Music.Theory.Metric.Polansky_1996

Larry Polansky. "Morphological Metrics". Journal of New Music Research, 25(4):289-368, 1996.

Documentation

type Interval a n = a -> a -> n

Distance function, ordinarily n below is in Num, Fractional or Real.

dif_i :: (Integral a, Num b) => a -> a -> b

fromIntegral . -.

dif_r :: (Real a, Fractional b) => a -> a -> b

realToFrac . -.

abs_dif :: Num n => Interval a n -> a -> a -> n

abs . f.

sqr :: Num a => a -> a

Square.

sqr_dif :: Num n => Interval a n -> a -> a -> n

sqr . f.

sqr_abs_dif :: Num n => Interval a n -> a -> a -> n

sqr . abs . f.

sqrt_abs_dif :: Floating c => Interval a c -> a -> a -> c

sqrt . abs . f.

city_block_metric :: Num n => Interval a n -> (a, a) -> (a, a) -> n

City block metric, p.296

 city_block_metric (-) (1,2) (3,5) == 2+3

euclidean_metric_2 :: Floating n => Interval a n -> (a, a) -> (a, a) -> n

Two-dimensional euclidean metric, p.297.

 euclidean_metric_2 (-) (1,2) (3,5) == sqrt (4+9)

euclidean_metric_l :: Floating c => Interval b c -> [b] -> [b] -> c

n-dimensional euclidean metric

 euclidean_metric_l (-) [1,2] [3,5] == sqrt (4+9)
 euclidean_metric_l (-) [1,2,3] [2,4,6] == sqrt (1+4+9)

cbrt :: Floating a => a -> a

Cube root.

 map cbrt [1,8,27] == [1,2,3]

nthrt :: Floating a => a -> a -> a

n-th root

 map (nthrt 4) [1,16,81] == [1,2,3]

minkowski_metric_2 :: Floating a => Interval t a -> a -> (t, t) -> (t, t) -> a

Two-dimensional Minkowski metric, p.297

 minkowski_metric_2 (-) 1 (1,2) (3,5) == 5
 minkowski_metric_2 (-) 2 (1,2) (3,5) == sqrt (4+9)
 minkowski_metric_2 (-) 3 (1,2) (3,5) == cbrt (8+27)

minkowski_metric_l :: Floating a => Interval t a -> a -> [t] -> [t] -> a

n-dimensional Minkowski metric

 minkowski_metric_l (-) 2 [1,2,3] [2,4,6] == sqrt (1+4+9)
 minkowski_metric_l (-) 3 [1,2,3] [2,4,6] == cbrt (1+8+27)

d_dx :: Interval a n -> [a] -> [n]

Integration with f.

 d_dx (-) [0,2,4,1,0] == [2,2,-3,-1]
 d_dx (-) [2,3,0,4,1] == [1,-3,4,-3]

d_dx_abs :: Num n => Interval a n -> [a] -> [n]

map abs . d_dx.

 d_dx_abs (-) [0,2,4,1,0] == [2,2,3,1]
 d_dx_abs (-) [2,3,0,4,1] == [1,3,4,3]

olm_no_delta' :: Fractional a => [a] -> [a] -> a

Ordered linear magnitude (no delta), p.300

 olm_no_delta' [0,2,4,1,0] [2,3,0,4,1] == 1.25

olm_general :: (Fractional a, Enum a, Fractional n) => Interval a n -> [a] -> [a] -> n

Ordered linear magintude (general form) p.302

 olm_general (abs_dif (-)) [0,2,4,1,0] [2,3,0,4,1] == 1.25
 olm_general (abs_dif (-)) [1,5,12,2,9,6] [7,6,4,9,8,1] == 4.6

type Delta n a = [n] -> Int -> a

Delta (Δ) determines an interval given a sequence and an index.

ix_dif :: Interval a t -> Delta a t

f at indices i and i+1 of x.

 map (ix_dif (-) [0,1,3,6,10]) [0..3] == [-1,-2,-3,-4]

abs_ix_dif :: Num n => Interval a n -> Delta a n

abs . ix_dif

 map (abs_ix_dif (-) [0,2,4,1,0]) [0..3] == [2,2,3,1]

sqr_abs_ix_dif :: Num n => Interval a n -> Delta a n

sqr . abs_ix_dif

 map (sqr_abs_ix_dif (-) [0,2,4,1,0]) [0..3] == [4,4,9,1]
 map (sqr_abs_ix_dif (-) [2,3,0,4,1]) [0..3] == [1,9,16,9]

type Psi a = a -> a -> a

Psi (Ψ) joins Delta equivalent intervals from morphologies m and n.

olm :: (Fractional a, Enum a) => Psi a -> Delta n a -> ([a] -> a) -> [n] -> [n] -> a

