A141598 Total number of all-interval rows for systems with 2n notes in the octave (2n-edo).

2, 8, 24, 192, 2880, 46272, 1250592, 44095488, 1865756160

Offset: 1

Author

Milan Gustar (artech(AT)noise.cz), Sep 03 2008

Keywords

• nonn
• more

Comments

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

Links

• E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence (see Q(n)). - _N. J. A. Sloane_, Mar 15 2014
• Milan Gustar, More information
• Milan Gustar, Programs

Crossrefs

Cf. A067601, A141599, A155914.

Formula

a(n) = 2*n*A141599(n). - Leo C. Stein, Nov 26 2016

Extensions

a(9) is calculated from A141599(9) after David V. Feldman. - Jud McCranie, Oct 07 2018