Music.Theory.Interval.Barlow_1987

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

Documentation

barlow :: (Integral a, Fractional b) => a -> b

Barlow's indigestibility function for prime numbers.

 map barlow [1,2,3,5,7,11,13] == [0,1,8/3,32/5,72/7,200/11,288/13]

factor :: Integral a => [a] -> a -> [a]

Generate list of factors of n from x.

 factor primes 315 == [3,3,5,7]

prime_factors :: Integral a => a -> [a]

factor n from primes.

 prime_factors 315 == [3,3,5,7]

multiplicities :: (Eq a, Integral n) => [a] -> [(a, n)]

Collect number of occurences of each element of a sorted list.

 multiplicities [1,1,1,2,2,3] == [(1,3),(2,2),(3,1)]

prime_factors_m :: Integral a => a -> [(a, a)]

multiplicities . prime_factors.

 prime_factors_m 315 == [(3,2),(5,1),(7,1)]

merge :: (Ord a, Num b, Eq b) => [(a, b)] -> [(a, b)] -> [(a, b)]

Merging function for rational_prime_factors_m.

rational_prime_factors_m :: Integral b => (b, b) -> [(b, b)]

Collect the prime factors in a rational number given as a numerator/ denominator pair (n,m). Prime factors are listed in ascending order with their positive or negative multiplicities, depending on whether the prime factor occurs in the numerator or the denominator (after cancelling out common factors).

 rational_prime_factors_m (16,15) == [(2,4),(3,-1),(5,-1)]
 rational_prime_factors_m (10,9) == [(2,1),(3,-2),(5,1)]
 rational_prime_factors_m (81,64) == [(2,-6),(3,4)]
 rational_prime_factors_m (27,16) == [(2,-4),(3,3)]
 rational_prime_factors_m (12,7) == [(2,2),(3,1),(7,-1)]

rational_prime_factors_t :: Integral b => Int -> (b, b) -> [b]

Variant of rational_prime_factors_m giving results in a table up to the nth prime.

 rational_prime_factors_t 6 (12,7) == [2,1,0,-1,0,0]

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

Compute the disharmonicity of the interval (p,q) using the prime valuation function pv.

 map (disharmonicity barlow) [(9,10),(8,9)] ~= [12.733333,8.333333]

harmonicity :: (Integral a, Fractional b) => (a -> b) -> (a, a) -> b

The reciprocal of disharmonicity.

 map (harmonicity barlow) [(9,10),(8,9)] ~= [0.078534,0.120000]

harmonicity_r :: (Integral a, Fractional b) => (a -> b) -> Ratio a -> b

Variant of harmonicity with Ratio input.

cents :: (Real a, Floating b) => a -> b

Interval ratio to cents.

 map cents [16%15,16%9] == [111.73128526977776,996.0899982692251]

to_rational :: Integral a => (a, a) -> Ratio a

uncurry (%).

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

Make numerator denominator pair of n.

type Table_2_Row = (Double, [Integer], Rational, Double)

Set of 1. interval size (cents), 2. intervals as product of powers of primes, 3. frequency ratio and 4. harmonicity value.

table_2 :: Double -> [Table_2_Row]

Table 2 (p.45)

 length (table_2 0.06) == 24

table_2_pp :: Table_2_Row -> String

Pretty printer for Table_2_Row values.

 mapM_ (putStrLn . table_2_pp) (table_2 0.06)

    0.000 |  0  0  0  0  0  0 |  1:1  | Infinity
  111.731 |  4 -1 -1  0  0  0 | 15:16 | 0.076531
  182.404 |  1 -2  1  0  0  0 |  9:10 | 0.078534
  203.910 | -3  2  0  0  0  0 |  8:9  | 0.120000
  231.174 |  3  0  0 -1  0  0 |  7:8  | 0.075269
  266.871 | -1 -1  0  1  0  0 |  6:7  | 0.071672
  294.135 |  5 -3  0  0  0  0 | 27:32 | 0.076923
  315.641 |  1  1 -1  0  0  0 |  5:6  | 0.099338
  386.314 | -2  0  1  0  0  0 |  4:5  | 0.119048
  407.820 | -6  4  0  0  0  0 | 64:81 | 0.060000
  435.084 |  0  2  0 -1  0  0 |  7:9  | 0.064024
  498.045 |  2 -1  0  0  0  0 |  3:4  | 0.214286
  519.551 | -2  3 -1  0  0  0 | 20:27 | 0.060976
  701.955 | -1  1  0  0  0  0 |  2:3  | 0.272727
  764.916 |  1 -2  0  1  0  0 |  9:14 | 0.060172
  813.686 |  3  0 -1  0  0  0 |  5:8  | 0.106383
  884.359 |  0 -1  1  0  0  0 |  3:5  | 0.110294
  905.865 | -4  3  0  0  0  0 | 16:27 | 0.083333
  933.129 |  2  1  0 -1  0  0 |  7:12 | 0.066879
  968.826 | -2  0  0  1  0  0 |  4:7  | 0.081395
  996.090 |  4 -2  0  0  0  0 |  9:16 | 0.107143
 1017.596 |  0  2 -1  0  0  0 |  5:9  | 0.085227
 1088.269 | -3  1  1  0  0  0 |  8:15 | 0.082873
 1200.000 |  1  0  0  0  0  0 |  1:2  | 1.000000