hmt-0.20: Music/Theory/Z/Forte_1973.hs
-- | Allen Forte. /The Structure of Atonal Music/.
-- Yale University Press, New Haven, 1973.
module Music.Theory.Z.Forte_1973 where
import Data.Bifunctor {- base -}
import Data.List {- base -}
import Data.Maybe {- base -}
import qualified Music.Theory.List as T {- hmt -}
import qualified Music.Theory.Set.List as S {- hmt -}
import Music.Theory.Unicode {- hmt -}
import Music.Theory.Z {- hmt -}
import Music.Theory.Z.Sro {- hmt -}
-- * Prime form
-- | T-related rotations of /p/, ie. all rotations tranposed to be at zero.
--
-- > z_t_rotations z12 [1,2,4] == [[0,1,3],[0,2,11],[0,9,10]]
z_t_rotations :: Integral i => Z i -> [i] -> [[i]]
z_t_rotations z p =
let r = T.rotations (sort p)
in map (z_sro_tn_to z 0) r
-- | T\/I-related rotations of /p/.
--
-- > ti_rotations z12 [0,1,3] == [[0,1,3],[0,2,11],[0,9,10],[0,9,11],[0,2,3],[0,1,10]]
z_ti_rotations :: Integral i => Z i -> [i] -> [[i]]
z_ti_rotations z p =
let q = z_sro_invert z 0 p
r = T.rotations (sort p) ++ T.rotations (sort q)
in map (z_sro_tn_to z 0) r
-- | Prime form rule requiring comparator, considering 't_rotations'.
z_t_cmp_prime :: Integral i => Z i -> ([i] -> [i] -> Ordering) -> [i] -> [i]
z_t_cmp_prime z f = T.minimumBy_or [] f . z_t_rotations z
-- | Prime form rule requiring comparator, considering 'ti_rotations'.
z_ti_cmp_prime :: Integral i => Z i -> ([i] -> [i] -> Ordering) -> [i] -> [i]
z_ti_cmp_prime z f = T.minimumBy_or [] f . z_ti_rotations z
-- | Forte comparison function (rightmost first then leftmost outwards).
--
-- > forte_cmp [0,1,3,6,8,9] [0,2,3,6,7,9] == LT
forte_cmp :: (Ord t) => [t] -> [t] -> Ordering
forte_cmp p q =
case (p,q) of
([],[]) -> EQ
([],_) -> LT
(_,[]) -> GT
_ -> let r = compare (last p) (last q)
in if r == EQ then compare p q else r
{- | Forte prime form, ie. 'z_ti_cmp_prime' of 'forte_cmp'.
> z_forte_prime z12 [0,1,3,6,8,9] == [0,1,3,6,8,9]
> z_forte_prime z5 [0,1,4] == [0,1,2]
> z_forte_prime z5 [0,1,1] -- ERROR
> S.set (map (z_forte_prime z5) (S.powerset [0..4]))
> S.set (map (z_forte_prime z7) (S.powerset [0..6]))
-}
z_forte_prime :: Integral i => Z i -> [i] -> [i]
z_forte_prime z x =
if nub x /= x || map (z_mod z) x /= x
then error "z_forte_prime: invalid input"
else z_ti_cmp_prime z forte_cmp x
-- | Transpositional equivalence prime form,
-- ie. 'z_t_cmp_prime' of 'forte_cmp'.
--
-- > (z_forte_prime z12 [0,2,3],z_t_prime z12 [0,2,3]) == ([0,1,3],[0,2,3])
z_t_prime :: Integral i => Z i -> [i] -> [i]
z_t_prime z = z_t_cmp_prime z forte_cmp
-- * ICV Metric
-- | Interval class of interval /i/.
--
-- > map (z_ic z12) [0..12] == [0,1,2,3,4,5,6,5,4,3,2,1,0]
-- > map (z_ic z7) [0..7] == [0,1,2,3,3,2,1,0]
-- > map (z_ic z5) [0..5] == [0,1,2,2,1,0]
-- > map (z_ic z12) [5,6,7] == [5,6,5]
-- > map (z_ic z12) [-13,-1,0,1,13] == [1,1,0,1,1]
z_ic :: Integral i => Z i -> i -> i
z_ic z i =
let j = z_mod z i
m = z_modulus z
in if j <= (m `div` 2) then j else m - j
-- | Forte notation for interval class vector.
