hmt-0.12: Music/Theory/Tuning.hs
-- | Tuning theory
module Music.Theory.Tuning where
import Data.List
import Data.Ratio
-- * Either/Maybe
-- | Maybe 'Left' of 'Either'.
fromLeft :: Either a b -> Maybe a
fromLeft e =
case e of
Left x -> Just x
_ -> Nothing
-- | Maybe 'Right' of 'Either'.
fromRight :: Either a b -> Maybe b
fromRight e =
case e of
Right x -> Just x
_ -> Nothing
-- * Types
-- | An approximation of a ratio.
type Approximate_Ratio = Double
-- | A real valued division of a tone into one hundred parts.
type Cents = Double
-- | A tuning specified 'Either' as a sequence of exact ratios, or as
-- a sequence of possibly inexact 'Cents'.
data Tuning = Tuning {ratios_or_cents :: Either [Rational] [Cents]
,octave_ratio :: Rational}
deriving (Eq,Show)
-- | Divisions of octave.
--
-- > divisions ditone == 12
divisions :: Tuning -> Int
divisions = either length length . ratios_or_cents
-- | 'Maybe' exact ratios of 'Tuning'.
ratios :: Tuning -> Maybe [Rational]
ratios = fromLeft . ratios_or_cents
-- | Possibly inexact 'Cents' of tuning.
cents :: Tuning -> [Cents]
cents = either (map to_cents_r) id . ratios_or_cents
-- | 'map' 'round' '.' 'cents'.
cents_i :: Integral i => Tuning -> [i]
cents_i = map round . cents
-- | Convert from cents invterval to frequency ratio.
--
-- > map cents_to_ratio [0,701.9550008653874,1200] == [1,3/2,2]
cents_to_ratio :: Floating a => a -> a
cents_to_ratio n = 2 ** (n / 1200)
-- | Convert from frequency ratio to cents interval.
--
-- > map ratio_to_cents [1,4/3,2] == [0.0,498.04499913461245,1200.0]
ratio_to_cents :: Floating a => a -> a
ratio_to_cents n = logBase 2 n * 1200
-- | Possibly inexact 'Approximate_Ratio's of tuning.
approximate_ratios :: Tuning -> [Approximate_Ratio]
approximate_ratios =
either (map approximate_ratio) (map cents_to_ratio) .
ratios_or_cents
-- | 'Maybe' exact ratios reconstructued from possibly inexact 'Cents'
-- of 'Tuning'.
--
-- > let r = [1,17/16,9/8,13/11,5/4,4/3,7/5,3/2,11/7,5/3,16/9,15/8]
-- > in reconstructed_ratios 1e-2 werckmeister_iii == Just r
reconstructed_ratios :: Double -> Tuning -> Maybe [Rational]
reconstructed_ratios epsilon =
fmap (map (reconstructed_ratio epsilon)) .
fromRight .
ratios_or_cents
-- | Convert from an 'Approximate_Ratio' to 'Cents'.
--
-- > round (to_cents (3/2)) == 702
to_cents :: Approximate_Ratio -> Cents
to_cents x = 1200 * logBase 2 x
-- | Convert from 'Rational' to 'Approximate_Ratio', ie. 'fromRational'.
approximate_ratio :: Rational -> Approximate_Ratio
approximate_ratio = fromRational
-- | 'to_cents' '.' 'approximate_ratio'.
to_cents_r :: Rational -> Cents
to_cents_r = to_cents . approximate_ratio
-- | Construct an exact 'Rational' that approximates 'Cents' to within
-- /epsilon/.
--
-- > map (reconstructed_ratio 1e-5) [0,700,1200] == [1,442/295,2]
--
-- > to_cents_r (442/295) == 699.9976981706735
reconstructed_ratio :: Double -> Cents -> Rational
reconstructed_ratio epsilon c = approxRational (cents_to_ratio c) epsilon
-- | Frequency /n/ cents from /f/.
--
-- > map (cps_shift_cents 440) [-100,100] == map octpc_to_cps [(4,8),(4,10)]
cps_shift_cents :: Floating a => a -> a -> a
cps_shift_cents f = (* f) . cents_to_ratio
-- | Interval in /cents/ from /p/ to /q/, ie. 'ratio_to_cents' of /p/
-- '/' /q/.
