Safe Haskell | None |
---|---|

Language | Haskell98 |

Polansky, Larry and Bassein, Richard
"Possible and Impossible Melody: Some Formal Aspects of Contour"
*Journal of Music Theory* 36/2, 1992 (pp.259-284)
(http://www.jstor.org/pss/843933)

- adjacent_indices :: Integral i => i -> [(i, i)]
- all_indices :: Integral i => i -> [(i, i)]
- type Matrix a = [[a]]
- matrix_f :: (a -> a -> b) -> [a] -> Matrix b
- contour_matrix :: Ord a => [a] -> Matrix Ordering
- data Contour_Half_Matrix = Contour_Half_Matrix {}
- half_matrix_f :: (a -> a -> b) -> [a] -> Matrix b
- contour_half_matrix :: Ord a => [a] -> Contour_Half_Matrix
- contour_half_matrix_str :: Contour_Half_Matrix -> String
- data Contour_Description = Contour_Description {}
- contour_description :: Ord a => [a] -> Contour_Description
- contour_description_str :: Contour_Description -> String
- half_matrix_to_description :: Contour_Half_Matrix -> Contour_Description
- contour_description_ix :: Contour_Description -> (Int, Int) -> Ordering
- uniform :: Contour_Description -> Bool
- no_equalities :: Contour_Description -> Bool
- all_contours :: Int -> [Contour_Description]
- implication :: (Ordering, Ordering) -> Maybe Ordering
- violations :: Contour_Description -> [(Int, Int, Int, Ordering)]
- is_possible :: Contour_Description -> Bool
- possible_contours :: Int -> [Contour_Description]
- impossible_contours :: Int -> [Contour_Description]
- contour_description_lm :: Integral a => a -> a
- contour_truncate :: Contour_Description -> Int -> Contour_Description
- contour_is_prefix_of :: Contour_Description -> Contour_Description -> Bool
- contour_eq_at :: Contour_Description -> Contour_Description -> Int -> Bool
- draw_contour :: Integral i => Contour_Description -> [i]
- contour_description_invert :: Contour_Description -> Contour_Description
- type Build_f st e = st -> Maybe (e, st)
- type Conforms_f e = Int -> [e] -> Bool
- build_f_n :: Build_f st e -> Build_f (Int, st) e
- build_sequence :: Int -> Build_f st e -> Conforms_f e -> Int -> st -> (Maybe [e], st)
- build_contour :: Ord e => Build_f st e -> Contour_Description -> Int -> st -> (Maybe [e], st)
- build_contour_retry :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> (Maybe [e], st)
- build_contour_set :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> [[e]]
- build_contour_set_nodup :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> [[e]]
- ex_1 :: [Rational]
- ex_2 :: [Integer]
- ex_3 :: [Integer]
- ex_4 :: Contour_Description

# Indices

adjacent_indices :: Integral i => i -> [(i, i)] Source #

Construct set of *n* `-`

`1`

adjacent indices, left right order.

adjacent_indices 5 == [(0,1),(1,2),(2,3),(3,4)]

all_indices :: Integral i => i -> [(i, i)] Source #

All *(i,j)* indices, in half matrix order.

all_indices 4 == [(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)]

# Matrix

matrix_f :: (a -> a -> b) -> [a] -> Matrix b Source #

Apply *f* to construct `Matrix`

from sequence.

matrix_f (,) [1..3] == [[(1,1),(1,2),(1,3)] ,[(2,1),(2,2),(2,3)] ,[(3,1),(3,2),(3,3)]]

# Half matrix

data Contour_Half_Matrix Source #

Half matrix notation for contour.

half_matrix_f :: (a -> a -> b) -> [a] -> Matrix b Source #

Half `Matrix`

of contour given comparison function *f*.

half_matrix_f (flip (-)) [2,10,6,7] == [[8,4,5],[-4,-3],[1]] half_matrix_f (flip (-)) [5,0,3,2] == [[-5,-2,-3],[3,2],[-1]] half_matrix_f compare [5,0,3,2] == [[GT,GT,GT],[LT,LT],[GT]]

contour_half_matrix :: Ord a => [a] -> Contour_Half_Matrix Source #

Construct `Contour_Half_Matrix`

(p.264)

contour_half_matrix_str :: Contour_Half_Matrix -> String Source #

`Show`

function for `Contour_Half_Matrix`

.

