[PD] more Fibonacci

[Charles Henry]

> I don't think 5-tone equal temperament is a substitute for the
> pentatonic scale.  The pentatonic scale is made up of decidedly uneven
> intervals.

Yes, it's true.  They're two different things.  I disagree with the
John S. Allen article that there's a connection between the pentatonic
scale, diatonic scale and fibonacci sequences.  One is a subset of the
other.  We're not get better approximations of just intonation as we
move from one to the other.  I'm saying that we get better and better
approximations, in a monotone decreasing series, as we move through
the equal tempered scales using the fibonacci sequence.  I'd trust in
the just intonation system, if it weren't so hard to modulate keys.

> Also, it is not true that all intervals are based on fifths.  Anceint
> Chinese theorists suggested this, but their anceint Chinese secret was
> that the Ch'in used intervals not derived this way, such as 8/7.
> Ptolemy and Didymos both suggested tuning thirds based on the 5th
> harmonic, 5/4, rather than Pythagoras' 81/64, which is more out of
> tune than 12-tone equal temperament.
> Islamic theorists used intervals like 18/17, 81/68, and 27/22,
> supposedly because it was simply easier to tell people where to put
> the frets that way.
> This part is more against the Jeans quote than the article, though.

The small-numbered ratios of frequencies create consonant intervals,
because of the harmonic overlap (when using a harmonic series).  These
are the local minima of dissonance as a function of interval.  I think
that what we hear in these more outlandish tuning ratios are only
approximations of the small-numbered ratios (don't ask, how small is
small-numbered? cause I really don't know...It's pretty hard to tell).

There's something strange about those mistuned harmonics that gives an
instrument its richness.  I know that Plomp studied this back in the
60s "Beats of mistuned consonances" in JASA.  When we detune two pure
tones away from a small-numbered ratio (like 3/2), there are beats,
that do not correspond to difference tones, cubic differences, or
other distortion products.  It's not a physical phenomenon, not part
of the cochlea, but something to do with the brain and how it groups
the harmonics together (could possibly be related to stochastic ghost
resonance, mentioned by Chialvo in Chaos, which has to do with the way
that harmonic ratios of impulses sync up along single neurons)

What is it with people and their morbid fascination with fibonacci sequences?

Chuck