--
-- > cps_difference_cents 440 (octpc_to_cps (5,2)) == 500
--
-- > let abs_dif i j = abs (i - j)
-- > in cps_difference_cents 440 (fmidi_to_cps 69.1) `abs_dif` 10 < 1e9
cps_difference_cents :: Floating a => a -> a -> a
cps_difference_cents p q = ratio_to_cents (q / p)
-- * Commas
-- | The Syntonic comma.
--
-- > syntonic_comma == 81/80
syntonic_comma :: Rational
syntonic_comma = 81 % 80
-- | The Pythagorean comma.
--
-- > pythagorean_comma == 3^12 / 2^19
pythagorean_comma :: Rational
pythagorean_comma = 531441 / 524288
-- | Mercators comma.
--
-- > mercators_comma == 3^53 / 2^84
mercators_comma :: Rational
mercators_comma = 19383245667680019896796723 / 19342813113834066795298816
-- | Calculate /n/th root of /x/.
--
-- > 12 `nth_root` 2 == twelve_tone_equal_temperament_comma
nth_root :: (Floating a,Eq 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))
-- | 12-tone equal temperament comma (ie. 12th root of 2).
--
-- > twelve_tone_equal_temperament_comma == 1.0594630943592953
twelve_tone_equal_temperament_comma :: (Floating a,Eq a) => a
twelve_tone_equal_temperament_comma = 12 `nth_root` 2
-- * 12-tone tunings
-- > let c = [0,114,204,294,408,498,612,702,816,906,996,1110]
-- > in map (round.to_cents_r) ditone_r == c
ditone_r :: [Rational]
ditone_r =
[1,2187/2048 {- 256/243 -}
,9/8,32/27
,81/64
,4/3,729/512
,3/2,6561/4096 {- 128/81 -}
,27/16,16/9
,243/128]
-- | Ditone/pythagorean tuning,
-- see <http://www.billalves.com/porgitaro/ditonesettuning.html>
--
-- > cents_i ditone == [0,114,204,294,408,498,612,702,816,906,996,1110]
ditone :: Tuning
ditone = Tuning (Left ditone_r) 2
-- > let c = [0,90,204,294,408,498,612,702,792,906,996,1110]
-- > in map (round.to_cents_r) pythagorean_r == c
pythagorean_r :: [Rational]
pythagorean_r =
[1,256/243 {- 2187/2048 -}
,9/8,32/27
,81/64
,4/3,729/512
,3/2,128/81 {- 6561/4096 -}
,27/16,16/9
,243/128]
-- | Pythagorean tuning.
--
-- > cents_i pythagorean == [0,90,204,294,408,498,612,702,792,906,996,1110]
pythagorean :: Tuning
pythagorean = Tuning (Left pythagorean_r) 2
-- > let c = [0,90,192,294,390,498,588,696,792,888,996,1092]
-- > in map (round.to_cents) werckmeister_iii_ar == c
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_c :: [Cents]
werckmeister_iii_c = map to_cents werckmeister_iii_ar
-- | Werckmeister III, Andreas Werckmeister (1645-1706)
--
-- > cents_i werckmeister_iii == [0,90,192,294,390,498,588,696,792,888,996,1092]
werckmeister_iii :: Tuning
werckmeister_iii = Tuning (Right werckmeister_iii_c) 2
-- > let c = [0,82,196,294,392,498,588,694,784,890,1004,1086]
-- > in map (round.to_cents) werckmeister_iv_ar == c
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_c :: [Cents]
werckmeister_iv_c = map to_cents werckmeister_iv_ar
-- | Werckmeister IV, Andreas Werckmeister (1645-1706)
--
-- > cents_i werckmeister_iv == [0,82,196,294,392,498,588,694,784,890,1004,1086]
werckmeister_iv :: Tuning
werckmeister_iv = Tuning (Right werckmeister_iv_c) 2
-- > let c = [0,96,204,300,396,504,600,702,792,900,1002,1098]
-- > in map (round.to_cents) werckmeister_v_ar == c
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_c :: [Cents]
werckmeister_v_c = map to_cents werckmeister_v_ar
-- | Werckmeister V, Andreas Werckmeister (1645-1706)
--
-- > cents_i werckmeister_v == [0,96,204,300,396,504,600,702,792,900,1002,1098]
werckmeister_v :: Tuning
werckmeister_v = Tuning (Right werckmeister_v_c) 2
-- > let c = [0,91,196,298,395,498,595,698,793,893,1000,1097]
-- > in map (round.to_cents_r) werckmeister_vi_r == c
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)
--
-- > cents_i werckmeister_vi == [0,91,196,298,395,498,595,698,793,893,1000,1097]
werckmeister_vi :: Tuning
werckmeister_vi = Tuning (Left werckmeister_vi_r) 2
-- > let c = [0,76,193,310,386,503,580,697,773,890,1007,1083]
-- > in map round pietro_aaron_1523_c == c
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]
