Music.Theory.Z.Read_1978

Ronald C. Read. "Every one a winner or how to avoid isomorphism search when cataloguing combinatorial configurations." /Annals of Discrete Mathematics/ 2:107–20, 1978.

Documentation

type Code = Word64

Coding.

code_len :: Num n => n

Number of bits at Code.

type Bit_Array = [Bool]

Bit array.

bit_array_complement :: Bit_Array -> Bit_Array

Logical complement.

bit_array_pp :: Bit_Array -> String

Pretty printer for Bit_Array.

bit_array_parse :: String -> Bit_Array

Parse PP of Bit_Array.

bit_array_parse "01001" == [False,True,False,False,True]

MSB (BIG-ENDIAN)

bit_array_to_code :: Bit_Array -> Code

Generate Code from Bit_Array, the coding is most to least significant.

map (bit_array_to_code . bit_array_parse) (words "000 001 010 011 100 101 110 111") == [0..7]
bit_array_to_code (bit_array_parse "1100100011100") == 6428

code_to_bit_array :: Int -> Code -> Bit_Array

Inverse of bit_array_to_code.

code_to_bit_array 13 6428 == bit_array_parse "1100100011100"

bit_array_to_set :: Integral i => Bit_Array -> [i]

Bit_Array to set.

bit_array_to_set (bit_array_parse "1100100011100") == [0,1,4,8,9,10]
set_to_code 13 [0,1,4,8,9,10] == 6428

set_to_bit_array :: Integral i => i -> [i] -> Bit_Array

Inverse of bit_array_to_set, z is the degree of the array.

set_to_code :: Integral i => i -> [i] -> Code

bit_array_to_code of set_to_bit_array.

set_to_code 12 [0,2,3,5] == 2880
map (set_to_code 12) (Sro.z_sro_ti_related (flip mod 12) [0,2,3,5])

bit_array_is_prime :: Bit_Array -> Bool

The prime form is the maximum encoding.

bit_array_is_prime (set_to_bit_array 12 [0,2,3,5]) == False

bit_array_augment :: Bit_Array -> [Bit_Array]

The augmentation rule adds 1 in each empty slot at end of array.

map bit_array_pp (bit_array_augment (bit_array_parse "01000")) == ["01100","01010","01001"]

enumerate_half :: (Bit_Array -> Bool) -> Int -> [(Int, [Bit_Array])]

Enumerate first half of the set-classes under given prime function. The second half can be derived as the complement of the first.

import Music.Theory.Z.Forte_1973
length scs == 224
map (length . scs_n) [0..12] == [1,1,6,12,29,38,50,38,29,12,6,1,1]

let z12 = map (fmap (map bit_array_to_set)) (enumerate_half bit_array_is_prime 12)
map (length . snd) z12 == [1,1,6,12,29,38,50]

This can become slow, edit z to find out. It doesn't matter about n. This can be edited so that small n would run quickly even for large z.

fmap (map bit_array_to_set) (lookup 5 (enumerate_half bit_array_is_prime 16))

LSB - LITTLE-ENDIAN

set_coding_validate :: [t] -> [t]

If the size of the set is > code_len then error, else id.

set_encode :: Integral i => [i] -> Code

Encoder for encode_prime.

map set_encode [[0,1,3,7,8],[0,1,3,6,8,9]] == [395,843]

map (set_to_code 12) [[0,1,3,7,8],[0,1,3,6,8,9]] == [3352,3372]

set_decode :: Integral i => Int -> Code -> [i]

Decoder for encode_prime.

map (set_decode 12) [395,843] == [[0,1,3,7,8],[0,1,3,6,8,9]]

set_encode_prime :: Integral i => Z i -> [i] -> [i]

Binary encoding prime form algorithm, equalivalent to Rahn.

set_encode_prime Z.z12 [0,1,3,6,8,9] == [0,2,3,6,7,9]
Music.Theory.Z.Rahn_1980.rahn_prime Z.z12 [0,1,3,6,8,9] == [0,2,3,6,7,9]