--- In tuning-math@y..., jpehrson@r... wrote:
> In arithmetic and mathematics is the *numerator* of a fraction ever
> considered "more important" than the *denominator?*

No. I don't think so. It's all completely dual.

> Or is that a silly question...?

No. Its a good question.

> It seems to me in simple
arithmetic,
> the numerator seems more "impressive..." maybe because the numbers
> are larger??

In ordinary (non-musical) usage the numerator is just as likely to be
smaller than the denominator.

> Just as in "otonal??" Hasn't the "otonal" series, on the overall,
> been considered *significantly* more important than the *utonal*
over
> the years??

Yes. But this doesn't make the numerator or denominator special. It
makes _the_smallest_of_the_two_numbers_ special. Several frequencies
having their fundamentals ocurring as if they are the harmonics of a
lower virtual fundamental, gives more consonance than several
frequencies that each have one harmonic corresponding to a higher
"guide-tone".

In the case of octave-equivalent pitches we have a convention to put
them in a form that is between 1/1 and 2/1 so they have positive
logarithms. But for non octave-equivalent pitches we can have 2/3
different from 3/2.

For intervals, octave equivalence doesn't matter. 2:3 describes
exactly the same interval as 3:2. I have argued before for a
convention of putting the small number first, as we do for "extended
ratios" such as 4:5:6. But when we want to take its logarithm (to
convert to cents) we will still enter it as 3/2, i.e. big number as
numerator, so that we are dealing with positive logarithms.

But remember these are only conventions or conveniences. The musical
specialness is in "big number versus little number", not "numerator
versus denominator".

Regards,
-- Dave Keenan