Now, musical notation is language-like in that it signifies things, just like words formed from the Roman alphabet. If you know how to read it, the significance of this is perfectly obvious:
To a guitarist this says "play a C major chord (probably in first position with your third LH finger on the low C, your 2nd finger on E and your first finger on the high C--the G is an open string) and hold it for four beats." Now that is certainly a musical instruction, but it is not at this point, music. Assume the guitarist plays the chord and holds it for four beats, what does a listener hear, musically? A major chord. But there is really not much of a musical effect unless it is followed by a whole bunch of other stuff. This other stuff will consist of various notes forming chords and melodies and they will be articulated in time with rhythms and a meter consisting of groups of measured pulses or beats. Perhaps we will end up with something like this:
That piece starts with the simple C major chord of my example and goes on to be a complete, simple, piece of music. Each measure of the notation signifies to the guitarist instructions such as I described above. The composer, guitarist and listener are all perceiving this as a piece of music, that is to say, a structure of sound in time. The musical work exists in different ways in different places: the composer's mind when he was writing it, the music notation, the guitarist's mind as he learns how to play it, the performance itself and finally in the mind or ear of the listener. Given that, perhaps we can take a stab at the kind of meaning there is at each stage. In this particular case, the composer was wanting to create something that would teach a guitar student something about different 'voices' and about harmony. The guitarist has to solve a number of technical problems, especially for the left hand, in order to realize the music. If he is successful, the listener hears a serene piece in C major with independent voices and nice suspensions.
Sorry for the lengthy example! We can derive from this the reasons why music has some language-like aspects, but is not a language. When a musician looks at the score to this piece he sees a precise notation of the sound structure. With it, he can perform the piece. But neither the instructions, nor the resulting performance can easily be put into words. You could describe what you do: "first I put my fingers on a C major chord and pluck with the right hand..." but it would be terribly lengthy and pointless. And what can a listener do to describe the performance? "There were a lot of chords and they seemed to fit together well and the guitarist was wearing a white shirt..." Also terribly lengthy and pointless!
So how is the music itself language-like? Not in any specific way, certainly. But it is hard to deny that communication is going on and and very clear communication at that if you are acquainted with this kind of music or with guitar-playing. Which brings me once again to a famous quote by Mendelssohn:
A piece of music that I love expresses thoughts to me that are not too imprecise to be framed in words, but too precise.I think that where music fails to be a language is in the correlation of signifier to signified. We have all those notes, but they don't point to something outside themselves like the word 'smoke' points to a plume of smoke. This gives them the freedom to have expressive power without being caught in a web of language. Now songs, of course, are a whole different story because they operate on both levels: musical expression and the significance of language. Here is the first song from Dichterliebe, a masterpiece where a great composer of song, Robert Schumann, sets sixteen poems by Heinrich Heine, a master poet.