Safe Haskell	None
Language	Haskell98

Music.Theory.Xenakis.S4

Contents

S4 notation
Rel
Face
Figures

Description

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

Synopsis

S4 notation

data Label Source

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] Source

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

type Seq = [Int] Source

Complete sequence (ie. #8).

complement :: Half_Seq -> Half_Seq Source

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 Source

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 Source

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 Source

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 Source

Seq of Label, inverse of label_of.

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

half_seq_of :: Label -> Seq Source

Half_Seq of Label, ie. half_seq . seq_of.

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

half_seq :: Seq -> Half_Seq Source

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 Source

Reverse table lookup.

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

label_of :: Seq -> Label Source

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 Source

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) Source

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

relate :: Half_Seq -> Half_Seq -> Rel Source

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 Source

Rel from Label p to q.

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

relations :: [Half_Seq] -> [Rel] Source

relate adjacent Half_Seq, see also relations_l.

relations_l :: [Label] -> [Rel] Source

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 Source

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] Source

Apply sequence of Rel to initial Half_Seq.

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

Variant of apply_relations.

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

Face

data Face Source

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)] Source

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] Source

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]] Source

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] Source

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] Source

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)] Source

Variant of viii_6b with Half_Seq.

viii_6b :: [(Label, Seq)] Source

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] Source

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] Source

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