hmt-0.15: Music/Theory/Interval.hs
-- | Common music notation intervals.
module Music.Theory.Interval where
import Data.List {- base -}
import Data.Maybe {- base -}
import Music.Theory.Pitch
import Music.Theory.Pitch.Note
-- | Interval type or degree.
data Interval_T = Unison | Second | Third | Fourth
| Fifth | Sixth | Seventh
deriving (Eq,Enum,Bounded,Ord,Show)
-- | Interval quality.
data Interval_Q = Diminished | Minor
| Perfect
| Major | Augmented
deriving (Eq,Enum,Bounded,Ord,Show)
-- | Common music notation interval. An 'Ordering' of 'LT' indicates
-- an ascending interval, 'GT' a descending interval, and 'EQ' a
-- unison.
data Interval = Interval {interval_type :: Interval_T
,interval_quality :: Interval_Q
,interval_direction :: Ordering
,interval_octave :: Octave}
deriving (Eq,Show)
-- | Interval type between 'Note_T' values.
--
-- > map (interval_ty C) [E,B] == [Third,Seventh]
interval_ty :: Note_T -> Note_T -> Interval_T
interval_ty n1 n2 = toEnum ((fromEnum n2 - fromEnum n1) `mod` 7)
-- | Table of interval qualities. For each 'Interval_T' gives
-- directed semitone interval counts for each allowable 'Interval_Q'.
-- For lookup function see 'interval_q', for reverse lookup see
-- 'interval_q_reverse'.
interval_q_tbl :: Integral n => [(Interval_T, [(n,Interval_Q)])]
interval_q_tbl =
[(Unison,[(11,Diminished)
,(0,Perfect)
,(1,Augmented)])
,(Second,[(0,Diminished)
,(1,Minor)
,(2,Major)
,(3,Augmented)])
,(Third,[(2,Diminished)
,(3,Minor)
,(4,Major)
,(5,Augmented)])
,(Fourth,[(4,Diminished)
,(5,Perfect)
,(6,Augmented)])
,(Fifth,[(6,Diminished)
,(7,Perfect)
,(8,Augmented)])
,(Sixth,[(7,Diminished)
,(8,Minor)
,(9,Major)
,(10,Augmented)])
,(Seventh,[(9,Diminished)
,(10,Minor)
,(11,Major)
,(12,Augmented)])]
-- | Lookup 'Interval_Q' for given 'Interval_T' and semitone count.
--
-- > interval_q Unison 11 == Just Diminished
-- > interval_q Third 5 == Just Augmented
-- > interval_q Fourth 5 == Just Perfect
-- > interval_q Unison 3 == Nothing
interval_q :: Interval_T -> Int -> Maybe Interval_Q
interval_q i n = lookup i interval_q_tbl >>= lookup n
-- | Lookup semitone difference of 'Interval_T' with 'Interval_Q'.
--
-- > interval_q_reverse Third Minor == Just 3
-- > interval_q_reverse Unison Diminished == Just 11
interval_q_reverse :: Interval_T -> Interval_Q -> Maybe Int
interval_q_reverse ty qu =
case lookup ty interval_q_tbl of
Nothing -> Nothing
Just tbl -> fmap fst (find ((== qu) . snd) tbl)
-- | Semitone difference of 'Interval'.
--
-- > interval_semitones (interval (Pitch C Sharp 4) (Pitch E Sharp 5)) == 16
-- > interval_semitones (interval (Pitch C Natural 4) (Pitch D Sharp 3)) == -9
interval_semitones :: Interval -> Int
interval_semitones (Interval ty qu dir oct) =
case interval_q_reverse ty qu of
Just n -> let o = 12 * oct
in if dir == GT then negate n - o else n + o
Nothing -> error "interval_semitones"
-- | Inclusive set of 'Note_T' within indicated interval. This is not
-- equal to 'enumFromTo' which is not circular.
