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Language	Haskell98

Music.Theory.Meter.Barlow_1987

Description

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

Synopsis

Documentation

traceShow :: a -> b -> b Source

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

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 Source

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 Source

Variant of mod with input constraints.

mod' (-1) 2 == 1

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

Specialised variant of fromIntegral.

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

Variant on div with input constraints.

type Stratification t = [t] Source

A stratification is a tree of integral subdivisions.

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

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 Source

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] Source

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 Source

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 Source

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]] Source

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 Source

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] Source

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] Source

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)] Source

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) Source

Type pairing a stratification and a tempo.

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

Variant of div that requires mod be 0.

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

Variant of quot that requires rem be 0.

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

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 Source

Arithmetic mean (average) of a list.

mean [0..5] == 2.5

square :: Num a => a -> a Source

Square of n.

square 5 == 25

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

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 Source

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 Source

The MPS limit equation given on p.58.

mps_limit 3 == 21 + 7/9

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

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 Source

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 Source

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