Re: [tuning-math] Digest Number 1567

[Paul Hj]:
> In the case of inverse complementation, it is sets that map into > themselves through TnI (transpose + mirror inversion). One thing I am > not settled on, is should it be transpose THEN inverse or inverse THEN > tranpose?

The usual practice is to perform the transforms from inside outwards, so T_3I, for example, is T_3 of the inversion (that is, the inversion around pitch-class 0). This is called "left orthography", where the operator appears to the right of the operand. TnI is then often written simply I_n, which means the inversion such that corresponding pitch-classes sum to n (i.e. one of the axes of symmetry is n/2). If you check it, you'll find that I_n accomplishes the same as TnI. So the 2n transformations in the dihedral group on a universe of n elements can be labelled T0, T1, T2... Tn-1 and I0, I1, I2... In-1.

Best --Jon

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> [Paul Hj]:
> > In the case of inverse complementation, it is sets that map into
> > themselves through TnI (transpose + mirror inversion). One thing
I am
> > not settled on, is should it be transpose THEN inverse or inverse
THEN
> > tranpose?
>
> The usual practice is to perform the transforms from inside
outwards, so
> T_3I, for example, is T_3 of the inversion (that is, the inversion
around
> pitch-class 0). This is called "left orthography", where the
operator
> appears to the right of the operand. TnI is then often written
simply I_n,
> which means the inversion such that corresponding pitch-classes sum
to n
> (i.e. one of the axes of symmetry is n/2). If you check it, you'll
find
> that I_n accomplishes the same as TnI. So the 2n transformations in
the
> dihedral group on a universe of n elements can be labelled T0, T1,
T2...
> Tn-1 and I0, I1, I2... In-1.
>
> Best --Jon
>
Thanks for the clarification. So sometimes an axis of symmetry is
a non-integer value? (11/2 for example)?

It's interesting that Polya's method gives the right values for
symmetrical sets, but not with the correct Tn values (If you create
a grid). However, transposition, complementability, and inverse-
complementability all pan out correctly --- It's in one of my posts
from a week or so ago - I guess its not terribly importantwhich sets
map into themselves under different TnI but I still think it would
be fun to find a formula that would do this correctly for symmetrical
sets...