Physical means of playing the saxophone
Saxophone technique refers to the physical means of playing the saxophone. It includes how to hold the instrument, how the embouchure is formed and the airstream produced, tone production, hands and fingering positions, and a number of other aspects. Instrumental technique and corresponding pedagogy is a topic of much interest to musicians and teachers and therefore has been subjected to personal opinions and differences in approach. Over the course of the saxophone's performance history, notable saxophonists have contributed much to the literature on saxophone technique.
Embouchure.
Saxophone embouchure is the position of the facial muscles and shaping of the lips to the mouthpiece when playing a saxophone.
Playing technique for the saxophone can derive from an intended style (classical, jazz, rock, funk, etc.) and the player's idealized sound. The design of the saxophone allows for a wide variety of different approaches to sound production. However, there is a basic underlying structure to most techniques.
The most common saxophone embouchures in modern music use are variants of the "single-lip embouchure", in which the mouthpiece position is stabilized with firm pressure from the upper teeth resting on the mouthpiece (sometimes padded with a thin strip of rubber known as a "bite-pad" or "mouthpiece-patch"). The lower lip is supported by the buccinator and chin muscles and rests in contact with the lower teeth, making contact with the reed. The mouthpiece is inserted at least to the "break" of the "facing curve" (the beginning of the curve from the plane of the "table" to the "aperture", or tip opening), but generally with the "beak" not taken more than halfway into the player's mouth. Specific aspects of single-lip embouchure technique are described in seminal works by Larry Teal and Joseph Allard. Santy Runyon was another influential educator on modern embouchure technique, having instructed many of the top saxophonists of the big band era and top jazz musicians including Charlie Parker, Paul Desmond, Harry Carney, Lee Konitz, and Sonny Stitt.
Individual approaches by notable pedagogues.
Many saxophonists and pedagogues have published material on the saxophone embouchure and tone production. Two of the works most influential on modern teaching were published by Joe Allard and Larry Teal.
Joe Allard.
Allard taught that the embouchure must conform to the mouthpiece. Frequently citing anatomy, Allard depicted that when the skull comes down, the larynx and the throat are constricted. He had his students think of keeping their heads straight when they played. Allard also recommended that saxophonists use very little pressure from the top teeth and lip and just let everything rest naturally. Allard described the proper lower lip position as slightly drawn in to rest against the lower teeth, as in pronouncing the letter "V," to cushion the reed without excessively dampening vibration. Pressure from the reed slightly spreads the relaxed lip, with the inner part slightly over the tops of the teeth and the outer part slightly protruding. Allard emphasized the role of jaw pressure in maintaining control of the reed, with slightly increased lip pressure toward the center of the reed optimal for tonal richness, and tongue position in controlling airflow (the soft "kihhhh"). Often quoting Douglas Stanly's "The Science of Voice," he said that keeping an open throat and a relaxed throat are contradictory. The summary of Allard's approach to saxophone is to keep everything as natural as possible being careful not to interfere with head position, tongue position, breathing, or embouchure.
Larry Teal.
Teal's "The Art of Saxophone Playing" has also been an influential work in modern saxophone embouchure technique. Teal placed relatively greater emphasis on lip tension in forming the "drawstring" or "ooo" embouchure with a good seal at the corners of the mouth for maintaining tonal control. Teal's concepts are influential in developing technique for subtoning.
The Teal and Allard works are complementary in describing different aspects of modern saxophone embouchure. Allard's approach is detailed on mechanics, while Teal's is more about feel and concept. There is no "Allard School" or "Teal School" of embouchure, as teachers mix and match concepts from both sources to achieve the best result in individual situations. In historical context, Allard and Teal presented their works at a time when the legacy of clarinet-derived embouchure teaching for saxophonists was still strong, although performance technique was rapidly expanding to realize the full tonal and dynamic potential of the instrument. They codified the new techniques being developed by their contemporaries such as Santy Runyon.
Alternate embouchure styles.
The "clarinet-style" embouchure is a variation of the "single-lip embouchure" with the lower lip rolled over the teeth and the corners of the mouth drawn back. It was regarded as standard technique into the first half of the Twentieth Century, when reed instrument pedagogy was geared almost entirely to the clarinet and saxophone specialists were rare. It is still sometimes used with the alto and smaller saxophones, particularly in classical technique. It is sometimes used for subtoning, in the technique of tenor saxophonist Stan Getz "switched on" by thrusting the jaw forward and drawing the corners of the mouth back. The clarinet and tenor saxophone player Jimmy Giuffre used a clarinet-style embouchure with a tenor saxophone with a specially-modified neck. It is still commonly, and controversially, taught to beginning students as a shortcut to a passable result in lieu of more sustained effort developing embouchure strength and technique.
