If you are in Madrid and you want to live some good moments with traditional music, I would recommend LA TABERNA DE ELISA (calle Santa Maria, 42) on Fridays and Saturdays. They also have sessions on Bluegrass on Mondays. There is also EL LEON DE ORO (calle del León, 10), having sessions those same days and also on Thursdays. Sometimes you will find traditional or folk music, but other times you will find styles that have nothing to do with that.
O el León de Oro, en la Calle del León, 10 los mismos días incluyendo el jueves. A veces te encuentras con música tradicional pero otras veces puedes encontrar mezclas que nada tiene que ver con lo citado anteriormente.