--
-- > note_span E B == [E,F,G,A,B]
-- > note_span B D == [B,C,D]
-- > enumFromTo B D == []
note_span :: Note_T -> Note_T -> [Note_T]
note_span n1 n2 =
let fn x = toEnum (x `mod` 7)
n1' = fromEnum n1
n2' = fromEnum n2
n2'' = if n1' > n2' then n2' + 7 else n2'
in map fn [n1' .. n2'']
-- | Invert 'Ordering', ie. 'GT' becomes 'LT' and vice versa.
--
-- > map invert_ordering [LT,EQ,GT] == [GT,EQ,LT]
invert_ordering :: Ordering -> Ordering
invert_ordering x =
case x of
LT -> GT
EQ -> EQ
GT -> LT
-- | Determine 'Interval' between two 'Pitch'es.
--
-- > interval (Pitch C Sharp 4) (Pitch D Flat 4) == Interval Second Diminished EQ 0
-- > interval (Pitch C Sharp 4) (Pitch E Sharp 5) == Interval Third Major LT 1
interval :: Pitch -> Pitch -> Interval
interval p1 p2 =
let c = compare p1 p2
(Pitch n1 _ o1) = p1
(Pitch n2 _ o2) = p2
p1' = pitch_to_pc p1
p2' = pitch_to_pc p2
st = (p2' - p1') `mod` 12
ty = interval_ty n1 n2
(Just qu) = interval_q ty (fromIntegral st)
o_a = if n1 > n2 then -1 else 0
in case c of
GT -> (interval p2 p1) { interval_direction = GT }
_ -> Interval ty qu c (o2 - o1 + o_a)
-- | Apply 'invert_ordering' to 'interval_direction' of 'Interval'.
--
-- > invert_interval (Interval Third Major LT 1) == Interval Third Major GT 1
invert_interval :: Interval -> Interval
invert_interval (Interval t qu d o) =
let d' = invert_ordering d
in Interval t qu d' o
-- | The signed difference in semitones between two 'Interval_Q'
-- values when applied to the same 'Interval_T'. Can this be written
-- correctly without knowing the Interval_T?
--
-- > quality_difference_m Minor Augmented == Just 2
-- > quality_difference_m Augmented Diminished == Just (-3)
-- > quality_difference_m Major Perfect == Nothing
quality_difference_m :: Interval_Q -> Interval_Q -> Maybe Int
quality_difference_m a b =
let rule (x,y) =
if x == y
then Just 0
else case (x,y) of
(Diminished,Minor) -> Just 1
(Diminished,Major) -> Just 2
(Diminished,Augmented) -> Just 3
(Minor,Major) -> Just 1
(Minor,Augmented) -> Just 2
(Major,Augmented) -> Just 1
(Diminished,Perfect) -> Just 1
(Perfect,Augmented) -> Just 1
_ -> Nothing
fwd = rule (a,b)
rvs = rule (b,a)
in case fwd of
Just n -> Just n
Nothing -> case rvs of
Just n -> Just (negate n)
Nothing -> Nothing
-- | Erroring variant of 'quality_difference_m'.
quality_difference :: Interval_Q -> Interval_Q -> Int
quality_difference a b =
let err = error ("quality_difference: " ++ show (a,b))
in fromMaybe err (quality_difference_m a b)
-- | Transpose a 'Pitch' by an 'Interval'.
--
-- > transpose (Interval Third Diminished LT 0) (Pitch C Sharp 4) == Pitch E Flat 4
pitch_transpose :: Interval -> Pitch -> Pitch
pitch_transpose i ip =
let (Pitch p_n p_a p_o) = ip
(Interval i_t i_q i_d i_o) = i
i_d' = if i_d == GT
then -1
else 1
p_n' = toEnum ((fromEnum p_n + (fromEnum i_t * i_d')) `mod` 7)
-- oa = octave alteration
oa = if p_n' > p_n && i_d == GT
then -1
else if p_n' < p_n && i_d == LT
then 1
else 0
ip' = Pitch p_n' p_a (p_o + i_o + oa)
st = if i_d == GT
then (pitch_to_pc ip - pitch_to_pc ip') `mod` 12
else (pitch_to_pc ip' - pitch_to_pc ip) `mod` 12
ty = if i_d == GT
then interval_ty p_n' p_n
else interval_ty p_n p_n'
qu = let err = error ("qu: " ++ show (ty,st))
in fromMaybe err (interval_q ty (fromIntegral st))
qd = quality_difference qu i_q * i_d'
p_a' = toEnum (fromEnum p_a + (qd * 2))
in ip' { alteration = p_a' }
-- | Make leftwards (perfect fourth) and and rightwards (perfect
-- fifth) circles from 'Pitch'.
