007: All-Interval Series

$
$ All interval problem in Essence'.
$
$
$ CSPLib problem number 7
$ http://www.csplib.org/Problems/prob007
$ """
$ Given the twelve standard pitch-classes (c, c$, d, ...), represented by 
$ numbers 0,1,...,11, find a series in which each pitch-class occurs exactly 
$ once and in which the musical intervals between neighbouring notes cover 
$ the full set of intervals from the minor second (1 semitone) to the major 
$ seventh (11 semitones). That is, for each of the intervals, there is a 
$ pair of neigbhouring pitch-classes in the series, between which this 
$ interval appears. The problem of finding such a series can be easily 
$ formulated as an instance of a more general arithmetic problem on Z_n, 
$ the set of integer residues modulo n. Given n in N, find a vector 
$ s = (s_1, ..., s_n), such that (i) s is a permutation of 
$ Z_n = {0,1,...,n-1}; and (ii) the interval vector 
$ v = (|s_2-s_1|, |s_3-s_2|, ... |s_n-s_{n-1}|) is a permutation of 
$ Z_n-{0} = {1,2,...,n-1}. A vector v satisfying these conditions is 
$ called an all-interval series of size n; the problem of finding such 
$ a series is the all-interval series problem of size n. We may also be 
$ interested in finding all possible series of a given size. 
$ """
$
$ Model created by Hakan Kjellerstrand, hakank@gmail.com
$ See also my Essence'/Tailor page: http://www.hakank.org/savile_row/
$
$
$ Licenced under CC-BY-4.0 : http://creativecommons.org/licenses/by/4.0/

language ESSENCE' 1.0

letting n be 12
letting range be domain int(1..n)
letting range2 be domain int(1..n-1)
find x: matrix indexed by [range] of range
find diffs: matrix indexed by [range2] of range2

such that
   allDiff(diffs),
   allDiff(x),
   forall k : range2 . diffs[k] = |x[k+1]-x[k]|,
   $ symmetry breaking
   x[1] < x[n-1],
   diffs[1] < diffs[2]