Labels for elements of the symmetric group P4.

Initial half of Seq (ie. #4). The complete Seq is formed by appending the complement of the Half_Seq.

Complete sequence (ie. #8).

Complement of a Half_Seq.

map complement [[4,1,3,2],[6,7,8,5]] == [[8,5,7,6],[2,3,4,1]]

Form Seq from Half_Seq.

full_seq [3,2,4,1] == [3,2,4,1,7,6,8,5]
label_of (full_seq [3,2,4,1]) == G2
label_of (full_seq [1,4,2,3]) == L

Lower Half_Seq, ie. complement or id.

map lower [[4,1,3,2],[6,7,8,5]] == [[4,1,3,2],[2,3,4,1]]

Application of Label p on q.

l_on Q1 I == Q1
l_on D Q12 == Q4
[l_on L L,l_on E D,l_on D E] == [L2,C,B]

Generalisation of Fibonnaci process, f is the binary operator giving the next element, p and q are the initial elements.

See discussion in: Carlos Agon, Moreno Andreatta, Gérard Assayag, and Stéphan Schaub. _Formal Aspects of Iannis Xenakis' "Symbolic Music": A Computer-Aided Exploration of Compositional Processes_. Journal of New Music Research, 33(2):145-159, 2004.

Note that the article has an error, printing Q4 for Q11 in the sequence below.

import qualified Music.Theory.List as T

let r = [D,Q12,Q4, E,Q8,Q2, E2,Q7,Q4, D2,Q3,Q11, L2,Q7,Q2, L,Q8,Q11]
(take 18 (fib_proc l_on D Q12) == r,T.duplicates r == [Q2,Q4,Q7,Q8,Q11])

Beginning E then G2 no Q nodes are visited.

let r = [E,G2,L2,C,G,D,E,B,D2,L,G,C,L2,E2,D2,B]
(take 16 (fib_proc l_on E G2) == r,T.duplicates r == [B,C,D2,E,G,L2])

let [a,b] = take 2 (T.segments 18 18 (fib_proc l_on D Q12)) in a == b

The prime numbers that are not factors of 18 are {1,5,7,11,13,17}. They form a closed group under modulo 18 multiplication.

let n = [5,7,11,13,17]
let r0 = [(5,7,17),(5,11,1),(5,13,11),(5,17,13)]
let r1 = [(7,11,5),(7,13,1),(7,17,11)]
let r2 = [(11,13,17),(11,17,7)]
let r3 = [(13,17,5)]
[(p,q,(p * q) `mod` 18) | p <- n, q <- n, p < q] == concat [r0,r1,r2,r3]

The article also omits the 5 after 5,1 in the sequence below.

let r = [11,13,17,5,13,11,17,7,11,5,1,5,5,7,17,11,7,5,17,13,5,11,1,11]
take 24 (fib_proc (\p q -> (p * q) `mod` 18) 11 13) == r

Seq of Label, inverse of label_of.

seq_of Q1 == [8,7,5,6,4,3,1,2]

Half_Seq of Label, ie. half_seq . seq_of.

half_seq_of Q1 == [8,7,5,6]

Half_Seq of Seq, ie. take 4.

complement (half_seq (seq_of Q7)) == [3,4,2,1]

Label of Seq, inverse of seq_of.

label_of [8,7,5,6,4,3,1,2] == Q1
label_of (seq_of Q4) == Q4

True if two Half_Seqs are complementary, ie. form a Seq.

complementary [4,2,1,3] [8,6,5,7] == True

Relation between to Half_Seq values as a (complementary,permutation) pair.

Determine Rel of Half_Seqs.

relate [1,4,2,3] [1,3,4,2] == (False,[0,3,1,2])
relate [1,4,2,3] [8,5,6,7] == (True,[1,0,2,3])

Rel from Label p to q.

relate_l L L2 == (False,[0,3,1,2])

relate adjacent Half_Seq, see also relations_l.

relate adjacent Labels.

relations_l [L2,L,A] == [(False,[0,2,3,1]),(False,[2,0,1,3])]

Apply Rel to Half_Seq.

apply_relation (False,[0,3,1,2]) [1,4,2,3] == [1,3,4,2]

Apply sequence of Rel to initial Half_Seq.

Variant of apply_relations.

apply_relations_l (relations_l [L2,L,A,Q1]) L2 == [L2,L,A,Q1]

Enumeration of set of faces of a cube.

Table indicating set of faces of cubes as drawn in Fig. VIII-6 (p.220).

lookup [1,4,6,7] faces == Just F_Left
T.reverse_lookup F_Right faces == Just [2,3,5,8]

Label sequence of Fig. VIII-6. Hexahedral (Octahedral) Group (p. 220)

let r = [I,A,B,C,D,D2,E,E2,G,G2,L,L2,Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,Q9,Q10,Q11,Q12]
in viii_6_lseq == r

Label sequence of Fig. VIII-7 (p.221)

let r = [I,A,B,C,D,D2,E,E2,G,G2,L,L2,Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8,Q9,Q10,Q11,Q12]
in viii_7_lseq == r

Fig. VIII-7 (p.221)

map (take 4) (take 4 viii_7) == [[I,A,B,C]
                                ,[A,I,C,B]
                                ,[B,C,I,A]
                                ,[C,B,A,I]]

import Music.Theory.Array.MD

let t = md_matrix_opt show (\x -> "_" ++ x ++ "_") (head viii_7,head viii_7) viii_7
putStrLn $ unlines $ md_table' t

Label sequence of Fig. VIII-6/b (p.221)

length viii_6b_l == length viii_6_l
take 8 viii_6b_l == [I,A,B,C,D2,D,E2,E]

Fig. VIII-6/b Half_Seq.

viii_6b_p' == map half_seq_of viii_6b_l
nub (map (length . nub) viii_6b_p') == [4]

Variant of viii_6b with Half_Seq.

Fig. VIII-6/b.

map (viii_6b !!) [0,8,16] == [(I,[1,2,3,4,5,6,7,8])
                             ,(G2,[3,2,4,1,7,6,8,5])
                             ,(Q8,[6,8,5,7,2,4,1,3])]

The sequence of Rel to give viii_6_l from L2.

apply_relations_l viii_6_relations L2 == viii_6_l
length (nub viii_6_relations) == 14

The sequence of Rel to give viii_6b_l from I.

apply_relations_l viii_6b_relations I == viii_6b_l
length (nub viii_6b_relations) == 10