Music.Theory.Meter.Barlow_1987

Clarence Barlow. "Two Essays on Theory". Computer Music Journal, 11(1):44-60, 1987. Translated by Henning Lohner.

Documentation

traceShow :: a -> b -> b

at :: Integral n => [a] -> n -> a

One indexed variant of genericIndex.

 map (at [11..13]) [1..3] == [11,12,13]

at' :: (Num a, Show a, Integral n, Show n, Show m) => m -> [a] -> n -> a

Variant of at with boundary rules and specified error message.

 map (at' 'x' [11..13]) [0..4] == [1,11,12,13,1]
 at' 'x' [0] 3 == undefined

mod' :: (Integral a, Show a) => a -> a -> a

Variant of mod with input constraints.

 mod' (-1) 2 == 1

type R = Double

Alias for Double (quieten compiler).

to_r :: (Integral n, Show n) => n -> R

Specialised variant of fromIntegral.

div' :: (Integral a, Show a) => String -> a -> a -> a

Variant on div with input constraints.

type Stratification t = [t]

A stratification is a tree of integral subdivisions.

indispensibilities :: (Integral n, Show n) => Stratification n -> [n]

Indispensibilities from stratification.

 indispensibilities [3,2,2] == [11,0,6,3,9,1,7,4,10,2,8,5]
 indispensibilities [2,3,2] == [11,0,6,2,8,4,10,1,7,3,9,5]
 indispensibilities [2,2,3] == [11,0,4,8,2,6,10,1,5,9,3,7]
 indispensibilities [3,5] == [14,0,9,3,6,12,1,10,4,7,13,2,11,5,8]

lower_psi :: (Integral a, Show a) => Stratification a -> a -> a -> a

The indispensibility measure (ψ).

 map (lower_psi [2] 1) [1..2] == [1,0]
 map (lower_psi [3] 1) [1..3] == [2,0,1]
 map (lower_psi [2,2] 2) [1..4] == [3,0,2,1]
 map (lower_psi [5] 1) [1..5] == [4,0,3,1,2]
 map (lower_psi [3,2] 2) [1..6] == [5,0,3,1,4,2]
 map (lower_psi [2,3] 2) [1..6] == [5,0,2,4,1,3]

reverse_primes :: (Integral n, Show n) => n -> [n]

The first nth primes, reversed.

 reverse_primes 14 == [43,41,37,31,29,23,19,17,13,11,7,5,3,2]

prime_stratification :: (Integral n, Show n) => n -> Stratification n

Generate prime stratification for n.

 map prime_stratification [2,3,5,7,11] == [[2],[3],[5],[7],[11]]
 map prime_stratification [6,8,9,12] == [[3,2],[2,2,2],[3,3],[3,2,2]]
 map prime_stratification [22,10,4,1] == [[11,2],[5,2],[2,2],[]]
 map prime_stratification [18,16,12] == [[3,3,2],[2,2,2,2],[3,2,2]]

upper_psi :: (Integral a, Show a) => a -> a -> a

Fundamental indispensibilities for prime numbers (Ψ).

 map (upper_psi 2) [1..2] == [1,0]
 map (upper_psi 3) [1..3] == [2,0,1]
 map (upper_psi 5) [1..5] == [4,0,3,1,2]
 map (upper_psi 7) [1..7] == [6,0,4,2,5,1,3]
 map (upper_psi 11) [1..11] == [10,0,6,4,9,1,7,3,8,2,5]
 map (upper_psi 13) [1..13] == [12,0,7,4,10,1,8,5,11,2,9,3,6]

thinning_table :: (Integral n, Show n) => Stratification n -> [[Bool]]

Table such that each subsequent row deletes the least indispensibile pulse.

 thinning_table [3,2] == [[True,True,True,True,True,True]
                         ,[True,False,True,True,True,True]
                         ,[True,False,True,False,True,True]
                         ,[True,False,True,False,True,False]
                         ,[True,False,False,False,True,False]
                         ,[True,False,False,False,False,False]]

thinning_table_pp :: (Integral n, Show n) => Stratification n -> String

Trivial pretty printer for thinning_table.

 putStrLn (thinning_table_pp [3,2])
 putStrLn (thinning_table_pp [2,3])

 ******   ******
 *.****   *.****
 *.*.**   *.**.*
 *.*.*.   *..*.*
 *...*.   *..*..
 *.....   *.....

relative_to_length :: (Real a, Fractional b) => [a] -> [b]

Scale values against length of list minus one.

