Breathing or aerating wine prior to serving is a slightly controversial topic. Some people swear by it others say it is at best just a show.
The idea is to allow some of the less attractive volatile smells out of the wine and to introduce a small amount of oxygen into the wine to allow the flavours to be enhanced. Oxygen of course will destroy wine given time, but some people argue that a little at the time of serving help
The simplest method is to just pull the cork and allow the wine to sit in the unopened bottle for a couple of hours. This is what most people mean when they refer top "breathing" the wine. Breathing this way has little effect as only a small area exposed to the air and probably does very little to enhance the wine
Swirling wine in the glass is the next simplest method. This is one reason why large wine glasses are preferred when serving wine. Wine tasters often do this to try to get a fuller appreciation of nuances in the bouquet of the wine.
Wine is often decanted, ie poured from the bottle into another container and perhaps back into the original bottle. This process is often done primarily to remove the sediment from an aged wine and the aeration is a side effect. Depending on the method of pouring there can be quite an effect in decanting.
Wine can also be aerated by pouring it through a specially made aerator.
Just a simple test. - Unscrew the bottle, pour some into two identical glasses, one directly from the bottle, one through the aerator. Evaluate each wine immediately and after a half an hour in the glass.
A couple more tests with a cleanskins 2006 Barbera and an 2004 Barossa shiraz shows that the aerator does have an effect, basically it softens the tannins making a smoother drink.
I would always try the wine first. Then taste some that has gone through the aerator
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