Rearrangement structures

Intervallic series

All-interval series 

All-interval serie  is a sequence of tones, in which each musical interval and tone (in octave) represented exactly once.
For Example the so-called serie of the Mallalieu  [C,Db,E,D,A,F,B,Eb,Ab,Bb,G,F#] includes intervals 1,3,10,7,8,6,4,5,2,9,11,(6).
The interval between the last tone F# and first tone C  is  6 (=tritonus, i.e. 6 halftones) - here is the repetition of the interval tolerated.
In general, numbers 1,2, ..., k-1 have the sum S (k) = k (k-1)/2 and starting with the tone 0 ends on the tone S(k) mod k,  
that is ends on the tone 0 (=the initial tone) for odd n, or on the tone n/2 (the median of the musical system of k) for even k, 
see table.
n 1 2 3 4 5 6 7 8 9 10 11 12
S(n) = n (n-1)/2 0 1 3 6 10 15 21 28 36 45 55 66
S(n) mod n 0 1 0 2 0 3 0 4 0 5 0 6

Generation

We generate the all-interval series by a ' brute force '- i.e. we create all the possible permutations of tones and we knock off of them, those that do not have the required properties. 

According to this:
1/ we generate a permutation of (k-1) elements.
2/ we calculate the partial sums (mod k) for each permutation.
3/ only such permutation, in which all the partial sums (mod k) give just numbers 0, ..., k-1 are written.
For example, for k = 6: the permutation (1, 4, 3, 2, 5) makes totals {0, 1, 5, 8, 10, 15}
and {0, 1, 5, 8, 10, 15} mod 6 = {0, 1, 5, 2, 4, 3}, therefore this permutation complies with the conditions and we write it.
 
Condition 3/(the existence of all the different tones) can be met only for even k.
The last member of the serie for odd k is always congruent with 0 (mod k).
For example, for k = 5 is S(5) =5 * 4/2 = 10 divisible by k = 5, which leads to repetition of the initial tone and to exclusion of the serie.

Enumeration

For particular k we get the following numbers of the resulting series:
k 2 4 6 8 10 12
Number 1 2 24  288  3856 
These numbers are very low comparing to (k-1)! and they are not in general divisors of (k-1)!.

Results

All-interval series of order k=2,4,6,8:
k=2
(1): 0,1

k=4
(1,2,3): 0,1,3,2    (3,2,1): 0,3,1,2

k=6
(1,4,3,2,5): 0,1,5,2,4,3    (5,2,3,4,1): 0,5,1,4,2,3
(2,5,3,1,4): 0,2,1,4,5,3    (4,1,3,5,2): 0,4,5,2,1,3

k=8
(1,2,3,4,5,6,7): 0,1,3,6,2,7,5,4    (7,6,5,4,3,2,1): 0,7,5,2,6,1,3,4
(1,5,7,6,4,3,2): 0,1,6,5,3,7,2,4    (7,3,1,2,4,5,6): 0,7,2,3,5,1,6,4
(1,6,3,4,5,2,7): 0,1,7,2,6,3,5,4    (7,2,5,4,3,6,1): 0,7,1,6,2,5,3,4
(1,6,4,3,7,5,2): 0,1,7,3,6,5,2,4    (7,2,4,5,1,3,6): 0,7,1,5,2,3,6,4
(2,1,3,7,4,6,5): 0,2,3,6,5,1,7,4    (6,7,5,1,4,2,3): 0,6,5,2,3,7,1,4
(2,3,4,6,7,5,1): 0,2,5,1,7,6,3,4    (6,5,4,2,1,3,7): 0,6,3,7,1,2,5,4
(2,5,7,3,4,6,1): 0,2,7,6,1,5,3,4    (6,3,1,5,4,2,7): 0,6,1,2,7,3,5,4
(2,7,4,6,3,1,5): 0,2,1,5,3,6,7,4    (6,1,4,2,5,7,3): 0,6,7,3,5,2,1,4
(3,2,1,4,7,6,5): 0,3,5,6,2,1,7,4    (5,6,7,4,1,2,3): 0,5,3,2,6,7,1,4
(3,2,4,1,5,7,6): 0,3,5,1,2,7,6,4    (5,6,4,7,3,1,2): 0,5,3,7,6,1,2,4
(3,6,1,4,7,2,5): 0,3,1,2,6,5,7,4    (5,2,7,4,1,6,3): 0,5,7,6,2,3,1,4
(3,7,5,2,4,1,6): 0,3,2,7,1,5,6,4    (5,1,3,6,4,7,2): 0,5,6,1,7,3,2,4

