Music.Theory.Math

Math functions.

Documentation

mod5 :: Integral i => i -> i

mod 5.

mod7 :: Integral i => i -> i

mod 7.

mod12 :: Integral i => i -> i

mod 12.

mod16 :: Integral i => i -> i

mod 16.

integral_and_fractional_parts :: (Integral i, RealFrac t) => t -> (i, t)

http://reference.wolfram.com/mathematica/ref/FractionalPart.html i.e. properFraction

integral_and_fractional_parts 1.5 == (1,0.5)

integer_and_fractional_parts :: RealFrac t => t -> (Integer, t)

Type specialised.

fractional_part :: RealFrac a => a -> a

http://reference.wolfram.com/mathematica/ref/FractionalPart.html

import Sound.SC3.Plot {- hsc3-plot -}
plot_p1_ln [map fractional_part [-2.0,-1.99 .. 2.0]]

real_floor :: (Real r, Integral i) => r -> i

floor of real_to_double.

real_floor_int :: Real r => r -> Int

Type specialised real_floor.

real_round :: (Real r, Integral i) => r -> i

round of real_to_double.

real_round_int :: Real r => r -> Int

Type specialised real_round.

round_int :: RealFrac t => t -> Int

Type specialised round

from_integral_to_int :: Integral i => i -> Int

Type-specialised fromIntegral

int_id :: Int -> Int

Type-specialised id

zero_to_precision :: Real r => Int -> r -> Bool

Is r zero to k decimal places.

map (flip zero_to_precision 0.00009) [4,5] == [True,False]
map (zero_to_precision 4) [0.00009,1.00009] == [True,False]

whole_to_precision :: Real r => Int -> r -> Bool

Is r whole to k decimal places.

map (flip whole_to_precision 1.00009) [4,5] == [True,False]

sawtooth_wave :: RealFrac a => a -> a

http://reference.wolfram.com/mathematica/ref/SawtoothWave.html

plot_p1_ln [map sawtooth_wave [-2.0,-1.99 .. 2.0]]

rational_simplifies :: Integral a => (a, a) -> Bool

Predicate that is true if n/d can be simplified, ie. where gcd of n and d is not 1.

map rational_simplifies [(2,3),(4,6),(5,7)] == [False,True,False]

rational_nd :: Integral t => Ratio t -> (t, t)

numerator and denominator of rational.

rational_whole :: Integral a => Ratio a -> Maybe a

Rational as a whole number, or Nothing.

rational_whole_err :: Integral a => Ratio a -> a

Erroring variant.

ratio_nd_sum :: Integral t => Ratio t -> t

Sum of numerator & denominator.

real_is_whole :: Real n => n -> Bool

Is n a whole (integral) value.

map real_is_whole [-1.0,-0.5,0.0,0.5,1.0] == [True,False,True,False,True]

floor_f :: (RealFrac a, Num b) => a -> b

fromInteger . floor.

round_to :: RealFrac n => n -> n -> n

Round b to nearest multiple of a.

map (round_to 0.25) [0,0.1 .. 1] == [0.0,0.0,0.25,0.25,0.5,0.5,0.5,0.75,0.75,1.0,1.0]
map (round_to 25) [0,10 .. 100] == [0,0,25,25,50,50,50,75,75,100,100]

recip_m :: (Eq a, Fractional a) => a -> Maybe a

Variant of recip that checks input for zero.

map recip [1,1/2,0,-1]
map recip_m [1,1/2,0,-1] == [Just 1,Just 2,Nothing,Just (-1)]

One-indexed

oi_mod :: Integral a => a -> a -> a

One-indexed mod function.

map (`oi_mod` 5) [1..10] == [1,2,3,4,5,1,2,3,4,5]

oi_divMod :: Integral t => t -> t -> (t, t)

One-indexed divMod function.

map (`oi_divMod` 5) [1,3 .. 9] == [(0,1),(0,3),(0,5),(1,2),(1,4)]

I = integral

i_square_root :: Integral t => t -> t

Integral square root (sqrt) function.

