Final answer:
The number of unique transpositions of the octatonic collection is 8! (8 factorial), which is equal to 40,320.
Step-by-step explanation:
An octatonic collection consists of eight different notes from a musical scale. A transposition is a rearrangement of these notes. To find the number of unique transpositions, we can use the concept of permutations.
In an octatonic collection, there are 8 choices for the first note, 7 choices for the second note, 6 choices for the third note, and so on. Therefore, the number of unique transpositions of the octatonic collection is 8! (8 factorial), which is equal to 40,320.