Music.Theory.Xenakis.S4

Symetric Group S4 as related to the composition "Nomos Alpha" by Iannis Xenakis. In particular in relation to the discussion in "Towards a Philosophy of Music", Formalized Music pp. 219 -- 221

S4 notation

data Label

Labels for elements of the symmetric group P4.

Constructors

A

B

C

D

D2

E

E2

G

G2

I

L

L2

Q1

Q2

Q3

Q4

Q5

Q6

Q7

Q8

Q9

Q10

Q11

Q12

Instances

Bounded Label

Enum Label

Eq Label

Ord Label

Show Label

type Half_Seq = [Int]

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

type Seq = [Int]

Complete sequence (ie. #8).

complement :: Half_Seq -> Half_Seq

Complement of a Half_Seq.

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

full_seq :: Half_Seq -> Seq

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 -> Half_Seq

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]]

l_on :: Label -> Label -> Label

Application of Label p on q.

 l_on Q1 I == Q1
 l_on D A == G
 [l_on L L,l_on E D,l_on D E] == [L2,C,B]

seq_of :: Label -> Seq

Seq of Label, inverse of label_of.

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

half_seq_of :: Label -> Seq

Half_Seq of Label, ie. half_seq . seq_of.

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

half_seq :: Seq -> Half_Seq

Half_Seq of Seq, ie. take 4.

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

reverse_lookup :: Eq a => a -> [(b, a)] -> Maybe b

Reverse table lookup.

 reverse_lookup 'b' (zip [1..] ['a'..]) == Just 2
 lookup 2 (zip [1..] ['a'..]) == Just 'b'

label_of :: Seq -> Label

Label of Seq, inverse of seq_of.

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

complementary :: Half_Seq -> Half_Seq -> Bool

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

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

Rel

type Rel = (Bool, Permute)

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

relate :: Half_Seq -> Half_Seq -> Rel

Determine Rel of Half_Seqs.

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

relate_l :: Label -> Label -> Rel

Rel from Label p to q.

 relate_l L L2 == (False,P.listPermute 4 [0,3,1,2])

relations :: [Half_Seq] -> [Rel]

relate adjacent Half_Seq, see also relations_l.

relations_l :: [Label] -> [Rel]

relate adjacent Labels.

 relations_l [L2,L,A] == [(False,P.listPermute 4 [0,2,3,1])
                         ,(False,P.listPermute 4 [2,0,1,3])]

apply_relation :: Rel -> Half_Seq -> Half_Seq

Apply Rel to Half_Seq.

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

apply_relations :: [Rel] -> Half_Seq -> [Half_Seq]

Apply sequence of Rel to initial Half_Seq.

apply_relations_l :: [Rel] -> Label -> [Label]

Variant of apply_relations.

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

Face

data Face

Enumeration of set of faces of a cube.

Constructors

F_Back

F_Front

F_Right

F_Left

F_Bottom

F_Top

Instances

Bounded Face

Enum Face

Eq Face

Ord Face

Show Face

faces :: [([Int], Face)]

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

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

Figures

viii_6_l :: [Label]

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

 length viii_6_l == 24
 take 7 viii_6_l == [L2,L,A,Q1,Q7,Q3,Q9]

viii_7 :: [[Label]]

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]]

viii_6b_l :: [Label]

Fig. VIII-6/b Labels (p.221)

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

viii_6b_p' :: [Half_Seq]

Fig. VIII-6/b Half_Seq.

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

viii_6b' :: [(Label, Half_Seq)]

Variant of viii_6b with Half_Seq.

viii_6b :: [(Label, 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])]

viii_6_relations :: [Rel]

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

viii_6b_relations :: [Rel]

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