map i_square_root [0,1,4,9,16,25,36,49,64,81,100] == [0 .. 10]
map i_square_root [4 .. 16] == [2,2,2,2,2,3,3,3,3,3,3,3,4]

Interval

in_open_interval :: Ord a => (a, a) -> a -> Bool

(0,1) = {x | 0 < x < 1}

in_closed_interval :: Ord a => (a, a) -> a -> Bool

0,1

= {x | 0 ≤ x ≤ 1}

in_left_half_open_interval :: Ord a => (a, a) -> a -> Bool

(p,q] (0,1] = {x | 0 < x ≤ 1}

in_right_half_open_interval :: Ord a => (a, a) -> a -> Bool

[p,q) [0,1) = {x | 0 ≤ x < 1}

nth_root :: (Floating a, Eq a) => a -> a -> a

Calculate nth root of x.

12 `nth_root` 2 == 1.0594630943592953

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

Arithmetic mean (average) of a list.

map arithmetic_mean [[-3..3],[0..5],[1..5],[3,5,7],[7,7],[3,9,10,11,12]] == [0,2.5,3,5,7,9]

ns_mean :: Floating a => [a] -> a

Numerically stable mean

map ns_mean [[-3..3],[0..5],[1..5],[3,5,7],[7,7],[3,9,10,11,12]] == [0,2.5,3,5,7,9]

square :: Num a => a -> a

Square of n.

square 5 == 25

totient :: Integral i => i -> i

The totient function phi(n), also called Euler's totient function.

import Sound.SC3.Plot {- hsc3-plot -}
plot_p1_stp [map totient [1::Int .. 100]]

farey :: Integral i => i -> [Ratio i]

The n-th order Farey sequence.

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

farey_length :: Integral i => i -> i

The length of the n-th order Farey sequence.

map farey_length [1 .. 12] == [2,3,5,7,11,13,19,23,29,33,43,47]
map (length . farey) [1 .. 12] == map farey_length [1 .. 12]

stern_brocot_tree_f :: Num n => [(n, n)] -> [[(n, n)]]

Function to generate the Stern-Brocot tree from an initial row. % normalises so 1/0 cannot be written as a Rational, hence (n,d).

stern_brocot_tree :: Num n => [[(n, n)]]

The Stern-Brocot tree from (01,10).

let t = stern_brocot_tree
t !! 0 == [(0,1),(1,0)]
t !! 1 == [(0,1),(1,1),(1,0)]
t !! 2 == [(0,1),(1,2),(1,1),(2,1),(1,0)]
t !! 3 == [(0,1),(1,3),(1,2),(2,3),(1,1),(3,2),(2,1),(3,1),(1,0)]

map length (take 12 stern_brocot_tree) == [2,3,5,9,17,33,65,129,257,513,1025,2049] -- A000051

stern_brocot_tree_lhs :: Num n => [[(n, n)]]

Left-hand (rational) side of the the Stern-Brocot tree, ie, from (01,11).

stern_brocot_tree_f_r :: Integral n => [Ratio n] -> [[Ratio n]]

stern_brocot_tree_f as Ratios, for finite subsets.

let t = stern_brocot_tree_f_r [0,1]
t !! 1 == [0,1/2,1]
t !! 2 == [0,1/3,1/2,2/3,1]
t !! 3 == [0,1/4,1/3,2/5,1/2,3/5,2/3,3/4,1]
t !! 4 == [0,1/5,1/4,2/7,1/3,3/8,2/5,3/7,1/2,4/7,3/5,5/8,2/3,5/7,3/4,4/5,1]

outer_product :: (a -> b -> c) -> [a] -> [b] -> [[c]]

Outer product of vectors represented as lists, c.f. liftM2

outer_product (*) [2..5] [2..5] == [[4,6,8,10],[6,9,12,15],[8,12,16,20],[10,15,20,25]]
liftM2 (*) [2..5] [2..5] == [4,6,8,10,6,9,12,15,8,12,16,20,10,15,20,25]