-- | Max Meyer. \"The musician's arithmetic: drill problems for an -- introduction to the scientific study of musical composition.\" The -- University of Missouri, 1929. p.22 module Music.Theory.Tuning.Meyer_1929 where import Data.List import Data.Ratio import qualified Music.Theory.Tuning as T -- | Odd numbers to /n/. -- -- > odd_to 7 == [1,3,5,7] odd_to :: (Num t, Enum t) => t -> [t] odd_to n = [1,3 .. n] -- | Generate initial row for /n/. -- -- > row 7 == [1,5/4,3/2,7/4] row :: Integral i => i -> [Ratio i] row = sort . map T.fold_ratio_to_octave . odd_to . (% 1) -- | Generate initial column for /n/. -- -- > column 7 == [1,8/5,4/3,8/7] column :: Integral i => i -> [Ratio i] column = map (T.fold_ratio_to_octave . recip) . row -- | 'T.fold_to_octave' '.' '*'. in_oct_mul :: Integral i => Ratio i -> Ratio i -> Ratio i in_oct_mul i j = T.fold_ratio_to_octave (i * j) -- | Given /row/ and /column/ generate matrix value at /(i,j)/. -- -- > inner (row 7,column 7) (1,2) == 6/5 inner :: Integral i => ([Ratio i],[Ratio i]) -> (i,i) -> Ratio i inner (r,c) (i,j) = in_oct_mul (r `genericIndex` j) (c `genericIndex` i) meyer_table_rck :: Integral i => i -> ([Ratio i],[Ratio i],i) meyer_table_rck n = let r = row n c = column n k = n `div` 2 in (r,c,k) -- | Meyer table in form /(r,c,n)/. -- -- > meyer_table_indices 7 == [(0,0,1/1),(0,1,5/4),(0,2,3/2),(0,3,7/4) -- > ,(1,0,8/5),(1,1,1/1),(1,2,6/5),(1,3,7/5) -- > ,(2,0,4/3),(2,1,5/3),(2,2,1/1),(2,3,7/6) -- > ,(3,0,8/7),(3,1,10/7),(3,2,12/7),(3,3,1/1)] meyer_table_indices :: Integral i => i -> [(i,i,Ratio i)] meyer_table_indices n = let (r,c,k) = meyer_table_rck n in [(i,j,inner (r,c) (i,j)) | i <- [0..k], j <- [0..k]] -- | Meyer table as set of rows. -- -- > meyer_table_rows 7 == [[1/1, 5/4, 3/2,7/4] -- > ,[8/5, 1/1, 6/5,7/5] -- > ,[4/3, 5/3, 1/1,7/6] -- > ,[8/7,10/7,12/7,1/1]] -- -- > let r = [[ 1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8] -- > ,[16/9, 1/1, 10/9, 11/9, 4/3, 13/9, 14/9, 5/3] -- > ,[ 8/5, 9/5, 1/1, 11/10, 6/5, 13/10, 7/5, 3/2] -- > ,[16/11, 18/11, 20/11, 1/1, 12/11, 13/11, 14/11, 15/11] -- > ,[ 4/3, 3/2, 5/3, 11/6, 1/1, 13/12, 7/6, 5/4] -- > ,[16/13, 18/13, 20/13, 22/13, 24/13, 1/1, 14/13, 15/13] -- > ,[ 8/7, 9/7, 10/7, 11/7, 12/7, 13/7, 1/1, 15/14] -- > ,[16/15, 6/5, 4/3, 22/15, 8/5, 26/15, 28/15, 1/1]] -- > in meyer_table_rows 15 == r meyer_table_rows :: Integral a => a -> [[Ratio a]] meyer_table_rows n = let (r,c,k) = meyer_table_rck n rn i = [inner (r,c) (i,j) | j <- [0..k]] in map rn [0..k] -- | Third element of three-tuple. t3_3 :: (t1,t2,t3) -> t3 t3_3 (_,_,i) = i -- | Set of unique ratios in /n/ table. -- -- > elements 7 == [1,8/7,7/6,6/5,5/4,4/3,7/5,10/7,3/2,8/5,5/3,12/7,7/4] -- -- > elements 9 == [1,10/9,9/8,8/7,7/6,6/5,5/4,9/7,4/3,7/5,10/7 -- > ,3/2,14/9,8/5,5/3,12/7,7/4,16/9,9/5] elements :: Integral i => i -> [Ratio i] elements = nub . sort . concat . meyer_table_rows -- | Number of unique elements at /n/ table. -- -- > map degree [7,9,11,13,15] == [13,19,29,41,49] degree :: Integral i => i -> i degree = genericLength . elements -- | <http://en.wikipedia.org/wiki/Farey_sequence> -- -- > let r = [[0,1/2,1] -- > ,[0,1/3,1/2,2/3,1] -- > ,[0,1/4,1/3,1/2,2/3,3/4,1] -- > ,[0,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,1] -- > ,[0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,1]] -- > in map farey_sequence [2..6] == r farey_sequence :: Integral a => a -> [Ratio a] farey_sequence k = 0 : nub (sort [n%d | d <- [1..k], n <- [1..d]])