Category:Rappers from New York City Category:People from New Rochelle, New York Category:People from Westchester County, New York Category:Pete Rock

de:CL Smooth es:CL Smooth fr:CL Smooth

Coordinates	52°10′″N9°45′″N
alt	A mid-twenties African American man wearing a sequined military jacket and dark sunglasses. He is walking while waving his right hand, which is adorned with a white glove. His left hand is bare.
background	solo_singer
birth name	Michael Joseph Jackson
alias	Michael Joe Jackson, MJ, King of Pop
birth date	August 29, 1958
birth place	Gary, Indiana, U.S.
death date	June 25, 2009
death place	Los Angeles, California, U.S.
instrument	vocals, guitar, drums, percussion, keyboards
genre	R&B;, pop, rock, soul, dance, funk, disco, new jack swing
occupation	Singer-songwriter, musician, composer, dancer, choreographer, record producer, actor, businessman, philanthropist
years active	1964–2009
label	Motown, Epic, Legacy
associated acts	The Jackson 5
relatives	Janet Jackson (sister)
website	130pxMichael Jackson's signature }}

Michael Joseph Jackson (August 29, 1958 – June 25, 2009) was an American recording artist, entertainer, and businessman. Often referred to as the King of Pop, or by his initials MJ, Jackson is recognized as the most successful entertainer of all time by Guinness World Records. His contribution to music, dance, and fashion, along with a much-publicized personal life, made him a global figure in popular culture for over four decades. The seventh child of the Jackson family, he debuted on the professional music scene along with his brothers as a member of The Jackson 5, then the Jacksons in 1964, and began his solo career in 1971.

In the early 1980s, Jackson became a dominant figure in popular music. The music videos for his songs, including those of "Beat It", "Billie Jean", and "Thriller", were credited with transforming the medium into an art form and a promotional tool, and the popularity of these videos helped to bring the relatively new television channel MTV to fame. Videos such as "Black or White" and "Scream" made him a staple on MTV in the 1990s. Through stage performances and music videos, Jackson popularized a number of complicated dance techniques, such as the robot and the moonwalk, to which he gave the name. His distinctive musical sound and vocal style have influenced numerous hip hop, post-disco, contemporary R&B;, pop and rock artists.

Jackson's 1982 album ''Thriller'' is the best-selling album of all time. His other records, including ''Off the Wall'' (1979), ''Bad'' (1987), ''Dangerous'' (1991), and ''HIStory'' (1995), also rank among the world's best-selling. Jackson is one of the few artists to have been inducted into the Rock and Roll Hall of Fame twice. He was also inducted into the Dance Hall of Fame as the first (and currently only) dancer from the world of pop and rock 'n' roll. Some of his other achievements include multiple Guinness World Records; 13 Grammy Awards (as well as the Grammy Legend Award and the Grammy Lifetime Achievement Award); 26 American Music Awards (more than any other artist, including the "Artist of the Century"); 13 number-one singles in the United States in his solo career (more than any other male artist in the Hot 100 era); and the estimated sale of over 750 million records worldwide. Jackson won hundreds of awards, which have made him the most-awarded recording artist in the history of popular music.

Life and career

Early life and The Jackson 5 (1958–1975)

Michael Jackson was born on August 29, 1958, the eighth of ten children in an African American working-class family who lived in a small 3-room house in Gary, Indiana, an industrial city near Chicago. His mother, Katherine Esther Scruse, was a devout Jehovah's Witness, and his father, Joseph Walter "Joe" Jackson, was a steel mill worker who performed with an R&B; band called The Falcons. Jackson had three sisters: Rebbie, La Toya, and Janet, and five brothers: Jackie, Tito, Jermaine, Marlon, and Randy. A sixth brother, Brandon, died shortly after birth.

Jackson had a troubled relationship with his father, Joe. In 1980, Jackson won three awards at the American Music Awards for his solo efforts: Favorite Soul/R&B; Album, Favorite Soul/R&B; Male Artist, and Favorite Soul/R&B; Single for "Don't Stop 'Til You Get Enough". That year, he also won Billboard Year-End for Top Black Artist and Top Black Album and a Grammy Award for Best Male R&B; Vocal Performance, also for "Don't Stop 'Til You Get Enough". Jackson again won at the American Music Awards in 1981 for Favorite Soul/R&B; Album and Favorite Soul/R&B; Male Artist. Despite its commercial success, Jackson felt ''Off the Wall'' should have made a much bigger impact, and was determined to exceed expectations with his next release. In 1980, he secured the highest royalty rate in the music industry: 37 percent of wholesale album profit.

