On 2 May 2010 11:39, Carl Lumma <carl@...
> Oh right; obviously we're constrained to the prime limit of the
> intervals we're mapping. So I withdrawal that criticism.
We need to choose some kind of limit, anyway.
>> There's also the problem that, for example, 81/80 is comparable
>> to 81*80. But 81/80 is clearly audible whereas 81*80 covers
>> over 12 octaves and even good human hearing is only about 10
>> octaves (16 Hz to 16.4 kHz).
> 81*80 is way beyond where Tenney height is claimed to work.
> The highest 9-odd-limit interval is, what, 14*9 = 126. That
> would be the upper limit of where I'd claim it worked.
> 128:1 is 7 octaves, which is audible.
You don't get anything very close to Tenney weighting until the
intervals get stupidly complex. Before that lower primes are given
systematically higher weights. But that assumes equal weighting of
intervals within a Tenney limit is what you wanted. If you don't know
exactly what you want Tenney weighting's a pretty good bet.
>> In real music, even ignoring aesthetic preferences, most
>> intervals will be small because they occur between notes in
>> chords. That makes them more important.
> If you go to the orchestra you'll hear notes all over the place.
> It's known that very large intervals are less important because
> less of their spectra overlap. I suppose you could say that
> Tenney weighting doesn't account for this. But we're regular-
> mapping so we need prime limits (though I suppose we could
> consider other fields that generate the rationals). So the best
> thing to do is bend them to be psychoacoustically relevant.
> I don't know of anything better than Tenney weighting for this.
Notes all over the place means the majority of intervals will be
small. And, yes, the spectra overlap less. So I count 14:9 as much
more important than 126:1. The good thing about octave stretching is
that we can ignore this issue. With TOP-Max you get the same solution
for any set of intervals. With TOP-RMS you're optimizing the largest
intervals but the symmetry means it doesn't matter.
> I was just bellyaching that you spent so much ink on pure-octave
That must be primerr.pdf because I didn't consider pure octaves with
the composite errors.