# Contour description

data Contour_Description Source #

*Description* notation of contour.

contour_description :: Ord a => [a] -> Contour_Description Source #

Construct `Contour_Description`

of contour (p.264).

let c = [[3,2,4,1],[3,2,1,4]] in map (show.contour_description) c == ["202 02 2","220 20 0"]

contour_description_str :: Contour_Description -> String Source #

`Show`

function for `Contour_Description`

(p.264).

half_matrix_to_description :: Contour_Half_Matrix -> Contour_Description Source #

Convert from `Contour_Half_Matrix`

notation to `Contour_Description`

.

contour_description_ix :: Contour_Description -> (Int, Int) -> Ordering Source #

Ordering from *i*th to *j*th element of sequence described at *d*.

contour_description_ix (contour_description "abdc") (0,3) == LT

uniform :: Contour_Description -> Bool Source #

`True`

if contour is all descending, equal or ascending.

let c = ["abc","bbb","cba"] in map (uniform.contour_description) c == [True,True,True]

all_contours :: Int -> [Contour_Description] Source #

Set of all contour descriptions.

map (length.all_contours) [3,4,5] == [27,729,59049]

implication :: (Ordering, Ordering) -> Maybe Ordering Source #

A sequence of orderings *(i,j)* and *(j,k)* may imply ordering
for *(i,k)*.

map implication [(LT,EQ),(EQ,EQ),(EQ,GT)] == [Just LT,Just EQ,Just GT]

violations :: Contour_Description -> [(Int, Int, Int, Ordering)] Source #

List of all violations at a `Contour_Description`

(p.266).

is_possible :: Contour_Description -> Bool Source #

Is the number of `violations`

zero.

possible_contours :: Int -> [Contour_Description] Source #

All possible contour descriptions

map (length.possible_contours) [3,4,5] == [13,75,541]

impossible_contours :: Int -> [Contour_Description] Source #

All impossible contour descriptions

map (length.impossible_contours) [3,4,5] == [14,654,58508]

contour_description_lm :: Integral a => a -> a Source #

Calculate number of contours of indicated degree (p.263).

map contour_description_lm [2..7] == [1,3,6,10,15,21]

let r = [3,27,729,59049,14348907] in map (\n -> 3 ^ n) (map contour_description_lm [2..6]) == r

contour_truncate :: Contour_Description -> Int -> Contour_Description Source #

Truncate a `Contour_Description`

to have at most *n* elements.

let c = contour_description [3,2,4,1] in contour_truncate c 3 == contour_description [3,2,4]

contour_is_prefix_of :: Contour_Description -> Contour_Description -> Bool Source #

Is `Contour_Description`

*p* a prefix of *q*.

let {c = contour_description [3,2,4,1] ;d = contour_description [3,2,4]} in d `contour_is_prefix_of` c == True

contour_eq_at :: Contour_Description -> Contour_Description -> Int -> Bool Source #

Are `Contour_Description`

s *p* and *q* equal at column *n*.

let {c = contour_description [3,2,4,1,5] ;d = contour_description [3,2,4,1]} in map (contour_eq_at c d) [0..4] == [True,True,True,True,False]

# Contour drawing

draw_contour :: Integral i => Contour_Description -> [i] Source #

Derive an `Integral`

contour that would be described by
`Contour_Description`

. Diverges for impossible contours.

draw_contour (contour_description "abdc") == [0,1,3,2]

contour_description_invert :: Contour_Description -> Contour_Description Source #