The "double-lip embouchure" involves curving the upper lip under the upper teeth, so that the lip comes between the upper teeth and the beak of the mouthpiece; and curving the lower lip over the top of the lower teeth, so that it comes between the lower teeth and the reed. It was an accepted technique during the earliest days of the saxophone but is no longer in common use. It gained some currency as a technique when tenor saxophonist John Coltrane used it to mitigate tooth pain while playing.
The "curved out double-lip no teeth embouchure", known by an even smaller number of saxophone players, involves taking the bottom lip and curving it out so that only a small part touches the teeth; resting just your lip on the top curved out, but with no teeth touching the mouthpiece; and putting your lips as far onto the mouthpiece as the reed and mouthpiece are still separated. 
Tone.
Tone refers to characteristics of the actual sound the saxophone produces. A player's "tonal concept" is the sound that they wish to create.
The tone produced is influenced by several factors: 
Vibrato.
Saxophone vibrato is much like a vocal or string vibrato, except the pitch variations are made using the jaw instead of the player's fingers or breathing organs. The jaw motions required for vibrato can be simulated by saying the syllables "wah-wah-wah" or "tai-yai-yai." Classical vibrato can vary between players (soft and subtle, wide and abrasive, or a combination thereof). Many classical players look to violinists as the models for their sound. It has been suggested that this follows the example of Marcel Mule of the Paris Conservatory, one of the early proponents of classical saxophone playing. Sigurd Rascher, an important German saxophone player, was known for the quicker style of vibrato which was opposite to Marcel Mule's. Jazz vibrato varies even more amongst players. Fast and wide vibrato is used by Swing music players, while some modern jazz saxophonists use almost no vibrato except in slow ballads. Typically, less vibrato is used at faster tempos.
Players just starting out with vibrato will usually start out slow with exaggerated jaw movements. As they progress, the vibrato becomes quicker until the desired speed is reached. Vibrato can also be produced by controlling the air stream with the tongue. Techniques alternative to jaw vibrato can be used to achieve a beautiful tone quality, but can also diverge noticeably from tone quality produced by classical jaw vibrato.
The lip vibrato, which is often confused with the jaw vibrato, is produced by moving the lips in something like a “wa-wa-wa---” motion. However, this is more difficult to control, as it causes a greater disturbance to the basic embouchure. This type of pulsation tends to dominate the tone so much than the listener hears more vibrato than tone.
The throat vibrato, which is seldom used any more, was at one time prevalent in wind instrument performance, especially among brass players. This is a type of “spasm” generated by tensing the throat muscles, and results in a sort of “quiver.” This vibrato has at various times been described disparagingly as the “whinny” or the “nanny-goat” type.
The diaphragm vibrato, sometimes called "breath vibrato", is predominantly an intensity vibrato. It is induced by a changing of the rate of the air pressure on the reed, and accomplished by moving the abdominal muscles, which in turn put pressure on the diaphragm, much as one would say “huh-huh-huh---.” This vibrato has proved to be quite satisfactory in a few cases, but its use is restricted, since it is difficult to attain a sensitive control of either the rate or the amplitude.
Extended techniques.
This list applies to techniques outside the basic ability to comfortably and easily play the saxophone. They would usually be learnt only after mastery of the basics and employed for unorthodox musical vocabulary. 
Electronic effects.
The use of electronic effects with the saxophone began with innovations such as the Varitone system, which Selmer introduced in 1965. The Varitone included a small microphone mounted on the saxophone neck, a set of controls attached to the saxophone's body, and an amplifier and loudspeaker mounted inside a cabinet. The Varitone's effects included echo, tremolo, tone control, and an octave divider. Two notable Varitone players were Eddie Harris and Sonny Stitt. Similar products included the Hammond Condor.
In addition to playing the Varitone, Eddie Harris experimented with looping techniques on his 1968 album "Silver Cycles".
David Sanborn and Traffic member Chris Wood employed effects such as wah-wah and delay on various recordings during the 1970s.
In more recent years, the term "saxophonics" has been used to describe the use of these techniques by saxophonists such as Skerik, who has used a wide variety of effects that are often associated with the electric guitar, and Jeff Coffin, who has made notable use of an envelope follower.
References.
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