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Cabochon and facet 
Cabochons, which are smooth, often domed, with flat backs. Agates and turquoise are usually cut this way, but precious stones such as rubies, emeralds and sapphires also may be. Many stones like star sapphires and moonstones must be cut this way in order to see the effects the stones have in them.
The other type of stone is generically called faceted, in which the stone has the general overall shape of the modern diamond, with a thin edge, called the girdle, the top angling up into what is called the crown, and the bottom angling down into what is called the pavilion. In the case of a cabochon stone, the side of the stone is usually cut at a shallow angle, so that when the bezel is pushed over the stone that angle permits it to hold the stone in place and keep it tight. In the case of faceted stones a shallow groove is cut into the side of the bezel into which the girdle of the stone is placed, and then metal is pushed over, holding the stone in place. Cabochons can also be set into prong settings of various kinds, but the idea is the same - it's the prongs going over the angle of the stone that creates the pressure that holds the stone in place.
Just as the angle of the sides of a cabochon creates the pressure to hold the stone in place, so there is an overlying principle in setting faceted stones. If one looks at a side view of a round diamond, for example, one will see that there is an outer edge, called the girdle, and the top angles up from there, and the bottom angles down from there. Faceted stones are set by "pinching" that angle with metal. If you imagine holding the girdle with the tips of your thumb and forefinger with both hands, that illustrates it fairly well. All of the styles of faceted stone setting use this concept in one way or another.
Types of stonesetting 
There are thousands of variations of setting styles, but there are several fundamental types:
Bezel setting 
The earliest known technique of attaching stones to jewelry was bezel setting. A bezel is a strip of metal bent into the shape and size of the stone and then soldered to the piece of jewelry. Then the stone is inserted into the bezel and the metal rubbed over the stone, holding it in place. This method works well for either cabochon or faceted stones.
Prong setting 
Prong setting is the simplest and most common type of setting, largely because it uses the least amount of metal to hold the stone, thus showing it off to its best advantage. Generally it is simply some number of wires, called prongs, which are of a certain size and shape, arranged in a shape and size to hold the given stone, and fixed at the base. Then a burr of the proper size, is used to cut what is known as a "bearing", which is a notch that corresponds to the angles of the stone. The burr most often used is called a "hart bur" that is angled and sized for the job of setting diamonds. That bearing is cut equally into all of the prongs and at the same height above the base. Then the stone is inserted so that it goes into all of the bearings, pliers or a pusher are used to bend the prongs gently over the crown of the stone, and the tops of the prongs are clipped off with snips, filed to an even height above the stone, and finished. Usually a "cup burr" is used to give the prong a nice round tip. A cup burr is in the shape of a hemisphere with teeth on the inside, for making rounded tips on wires and prongs. There are many variations of prong settings - 2 prongs up to 24 or more, many variations involving decoration, size and shapes of the prongs themselves, and how they are fixed or used in jewelry. But the method of setting is generally the same for all of them.
Channel setting 
Channel setting is a method whereby stones are suspended between two bars or strips of metal, called channels. Often when setting small stones and the bars go in a linear line with the design it is called channel setting, and when the bars cross the lines of the design, it's called bar set. The idea is the same, though. The channel is some variation of a "U" shape, with two sides and a bottom. The sides are made just a bit narrower than the width of the stone or stones to be set, and then, using the same burs as in prong setting, a small notch, which is again called a bearing, is cut into each wall. The stone is put in place in those notches, and the metal on top is pushed down, tightening the stone in place. The proper way to set a channel is to cut a notch for each stone, but for cheaper production work sometimes a groove is cut along each channel. Also, since the metal can be very stiff and strong, this is a situation where a reciprocating hammer, which is like a jackhammer but jewelry sized, might be used to hammer down the metal, as it can be difficult to do by hand. Then, as always, the metal is filed down and finished, and the inner edge near the stones cleaned up and straightened as necessary. As with all jewelry, there can be many variations of channel work. At times the walls will be raised - sometimes a center stone will be set between two bars that rise high from the base ring - or the channel might just be cut directly into some surface, making the stones flush with the metal. It is still channel setting, though.
Bead setting 
Bead setting is a generic term for setting a stone directly into metal using gravers, also called burins, which are essentially tiny chisels. A hole is drilled directly into the metal surface, and then a ball burr is used to make a concave depression just the size of the stone. Some setters will set the stone into that concave depression, and some will use a hart burr to cut a bearing around the edge. Then the stone is inserted into that space, and the gravers or burins are used to lift and push a tiny bit of the metal into and over the edge of the stone. Then a beading tool, which is simply a steel shaft with a concave dimple cut into the tip, is pushed onto the bit of metal, rounding and smoothing it, pushing it firmly onto the stone, and creating a "bead". That is the essential method, but there are many types of setting that use the technique. When many stones are set in this fashion very closely together, covering a surface, that is called "pavé" - from the French for paved or cobblestoned. When a long line is engraved into the metal going up to each of the beads, that is "star set", because of the look. The other common usage is called "bead and bright", "grain setting" or "threading" in Europe, and other names at times. This is when, after the stone is set as described above, the background metal around the stone is cut away, usually in geometric shapes. In the end what is left is the stone with four beads in a lowered box shape with an edge around it. Often it is a row of stones, so it will be in a long shape with a raised edge and a row of stones and beads down the center. This type of setting is still used often, but it was very common in the early to middle 20th century.
Burnish setting 
Burnish setting, flush setting or gypsy setting (The term gypsy setting is used less often today because the word gypsy is seen as derogatory) is similar to bead setting, but after the stone is inserted into the space, instead of using a graver to lift beads, a burnishing tool is used to push the metal all around the stone. The stone will be roughly flush with the surface, with a burnished or rubbed edge around it. This type of setting has a long history but is gaining a resurgence in contemporary jewelry. Sometimes the metal is finished using sandblasting, as it shows off the work very well.