--
-- > z_icv z12 [0,1,2,4,7,8] == [3,2,2,3,3,2]
z_icv :: (Integral i, Num n) => Z i -> [i] -> [n]
z_icv z s =
let i = map (z_ic z . z_mod z . uncurry (-)) (S.pairs s)
f l = (head l,genericLength l)
j = map f (group (sort i))
k = map (`lookup` j) [1 .. z_modulus z `div` 2]
in map (fromMaybe 0) k
-- * BIP Metric
-- | Basic interval pattern, see Allen Forte \"The Basic Interval Patterns\"
-- /JMT/ 17/2 (1973):234-272
--
-- >>> bip 0t95728e3416
-- 11223344556
--
-- > z_bip z12 [0,10,9,5,7,2,8,11,3,4,1,6] == [1,1,2,2,3,3,4,4,5,5,6]
z_bip :: Integral i => Z i -> [i] -> [i]
z_bip z = sort . map (z_ic z . z_mod z) . T.d_dx
{- | Generate SC universe, though not in order of the Forte table.
> length (z_sc_univ z7) == 18
> sort (z_sc_univ z12) == sort (map snd sc_table)
> zipWith (\p q -> (p == q,p,q)) (z_sc_univ z12) (map snd sc_table)
-}
z_sc_univ :: Integral i => Z i -> [[i]]
z_sc_univ z =
T.sort_by_two_stage_on length id $
nub $
map (z_forte_prime z) $
S.powerset (z_univ z)
-- * Forte Names (Z12)
-- | Synonym for 'String'.
type SC_Name = String
-- | Table of (SC-NAME,PCSET).
type SC_Table n = [(SC_Name,[n])]
-- | The Z12 set-class table (Forte prime forms).
--
-- > length sc_table == 224
sc_table :: Num n => SC_Table n
sc_table =
[("0-1",[])
,("1-1",[0])
,("2-1",[0,1])
,("2-2",[0,2])
,("2-3",[0,3])
,("2-4",[0,4])
,("2-5",[0,5])
,("2-6",[0,6])
,("3-1",[0,1,2])
,("3-2",[0,1,3])
,("3-3",[0,1,4])
,("3-4",[0,1,5])
,("3-5",[0,1,6])
,("3-6",[0,2,4])
,("3-7",[0,2,5])
,("3-8",[0,2,6])
,("3-9",[0,2,7])
,("3-10",[0,3,6])
,("3-11",[0,3,7])
,("3-12",[0,4,8])
,("4-1",[0,1,2,3])
,("4-2",[0,1,2,4])
,("4-3",[0,1,3,4])
,("4-4",[0,1,2,5])
,("4-5",[0,1,2,6])
,("4-6",[0,1,2,7])
,("4-7",[0,1,4,5])
,("4-8",[0,1,5,6])
,("4-9",[0,1,6,7])
,("4-10",[0,2,3,5])
,("4-11",[0,1,3,5])
,("4-12",[0,2,3,6])
,("4-13",[0,1,3,6])
,("4-14",[0,2,3,7])
,("4-Z15",[0,1,4,6])
,("4-16",[0,1,5,7])
,("4-17",[0,3,4,7])
,("4-18",[0,1,4,7])
,("4-19",[0,1,4,8])
,("4-20",[0,1,5,8])
,("4-21",[0,2,4,6])
,("4-22",[0,2,4,7])
,("4-23",[0,2,5,7])
,("4-24",[0,2,4,8])
,("4-25",[0,2,6,8])
,("4-26",[0,3,5,8])
,("4-27",[0,2,5,8])
,("4-28",[0,3,6,9])
,("4-Z29",[0,1,3,7])
,("5-1",[0,1,2,3,4])
,("5-2",[0,1,2,3,5])
,("5-3",[0,1,2,4,5])
,("5-4",[0,1,2,3,6])
