Re: a question [for Dan S]

Dan wrote:

> Here's a math-music creative problem solving type question that's
> interested me for a while.
>
> Given any scale, and be that scale rational, equidistant, or what have
> you, what methods have folks worked with (if any) that could
> generalize and stamp one single rotation as something that could in
> one way or another be called the standard rotation?

Hi Dan - this is the problem that "regular" music theory deals with by
using prime-forms for each set-class. E.g., all 12-tet diatonic modes form
a set-class, and the "prime form" selected to represent the set-class is
[013568t], or the B-B (Locrian) mode, because it is most compact and
"left-packed". With 12-tet harmonic (major and minor) triads, the
set-class is represented by [037], the root-position minor triad, for the
same reasons.

The algorithm is: first find the modes that span the shortest distance.
(In the case of diatonic modes, this allows you to reject the C-C and F-F
modes only. In the case of triads you reject first- and second-inversion
triads, since only root-position ones span a fifth). Then, from these,
find the one whose notes are most "left-packed" - that is, whose interval
sequence has its larger intervals nearest the upper end. There is a formal
long-winded way to define this but it's probably clear intuitively.

This doesn't generally select the most "natural" mode to represent the
scale or set-class, depending on what you judge natural, but it is
straight-forward to implement, and two people using it will agree on the
same answer. It is very easy to generalise for other tuning systems, since
nothing about it is specific to 12-tet. And once "professional" music
theorists get around to cataloguing microtonal entities, this will almost
undoubtedly be a basis for it, given set-theory's importance in academia.

cheers --jon

--- In tuning@y..., jon wild <wild@f...> wrote:

/tuning/topicId_24253.html#24253
>
> Dan wrote:
>
> > Given any scale, ... what methods have folks worked with
> (if any) that could generalize and stamp one single rotation
> as something that could in one way or another be called the
> standard rotation?
>
> Hi Dan - this is the problem that "regular" music theory
> deals with by using prime-forms for each set-class.
>
> ...
>
> This doesn't generally select the most "natural" mode to
> represent the scale or set-class, depending on what you judge
> natural, but it is straight-forward to implement, and two
> people using it will agree on the same answer. It is very
> easy to generalise for other tuning systems, since nothing
> about it is specific to 12-tet. And once "professional" music
> theorists get around to cataloguing microtonal entities, this
> will almost undoubtedly be a basis for it, given set-theory's
> importance in academia.

I agree with you here, Jon. For the same reasons that I use
compact notations like "3^(-2...3) * 5^(-1...1)" and Erv
Wilsons notations like "2,4 (3 5 7 11)" to describe JI
systems, I like the prime-form method of describing scales
in set-class theory. I agree too that it works just as well
for any other EDO as it does for 12.

Answers should probably go to the tuning-math list, and
so perhaps so should this question... oh well... I'd like
to see more on the mathematics of manipulating sets of
exponents in a JI tuning system, similar to the way
academic theorists use set-theory in 12-EDO. Any ideas
on this?

-monz
http://www.monz.org
"All roads lead to n^0"