''Thriller'' and ''Motown 25'' (1982–83)

In 1982, Jackson contributed the song "Someone In the Dark" to the storybook for the film ''E.T. the Extra-Terrestrial''; the record won a Grammy for Best Recording for Children in 1984. In the same year he won another seven Grammys and eight American Music Awards (including the Award of Merit, the youngest artist to win it), making him the most awarded in one night for both award shows. The album topped the ''Billboard'' 200 chart for 37 weeks and was in the top 10 of the 200 for 80 consecutive weeks. It was the first album to have seven ''Billboard'' Hot 100 top 10 singles, including "Billie Jean", "Beat It," and "Wanna Be Startin' Somethin'." ''Thriller'' was certified for 29 million shipments by the RIAA, giving it Double Diamond status in the United States. The album won also another Grammy for Best Engineered Recording – Non Classical in 1984, awarding Bruce Swedien for his work. Thanks to the ''Bad'' album, Bruce Swedien and Humberto Gatica won one Grammy in 1988 for Best Engineered Recording – Non Classical and Michael Jackson won one Grammy for Best Music Video, Short Form for "Leave Me Alone" in 1989. ''HIStory'' received a Grammy nomination for Album of the Year. However, the sales for ''Invincible'' were lower than those of his previous releases, due in part to a lack of promotion, no supporting world tour and the label dispute. The album also came out at a bad time for the music industry in general. Jackson rehearsed in Los Angeles in the weeks leading up to the tour under the direction of choreographer Kenny Ortega. Most of these rehearsals took place at the Staples Center, which was owned by AEG. With ''Thriller'', Christopher Connelly of ''Rolling Stone'' commented that Jackson developed his long association with the subliminal theme of paranoia and darker imagery. Allmusic's Stephen Thomas Erlewine noted this is evident on the songs "Billie Jean" and "Wanna Be Startin' Somethin'". In "Billie Jean", Jackson sings about an obsessive fan who alleges he has fathered a child of hers. In "Wanna Be Startin' Somethin'" he argues against gossip and the media. "Beat It" decried gang violence in an homage to ''West Side Story'', and was Jackson's first successful rock cross-over piece, according to Huey. He also observed that the title track "Thriller" began Jackson's interest with the theme of the supernatural, a topic he revisited in subsequent years. In 1985, Jackson co-wrote the charity anthem "We Are the World"; humanitarian themes later became a recurring theme in his lyrics and public persona.

In ''Bad'', Jackson's concept of the predatory lover can be seen on the rock song "Dirty Diana". The lead single "I Just Can't Stop Loving You" is a traditional love ballad, while "Man in the Mirror" is an anthemic ballad of confession and resolution. "Smooth Criminal" was an evocation of bloody assault, rape and likely murder. Allmusic's Stephen Thomas Erlewine states that ''Dangerous'' presents Jackson as a very paradoxical individual. He comments the album is more diverse than his previous ''Bad'', as it appeals to an urban audience while also attracting the middle class with anthems like "Heal the World". The first half of the record is dedicated to new jack swing, including songs like "Jam" and "Remember the Time". The album is Jackson's first where social ills become a primary theme; "Why You Wanna Trip on Me", for example, protests against world hunger, AIDS, homelessness and drugs. ''Dangerous'' contains sexually charged efforts such as the multifaceted love song, "In the Closet". The title track continues the theme of the predatory lover and compulsive desire. The second half includes introspective, pop-gospel anthems such as "Will You Be There", "Heal the World" and "Keep the Faith"; these songs show Jackson opening up about various personal struggles and worries. In the ballad "Gone Too Soon", Jackson gives tribute to his friend Ryan White and the plight of those with AIDS.

''HIStory'' creates an atmosphere of paranoia. Its content focuses on the hardships and public struggles Jackson went through just prior to its production. In the new jack swing-funk-rock efforts "Scream" and "Tabloid Junkie", along with the R&B; ballad "You Are Not Alone", Jackson retaliates against the injustice and isolation he feels, and directs much of his anger at the media. In the introspective ballad "Stranger in Moscow", Jackson laments over his "fall from grace", while songs like "Earth Song", "Childhood", "Little Susie" and "Smile" are all operatic pop pieces. In the track "D.S.", Jackson launched a verbal attack against Tom Sneddon. He describes Sneddon as an antisocial, white supremacist who wanted to "get my ass, dead or alive". Of the song, Sneddon said, "I have not—shall we say—done him the honor of listening to it, but I've been told that it ends with the sound of a gunshot". ''Invincible'' found Jackson working heavily with producer Rodney Jerkins. It is a record made up of urban soul like "Cry" and "The Lost Children", ballads such as "Speechless", "Break of Dawn" and "Butterflies" and mixes hip-hop, pop and R&B; in "2000 Watts", "Heartbreaker" and "Invincible".

