Even simple hunting horns without valves (which give modern concert horns the ability to play precise notes) can be played with various notes. Compare that to the bugle in military use - every soldier knows whether it is sounding attack, retreat, reveille, or taps. In LotR we also have examples of brass instruments playing signals that signify persons - Beregond identified the "sound of a trumpet ending on a long high note" (RotK) as Faramir's call. No magic involved there either.
Tolkien had military experience, something that a majority of his readers has not had. The most logical explanation for the warning in the Shire and all others uses of horns, trumpets, etc. is that the melody they played had a specific meaning, and each warning had a different sequence of notes.
I remember weekly signal "rehearsals" of the emergency warning signals in my home town when I was young - three long meant something different than one continuous, for example. Even with only one note, the rhythm denoted various meanings.
It could be that each of the words (fear, fire, foes, awake) had its own signal, and when combined, the highest level of warning was sounded.
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth.. .'
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