```module Music.Theory.Tuning where

import Data.List
import Data.Ratio

type Approximate_Ratio = Double
type Cents = Double

-- | Harmonic series (folded)
harmonic_series_folded :: Integer -> [Rational]
harmonic_series_folded n =
let hs = (zipWith (%) (repeat 1) [1..n])
fold x = if x >= 0.5
then x
else fold (x * 2)
in nub (sort (map fold hs))

-- | Pythagorean tuning
pythagorean_r :: [Rational]
pythagorean_r =
[1%1,243%256 {- 2048%2187 -}
,8%9,27%32
,64%81
,3%4,512%729
,2%3,81%128
,16%27,9%16
,128%243
,1%2]

-- | Pythagorean tuning
pythagorean_c :: [Cents]
pythagorean_c = map (to_cents.approximate_ratio) pythagorean_r

-- | Werckmeister III, Andreas Werckmeister (1645-1706)
werckmeister_iii_ar :: [Approximate_Ratio]
werckmeister_iii_ar =
let c0 = 2 ** (1/2)
c1 = 2 ** (1/4)
c2 = 8 ** (1/4)
in [1,256/243
,64/81 * c0,32/27
,256/243 * c1
,4/3,1024/729
,8/9 * c2,128/81
,1024/729 * c1,16/9
,128/81 * c1]

-- | Werckmeister III, Andreas Werckmeister (1645-1706)
werckmeister_iii_c :: [Cents]
werckmeister_iii_c = map to_cents werckmeister_iii_ar

-- | Werckmeister IV, Andreas Werckmeister (1645-1706)
werckmeister_iv_ar :: [Approximate_Ratio]
werckmeister_iv_ar =
let c0 = 2 ** (1/3)
c1 = 4 ** (1/3)
in [1,16384/19683 * c0
,8/9 * c0,32/27
,64/81 * c1
,4/3,1024/729
,32/27 * c0,8192/6561 * c0
,256/243 * c1,9/(4*c0)
,4096/2187]

-- | Werckmeister IV, Andreas Werckmeister (1645-1706)
werckmeister_iv_c :: [Cents]
werckmeister_iv_c = map to_cents werckmeister_iv_ar

-- | Werckmeister V, Andreas Werckmeister (1645-1706)
werckmeister_v_ar :: [Approximate_Ratio]
werckmeister_v_ar =
let c0 = 2 ** (1/4)
c1 = 2 ** (1/2)
c2 = 8 ** (1/4)
in [1,8/9 * c0
,9/8,c0
,8/9 * c1
,9/8 * c0,c1
,3/2,128/81
,c2,3/c2
,4/3 * c1]

-- | Werckmeister V, Andreas Werckmeister (1645-1706)
werckmeister_v_c :: [Cents]
werckmeister_v_c = map to_cents werckmeister_v_ar

-- | Werckmeister VI, Andreas Werckmeister (1645-1706)
werckmeister_vi_r :: [Rational]
werckmeister_vi_r =
[1,98%93
,28%25,196%165
,49%39
,4%3,196%139
,196%131,49%31
,196%117,98%55
,49%26]

-- | Werckmeister VI, Andreas Werckmeister (1645-1706)
werckmeister_vi_c :: [Cents]
werckmeister_vi_c = map (to_cents.approximate_ratio) werckmeister_vi_r

-- | Pietro Aaron (1523) - Meantone temperament
pietro_aaron_1523_c :: [Cents]
pietro_aaron_1523_c =
[0,76.0
,193.2,310.3
,386.3
,503.4,579.5
,696.8,772.6
,889.7,1006.8
,1082.9
,1200]

-- | Thomas Young (1799) - Well Temperament
thomas_young_1799_c :: [Cents]
thomas_young_1799_c =
[0,93.9
,195.8,297.8
,391.7
,499.9,591.9
,697.9,795.8
,893.8,999.8
,1091.8
,1200]

-- | Five-limit tuning
five_limit_tuning_r :: [Rational]
five_limit_tuning_r =
[1%1,15%16
,8%9,5%6
,4%5
,3%4,32%45
,2%3,5%8
,3%5,9%16
,8%15
,1%2]

five_limit_tuning_c :: [Cents]
five_limit_tuning_c = map (to_cents.approximate_ratio) five_limit_tuning_r

equal_temperament_c :: [Cents]
equal_temperament_c = [0, 100 .. 1200]

mk_isomorphic_layout :: Integral a => a -> a -> (a,a) -> [[(a,a)]]
mk_isomorphic_layout n_row n_col top_left =
let (a,b) `plus` (c,d) = (a+c,b+d)
mk_seq 0 _ _ = []
mk_seq n i z = z : mk_seq (n-1) i (z `plus` i)
left = mk_seq n_row (-1,1) top_left
in map (\i -> mk_seq n_col (-1,2) i) left

rank_two_regular_temperament :: Integral a => a -> a -> [(a,a)] -> [a]
rank_two_regular_temperament a b =
map (\(a', b') -> a * a' + b * b')

mk_syntonic_tuning :: Int -> [Cents]
mk_syntonic_tuning b =
let l = mk_isomorphic_layout 5 7 (3,-4)
t = map (rank_two_regular_temperament 1200 b) l
in nub (sort (map (\x -> fromIntegral (x `mod` 1200)) (concat t)))

syntonic_697_c :: [Cents]
syntonic_697_c = mk_syntonic_tuning 697

syntonic_702_c :: [Cents]
syntonic_702_c = mk_syntonic_tuning 702

syntonic_comma :: Rational
syntonic_comma = 81 % 80

-- ie. 3^12 % 2^19
pythagorean_comma :: Rational
pythagorean_comma = 531441 % 524288

-- ie. 3^53 % 2^84
mercators_comma :: Rational
mercators_comma = 19383245667680019896796723 % 19342813113834066795298816

approximate_ratio :: Rational -> Approximate_Ratio
approximate_ratio = fromRational

to_cents :: Approximate_Ratio -> Cents
to_cents x = 1200 * logBase 2 x

nth_root :: (Floating a) => a -> a -> a
nth_root n x =
let f (_,x0) = (x0, ((n-1)*x0+x/x0**(n-1))/n)
e = uncurry (==)
in fst (until e f (x, x/n))

twelve_tone_equal_temperament_comma :: (Floating a) => a
twelve_tone_equal_temperament_comma = 12 `nth_root` 2

minimal_isomorphic_note_layout :: [[(Int,Int)]]
minimal_isomorphic_note_layout =
[[(3,-4),(2,-2),(1,0),(0,2),(-1,4)]
,[(2,-3),(1,-1),(0,1),(-1,3)]
,[(2,-4),(1,-2),(0,0),(-1,2),(-2,4)]]
```