Reply to Dave Keenan

>In another thread there was discussion of whether something was a
45/32. Now this may be significant in its >relationship with other notes
of the scale but when played with the 1/1 it is surely more significant
that this is 7/5 >+7.7c.

Dave, I strongly believe this 45/32 is only significant in its
relationship with the other notes in the scale. It is a 5:4 above the
9/8. Played alone wih the 1/1, it certainly evokes a 7/5 harmonically.
It also has purely melodic significance which doesn't depend as much on
tuning.

>I understand there is a kind of "uncertainty principle" built into
Fourier (or Laplace or wavelet) analysis and >presumably also into
whatever the ear+brain does. You can't give a definite frequency to any
wave that you only >sample for a finite duration. The shorter it lasts,
the greater the uncertainty in its frequency. delta_f = 1/delta_t. So
>if a chord only lasts for a second it's impossible to say whether it
had zero beats. It might have had a beat slower >than one per second and
you wouldn't (couldn't?) notice it.

Absolutely. I've even read that the ear+brain will adjust its sampling
duration depending on the type of music (or speech) being played, in
order to extract more information than a fixed sampling duration would
allow.

>For example, Jazz appears to me to use 12-tET as if it were
approximating ratios of 7 (among others). 12-tET is >very poor at this
(up to 31 cent errors) but long sustained chords are notably absent from
Jazz, so it "works". I'm >sure people can train themselves to recognise
more complex ratios, but there will still be some limit on the
>complexity, and I see no way around requiring a longer time to do the
recognition.

This is why LaMonte Young keeps his high-limit chords going for minutes
or hours! (Not that I totally disagree with your skepticism of the very
high numbers he uses.)

Dave -- those chords are two-dimensional if you adopt a 7-limit standard of
consonance. The 7:9 and 5:9 interval classes are, I feel, noticeably more
dissonant than 4:7 and 6:7. So it's meaningful to draw the line between
consonance and dissonance between the ratios of 7 and ratios of 9 if one wishes.
Of course, it could be just as meaningful to adopt a 9-limit standard of
consonance, if a particular musical piece, composer, or style warrants it.

So I misunderstood you on the tetrahedron issue. Thanks for clarifying.