--
-- > let c = circle_of_fifths (Pitch F Sharp 4)
-- > in map pitch_to_pc (snd c) == [6,1,8,3,10,5,12,7,2,9,4,11]
circle_of_fifths :: Pitch -> ([Pitch], [Pitch])
circle_of_fifths x =
let p4 = Interval Fourth Perfect LT 0
p5 = Interval Fifth Perfect LT 0
mk y = take 12 (iterate (pitch_transpose y) x)
in (mk p4,mk p5)
-- | Parse a positive integer into interval type and octave
-- displacement.
--
-- > mapMaybe parse_interval_type (map show [1 .. 15])
parse_interval_type :: String -> Maybe (Interval_T,Octave)
parse_interval_type n =
case reads n of
[(n',[])] -> if n' == 0
then Nothing
else let (o,t) = (n' - 1) `divMod` 7
in Just (toEnum t,fromIntegral o)
_ -> Nothing
-- | Parse interval quality notation.
--
-- > mapMaybe parse_interval_quality "dmPMA" == [minBound .. maxBound]
parse_interval_quality :: Char -> Maybe Interval_Q
parse_interval_quality q =
let c = zip "dmPMA" [0..]
in fmap toEnum (lookup q c)
-- | Degree of interval type and octave displacement. Inverse of
-- 'parse_interval_type'.
--
-- > map interval_type_degree [(Third,0),(Second,1),(Unison,2)] == [3,9,15]
interval_type_degree :: (Interval_T,Octave) -> Int
interval_type_degree (t,o) = fromEnum t + 1 + (fromIntegral o * 7)
-- | Inverse of 'parse_interval_quality.
interval_quality_pp :: Interval_Q -> Char
interval_quality_pp q = "dmPMA" !! fromEnum q
-- | Parse standard common music interval notation.
--
-- > let i = mapMaybe parse_interval (words "P1 d2 m2 M2 A3 P8 +M9 -M2")
-- > in unwords (map interval_pp i) == "P1 d2 m2 M2 A3 P8 M9 -M2"
--
-- > mapMaybe (fmap interval_octave . parse_interval) (words "d1 d8 d15") == [-1,0,1]
parse_interval :: String -> Maybe Interval
parse_interval i =
let unisons = [(Perfect,Unison)
,(Diminished,Second)
,(Augmented,Seventh)]
f q n = case (parse_interval_quality q,parse_interval_type n) of
(Just q',Just (n',o)) ->
let o' = if (q',n') == (Diminished,Unison)
then o - 1
else o
d = if o' == 0 && (q',n') `elem` unisons
then EQ
else LT
in Just (Interval n' q' d o')
_ -> Nothing
in case i of
'-':q:n -> fmap invert_interval (f q n)
'+':q:n -> f q n
q:n -> f q n
_ -> Nothing
-- | Pretty printer for intervals, inverse of 'parse_interval'.
interval_pp :: Interval -> String
interval_pp (Interval n q d o) =
let d' = if d == GT then ('-' :) else id
in d' (interval_quality_pp q : show (interval_type_degree (n,o)))
-- | Standard names for the intervals within the octave, divided into
-- perfect, major and minor at the left, and diminished and augmented
-- at the right.
--
-- > let {bimap f (p,q) = (f p,f q)
-- > ;f = mapMaybe (fmap interval_semitones . parse_interval)}
-- > in bimap f std_interval_names
std_interval_names :: ([String],[String])
std_interval_names =
let pmM = "P1 m2 M2 m3 M3 P4 P5 m6 M6 m7 M7 P8"
dA = "d2 A1 d3 A2 d4 A3 d5 A4 d6 A5 d7 A6 d8 A7"
in (words pmM,words dA)