Invert `Contour_Description`

.

let c = contour_description "abdc" in draw_contour (contour_description_invert c) == [3,2,0,1]

# Construction

type Build_f st e = st -> Maybe (e, st) Source #

Function to perhaps generate an element and a new state from an
initial state. This is the function provided to `unfoldr`

.

type Conforms_f e = Int -> [e] -> Bool Source #

Function to test is a partial sequence conforms to the target sequence.

build_f_n :: Build_f st e -> Build_f (Int, st) e Source #

Transform a `Build_f`

to produce at most *n* elements.

let f i = Just (i,succ i) in unfoldr (build_f_n f) (5,'a') == "abcde"

build_sequence :: Int -> Build_f st e -> Conforms_f e -> Int -> st -> (Maybe [e], st) Source #

Attempt to construct a sequence of *n* elements given a `Build_f`

to generate possible elements, a `Conforms_f`

that the result
sequence must conform to at each step, an `Int`

to specify the
maximum number of elements to generate when searching for a
solution, and an initial state.

let {b_f i = Just (i,i+1) ;c_f i x = odd (sum x `div` i)} in build_sequence 6 b_f c_f 20 0 == (Just [1,2,6,11,15,19],20)

build_contour :: Ord e => Build_f st e -> Contour_Description -> Int -> st -> (Maybe [e], st) Source #

Attempt to construct a sequence that has a specified contour.
The arguments are a `Build_f`

to generate possible elements, a
`Contour_Description`

that the result sequence must conform to, an
`Int`

to specify the maximum number of elements to generate when
searching for a solution, and an initial state.

import System.Random

let {f = Just . randomR ('a','z') ;c = contour_description "atdez" ;st = mkStdGen 2347} in fst (build_contour f c 1024 st) == Just "nvruy"

build_contour_retry :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> (Maybe [e], st) Source #

A variant on `build_contour`

that retries a specified number of
times using the final state of the failed attempt as the state for
the next try.

let {f = Just . randomR ('a','z') ;c = contour_description "atdezjh" ;st = mkStdGen 2347} in fst (build_contour_retry f c 64 8 st) == Just "nystzvu"

build_contour_set :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> [[e]] Source #

A variant on `build_contour_retry`

that returns the set of all
sequences constructed.

let {f = Just . randomR ('a','z') ;c = contour_description "atdezjh" ;st = mkStdGen 2347} in length (build_contour_set f c 64 64 st) == 60

build_contour_set_nodup :: Ord e => Build_f st e -> Contour_Description -> Int -> Int -> st -> [[e]] Source #

Variant of `build_contour_set`

that halts when an generated
sequence is a duplicate of an already generated sequence.

let {f = randomR ('a','f') ;c = contour_description "cafe" ;st = mkStdGen 2346836 ;r = build_contour_set_nodup f c 64 64 st} in filter ("c" `isPrefixOf`) r == ["cafe","cbed","caed"]

# Examples

Example from p.262 (quarter-note durations)

ex_1 == [2,3/2,1/2,1,2] compare_adjacent ex_1 == [GT,GT,LT,LT] show (contour_half_matrix ex_1) == "2221 220 00 0" draw_contour (contour_description ex_1) == [3,2,0,1,3]

let d = contour_description_invert (contour_description ex_1) in (show d,is_possible d) == ("0001 002 22 2",True)

Example on p.265 (pitch)

ex_2 == [0,5,3] show (contour_description ex_2) == "00 2"

Example on p.265 (pitch)

ex_3 == [12,7,6,7,8,7] show (contour_description ex_3) == "22222 2101 000 01 2" contour_description_ix (contour_description ex_3) (0,5) == GT is_possible (contour_description ex_3) == True

ex_4 :: Contour_Description Source #

Example on p.266 (impossible)

show ex_4 == "2221 220 00 1" is_possible ex_4 == False violations ex_4 == [(0,3,4,GT),(1,3,4,GT)]