-- | Pietro Aaron (1523) meantone temperament, see
-- <http://www.kylegann.com/histune.html>
--
-- > cents_i pietro_aaron_1523 == [0,76,193,310,386,503,580,697,773,890,1007,1083]
pietro_aaron_1523 :: Tuning
pietro_aaron_1523 = Tuning (Right pietro_aaron_1523_c) 2
-- > let c = [0,94,196,298,392,500,592,698,796,894,1000,1092]
-- > in map round thomas_young_1799_c == c
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]
-- | Thomas Young (1799) - Well Temperament
--
-- > cents_i thomas_young_1799 == [0,94,196,298,392,500,592,698,796,894,1000,1092]
thomas_young_1799 :: Tuning
thomas_young_1799 = Tuning (Right thomas_young_1799_c) 2
-- > let c = [0,112,204,316,386,498,590,702,814,884,996,1088]
-- > in map (round.to_cents_r) five_limit_tuning_r == c
five_limit_tuning_r :: [Rational]
five_limit_tuning_r =
[1,16/15
,9/8,6/5
,5/4
,4/3,45/32
,3/2,8/5
,5/3,16/9 {- 9/5 -}
,15/8]
-- | Five-limit tuning (five limit just intonation).
--
-- > cents_i five_limit_tuning == [0,112,204,316,386,498,590,702,814,884,996,1088]
five_limit_tuning :: Tuning
five_limit_tuning = Tuning (Left five_limit_tuning_r) 2
-- > equal_temperament_c == [0,100..1100]
equal_temperament_c :: [Cents]
equal_temperament_c = [0, 100 .. 1100]
-- | Equal temperament.
--
-- > cents equal_temperament == [0,100..1100]
equal_temperament :: Tuning
equal_temperament = Tuning (Right equal_temperament_c) 2
-- > let c = [0,112,204,316,386,498,583,702,814,884,1018,1088]
-- > in map (round.to_cents_r) septimal_tritone_just_intonation == c
septimal_tritone_just_intonation_r :: [Rational]
septimal_tritone_just_intonation_r =
[1,16/15
,9/8,6/5
,5/4
,4/3,7/5
,3/2,8/5
,5/3,9/5
,15/8]
-- > cents_i septimal_tritone_just_intonation == [0,112,204,316,386,498,583,702,814,884,1018,1088]
septimal_tritone_just_intonation :: Tuning
septimal_tritone_just_intonation = Tuning (Left septimal_tritone_just_intonation_r) 2
-- > let c = [0,112,204,316,386,498,583,702,814,884,969,1088]
-- > in map (round.to_cents_r) seven_limit_just_intonation == c
seven_limit_just_intonation_r :: [Rational]
seven_limit_just_intonation_r =
[1,16/15
,9/8,6/5
,5/4
,4/3,7/5
,3/2,8/5
,5/3,7/4
,15/8]
-- > cents_i seven_limit_just_intonation == [0,112,204,316,386,498,583,702,814,884,969,1088]
seven_limit_just_intonation :: Tuning
seven_limit_just_intonation = Tuning (Left seven_limit_just_intonation_r) 2
-- > let c = [0,90,193,294,386,498,590,697,792,890,996,1088]
-- > in map (round.to_cents) kirnberger_iii_ar == c
kirnberger_iii_ar :: [Approximate_Ratio]
kirnberger_iii_ar =
[1,256/243
,sqrt 5 / 2,32/27
,5/4
,4/3,45/32
,5 ** 0.25,128/81
,(5 ** 0.75)/2,16/9
,15/8]
-- > cents_i kirnberger_iii == [0,90,193,294,386,498,590,697,792,890,996,1088]
kirnberger_iii :: Tuning
kirnberger_iii = Tuning (Right (map to_cents kirnberger_iii_ar)) 2
-- > let c = [0,94,196,298,392,502,592,698,796,894,1000,1090]
-- > in map round vallotti_c == c
vallotti_c :: [Cents]
vallotti_c =
[0.0,94.135
,196.09,298.045
,392.18
,501.955,592.18
,698.045,796.09
,894.135,1000.0
,1090.225]
-- > cents_i vallotti == [0,94,196,298,392,502,592,698,796,894,1000,1090]
vallotti :: Tuning
vallotti = Tuning (Right vallotti_c) 2
-- > let c = [0,128,139,359,454,563,637,746,841,911,1072,1183]
-- > in map (round.to_cents_r) mayumi_reinhard == c
mayumi_reinhard_r :: [Rational]
mayumi_reinhard_r =
[1,14/13
,13/12,16/13
,13/10
,18/13,13/9
,20/13,13/8
,22/13,13/7
,208/105]
-- > cents_i mayumi_reinhard == [0,128,139,359,454,563,637,746,841,911,1072,1183]
mayumi_reinhard :: Tuning
mayumi_reinhard = Tuning (Left mayumi_reinhard_r) 2
-- > let c = [0,177,204,240,471,444,675,702,738,969,942,1173]
-- > in map (round.to_cents_r) la_monte_young_r == c
la_monte_young_r :: [Rational]
la_monte_young_r =
[1,567/512
,9/8,147/128
,21/16
,1323/1024,189/128
,3/2,49/32
,7/4,441/256
,63/32]
-- | La Monte Young's \"The Well-Tuned Piano\", see
-- <http://www.kylegann.com/tuning.html>.
