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An automatic accompaniment generator for infinite melodies, applied to a simple mapping of the decimal digits of π to an A–harmonic-minor scale.
module PiMelody where
import Data.List
data MelodyNote = Gs | A | B | C' | D' | E' | F' | Gs' | A' | B'
deriving (Eq, Show, Enum)
type Melody = [MelodyNote] -- Assume simple all-quavers rythm.
piMelody :: Melody
piMelody = map toEnum piDigits
data Chord = Am | Dm | C | G7 | E
deriving (Eq, Show, Enum)
type Composition = [(Melody, Chord)]
-- (Infinite) list of pairs: a melody chunk, and what chord to go with it.
chordMNotes :: Chord -> [MelodyNote] -- Without suspensions.
chordMNotes Am = [A , C', E', A']
chordMNotes Dm = [A , D', F', A']
chordMNotes E = [Gs, B , D', E', Gs', B', F'] -- Minor dominant may also be diminished-7th.
chordMNotes C = [C', E']
chordMNotes G7 = [B , D', F', B']
resolves :: Chord -> [Chord]
resolves E = [Am, E] -- Dominants should resolve to their tonic, if at all.
resolves G7 = [C, G7, Am, E] -- For major dominant, allow also resolving to minor parallels.
resolves _ = [Am .. E] -- Non-dominant chord can resolve to anything.
accompany :: Melody -> Composition -- Choose suitable chords for a melody.
accompany melody = acc Am melody
where acc :: Chord -> Melody -> Composition
acc lastChord (n1:n2:ml) -- Try to find a chord that fits over two melody notes
| (Just nextChord) -- and works with the previous (possibly dominant) chord.
<- find (\ch -> all(`elem` chordMNotes ch) [n1,n2]) $ resolves lastChord
= ([n1,n2], nextChord) : acc nextChord ml
-- If two melody notes don't fit in one chord, use two.
| (Just c1) <- find ((n1`elem`) . chordMNotes) $ resolves lastChord
, (Just c2) <- find ((n2`elem`) . chordMNotes) $ resolves c1
= ([n1] , c1) : ([n2] , c2) : acc c2 ml
piDigits :: [Int] -- Infinite list of decimal digits of π. Algorithm taken from:
-- Jeremy Gibbons, "Unbounded Spigot Algorithms for the Digits of Pi";
-- The Mathematical Association of America 2005.
piDigits = map fromInteger $ g(1,0,1,1,3,3)
where g(q,r,t,k,n,l) = if 4*q+r-t<n*t
then n : g(10*q,10*(r-n*t),t,k,div(10*(3*q+r))t-10*n,l)
else g(q*k,(2*q+r)*l,t*l,k+1,div(q*(7*k+2)+r*l)(t*l),l+2)
@leftaroundabout

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leftaroundabout commented Sep 15, 2013

Generate a sequence of melody tones from the decimal digits of π by mapping them onto a range of notes from the A–harmonic-major scale; then create an accompaniment for this melody, by way of dominants resolving in the usual way.

This being a reply to this question on music.stackexchange.com.

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