Music.Theory.Tuning.Meyer_1929

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

Documentation

odd_to :: (Num t, Enum t) => t -> [t]

Odd numbers to n.

 odd_to 7 == [1,3,5,7]

row :: Integral i => i -> [Ratio i]

Generate initial row for n.

 row 7 == [1,5/4,3/2,7/4]

column :: Integral i => i -> [Ratio i]

Generate initial column for n.

 column 7 == [1,8/5,4/3,8/7]

in_oct_mul :: Integral i => Ratio i -> Ratio i -> Ratio i

fold_to_octave . *.

inner :: Integral i => ([Ratio i], [Ratio i]) -> (i, i) -> Ratio i

Given row and column generate matrix value at (i,j).

 inner (row 7,column 7) (1,2) == 6/5

meyer_table_rck :: Integral i => i -> ([Ratio i], [Ratio i], i)

meyer_table_indices :: Integral i => i -> [(i, i, Ratio i)]

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_rows :: Integral a => a -> [[Ratio a]]

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

t3_3 :: (t1, t2, t3) -> t3

Third element of three-tuple.

elements :: Integral i => i -> [Ratio 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]

degree :: Integral i => i -> i

Number of unique elements at n table.

 map degree [7,9,11,13,15] == [13,19,29,41,49]

farey_sequence :: Integral a => a -> [Ratio a]

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