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Number of all-interval rows for systems with 2n notes in the octave (2n-edo).

As determined by direct enumeration up to n=6, a(n) is the number of circular permutations of the integers from 0 to 2n-1 in which all "stepping-on" sequences terminate and one is complete. For example, 07531642 is one of the 24 such permutations for n=4, as starting at 1 and moving to the right by the number of steps indicated gives the complete sequence 1, 6, 3, 4, 5, 2, 7, 0. - Ian Duff, Oct 07 2018

No permutations of the integers from 0 to 2n can generate such a complete sequence. - Ian Duff, Dec 25 2018

This sequence and A141599 are based on the idea of "all-interval 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 n2-n1 modulo 12. For example, if note 1 is c# (1) and note 2 is f (5) the interval is 5-1=4, interval between 5 and 1 is 8 (1-5=-4, -4=8 mod 12), etc.

The "all-interval" 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 all-interval 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) "non-equivalent" unique all-interval 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 all-interval 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