Vocal style

Jackson sang from childhood, and over time his voice and vocal style changed noticeably. Between 1971 and 1975, Jackson's voice descended from boy soprano to high tenor. Jackson first used a technique called the "vocal hiccup" in 1973, starting with the song "It's Too Late to Change the Time" from The Jackson 5's ''G.I.T.: Get It Together'' album. Jackson did not use the hiccup technique— somewhat like a gulping for air or gasping— fully until the recording of ''Off the Wall'': it can be seen in full force in the "Shake Your Body (Down to the Ground)" promotional video. With the arrival of ''Off the Wall'' in the late 1970s, Jackson's abilities as a vocalist were well regarded. At the time, ''Rolling Stone'' compared his vocals to the "breathless, dreamy stutter" of Stevie Wonder. Their analysis was also that "Jackson's feathery-timbred tenor is extraordinarily beautiful. It slides smoothly into a startling falsetto that's used very daringly". 1982 saw the release of ''Thriller'', and ''Rolling Stone'' was of the opinion that Jackson was then singing in a "fully adult voice" that was "tinged by sadness".

A distinctive deliberate mispronunciation of "come on", used frequently by Jackson, occasionally spelled "cha'mone" or "shamone", is also a staple in impressions and caricatures of him. The turn of the 1990s saw the release of the introspective album ''Dangerous''. ''The New York Times'' noted that on some tracks, "he gulps for breath, his voice quivers with anxiety or drops to a desperate whisper, hissing through clenched teeth" and he had a "wretched tone". When singing of brotherhood or self-esteem the musician would return to "smooth" vocals. When commenting on ''Invincible'', ''Rolling Stone'' were of the opinion that—at the age of 43—Jackson still performed "exquisitely voiced rhythm tracks and vibrating vocal harmonies". Nelson George summed up Jackson's vocals by stating "The grace, the aggression, the growling, the natural boyishness, the falsetto, the smoothness—that combination of elements mark him as a major vocalist".

Music videos and choreography

Referred to as the King of Music Videos, Steve Huey of Allmusic observed how Jackson transformed the music video into an art form and a promotional tool through complex story lines, dance routines, special effects and famous cameo appearances; simultaneously breaking down racial barriers. Before ''Thriller'', Jackson struggled to receive coverage on MTV, allegedly because he was African American. Pressure from CBS Records persuaded MTV to start showing "Billie Jean" and later "Beat It", leading to a lengthy partnership with Jackson, also helping other black music artists gain recognition. MTV employees deny any racism in their coverage, or pressure to change their stance. MTV maintains that they played rock music, regardless of race. The popularity of his videos on MTV helped to put the relatively young channel "on the map"; MTV's focus shifted in favor of pop and R&B.; His performance on ''Motown 25: Yesterday, Today, Forever'' changed the scope of live stage show; "That Jackson lip-synced 'Billie Jean' is, in itself, not extraordinary, but the fact that it did not change the impact of the performance is extraordinary; whether the performance was live or lip-synced made no difference to the audience" thus creating an era in which artists re-create the spectacle of music video imagery on stage. Short films like ''Thriller'' largely remained unique to Jackson, while the group dance sequence in "Beat It" has frequently been imitated. The choreography in ''Thriller'' has become a part of global pop culture, replicated everywhere from Indian films to prisons in the Philippines. The ''Thriller'' short film marked an increase in scale for music videos, and has been named the most successful music video ever by the ''Guinness World Records''.

In the 19-minute music video for "Bad"—directed by Martin Scorsese—Jackson began using sexual imagery and choreography not previously seen in his work. He occasionally grabbed or touched his chest, torso and crotch. When asked by Oprah in the 1993 interview about why he grabbed his crotch, he replied, "I think it happens subliminally" and he described it as something that was not planned, but rather, as something that was compelled by the music. "Bad" garnered a mixed reception from both fans and critics; ''Time'' magazine described it as "infamous". The video also featured Wesley Snipes; in the future Jackson's videos would often feature famous cameo roles.

}} ;Bibliography

George, Nelson (2004). ''Michael Jackson: The Ultimate Collection'' booklet. Sony BMG.

Further reading

External links

Michael Jackson Live: Best Pics – slideshow by ''Life magazine''

Michael Jackson at MTV

FBI file on Michael Jackson

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