Graphing chord connections in equal temperaments

I've put graphs of the triads in 7, 12, 19 and 22 equal, with an edge
drawn whenever the triads share an interval. Unfortunately Maple did
not draw 19 and 22 in a way which shows the symmetry. The 7-et graph
can also be thought of as a diatonic graph.

I also put up a graph for tetrads in 12-et. Checking the
characteristic polynomial of this, I find that there are 240 chord
triangles for 12-et tetrads, where a chord triangle means three
chords, each of which shares an interval with the other chords.

The graphs can be found in the "chord connection graphs" album in the
"Photos" for this group.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put graphs of the triads in 7, 12, 19 and 22 equal, with an
edge
> drawn whenever the triads share an interval. Unfortunately Maple did
> not draw 19 and 22 in a way which shows the symmetry. The 7-et graph
> can also be thought of as a diatonic graph.
>
> I also put up a graph for tetrads in 12-et. Checking the
> characteristic polynomial of this, I find that there are 240 chord
> triangles for 12-et tetrads, where a chord triangle means three
> chords, each of which shares an interval with the other chords.
>
> The graphs can be found in the "chord connection graphs" album in
the
> "Photos" for this group.

These are great! Since I can't read the labels on the nodes (too
small and faint) I need to ask you why there are 24 nodes apiece for
12-et triads and 12-et tetrads. Thanks!!!

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > I've put graphs of the triads in 7, 12, 19 and 22 equal, with an
> edge
> > drawn whenever the triads share an interval. Unfortunately Maple
did
> > not draw 19 and 22 in a way which shows the symmetry. The 7-et
graph
> > can also be thought of as a diatonic graph.
> >
> > I also put up a graph for tetrads in 12-et. Checking the
> > characteristic polynomial of this, I find that there are 240 chord
> > triangles for 12-et tetrads, where a chord triangle means three
> > chords, each of which shares an interval with the other chords.
> >
> > The graphs can be found in the "chord connection graphs" album in
> the
> > "Photos" for this group.
>
> These are great! Since I can't read the labels on the nodes (too
> small and faint) I need to ask you why there are 24 nodes apiece
for
> 12-et triads and 12-et tetrads. Thanks!!!

seems obvious to me -- there are 24 5-limit triads, and 24 7-limit
tetrads, approximated in 12-equal. the former are the conventional
major and minor triads, the latter are the conventional dominant
seventh and half-diminished seventh chords. none of these chords are
of the "limited transposition" variety, so there are 12 of each type
of triad (24 total) and 12 of each type of tetrad (24 total).

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > I've put graphs of the triads in 7, 12, 19 and 22 equal, with
an
> > edge
> > > drawn whenever the triads share an interval. Unfortunately
Maple
> did
> > > not draw 19 and 22 in a way which shows the symmetry. The 7-et
> graph
> > > can also be thought of as a diatonic graph.
> > >
> > > I also put up a graph for tetrads in 12-et. Checking the
> > > characteristic polynomial of this, I find that there are 240
chord
> > > triangles for 12-et tetrads, where a chord triangle means three
> > > chords, each of which shares an interval with the other chords.
> > >
> > > The graphs can be found in the "chord connection graphs" album
in
> > the
> > > "Photos" for this group.
> >
> > These are great! Since I can't read the labels on the nodes (too
> > small and faint) I need to ask you why there are 24 nodes apiece
> for
> > 12-et triads and 12-et tetrads. Thanks!!!
>
> seems obvious to me -- there are 24 5-limit triads, and 24 7-limit
> tetrads, approximated in 12-equal. the former are the conventional
> major and minor triads, the latter are the conventional dominant
> seventh and half-diminished seventh chords. none of these chords
are
> of the "limited transposition" variety, so there are 12 of each
type
> of triad (24 total) and 12 of each type of tetrad (24 total).
Thanks! It would also be cool to study ALL the triad types - There
are 19, (not just major/minor) and all 43 tetrad types, both reduced
for transposition. Of course with transposition there are C{12,3}
triads and C{12,4}tetrads equalling 220 and 495 respectively.

> I've put graphs of the triads in 7, 12, 19 and 22 equal, with an edge
> drawn whenever the triads share an interval. Unfortunately Maple did
> not draw 19 and 22 in a way which shows the symmetry. The 7-et graph
> can also be thought of as a diatonic graph.
>
> I also put up a graph for tetrads in 12-et. Checking the
> characteristic polynomial of this, I find that there are 240 chord
> triangles for 12-et tetrads, where a chord triangle means three
> chords, each of which shares an interval with the other chords.
>
> The graphs can be found in the "chord connection graphs" album in the
> "Photos" for this group.

Oh, that sounds really interesting. Unfortunately, there seems to be a
problem with signing in to YaHoo at the moment. Any chance to see these
without having to sign in? I can see the "Files section but not the "Photos"...