Smallness/Primeness? [was: Tonal Affini

--- In tuning@y..., Robert C Valentine wrote:
>
> > > Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
> > >
> > > --- In tuning@y..., "monz" wrote:
> > >
> > > > I would argue that the reasons *why* those particular intervals
> > > > have been chosen, all around the earth, time and time again, are
> > > > because of properties of affinity which result from the low
> > > > primeness of these intervals:
> > >
> > > I would argue that it has nothing to do with the "low primeness", but
> > > rather with the fact that these are the simplest possible ratios, the
> > > ratios that involve the smallest numbers.
> >
> >
> > Hmmm... well, I can go along with you to some extent.
> >
> > But 5 is only one more than 4, so why is there so much difference
> > and inconsistency in the way the world's musical cultures have
> > treated 5-limit intervals, when there's so much similarity and
> > consistency with the way those with factors of 2 and 3 have been
> > treated? Clearly, there is some very special difference between
> > 3-limit and 5-limit ratios, more than simply the size of the numbers.
>
> Or look at meter. 3 and 2, and their "octave equivalents, 4, 6, 8 (and
> to a lesser extent 9 and 12) dominate musics which have a metric
> orientation. The more comlicated prime rhythm musics (Balkan and Indian
> musics) tend to heirarchically arrive at those primes by summing 2's, 3's
> and 4's.
>
> Its not just that small numbers are "good", but very small numbers are
> "good". As has been pointed out, due to the definition of prime numbers,
> the population of primes is more dense in the population of small
> numbers.
>
> Bob Valentine

J Gill: It also seems that -

The business of any differences arising out of the "in-divisibility" or
"divisibility" of a number (this being the difference between primes and their
multiples) still seems murky (despite the valid "octave-equivalence" which has
been applied by some).What about pitch multiplications/divisions of the numbers
3, 5, and 7?

And what do you think of a concept of "rooted-ness" occuring where powers of the
prime 2 appear in the numerator or the denominator (which, if I have interpreted
Dave Keenan correctly, may impart a character unique due to "octave-equivalence"
effects)?

(1) IF it is true that it *is* the integers "smallness" (and not its "prime
status") which is the cogent characteristic giving rise (at least in part) to a
sort of "recognizability" (and thus, perhaps, distinct tonal "personality") in
certain (but likely not *all*) contexts (primarily related, then, to a small
numerical vaue, as opposed to whether the value is either divisible or
in-divisible); AND

(2) IF it is true that listeners (perhaps even untrained ones, to a certian
extent) are, upon hearing "complex" tones sounded in unison with a "complex"
tone at 1/1, are able to "recognize and categorize" various levels of the
pitch-heights of the secondary "complex" tone of such a "complex" dyad; AND

(3) BEATS (whether they be perceived as benefical, beautiful, or maddening, or
familiar - as Paul Erlich pointed out in relation to the "beats" created by
12-tone ET which are a familiar and *desired* spectral characteristic "absent"
in JI tunings), being a (necessary) blessing/curse in any practical system; AND

(4) We think of, speak of, and appear to (in some cases) "aurally perceive"
various possible combinations of such "complex" tones formed into dyads [dyads
as described above [in (2)] as representing a (simultaneously sounded, or
"harmonious") tonal "affinity" of sorts, etc; THEN

(5) WHY(?) would one tend to minimize the fact that:

The *lower* in value of the size of the number, the lower the harmonic number
(of a 1/1 reference pitch) at which the *first* "harmonic coincidence" takes
place, and (correspondingly) the *more* "harmonic coincidences" *occur* (per
unit bandwidth) throughout the resulting frequency spectrum...

Thus it would (to me) seem that - while this (historically widely discussed)
phenomena may well *not*, in itself, "sum-up" perceptual metaphors such as
"consonance", or "con-cordance", or "affinity", etc, is it not (at least, one
of) the most tangible and widely applicable (to multiple tones, in addition to
dyads) descriptor of "physically" (as opposed to "psychically") describable and
characterizable tonal inter-relationships between members of groups of "complex"
(fundamental plus harmonic content possible) tones in unison.

Curiously, J Gill