Re: [tuning-math] Digest Number 1568

[Paul Hj]:

>> 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).
>>
> Thanks for the clarification. So sometimes an axis of symmetry is
> a non-integer value? (11/2 for example)?

That's right - whenever no pitch-class maps onto itself under the inversion in question, you will have non-integer axes of symmetry. There are always two axes of symmetry, half an octave apart. So in 12-edo if Ab inverts into Eb, giving an index of inversion of 11, you can find one axis of inversion at 11/2 and another at 23/2. That is, one that bissects the fourth Eb-Ab and another bissecting the fifth Ab-Eb.

Best --Jon

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> [Paul Hj]:
>
> >> 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).
> >>
> > Thanks for the clarification. So sometimes an axis of symmetry is
> > a non-integer value? (11/2 for example)?
>
> That's right - whenever no pitch-class maps onto itself under the
> inversion in question, you will have non-integer axes of symmetry.
There
> are always two axes of symmetry, half an octave apart. So in 12-edo
if Ab
> inverts into Eb, giving an index of inversion of 11, you can find
one axis
> of inversion at 11/2 and another at 23/2. That is, one that
bissects the
> fourth Eb-Ab and another bissecting the fifth Ab-Eb.
>
> Best --Jon
>
Cool. Symmetry is my obsession! I wonder if there is any significance
to 216 sets that map into themselves under T0I in 12-tET...still
trying to derive the formula. Using Polya, you can only find
the results of symmetrical sets, which is 1,1,5,15,10,20,10,15,5,1,1
as you know for 12tET...still quite amazing.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Jon Wild <wild@> wrote:
> >
> >
> > [Paul Hj]:
> >
> > >> 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).
> > >>
> > > Thanks for the clarification. So sometimes an axis of symmetry
is
> > > a non-integer value? (11/2 for example)?
> >
> > That's right - whenever no pitch-class maps onto itself under the
> > inversion in question, you will have non-integer axes of
symmetry.
> There
> > are always two axes of symmetry, half an octave apart. So in 12-
edo
> if Ab
> > inverts into Eb, giving an index of inversion of 11, you can find
> one axis
> > of inversion at 11/2 and another at 23/2. That is, one that
> bissects the
> > fourth Eb-Ab and another bissecting the fifth Ab-Eb.
> >
> > Best --Jon
> >
> Cool. Symmetry is my obsession! I wonder if there is any
significance
> to 216 sets that map into themselves under T0I in 12-tET...still
> trying to derive the formula. Using Polya, you can only find
> the results of symmetrical sets, which is 1,1,5,15,10,20,10,15,5,1,1
> as you know for 12tET...still quite amazing.
>
Oops its really (1,1,6,5,15,10,20,10,15,5,6,1,1) = 96