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S(T) is a monomial. If it is S+T no.
It is the expression t - s
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There are much more than three types of scales, but the 3 basic scales are the common major scale (T T S T T T S), the harmonic minor (T S T T S T1/2 S) and the melodic minor (T S T T T T S ascending - T S T T S T T descending). In brackets noticed I labelled the structure of the scale in tones and semitones. T = tone S = semitone T1/2 = minor 3rd/augmented 2nd interval (3 semitones)
The proof of this theorem is by contradiction. Suppose for convex sets S and T there are elements a and b such that a and b both belong to S∩T, i.e., a belongs to S and T and b belongs to S and T and there is a point c on the straight line between a and b that does not belong to S∩T. This would mean that c does not belong to one of the sets S or T or both. For whichever set c does not belong to this is a contradiction of that set's convexity, contrary to assumption. Thus no such c and a and b can exist and hence S∩T is convex.