 relative_to_length [0..5] == [0.0,0.2,0.4,0.6,0.8,1.0]

relative_indispensibilities :: (Integral n, Show n) => Stratification n -> [R]

Variant of indispensibilities that scales value to lie in (0,1).

relative_indispensibilities [3,2] == [1,0,0.6,0.2,0.8,0.4]

align_meters :: (t -> [b]) -> t -> t -> [(b, b)]

Align two meters (given as stratifications) to least common multiple of their degrees. The indispensibilities function is given as an argument so that it may be relative if required. This generates Table 7 (p.58).

 let r = [(5,5),(0,0),(2,3),(4,1),(1,4),(3,2)]
 in align_meters indispensibilities [2,3] [3,2] == r

 let r = [(1,1),(0,0),(0.4,0.6),(0.8,0.2),(0.2,0.8),(0.6,0.4)]
 in align_meters relative_indispensibilities [2,3] [3,2] == r

 align_meters indispensibilities [2,2,3] [3,5]
 align_meters relative_indispensibilities [2,2,3] [3,5]

type S_MM t = ([t], t)

Type pairing a stratification and a tempo.

whole_div :: Integral a => a -> a -> a

Variant of div that requires mod be 0.

whole_quot :: Integral a => a -> a -> a

Variant of quot that requires rem be 0.

prolong_stratifications :: (Integral n, Show n) => S_MM n -> S_MM n -> ([n], [n])

Rule to prolong stratification of two S_MM values such that pulse at the deeper level are aligned. (Paragraph 2, p.58)

 let x = ([2,2,2],1)
 in prolong_stratifications x x == (fst x,fst x)

 let r = ([2,5,3,3,2],[3,2,5,5])
 in prolong_stratifications ([2,5],50) ([3,2],60) == r

 prolong_stratifications ([2,2,3],5) ([3,5],4) == ([2,2,3],[3,5])

mean :: Fractional a => [a] -> a

Arithmetic mean (average) of a list.

 mean [0..5] == 2.5

square :: Num a => a -> a

Square of n.

 square 5 == 25

align_s_mm :: (Integral n, Show n) => ([n] -> [t]) -> S_MM n -> S_MM n -> [(t, t)]

Composition of prolong_stratifications and align_meters.

 align_s_mm indispensibilities ([2,2,3],5) ([3,5],4)

upper_psi' :: (Integral a, Show a) => a -> a -> a

An attempt at Equation 5 of the CMJ paper. When n is h-1 the output is incorrect (it is the product of the correct values for n at h-1 and h).

 map (upper_psi' 5) [1..5] /= [4,0,3,1,2]
 map (upper_psi' 7) [1..7] /= [6,0,4,2,5,1,3]
 map (upper_psi' 11) [1..11] /= [10,0,6,4,9,1,7,3,8,2,5]
 map (upper_psi' 13) [1..13] /= [12,0,7,4,10,1,8,5,11,2,9,3,6]

mps_limit :: Floating a => a -> a

The MPS limit equation given on p.58.

 mps_limit 3 == 21 + 7/9

mean_square_product :: Fractional n => [(n, n)] -> n

The square of the product of the input sequence is summed, then divided by the square of the sequence length.

 mean_square_product [(0,0),(1,1),(2,2),(3,3)] == 6.125
 mean_square_product [(2,3),(4,5)] == (6^2 + 20^2) / 2^2

metrical_affinity :: (Integral n, Show n) => [n] -> n -> [n] -> n -> R

An incorrect attempt at the description in paragraph two of p.58 of the CMJ paper.

 let p ~= q = abs (p - q) < 1e-4
 metrical_affinity [2,3] 1 [3,2] 1 ~= 0.0324
 metrical_affinity [2,2,3] 20 [3,5] 16 ~= 0.0028

metrical_affinity' :: (Integral t, Show t) => [t] -> t -> [t] -> t -> R

An incorrect attempt at Equation 6 of the CMJ paper, see omega_z.

 let p ~= q = abs (p - q) < 1e-4
 metrical_affinity' [2,2,2] 1 [2,2,2] 1 ~= 1.06735
 metrical_affinity' [2,2,2] 1 [2,2,3] 1 ~= 0.57185
 metrical_affinity' [2,2,2] 1 [2,3,2] 1 ~= 0.48575
 metrical_affinity' [2,2,2] 1 [3,2,2] 1 ~= 0.45872

 metrical_affinity' [3,2,2] 3 [2,2,3] 2 ~= 0.10282