The five aspects Ptolemy admits, conjunction, sextile, square, trine, opposition, are the ones the twelve signs can generate. Divide the circle by one, six, four, three and two and every result is a whole number of signs. That is not a coincidence; it is the reason those five and no others.
Johannes Kepler thought the reasoning was backwards. In the Harmony of the World of 1619 he argued that aspects work because certain angular ratios are harmonious in the same way musical intervals are, and that the sign-based derivation was an accident of an inherited scheme rather than a cause. If the ratios do the work, then divisions the signs cannot produce should also count. He proposed the quintile at seventy-two degrees and its double the biquintile at a hundred and forty-four, the semi-square at forty-five, and the sesquiquadrate at a hundred and thirty-five.
This is a genuinely awkward fact about the history. Kepler is the figure most often cited to show that serious astronomers once took astrology seriously, and the thing he actually contributed to astrology is the set of aspects traditional astrologers are most likely to throw out. He was also, in the same work, dismissive of most of the rest of the practice.
The argument was extended in the twentieth century by John Addey, whose Harmonics in Astrology of 1976 treated the whole aspect doctrine as a special case of dividing the circle by successive integers, generating septiles, noviles and beyond. Addey's scheme is internally consistent and produces an unbounded number of aspects, which is either its strength or the reduction that refutes it depending on who is describing it.
Positions currently held. Traditional and Hellenistic-revival practitioners generally use the Ptolemaic five and nothing else, on the grounds that the others cannot be sign relations. Mainstream modern astrology uses the five plus the quincunx, and often the semi-square and sesquiquadrate. Harmonic astrologers use as many divisions as the chart will support. The quintile family sits on the boundary, admitted by many and treated by others as arithmetic with a delineation attached to it after the fact.