--
-- > cents_i la_monte_young == [0,177,204,240,471,444,675,702,738,969,942,1173]
la_monte_young :: Tuning
la_monte_young = Tuning (Left la_monte_young_r) 2
-- > let c = [0,105,204,298,386,471,551,702,841,906,969,1088]
-- > in map (round.to_cents_r) ben_johnston_r == c
ben_johnston_r :: [Rational]
ben_johnston_r =
[1,17/16
,9/8,19/16
,5/4
,21/16,11/8
,3/2,13/8
,27/16,7/4
,15/8]
-- | Ben Johnston's \"Suite for Microtonal Piano\" (1977), see
-- <http://www.kylegann.com/tuning.html>
--
-- > cents_i ben_johnston == [0,105,204,298,386,471,551,702,841,906,969,1088]
ben_johnston :: Tuning
ben_johnston = Tuning (Left ben_johnston_r) 2
-- > let c = [0,112,182,231,267,316,386,498,603,702,814,884,933,969,1018,1088]
-- > in map (round.to_cents_r) lou_harrison_16_r == c
lou_harrison_16_r :: [Rational]
lou_harrison_16_r =
[1,16/15
,10/9,8/7
,7/6,6/5,5/4
,4/3
,17/12
,3/2
,8/5,5/3,12/7
,7/4,9/5,15/8]
-- | Lou Harrison 16 tone Just Intonation scale, see
-- <http://www.microtonal-synthesis.com/scale_harrison_16.html>
--
-- > cents_i lou_harrison_16 == [0,112,182,231,267,316,386,498,603,702,814,884,933,969,1018,1088]
lou_harrison_16 :: Tuning
lou_harrison_16 = Tuning (Left lou_harrison_16_r) 2
partch_43_r :: [Rational]
partch_43_r =
[1,81/80,33/32,21/20,16/15,12/11,11/10,10/9,9/8,8/7
,7/6,32/27,6/5,11/9,5/4,14/11,9/7
,21/16,4/3,27/20
,11/8,7/5,10/7,16/11
,40/27,3/2,32/21,14/9,11/7,8/5,18/11,5/3,27/16,12/7
,7/4,16/9,9/5,20/11,11/6,15/8,40/21,64/33,160/81]
-- | Harry Partch 43 tone scale, see
-- <http://www.microtonal-synthesis.com/scale_partch.html>
--
-- > cents_i partch_43 == [0,22,53,84,112,151,165
-- > ,182,204,231,267,294,316
-- > ,347,386,418,435
-- > ,471,498,520,551,583,617,649
-- > ,680,702,729,765,782,814,853,884,906,933
-- > ,969,996,1018,1035,1049,1088,1116,1147,1178]
partch_43 :: Tuning
partch_43 = Tuning (Left partch_43_r) 2
-- * Syntonic tuning
-- | Construct an isomorphic layout of /r/ rows and /c/ columns with
-- an upper left value of /(i,j)/.
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 (mk_seq n_col (-1,2)) left
-- | A minimal isomorphic note layout.