,("5-5",[0,1,2,3,7])
,("5-6",[0,1,2,5,6])
,("5-7",[0,1,2,6,7])
,("5-8",[0,2,3,4,6])
,("5-9",[0,1,2,4,6])
,("5-10",[0,1,3,4,6])
,("5-11",[0,2,3,4,7])
,("5-Z12",[0,1,3,5,6])
,("5-13",[0,1,2,4,8])
,("5-14",[0,1,2,5,7])
,("5-15",[0,1,2,6,8])
,("5-16",[0,1,3,4,7])
,("5-Z17",[0,1,3,4,8])
,("5-Z18",[0,1,4,5,7])
,("5-19",[0,1,3,6,7])
,("5-20",[0,1,3,7,8])
,("5-21",[0,1,4,5,8])
,("5-22",[0,1,4,7,8])
,("5-23",[0,2,3,5,7])
,("5-24",[0,1,3,5,7])
,("5-25",[0,2,3,5,8])
,("5-26",[0,2,4,5,8])
,("5-27",[0,1,3,5,8])
,("5-28",[0,2,3,6,8])
,("5-29",[0,1,3,6,8])
,("5-30",[0,1,4,6,8])
,("5-31",[0,1,3,6,9])
,("5-32",[0,1,4,6,9])
,("5-33",[0,2,4,6,8])
,("5-34",[0,2,4,6,9])
,("5-35",[0,2,4,7,9])
,("5-Z36",[0,1,2,4,7])
,("5-Z37",[0,3,4,5,8])
,("5-Z38",[0,1,2,5,8])
,("6-1",[0,1,2,3,4,5])
,("6-2",[0,1,2,3,4,6])
,("6-Z3",[0,1,2,3,5,6])
,("6-Z4",[0,1,2,4,5,6])
,("6-5",[0,1,2,3,6,7])
,("6-Z6",[0,1,2,5,6,7])
,("6-7",[0,1,2,6,7,8])
,("6-8",[0,2,3,4,5,7])
,("6-9",[0,1,2,3,5,7])
,("6-Z10",[0,1,3,4,5,7])
,("6-Z11",[0,1,2,4,5,7])
,("6-Z12",[0,1,2,4,6,7])
,("6-Z13",[0,1,3,4,6,7])
,("6-14",[0,1,3,4,5,8])
,("6-15",[0,1,2,4,5,8])
,("6-16",[0,1,4,5,6,8])
,("6-Z17",[0,1,2,4,7,8])
,("6-18",[0,1,2,5,7,8])
,("6-Z19",[0,1,3,4,7,8])
,("6-20",[0,1,4,5,8,9])
,("6-21",[0,2,3,4,6,8])
,("6-22",[0,1,2,4,6,8])
,("6-Z23",[0,2,3,5,6,8])
,("6-Z24",[0,1,3,4,6,8])
,("6-Z25",[0,1,3,5,6,8])
,("6-Z26",[0,1,3,5,7,8])
,("6-27",[0,1,3,4,6,9])
,("6-Z28",[0,1,3,5,6,9])
,("6-Z29",[0,1,3,6,8,9])
,("6-30",[0,1,3,6,7,9])
,("6-31",[0,1,3,5,8,9])
,("6-32",[0,2,4,5,7,9])
,("6-33",[0,2,3,5,7,9])
,("6-34",[0,1,3,5,7,9])
,("6-35",[0,2,4,6,8,10])
,("6-Z36",[0,1,2,3,4,7])
,("6-Z37",[0,1,2,3,4,8])
,("6-Z38",[0,1,2,3,7,8])
,("6-Z39",[0,2,3,4,5,8])
,("6-Z40",[0,1,2,3,5,8])
,("6-Z41",[0,1,2,3,6,8])
,("6-Z42",[0,1,2,3,6,9])
,("6-Z43",[0,1,2,5,6,8])
,("6-Z44",[0,1,2,5,6,9])
,("6-Z45",[0,2,3,4,6,9])
,("6-Z46",[0,1,2,4,6,9])
,("6-Z47",[0,1,2,4,7,9])
,("6-Z48",[0,1,2,5,7,9])
,("6-Z49",[0,1,3,4,7,9])
,("6-Z50",[0,1,4,6,7,9])
,("7-1",[0,1,2,3,4,5,6])
,("7-2",[0,1,2,3,4,5,7])
,("7-3",[0,1,2,3,4,5,8])
,("7-4",[0,1,2,3,4,6,7])
,("7-5",[0,1,2,3,5,6,7])
,("7-6",[0,1,2,3,4,7,8])
,("7-7",[0,1,2,3,6,7,8])
,("7-8",[0,2,3,4,5,6,8])
,("7-9",[0,1,2,3,4,6,8])
,("7-10",[0,1,2,3,4,6,9])
