new Encyclopaedia entry: Tonnetz

Hugo Riemann's historically important _Tonnetz_,
the precursor to the triangular lattices we use
all the time:

http://tonalsoft.com/enc/tonnetz.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
>
> Hugo Riemann's historically important _Tonnetz_,
> the precursor to the triangular lattices we use
> all the time:
>
> http://tonalsoft.com/enc/tonnetz.htm

I would say "octave equivalent" rather than "octave invariant". You
might also mention that you get lattices in higher prime limits, and
that in the 7-limit you can similarly make the lattice symmetrical.

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> >
> > Hugo Riemann's historically important _Tonnetz_,
> > the precursor to the triangular lattices we use
> > all the time:
> >
> > http://tonalsoft.com/enc/tonnetz.htm
>
> I would say "octave equivalent" rather than "octave invariant".

i've never understood the difference. perhaps you can clarify.

> You might also mention that you get lattices in higher
> prime limits,

if anyone reads the links to "lattice", they'll be able
to figure that out. is there some particular reason why
i would want to mention it here? the _Tonnetz_ is
specifically a 5-limit JI structure.

> and that in the 7-limit you can similarly
> make the lattice symmetrical.

lattices of any dimension are symmetrical ... we just
have trouble seeing it if it's >3-D.

;-)

you can easily see the symmetry in my old lattices
all the way up to the 13-limit:

http://tonalsoft.com/monzo/lattices/lattices.htm

i think pretending that the planes of the 11-limit
lattice are translucent glass helps even more to
see the 4-D symmetry.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> > I would say "octave equivalent" rather than "octave invariant".

> i've never understood the difference. perhaps you can clarify.

Invariant suggests something is being left unchanged, equivalent that
two things can in some way be treated alike. It seems to me when we
say 5/4, 5/2 and 5 are in some sense alike we are saying they are
equivalent. In any case pitch sets are equivalences classes
mathematically speaking.

> lattices of any dimension are symmetrical ... we just
> have trouble seeing it if it's >3-D.

That's like saying all triangles are equilateral. Lattices are regular
and so have radial symmetry, but that's it; in general the symmetry
they have is the symmetry of a parallelopided, which is a long way
from that of a cube or tetrahedron or similar such figures.

> you can easily see the symmetry in my old lattices
> all the way up to the 13-limit:
>
> http://tonalsoft.com/monzo/lattices/lattices.htm

I'm not seeing any symmetry but radial in these diagrams.