--
-- > let [i,j,k] = mk_isomorphic_layout 3 5 (3,-4)
-- > in [i,take 4 j,(2,-4):take 4 k] == minimal_isomorphic_note_layout
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)]]
-- | Make a rank two regular temperament from a list of /(i,j)/
-- positions by applying the scalars /a/ and /b/.
rank_two_regular_temperament :: Integral a => a -> a -> [(a,a)] -> [a]
rank_two_regular_temperament a b = let f (i,j) = i * a + j * b in map f
-- | Syntonic tuning system based on 'mk_isomorphic_layout' of @5@
-- rows and @7@ columns starting at @(3,-4)@ and a
-- 'rank_two_regular_temperament' with /a/ of @1200@ and indicated
-- /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)))
-- | 'mk_syntonic_tuning' of @697@.
--
-- > divisions syntonic_697 == 17
-- > cents_i syntonic_697 == [0,79,194,273,309,388,467,503,582,697,776,812,891,970,1006,1085,1164]
syntonic_697 :: Tuning
syntonic_697 = Tuning (Right (mk_syntonic_tuning 697)) 2
-- | 'mk_syntonic_tuning' of @702@.
--
-- > divisions syntonic_702 == 17
-- > cents_i syntonic_702 == [0,24,114,204,294,318,408,498,522,612,702,792,816,906,996,1020,1110]
syntonic_702 :: Tuning
syntonic_702 = Tuning (Right (mk_syntonic_tuning 702)) 2
-- * Harmonic series
-- | Raise or lower the frequency /q/ by octaves until it is in the
-- octave starting at /p/.
--
-- > fold_to_octave_of 55 392 == 98
fold_cps_to_octave_of :: (Ord a, Fractional a) => a -> a -> a
fold_cps_to_octave_of p q =
if q > p * 2
then fold_cps_to_octave_of p (q / 2)
else if q < p
then fold_cps_to_octave_of p (q * 2)
else q
-- | Harmonic series on /n/.
harmonic_series_cps :: (Num t, Enum t) => t -> [t]
harmonic_series_cps n = [n,n * 2 ..]
-- | /n/ elements of 'harmonic_series_cps'.
--
-- > harmonic_series_cps_n 14 55 == [55,110,165,220,275,330,385,440,495,550,605,660,715,770]
harmonic_series_cps_n :: (Num a, Enum a) => Int -> a -> [a]
harmonic_series_cps_n n = take n . harmonic_series_cps
-- | /n/th partial of /f1/, ie. one indexed.
--
-- > map (partial 55) [1,5,3] == [55,275,165]
partial :: (Num a, Enum a) => a -> Int -> a
partial f1 k = harmonic_series_cps f1 !! (k - 1)
-- | Fold ratio until within an octave, ie. @1@ '<' /n/ '<=' @2@.
fold_ratio_to_octave :: Integral i => Ratio i -> Ratio i
fold_ratio_to_octave n =
if n >= 2
then fold_ratio_to_octave (n / 2)
else if n < 1
then fold_ratio_to_octave (n * 2)
else n
-- | Derivative harmonic series, based on /k/th partial of /f1/.
--
-- > let {r = [52,103,155,206,258,309,361,412,464,515,567,618,670,721,773]
-- > ;d = harmonic_series_cps_derived 5 (octpc_to_cps (1,4))}
-- > in map round (take 15 d) == r
harmonic_series_cps_derived :: (Ord a, Fractional a, Enum a) => Int -> a -> [a]
harmonic_series_cps_derived k f1 =
let f0 = fold_cps_to_octave_of f1 (partial f1 k)
in harmonic_series_cps f0
-- | Harmonic series to /n/th harmonic (folded).
--
-- > harmonic_series_folded 17 == [1,17/16,9/8,5/4,11/8,3/2,13/8,7/4,15/8]
harmonic_series_folded :: Integer -> [Rational]
harmonic_series_folded n =
nub (sort (map fold_ratio_to_octave [1 .. n%1]))
-- | 'to_cents_r' variant of 'harmonic_series_folded'.
--
-- > map round (harmonic_series_folded_c 21) == [0,105,204,298,386,471,551,702,841,969,1088]
harmonic_series_folded_c :: Integer -> [Cents]
harmonic_series_folded_c = map to_cents_r . harmonic_series_folded
-- | @12@-tone tuning of first @21@ elements of the harmonic series.
--
-- > cents_i harmonic_series_folded_21 == [0,105,204,298,386,471,551,702,841,969,1088]
harmonic_series_folded_21 :: Tuning
harmonic_series_folded_21 = Tuning (Left (harmonic_series_folded 21)) 2
-- Local Variables:
-- truncate-lines:t
-- End: