Scales again and Crystals

Nice to hear such comment about rules and scales. Maybe I was stretching the
point about coherency. I am intending to make a more thorough study of
scales in grids and lattices in the near future, so 'rules' be they of thumb
or hard computation will need to be looked at again in detail. I understand
that math is important for the comprhension of sale structure, and I will
need to learn more to perform the study I intend, especially some of the
finer aspects of matrices - it is a long time since my degree, when I had to
find determinants and eigenvalues and eigenvectors for myself by hand (try
it on an 8x8 for a long evening's entertainment!)

As for music - I have a few works written in microtonal scales but only one
has been performed and that was in 1993. I am currently seeking ways to
realize the scales I have discovered thus far. There are many, many more,
and my study will (hopefully) be an unbiased survey of a good number.

I am currently essaying ways to make composition easier using ET>12 scales
and JI ratio sets. For example I am trying to conceive of a keyboard
somewhat similar in concept to a Bosanquet, but with a 'pallette' like
interface, so that non-standard ratio-sets can also be laid out. This stems
from an Idea I had after reading Bamberger, Jeanne, 1986, Cognitive issues
in the development of Musically Gifted Children. Chap 17 in Conceptions of
Giftedness, ed. Robert J. Sternberg and Janet E. Davidson, 388-413. A brief
summary can also be found in Lewin, Musical Form and Transformation, Yale
Univ. Press 1993, pp 45-53. Essentially, it concerns how children, when
presented with a tune or melody and a set of moveable Montessori bells,
organise the bells to better play the melody. It offers interesting insights
into the relationship of organised tone sequences and abstract tone set
presentation. I thought it might be interesting if the user could position
the 'keys' of a 'virtual keyboard' where they liked. And of course be able
to save and load different sets.

As for an interesting question made to me by a mechanical engineering friend
(from my own days as a mechanical eng), I quote:

"These [ratio] lattices and periodicity blocks, do they resemble those
vector descriptions of crystal lattices?"

At the time we were wondering if that, as a periodicity block approached as
sphere or circle in shape, and intersected it with a lattice, the radius and
centre could locate scale formations in FCC or HCP combinations of
spheres/circles. We also wondered what other solid equivalents to penrose
tilings would also describe scales, and whether these (if for example it was
a space-filling tiling of two different solids) would describe a 'diatonic'
and a 'pentatonic' simultaneously.

At that point the conversation wandered onto considering if scales could be
constructed on a line between 1/1 and 2/1 as Cantor sets like Cantor Dust.

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

>I am currently seeking ways to
> realize the scales I have discovered thus far. There are many, many more,
> and my study will (hopefully) be an unbiased survey of a good number.

If you like, I think people would be interested in seeing what you've done so far. The feedback you get will almost certainly be good for your paper, if you want to subject yourself to it.

> "These [ratio] lattices and periodicity blocks, do they resemble those
> vector descriptions of crystal lattices?"

You bet. Of course as an unreconstructed mathematician, I prefer to use the word "lattice" only when the vectors form a group--that is, when the some of any two vectors is always another vector in the given set.

> At the time we were wondering if that, as a periodicity block approached as
> sphere or circle in shape, and intersected it with a lattice, the radius and
> centre could locate scale formations in FCC or HCP combinations of
> spheres/circles.

I don't know what a FCC or HCP combination is, but I posted something about spheres and 7-limit lattices on the new tuning group, and was planning on following it up today with something on the 5-limit.

> At that point the conversation wandered onto considering if scales could be
> constructed on a line between 1/1 and 2/1 as Cantor sets like Cantor Dust.

Hmmm. Have you ever seen Farey circles? If you take the rational numbers between 1 and 2, and for each p/q make a circle of radius 1/2q^2 and center [p/q, 1/2q^2], you get a fractal collection of circles which never intersect, but where the circles for adjacent Farey fractions always touch. Maybe that could inspire something.