In music, an all-interval twelve-tone sequence is a sequence that contains a row of 12 distinct notes such that it contains one instance of each interval within the octave, 1 through 11. The more mathematical and general definition is as follows:
Definition: A sequence $x_1,\ldots, x_n$ is said to be an all-interval sequence mod $n$ if $x_i \not \equiv x_j \pmod n$ for all $1\leq i<j \leq n$ and $x_{j+1}-x_j \not \equiv x_{i+1}-x_i \pmod n$ for all $1\leq i<j \leq n-1$.
For example, the sequence $0,5,1,4,2,3$ is an all-interval sequence mod 6. There are exactly 2 such sequences mod 4, 4 such sequences mod 6, 24 of them mod 8, 288 of them mod 10, and they are given by the sequence A141599.
I could not find much about this subject online. I have found a reference Latin Squares ... which connects this definition to Latin squares of certain type.
It can be easily shown that if $n$ is odd, there are no all-interval sequences mod $n$. For $n$ even, using a computer one can compute the number of all-interval sequences mod $n$ for small values of $n$ but one quickly reaches the computing power of a regular computer around $n=20$.
I would like to explore this topic and would appreciate any references you can share. I am in particular interested in reasonable lower bounds on the number of all-interval sequences mod $n$. Currently I can show that there are at least $\phi(n)$ all-interval sequences mod $n$ but this is a miserable lower bound compared to the actual values.