All-interval series of order k = 10

 
All-interval series of order k = 12

Reverse permutations

Every permutation (for k>2) have corresponding 'reverse' permutation
and also corresponding 'complementary' permutation.
E.g. for k=8  permutation (1,6,4,3,7,5,2) have reverse  (2,5,7,3,4,6,1)  [i.e. numbers written in reverse order]
and  complement (7,2,4,5,1,3,6)  [differences from the number k].

If the permutation is all-interval serie, then its reverse and complementary permutation is also such a serie.
Number of all-interval series  is therefore (for k>2)  even number.

Sometimes is reverse permutation the same as complementary permutation, e.g. (1,6,3,4,5,2,7) is this case.

Nested permutations

Permutation, that makes all-interval serie have certain hidden structure,
e.g. for k=8 the following pairs  make permutations of numbers 2-6:
  (1,2,3,4,5,6,7)      (3,6,1,4,7,2,5)
  (1,6,3,4,5,2,7)      (3,2,1,4,7,6,5)

 Similarly this pair makes permutations of numbers 1-3 and 5-7
  (1,2,3,4,5,6,7)
  (3,2,1,4,7,6,5)

Other such cycles (1-3,2-6,5-7) occur in pairs:
  (1,5,7,6,4,3,2)     (2,1,3,7,4,6,5)      (1,6,4,3,7,5,2)
  (3,7,5,2,4,1,6)     (6,3,1,5,4,2,7)      (3,2,4,1,5,7,6) .

Primitive roots 

The previous paragraph reminds permutations of primitive roots - see restricted systems R(n,k,r),  e.g. for k = 12:
R( 2,12,13):  1   2   4   8   3   6  12  11   9   5  10   7      
R( 6,12,13):  1   6  10   8   9   2  12   7   3   5   4  11        
R( 7,12,13):  1   7  10   5   9  11  12   6   3   8   4   2
R(11,12,13):  1  11   4   5   3   7  12   2   9   8  10   6              

Here - total number 1!1!2!2!2!4!=192 of series arises from permutations of observed cycles [1][12][3,9][5,8][4,10][2,6,11,7].

Euler totient function

Other observation (Caleb Morgan): 
there are exactly 12 primitive roots mod 37:  2,5,13,15,17,18,19,20,22,24,32,35.

These relations hold  (mod 37) :
 2^1=2, 5^11=2, 13^23=2, 15^25=2, 17^31=2, 18^17=2,
19^35=2, 20^13=2, 22^7=2, 24^5=2, 32^29=2, and 35^19=2. 

We take only the exponents: 1,11,23,25,31,17,35,13,7,5,29,19.
and if we rank them (ascending):
we get order 1(0),11(3),23(7),25(8),31(9),17(5),35(10),13(4),7(2),5(1),29(9),19(6).

After transformation to tones, we get all-interval serie:
[C(0),Eb(3),G(7),Ab(8),Bb(9),F(5),B(10),E(4),D(2),C#(1),A(9),F#(6)].

The exponents 1,11,23,25,31,17,35,13,7,5,29,19 in case mod 37 are numbers having no common divisor with number 36.
Count of such numbers is determined by Euler totient function phi.
But result of Euler function can not make any integer, so it is not possible to use this principle  to make all-interval series for each k; e.g. it is not possible for k=14, because there exists no number with Euler function 14 ...


Schematic algebra