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A141598 Total number of all-interval rows for systems with 2n notes in the octave (2n-edo). 4
2, 8, 24, 192, 2880, 46272, 1250592, 44095488, 1865756160 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
FORMULA
a(n) = 2*n*A141599(n). - Leo C. Stein, Nov 26 2016
CROSSREFS
Sequence in context: A088994 A330505 A214849 * A071599 A047695 A093842
KEYWORD
nonn,more
AUTHOR
Milan Gustar (artech(AT)noise.cz), Sep 03 2008
EXTENSIONS
a(9) is calculated from A141599(9) after David V. Feldman. - Jud McCranie, Oct 07 2018
STATUS
approved

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Last modified July 23 18:56 EDT 2024. Contains 374553 sequences. (Running on oeis4.)