Can't we just imagine the sound without singing?
closed as not a real question by American Luke, nonpop, Dr Mayhem♦, luser droog, Andrew Mar 4 '13 at 21:36
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Imagining the sound is a very useful skill called audiation, but if you are only listening inside your head, how are you going to know if you are right or wrong?
Singing is essential because it is how you can check yourself and how your teacher can check you. The voice is an instrument that requires no extra hardware, and most musicians find it very easy to express pitches with their voice without having to develop technique at all.
Now, there are exceptions to the rule--and while I would say singing is essential/crucial, it's not flat-out required. Some people just aren't good at matching pitch with their voice, and some have other means of expression that they are more comfortable with. I knew a guy who was a wizard on the tenor saxophone--a really great jazz musician and improviser (which necessitates good aural skills/ear training), but he couldn't sing worth a darn.
However, learning to use the voice is a really great skill to learn in the first place, so out of hand I recommend all musicians learn how to sing even if it's not absolutely required for aural skills. Although, your teacher may not want a saxophone in ear training class, and if you don't have enough instrumental technique to fluently express yourself to begin with, learning how to do so is going to be MUCH easier with the voice.
Yes, it is a technique that has been used everywhere, forever. Ultimately you do need to hear intervals silently inside your head, and you should. But in order to learn to do this correctly, singing pitches out loud is the way to start. You don't need to develop singing technique to do this; you just have to start making sounds and learn how to make them in tune.
I would also point out that learning to sing the phrases of a melody can help you play them more expressively on any instrument, because it develops the concept of "phrasing". Good piano players or guitarists know how to breath along with their melodies and where to take "breaths" or pauses in the melody to punctuate phrases, just like a sax or trumpet player or singer would do.