,("7-11",[0,1,3,4,5,6,8])
,("7-Z12",[0,1,2,3,4,7,9])
,("7-13",[0,1,2,4,5,6,8])
,("7-14",[0,1,2,3,5,7,8])
,("7-15",[0,1,2,4,6,7,8])
,("7-16",[0,1,2,3,5,6,9])
,("7-Z17",[0,1,2,4,5,6,9])
,("7-Z18",[0,1,2,3,5,8,9])
,("7-19",[0,1,2,3,6,7,9])
,("7-20",[0,1,2,4,7,8,9])
,("7-21",[0,1,2,4,5,8,9])
,("7-22",[0,1,2,5,6,8,9])
,("7-23",[0,2,3,4,5,7,9])
,("7-24",[0,1,2,3,5,7,9])
,("7-25",[0,2,3,4,6,7,9])
,("7-26",[0,1,3,4,5,7,9])
,("7-27",[0,1,2,4,5,7,9])
,("7-28",[0,1,3,5,6,7,9])
,("7-29",[0,1,2,4,6,7,9])
,("7-30",[0,1,2,4,6,8,9])
,("7-31",[0,1,3,4,6,7,9])
,("7-32",[0,1,3,4,6,8,9])
,("7-33",[0,1,2,4,6,8,10])
,("7-34",[0,1,3,4,6,8,10])
,("7-35",[0,1,3,5,6,8,10])
,("7-Z36",[0,1,2,3,5,6,8])
,("7-Z37",[0,1,3,4,5,7,8])
,("7-Z38",[0,1,2,4,5,7,8])
,("8-1",[0,1,2,3,4,5,6,7])
,("8-2",[0,1,2,3,4,5,6,8])
,("8-3",[0,1,2,3,4,5,6,9])
,("8-4",[0,1,2,3,4,5,7,8])
,("8-5",[0,1,2,3,4,6,7,8])
,("8-6",[0,1,2,3,5,6,7,8])
,("8-7",[0,1,2,3,4,5,8,9])
,("8-8",[0,1,2,3,4,7,8,9])
,("8-9",[0,1,2,3,6,7,8,9])
,("8-10",[0,2,3,4,5,6,7,9])
,("8-11",[0,1,2,3,4,5,7,9])
,("8-12",[0,1,3,4,5,6,7,9])
,("8-13",[0,1,2,3,4,6,7,9])
,("8-14",[0,1,2,4,5,6,7,9])
,("8-Z15",[0,1,2,3,4,6,8,9])
,("8-16",[0,1,2,3,5,7,8,9])
,("8-17",[0,1,3,4,5,6,8,9])
,("8-18",[0,1,2,3,5,6,8,9])
,("8-19",[0,1,2,4,5,6,8,9])
,("8-20",[0,1,2,4,5,7,8,9])
,("8-21",[0,1,2,3,4,6,8,10])
,("8-22",[0,1,2,3,5,6,8,10])
,("8-23",[0,1,2,3,5,7,8,10])
,("8-24",[0,1,2,4,5,6,8,10])
,("8-25",[0,1,2,4,6,7,8,10])
,("8-26",[0,1,2,4,5,7,9,10])
,("8-27",[0,1,2,4,5,7,8,10])
,("8-28",[0,1,3,4,6,7,9,10])
,("8-Z29",[0,1,2,3,5,6,7,9])
,("9-1",[0,1,2,3,4,5,6,7,8])
,("9-2",[0,1,2,3,4,5,6,7,9])
,("9-3",[0,1,2,3,4,5,6,8,9])
,("9-4",[0,1,2,3,4,5,7,8,9])
,("9-5",[0,1,2,3,4,6,7,8,9])
,("9-6",[0,1,2,3,4,5,6,8,10])
,("9-7",[0,1,2,3,4,5,7,8,10])
,("9-8",[0,1,2,3,4,6,7,8,10])
,("9-9",[0,1,2,3,5,6,7,8,10])
,("9-10",[0,1,2,3,4,6,7,9,10])
,("9-11",[0,1,2,3,5,6,7,9,10])
,("9-12",[0,1,2,4,5,6,8,9,10])
,("10-1",[0,1,2,3,4,5,6,7,8,9])
,("10-2",[0,1,2,3,4,5,6,7,8,10])
,("10-3",[0,1,2,3,4,5,6,7,9,10])
,("10-4",[0,1,2,3,4,5,6,8,9,10])
,("10-5",[0,1,2,3,4,5,7,8,9,10])
,("10-6",[0,1,2,3,4,6,7,8,9,10])
,("11-1",[0,1,2,3,4,5,6,7,8,9,10])
,("12-1",[0,1,2,3,4,5,6,7,8,9,10,11])]
-- | Unicode (non-breaking hyphen) variant.
