Using the function Gradual(Omap())

The function of Mapped Opposite enables the musician to choose among sets of series sensitive to their permutational properties (Cf. J.Palfi et al., 2008, Le dodécaphonisme comme outil d’analyse – in French).

To say things quickly, the mapped opposite uses what group theory calls an invariant, in order to have a suite transposed permutationnally. If you do the gradual suite of the opposite of any series S, mapped onto S, you get a suite G that is the same for each transposition of S. Now if you unmap each element of G from S+1 you get a transposed suite Q+1, and you can get Q+2 by unmapping G from S+2. Capisce?

Example:
(the command lines below can be sent from the Series tab)

print Csgrouper::Gradual(Csgrouper::Omap(“32014″),’a’)
   12340 23401 34012 40123
# The gradual suite made upon the mapped opposite of 32014.

print Csgrouper::Gradual(Csgrouper::Omap(Csgrouper::Transpose("32014","00000",1,5,'')),'a')
   12340 23401 34012 40123 
# We see that however transposed, the result is the same.

print Csgrouper::Gradomap("32014",'a')
   20143 01432 14320 43201 
# This is the gradual suite made of the mapped opposite, where each row has been unmapped from 32014.

print Csgrouper::Gradomap(Csgrouper::Transpose("32014","00000",1,5,''),'a')
   31204 12043 20431 04312
# Here each row from the previous suite is transposed, permutationally.

The only thing to care about is that these gradual suites – as they are produced by Gradomap() in the Csgrouper sequence creation process (and not as in here with the command line) – will contain an additionnal row in the beggining, which is  the original row S itself, or S+n if transposed. This allows us to keep in touch with our starting point musically, and it is not unsound permutationally, since S is the last element in the cyclic suite (we could have made it last too). The difference, with normal, not unmapped, Gradual suites – which contain one row less – comes from the fact that in normal Gradual suites, as in the above example, Csgrouper does not add the last element which is always equivalent to Natural(S) i.e. the chromatic scale (for this has rarely a nice effect in music).

Note: 

In the preceding example Gradual(Omap(S)) is the original function without unmapping, unlike Gradomap(S) which proceeds to the unmapping. However, in Csgrouper’s Sequence Details tab, where one can choose functions by menu, the menu item Gradual(Omap(S)) implies automatic unmapping and in fact calls internally Gradomap(). This is due to the fact that musically speaking, Gradomap() is much more interesting as it yields transposed suites. If a user wants to create a bare Gradual(Omap(S)) as above, she will have to do it through command line from the Series tab and then paste the result as a Suite() into the chosen Sequence window. So all this naming is fallacious, and I should have named the menu item Unmap(Gradual(Omap(A)),A) and Gradomap() Ugradomap() or something. 

 

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