are we rational or irrational [math question]

I must admit my math is a little "rusty," but I've really enjoyed
getting it back a tiny bit "up to speed" by reading this list!!!

The question I have pertains to the relationship of just intonation
to "rational" numbers.

According to the dictionary (an obviously accelerated source for math
wisdom) a "rational number" is any number that can be expressed as a
"quotient of integers."

This means, of course, that ALL FRACTIONS are "rational numbers,"
correct??

Given that fact, and given the fact that just intonation is the
expression of small integer "rational" fractions, does that mean that
ANY tuning system with "rational" fractions IS just intonation??

In other words, is there a place where the line is drawn and
fractions no longer become just?? i.e. a "limit" using integers 11,
13 or whatever??

You can see I'm mightily confused about this. Might as well be
public about it...

If, then, it is true that ANY system with rational fractions IS just
intonation REGARDLESS of the size of the integers of the fractions,
then the only systems that are NOT just are systems which use
"irrational" numbers, yes??

The "irrational" numbers being decimals which are multiplied together
to get a constant... such as the 12th root of 2, etc. These
"irrational" numbers can NEVER be expressed as a "quotient of
integers" so are NEITHER "rational" NOR "just."

Am I confused, or what...
__________ ____ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> http://www.egroups.com/message/tuning/12436
>
> This means, of course, that ALL FRACTIONS are "rational numbers,"
> correct??

Correct. A fraction is an expression of a relation, or ratio,
between two quantities.

> Given that fact, and given the fact that just intonation is the
> expression of small integer "rational" fractions, does that mean
> that ANY tuning system with "rational" fractions IS just
> intonation??
>
> In other words, is there a place where the line is drawn and
> fractions no longer become just?? i.e. a "limit" using integers 11,
> 13 or whatever??

Joe, you have a basic grasp of this; the main reason you're
confused is because there is no complete consensus on where to
draw the line in describing a tuning as 'just-intonation'.

Traditionally, the term has meant specifically 5-limit JI.

I believe that it's mainly thru Partch's work that it has been
extended at least to 11- or 13-limit. But while Partch did
refer to his work as being in 'just-intonation', note that he
was careful enough to coin the name 'Monophony' for his own
tuning system.

Ezra Sims (I know you don't like his music, but I'm crazy about
his Clarinet Quintet and Night Piece) uses 72-tET notation which
represents a basic 37-limit JI scale as well as many more complex
ratios when he modulates. He himself has said that he doesn't hear
all that much difference between actual 72-tET versions and JI
versions of his music, and in a letter to me has said that he
doesn't particularly care about the 'rationality' of his small
divisions of the pitch-continuum. (I'm paraphrasing, not quoting)

I coined the term 'JustMusic' to refer to *any* (including very
complex) rational tuning systems, but I've become more conservative
lately.

In general, I think it's OK to equate 'just' with 'rational',
but popular usage tends to keep 'just' limited to either 5-limit
JI, or to higher-prime but otherwise low-integer JI (i.e., Partch).
For that reason, I often use 'rational' to refer to rational
tunings that are more complex than these.

My 'just intonation' Dictionary entry refers to some of this.

http://www.ixpres.com/interval/dict/just.htm

Other opinions would be most welcome.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/12438

> Ezra Sims (I know you don't like his music, but I'm crazy about
> his Clarinet Quintet and Night Piece) uses 72-tET notation which
> represents a basic 37-limit JI scale as well as many more complex
> ratios when he modulates.

This is an excellent example of the folly of publicly posting on the
list "I don't like 'so and so's' music, blah, blah, blah..."

I actually listened again to Ezra Sims' CD which is out on CRI and
had an ENTIRELY different reaction. It is true that it is not the
kind of thing I would be interested in doing with alternate tunings,
PERSONALLY, and it IS somewhat related to previous "academic"
practices, but I was able to step OUT OF MY OWN SKIN for just a
milisecond, and I realized the great quality and ability that is
shown in Ezra Sims' music. 'Big mouth' will not make that same
mistake again..

> In general, I think it's OK to equate 'just' with 'rational',
> but popular usage tends to keep 'just' limited to either 5-limit
> JI, or to higher-prime but otherwise low-integer JI (i.e., Partch).
> For that reason, I often use 'rational' to refer to rational
> tunings that are more complex than these.
>

That totally "sums up" (or maybe "multiplies") the question that I
had... Thanks for the clarification!!!!!!!!!!!!!!!!

joetoo
_______ ______ __ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> http://www.egroups.com/message/tuning/12441
>
> This is an excellent example of the folly of publicly posting on
> the list "I don't like 'so and so's' music, blah, blah, blah..."
>
> I actually listened again to Ezra Sims' CD which is out on CRI and
> had an ENTIRELY different reaction. It is true that it is not the
> kind of thing I would be interested in doing with alternate
> tunings, PERSONALLY, and it IS somewhat related to previous
> "academic" practices, but I was able to step OUT OF MY OWN SKIN
> for just a milisecond, and I realized the great quality and
> ability that is shown in Ezra Sims' music. 'Big mouth' will not
> make that same mistake again..

Hmmm... this is really interesting to me, because you're not the
first person who told me that initially he didn't like Sims's
music, then after I expressed my enthusiasm, gave it another
listen and recanted that position... it's happened several times.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

http://www.egroups.com/message/tuning/12436

>
> This means, of course, that ALL FRACTIONS are "rational numbers,"
> correct??
>
> Given that fact, and given the fact that just intonation is the
> expression of small integer "rational" fractions, does that mean
that ANY tuning system with "rational" fractions IS just intonation??

Continuing this discussion just a little bit... it is true then that
musicians like La Monte Young who use ratios comprised of very high
integers believe they are composing in just intonation (??)

And, what happens in the Margo Schulter example... when the large
ratio actually approaches and is indistinguishable from a "tempered"
value... or of 12-tET itself??
______________ _______ __ __ _
Joseph Pehrson