sc_table_unicode :: Num n => SC_Table n
sc_table_unicode =
let f = map (\c -> if c == '-' then non_breaking_hypen else c)
in map (first f) sc_table
-- | Lookup name of prime form of set class. It is an error for the
-- input not to be a forte prime form.
--
-- > forte_prime_name [0,1,4,6] == ("4-Z15",[0,1,4,6])
forte_prime_name :: (Num n,Eq n) => [n] -> (SC_Name,[n])
forte_prime_name p = fromMaybe (error "forte_prime_name") (find (\(_,q) -> p == q) sc_table)
-- | Lookup entry for set in table.
sc_tbl_lookup :: Integral i => SC_Table i -> [i] -> Maybe (SC_Name,[i])
sc_tbl_lookup tbl p = find (\(_,q) -> z_forte_prime z12 p == q) tbl
-- | Erroring variant
sc_tbl_lookup_err :: Integral i => SC_Table i -> [i] -> (SC_Name,[i])
sc_tbl_lookup_err tbl = fromMaybe (error "sc_tbl_lookup") . sc_tbl_lookup tbl
-- | 'fst' of 'sc_tbl_lookup_err'
sc_name_tbl :: Integral i => SC_Table i -> [i] -> SC_Name
sc_name_tbl tbl = fst . sc_tbl_lookup_err tbl
-- | Lookup a set-class name. The input set is subject to
-- 'forte_prime' of 'z12' before lookup.
--
-- > sc_name [0,2,3,6,7] == "5-Z18"
-- > sc_name [0,1,4,6,7,8] == "6-Z17"
sc_name :: Integral i => [i] -> SC_Name
sc_name = sc_name_tbl sc_table
-- | Long name (ie. with enumeration of prime form).
--
-- > sc_name_long [0,1,4,6,7,8] == "6-Z17[012478]"
sc_name_long :: Integral i => [i] -> SC_Name
sc_name_long p =
let (nm,p') = sc_tbl_lookup_err sc_table p
in nm ++ z16_vec_pp p'
-- | Unicode (non-breaking hyphen) variant.
sc_name_unicode :: Integral i => [i] -> SC_Name
sc_name_unicode = sc_name_tbl sc_table_unicode
-- | Lookup a set-class given a set-class name.
--
-- > sc "6-Z17" == [0,1,2,4,7,8]
sc :: Num n => SC_Name -> [n]
sc n = snd (fromMaybe (error "sc") (find (\(m,_) -> n == m) sc_table))
-- | The set-class table (Forte prime forms), ie. 'snd' of 'sc_table'.
scs :: Num n => [[n]]
scs = map snd sc_table
-- | Cardinality /n/ subset of 'scs'.
--
-- > map (length . scs_n) [1..11] == [1,6,12,29,38,50,38,29,12,6,1]
scs_n :: (Integral i, Num n) => i -> [[n]]
scs_n n = filter ((== n) . genericLength) scs
-- | Vector indicating degree of intersection with inversion at each transposition.
--
-- > tics z12 [0,2,4,5,7,9] == [3,2,5,0,5,2,3,4,1,6,1,4]
-- > map (tics z12) scs
tics :: Integral i => Z i -> [i] -> [Int]
tics z p =
let q = z_sro_t_related z (z_sro_invert z 0 p)
in map (length . intersect p) q
-- * Z-relation
-- | Locate /Z/ relation of set class.
--
-- > fmap sc_name (z_relation_of (sc "7-Z12")) == Just "7-Z36"
z_relation_of :: Integral i => [i] -> Maybe [i]
z_relation_of x =
let n = length x
eq_i :: [Integer] -> [Integer] -> Bool
eq_i = (==)
f y = (x /= y) && (z_icv z12 x `eq_i` z_icv z12 y)
in case filter f (scs_n n) of
[] -> Nothing
[r] -> Just r
_ -> error "z_relation_of"