Compact scales

There's some talk about higher-rank scales again, so I'll
take the opportunity to explain this idea that doesn't have
much to do with what everybody else was talking about. It
may do nothing to bring about generalized Myhill's
property, but it looks like an advance on hobbits, as I
understand them.

A compact scale is defined in a given temperament (class).
It repeats every octave (or equivalence interval, or maybe
even period). There's a one-to-one mapping between pitches
in the scale and an equal temperament belonging to the
temperament class you're dealing with. Each pitch is
closer to a given point than any pitch it could have been
replaced with. Where it differs from a hobbit is that
these distances are measured relative to the lattice
midpoint of the scale.

Maybe somebody already defined "compact scales". Maybe
that definition matches mine. Maybe not. I'll keep saying
"compact scales" until I find out.

I'll look at 5-limit JI diatonics because I have a good
idea what results to expect with them. Distances are
measured according to plain Tenney-Euclidean (TE)
complexity. With prime generators, the mapping matrix is
simply the identity matrix, and so temperamental complexity
defaults to Tenney weighting. Temperamental complexity
still carries its weight if you choose your generators
differently, though. The compactness of a scale is
independent of the choice of generators.

Compactness is a function of the scale, mapping and JI
setup. It's the sum or mean of the distances of pitches
from the midpoint. A compact scale will have lower
compactness than any alternatives that meet the other
criteria. I don't have an implementation to provide me
with compact scales: I have to plug in different scales and
see which is has the lowest compactness.

I'll also use octave-specific complexity. Octave
equivalence is a complication I'll leave for another day.
Of course, there are other complications associated with
octave specificity, so be warned that those are the
complications I'll be dealing with today.

There are two plausible justifications of the C major
scale. They are, on a 5-limit lattice,

(1)
A-E-B
F-C-G-D

(2)
D-A-E-B
F-C-G

(Maybe the lattices don't come out right. You can probably
re-construct them. At least, note that (1) has C-D as 9/8
but (2) has C-D as 10/9.)

(1) is what I think we'd all prefer. (2) happens to be the
more compact scale (taking pitches in the octave above C).
(2) also has simpler intervals relative to the tonic, given
that 9:8 has a higher TE complexity than 10:9. I don't
think that matters, but that (1) has more otonalities than
(2) probably does. Maybe somebody can work out how to bias
the scales towards otonalities. I believe dwarfs do it for
just intonation, but don't generalize to regular
temperaments.

It happens that C major as scale (1) has the same
compactness as F lydian as scale (2). You may think I
abstracted out the choice of tonic by taking the midpoint,
but in fact it does still matter because the ratios are
defined in the octave above the tonic. Because it's nice
to have a uniquely compact scale, I added a fudge to move
the midpoint a little closer to the tonic. Given this
fudge, C major as scale (1) is the most compact scale I
tried, but I didn't try everything.

One scale that certainly isn't as compact as either C major
scale is this:

D-A
F-C-G
EbBb

It's interesting in that it's the smallest I can find
measuring the distance from C. That is, it's the hobbit,
according to this octave-specific complexity. With a naïve
octave-equivalent complexity, I believe this is the hobbit:

E-B
F-C-G
DbAb

Either way, the hobbit doesn't have as many chords as the
compact scale. Measuring everything relative to the
midpoint gives all intervals, including A-E, an influence
on the result. Because the midpoint is generally not a
lattice point, it can't be replaced with the tonic to give
a hobbit. This is why I prefer compact scales to hobbits.

Pari/GP code for all this is at
http://x31eq.com/parametric.gp

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> Where it differs from a hobbit is that
> these distances are measured relative to the lattice
> midpoint of the scale.

How do you find this lattice midpoint if you are not assuming octave equivalence? How do you define it if you do?

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:
>
>
> --- In tuning-math@yahoogroups.com, Graham Breed
> <gbreed@...> wrote:
> > Where it differs from a hobbit is that
> > these distances are measured relative to the lattice
> > midpoint of the scale.
>
> How do you find this lattice midpoint if you are not
> assuming octave equivalence? How do you define it if you
> do?

Without octave equivalence, by averaging the position
vectors of the pitches in the scale (intervals relative to
the tonic). Everything is smaller than an octave and no
smaller than a unison. With octave equivalence, I think
I'd still average the position vectors. The difficult part
is finding a suitable metric.

Graham