Prime/Odd and harmonic complexity

After describing a list of ratios between 399.1 and 415.7 cents,
with their prime limits, Paul Erlich wrote:

> If you have a way of tuning intervals with 0.1 cent
> precision, hold one note constant and sweep the other one
> through this range. Without looking, stop the sweep where you
> hear 3-ness. Do the same for 7-ness,...

As I stated in a post a few weeks ago, I think you're missing
the point here, Paul. I agree with you totally here: to my
ears, it's difficult if not _impossible_ to isolate _exact_
prime qualities in a _dyad_. But add some more notes, and it
becomes fairly easy to compare the qualities heard in comparing
various intervals.

To my ears, fairly subtle pitch shadings can be distinguished
in a complex harmonic environment, and the only thing to which
I can attribute this ability to distinguish, is prime
factorization. I agree with what most subscribers have said
(and you have said that you can agree with this too), that in
a _very general_ sense, humanity has historically
expanded its familiarity with prime qualities in music in an
evolving manner, certainly beginning with prime-base 2, then
continuing with 3, much later 5, and more recently 7 and higher
primes. But these prime qualities don't mean much if you're
talking about a harmonic context of less than 3 notes.

It seems that, again in a general sense, people's aural
sensibilities reach a point where they're ready to embrace
the particular quality exhibited by a "new" prime. Recognizing
the unique quality of 2 gives us a scale of octaves -- not
very useful musically. Stacking powers of 3 gives a scale
of 12 obviously different notes, and if continued, new notes
which are only a small distance (a "Pythagorean Comma" or
23.46 cents) away from those 12. Of course, since 9=3^2, this
3-Limit scale gives exact harmonic 9-identities.

Prime-base 5 became accepted as a harmonic interval because
it gave a new sound to the musical interval of a "3rd", a
sound which was "sweeter", "softer", "more mellow", but at
any rate very different, from the sound of the 81/64 or
3-Limit "3rd". In expanding pitch resources, rather than
create new scales by stacking powers of 5, a few powers of 5
were used as bases from which were created other new scales
by stacking powers of 3.

At this point in history (today), I don't think anyone will dispute
that 7 is implied very strongly in most new tonal music in our
culture. This is because of a similar process of prime-base
7 giving a harmonic "minor 7th" which is much more consonant,
in contexts where the minor 7th is used in tbe manner of a
consonance (as in the blues). Certainly, there is still a
place, even in contemporary music, for 3- and 5-limit minor 7ths
and minor 3rds, but just as certainly, 7/4 and the "septimal minor
3rd" 7/6 have been enthusiastically accepted.

Harry Partch put 11 out there, Jon Catler's using 13, Ben Johnston
has gone as far as 31, Ezra Sims implies up to 37, and La Monte
Young's current Dream House goes up to 283, but alas, all this
music is so little known...

I'm coming to think more and more that this prime-factor
interplay is most expressive when it clashes against 12-Eq
accompaniment. Perhaps this is why I'm so attracted to
certain blues performers.

Robert Palmer, in his book "Deep Blues", characterizes the
"deep blues" of the title as that blues style deriving from the
(Mississippi) "Delta" (the area south of Memphis). Palmer
emphasizes again and again throughout the book that one of the
primary ingredients that sets this style apart from other blues
styles is its best performers's (Robert Johnson, Muddy Waters,
Otis Rush) subtle use of microtonality in the high bottleneck
guitar lines and especially in the vocals.

What's important to remember is that the rest of this music,
meaning the strummed guitar notes, the bass lines, and piano if
present, would be in 12-Eq (or something very close, depending on
how well-tuned the instruments were). This clash also figures
importantly in the style of Hendrix, and of course, is at the
bottom of the dispute to Schoenberg's claims of the 12-Eq scale's
implied ratios.

I'm beginning to have a lot more respect for Johnny Reinhard's
concept of "polymicrotonality", partly for this reason (and partly
because Reinhard's own compositions exploit the idea so
effectively, regardless of what Anthony Tommasini thinks).

The point, as far as a response to Erlich, is that _complexity_
of harmonic resources gives lots of intervals, odd numbers, and
primes to compare, and I think that as the harmonic resources
become more complex, we rely more and more on prime factorization
to delineate the differences, to "put together what goes
together".

(I'd really like to see more feedback about microtonality in
the blues.)

Joseph L. Monzo
monz@juno.com

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