Marvellous ellipses

If |a b c> is a 5-limit monzo, we may define a quadratic form on
note-classes by ell(|a b c>) = 4b^2 - 7bc + 4c^2. The discriminant of
this is (-7)^2 - 4*4*4 = -15, which is negative, so setting ell to a
constant gives us ellipses. ell(3) = ell(5) = ell(225/32) = 4; the
last is the 5-limit marvel projection of 7, and so one of the ellipses
has the class for the marvel versions of 3, 5, and 7 on its boundry.

A marvellous ellipse is simply a set of notes 1 <= q < 2 such that
ell(q) <= N for some bounding value N. These turn out to define
interesting scales. It is clear from the manner of their construction
that they are not only inversely symmetric but also have a 3<==>5
symmetry. They also have an odd number of notes to an octave. What
isn't clear, but which seems to be true, is that they have a tendency
to be permutation epimorphic. It might also be noted that attempting
to do the same for the 9-limit, and have an ellipse with 5/4, 225/128
and 9/8 on the boundry, will not work, since 5/4, 9/8, and 256/225 lie
along a line.

I've uploaded some marvelous ellipses--of sizes 9, 13, 15, 21, and
31--to the files section:

/tuning-math/files/marvell/

The 9-note scale is epimorphic, and 13, 15 and 19 are all permutation
epimorphic.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The 9-note scale is epimorphic, and 13, 15 and 19 are all permutation
> epimorphic.

I don't know the reason for it, but it is clear there is one. I looked
at using 2401/2400 with {5,7} note classes, and 3 represented by
2401/800. Trying to get 5, 7, and 2401/800 on the boundry of a conic
section centered at 1 leads to a hyperbola, so I tried
8x^2 + 15xy + 8y^2 instead. This didn't net me as many complete
tetrads as I might like (though of course very many 4:5:7 chords and
the like) but it did, once again, turn up with a lot of epimorphic and
permuation epimorphic offerings. Perhaps some theory for ellipsoids
could be developed to go with Fokker blocks.