
COMMENTS

This sequence and A141599 are based on the idea of "allinterval rows" from a musical techniques called dodecaphony and serialism.
Twelve tones from an octave (c, c#, d, d#, e, f, ..., b) are marked by numbers 0, 1, ..., 11 (c is 0, c# is 1, etc.).
The "interval" between two notes n1 and n2 is calculated as the difference n2n1 modulo 12. For example, if note 1 is c# (1) and note 2 is f (5) the interval is 51=4, interval between 5 and 1 is 8 (15=4, 4=8 mod 12), etc.
The "allinterval" row is any sequence of twelve notes containing all notes of an octave (0..11) and all intervals (1..11) between adjacent positions. For example, the row 0 1 3 2 7 10 8 4 11 5 9 6 has intervals 1 2 11 5 3 10 8 7 6 4 9, i.e., it is an allinterval row.
There are 46272 such rows from all possible 479001600 (12!) permutations.
Rows with the same interval structure are equivalent in dodecaphony, for example the rows 0, 1, ..., 10, 11 and 1, 2, ..., 11, 0 both have the same intervals (all 1s), the second row is only transposed (moved) one step higher. There are 12 possible transpositions of one row, therefore there are 3856 (46272/12) "nonequivalent" unique allinterval rows.
My generalization is an extension of this principle to microtonal systems  equal divisions of octave, EDO. Rows can be constructed for the tuning systems with any number of notes in the octave, not only 12. As it can be easily proved, the allinterval rows exist only in systems with even number of notes in the octave.
Also the number of permutations of 1..2n which have distinct differences [Gilbert].  N. J. A. Sloane, Mar 15 2014