Ordered linear magintude (generalised-interval form) p.305

 olm (abs_dif dif_r) (abs_ix_dif dif_r) (const 1) [1,5,12,2,9,6] [7,6,4,9,8,1] == 4.6
 olm (abs_dif dif_r) (abs_ix_dif dif_r) maximum [1,5,12,2,9,6] [7,6,4,9,8,1] == 0.46

olm_no_delta :: (Real a, Real n, Enum n, Fractional n) => [a] -> [a] -> n

olm_no_delta_squared :: (Enum a, Floating a) => [a] -> [a] -> a

second_order :: Num n => ([n] -> [n] -> t) -> [n] -> [n] -> t

olm_no_delta_second_order :: (Real a, Enum a, Fractional a) => [a] -> [a] -> a

olm_no_delta_squared_second_order :: (Enum a, Floating a) => [a] -> [a] -> a

second_order_binonial_coefficient :: Fractional a => a -> a

Second order binomial coefficient, p.307

 map second_order_binonial_coefficient [2..10] == [1,3,6,10,15,21,28,36,45]

direction_interval :: Ord i => [i] -> [Ordering]

d_dx of flip compare.

 direction_interval [5,9,3,2] == [LT,GT,GT]
 direction_interval [2,5,6,6] == [LT,LT,EQ]

ord_hist :: Integral t => [Ordering] -> (t, t, t)

Histogram of list of Orderings.

 ord_hist [LT,GT,GT] == (1,0,2)

direction_vector :: Integral i => Ord a => [a] -> (i, i, i)

Histogram of directions of adjacent elements, p.312.

 direction_vector [5,9,3,2] == (1,0,2)
 direction_vector [2,5,6,6] == (2,1,0)

uld :: (Integral n, Ord a) => [a] -> [a] -> Ratio n

Unordered linear direction, p.311 (Fig. 5)

 uld [5,9,3,2] [2,5,6,6] == 2/3
 uld [5,3,6,1,4] [3,6,1,4,2] == 0

old :: (Ord i, Integral a) => [i] -> [i] -> Ratio a

Ordered linear direction, p.312

 direction_interval [5,3,6,1,4] == [GT,LT,GT,LT]
 direction_interval [3,6,1,4,2] == [LT,GT,LT,GT]
 old [5,3,6,1,4] [3,6,1,4,2] == 1

ocd :: (Ord a, Integral i) => [a] -> [a] -> Ratio i

Ordered combinatorial direction, p.314

 ocd [5,9,3,2] [2,5,6,6] == 5/6
 ocd [5,3,6,1,4] [3,6,1,4,2] == 4/5

ucd :: (Integral n, Ord a) => [a] -> [a] -> Ratio n

Unordered combinatorial direction, p.314

 ucd [5,9,3,2] [2,5,6,6] == 5/6
 ucd [5,3,6,1,4] [3,6,1,4,2] == 0
 ucd [5,3,7,6] [2,1,2,1] == 1/2
 ucd [2,1,2,1] [8,3,5,4] == 1/3
 ucd [5,3,7,6] [8,3,5,4] == 1/3

combinatorial_magnitude_matrix :: Interval a n -> [a] -> [[n]]

half_matrix_f, Fig.9, p.318

 let r = [[2,3,1,4]
           ,[1,3,6]
             ,[4,7]
               ,[3]]
 in combinatorial_magnitude_matrix (abs_dif (-)) [5,3,2,6,9] == r

ulm_simplified :: Fractional n => Interval a n -> [a] -> [a] -> n

Unordered linear magnitude (simplified), p.320-321

 let r = abs (sum [5,4,3,6] - sum [12,2,11,7]) / 4
 in ulm_simplified (abs_dif (-)) [1,6,2,5,11] [3,15,13,2,9] == r

 ulm_simplified (abs_dif (-)) [1,5,12,2,9,6] [7,6,4,9,8,1] == 3

ocm_zcm :: (Fractional n, Num a) => Interval a n -> [a] -> [a] -> (n, n, [n])

ocm :: (Fractional a, Enum a, Fractional n) => Interval a n -> [a] -> [a] -> n

Ordered combinatorial magnitude (OCM), p.323

 ocm (abs_dif (-)) [1,6,2,5,11] [3,15,13,2,9] == 5.2
 ocm (abs_dif (-)) [1,5,12,2,9,6] [7,6,4,9,8,1] == 3.6

ocm_absolute_scaled :: (Ord a, Fractional a, Enum a, Ord n, Fractional n) => Interval a n -> [a] -> [a] -> n

Ordered combinatorial magnitude (OCM), p.323

 ocm_absolute_scaled (abs_dif (-)) [1,6,2,5,11] [3,15,13,2,9] == 0.4
 ocm_absolute_scaled (abs_dif (-)) [1,5,12,2,9,6] [7,6,4,9,8,1] == 54/(15*11)