This may have worked only for me, or it may be what everybody does, but I'll put in my two cents anyway. (The following should be considered an amateur opinion.)
The best way to hear beats is to know what note you should be listening for when you try to hear the beats.
If, for example, you are tuning a fifth -- say middle C and the G above middle C -- the "pure" ratio is 3 to 2. So you figure out which note has 3 times the frequency of the bottom note (middle C) and 2 times the frequency of the top note (G above middle C). The desired note is then, of course, the G an octave above.
Listen for this note when you strike your C-G combination. Do not actually strike the G an octave above. If you can hear this high G when you strike the other two notes, you should also be able to hear the beats you are looking for.
Or, suppose you are tuning a major third, such as middle C and the E just above. (I would not recommend tuning by thirds if you are tuning equal temperament, but it works well for meantone.) A pure third is a 5 to 4 ratio, so the note you should be listening for is the E two octaves above the E you are tuning (5 times the middle C, and 4 times the middle E). If you can hear this two-octave E when you strike the middle C and middle E, you should be able to hear the beats as well.
Bill Smythe
