Just set theory?

I'm wondering if there has ever been an attempt to create a just set
theory, giving numbers to all intervals, triads etc. Obviously there
can be no full "complementary" sets nor can they be all-encompassing
as the amount of just intervals is infinite, but they could certainly
be useful in analyzing scales, looking for self-similarities, counting
the occurence of each interval and triad within it. How would you go
about with deciding which intervals are to be considered "higher" or
"more advanced" than others?

/Ö

--- In tuning@yahoogroups.com, Mats Ö¬jare <oljare@...> wrote:
>
> I'm wondering if there has ever been an attempt to create a just set
> theory, giving numbers to all intervals, triads etc.

The first thing to say about musical set theory is that it isn't set
theory, and the basic structure being analyzed is a group (in the usual
case, the cyclic group with twelve elements, C12.) In the case of p-
limit JI modulo octaves, you get a very different group--the free group
of odd-ratio members of the p-limit, with a unique reduced odd-number
ratio representing each interval class. This, therefore, is what you
would take n-element subsets of, but unlike the case with equal
temperament, you don't get a finite number of chords. Hence you
probably want to bound the results somehow. One way is to confine
yourself to chords of a scale, another is to bound heights.

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

> The first thing to say about musical set theory is that it isn't
set
> theory, and the basic structure being analyzed is a group (in the
usual
> case, the cyclic group with twelve elements, C12.)

It seems to me that more fruitful than JI musical set theory might be
mst for rank one temperaments, particularly linear temperaments such
as meantone or miracle. Here the group is isomorphic to the integers,
and an abstract chord can be equated to a finite set of integers.

The span of such a chord can be defined as the maximum minus the
minimum element, and we get a finite set of chord types by bounding
the span. Chords can be nornalized by setting the least integer to
zero. The resulting theory would actually make more sense of common
practice music through the 18th century at least than the usual
musical set theory, so some enterprising soul should rush out and
publish it in a mainstream journal. "Diatonic set theory" has been
fumbling about in this direction anyway.

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

> It seems to me that more fruitful than JI musical set theory might be
> mst for rank one temperaments, particularly linear temperaments such
> as meantone or miracle.

I followed this idea up on tuning-math, with a listing of 45 "basic"
meantone triads, which would certainly be a good start for a meantone
version of musical set theory:

/tuning-math/message/15987

On 30/01/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
>
> It seems to me that more fruitful than JI musical set theory might be
> mst for rank one temperaments, particularly linear temperaments such
> as meantone or miracle. Here the group is isomorphic to the integers,
> and an abstract chord can be equated to a finite set of integers.

How did linear temperaments get to be rank one?

Graham

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

> The first thing to say about musical set theory is that it isn't set
> theory, and the basic structure being analyzed is a group (in the
>usual
> case, the cyclic group with twelve elements, C12.)

I'm glad you pointed that out Gene, it's one of the things on the long
list of peas under my mattress pile in music theory. A 1970s book I'm
reading at the moment refers to musical "set theory" as combinatorics,
which I believe is not only nicely accurate, but ties in with the long
history of combinatorics in music theory.

-Cameron Bobro

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> How did linear temperaments get to be rank one?

Paul reversed course and decided that Erv meant for "linear" to mean
octave generators only. Deciding what Erv means is a chancy business,
but I latched onto this since I think the distinction is useful. Hence,
these days I mean by a linear temperament a rank one temperament with
an octave generator.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> I'm glad you pointed that out Gene, it's one of the things on the
long
> list of peas under my mattress pile in music theory. A 1970s book I'm
> reading at the moment refers to musical "set theory" as
combinatorics,
> which I believe is not only nicely accurate, but ties in with the
long
> history of combinatorics in music theory.

Combinatorics is where mathematicians would place it, in case that
counts.

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

> Combinatorics is where mathematicians would place it, in case that
> counts.

I think it does count- if you're going to borrow from other sciences,
you might as well do it right. Maybe being way off base can produce
some fine art, though- a research scientist (cancer) friend of mine
once bemoaned the fact that he could spend a lifetime doing fine and
careful work, and be on a wrong track all along, while an artist might
base his work on scientifically ridiculous premeses and still create
something of value to society.

-Bobro

> How did linear temperaments get to be rank one?
>
>
> Graham

I was just going to say that. -Carl

On 30/01/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
>
> --- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> > How did linear temperaments get to be rank one?
>
> Paul reversed course and decided that Erv meant for "linear" to mean
> octave generators only. Deciding what Erv means is a chancy business,
> but I latched onto this since I think the distinction is useful. Hence,
> these days I mean by a linear temperament a rank one temperament with
> an octave generator.

I've been going around calling them rank two and nobody called me on
it. I hope you made a double typo and don't really mean to call them
rank one. I'm dubious about the validity of the restricted sense of
"linear temperament" as well, but that's another story.

Graham

--- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> I've been going around calling them rank two and nobody called me on
> it. I hope you made a double typo and don't really mean to call them
> rank one. I'm dubious about the validity of the restricted sense of
> "linear temperament" as well, but that's another story.

Sorry about that, I meant rank two of course.

Why do you think the "linear" restriction is dubious?

On 31/01/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
>
> --- In tuning@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> > I've been going around calling them rank two and nobody called me on
> > it. I hope you made a double typo and don't really mean to call them
> > rank one. I'm dubious about the validity of the restricted sense of
> > "linear temperament" as well, but that's another story.
>
> Sorry about that, I meant rank two of course.

That's good, panic over.

> Why do you think the "linear" restriction is dubious?

Like with "MOS" and friends, I don't think anybody had been thinking
of generic rank 2 temperaments in a linear temperament-like way before
some of us did and started calling them linear temperaments. As you
say, it's always difficult to infer precisely what Erv means. And I
vaguely remember Paul reversing course about reversing course.

The phrase "linear temperament"'s also nicer than "rank two
temperament". The former may have more syllables, but doesn't have
two "t"s close together and a "k" and "t" with no vowel intervening.
And it's less likely you'd have said "equal" when you meant "linear"
than "one" when you meant "two".

I'd prefer to say "strict linear temperaments" when I want a
short-hand for "rank 2 regular temperaments with an octave period".
But for now I stay clear of the whole area as a terminological mess.

Graham

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
>
> > Combinatorics is where mathematicians would place it, in case that
> > counts.
>
> I think it does count- if you're going to borrow from other sciences,
> you might as well do it right. Maybe being way off base can produce
> some fine art, though- a research scientist (cancer) friend of mine
> once bemoaned the fact that he could spend a lifetime doing fine and
> careful work, and be on a wrong track all along, while an artist might
> base his work on scientifically ridiculous premeses and still create
> something of value to society.
>
> -Bobro

As one of those artists, I can quite see his point.

-Patrick

--- In tuning@yahoogroups.com, "p_heddles" <p_heddles@...> wrote:

> As one of those artists, I can quite see his point.

Hahaha! Of course "religion" is probably often considered
scientifically ridiculous but an immense body of art is based on
religious ideas. And of course science itself is full of poetic
explanations, like string theory, and the social sciences are full
of things prone to embarassing parochialism or blatant propaganda,
social Darwinism for example. And anything can be viewed
metaphorically- the geocentric solar system is a very accurate
description of the reality of those who believed in it.

-Cameron Bobro