Evaluating the error of 8:9:10:12 in 15-tet vs 20-tet

A long time ago, Igs pointed out that Blackwood sounded a lot better
in 20-tet than 15-tet. Something about the 180 cent minor whole tone
makes for a much more pleasant melodic interval than the 160 cent
minor whole tone in 15-tet.

I think then, that the reason that 20-tet sounds so much brighter than
15-tet is because 8:9:10:12 is more accurate than 20-tet, and it
causes the virtual 1/2/4 to pop out that much more strongly - hence
it's somehow "brighter" sounding. Even though you might not actually
play 8:9:10:12 all the time in Blackwood, I think that you'll probably
play it indirectly if you ever decide to use the 8->9->10 melodic
motif, which you probably will.

We also know that arpeggiating a triad can cause the VF mechanism to
fire indirectly due to priming, so there's some psychoacoustic basis
for the above statement.

So is there any metric to evaluate the error of tetrads, so that we
can compare 8:9:10:12 in 20-tet to 8:9:10:12 in 15-tet?

-Mike

On Mon, Jan 17, 2011 at 10:16 PM, Mike Battaglia <battaglia01@...> wrote:
>
> We also know that arpeggiating a triad can cause the VF mechanism to
> fire indirectly due to priming, so there's some psychoacoustic basis
> for the above statement.
>
> So is there any metric to evaluate the error of tetrads, so that we
> can compare 8:9:10:12 in 20-tet to 8:9:10:12 in 15-tet?

I also just realized that the "unfair" 25-tet version also sounds
really good, although the 8:9:10:12 tetrads start to sound so bright
that they almost lose their color. Take away the 9, though, so you're
left with 4:5:6, and the whole thing starts to sound less pleasant.

Maybe the way to find the optimal blackwood temperament is to do some
kind of least-squares optimization whereby the error of 8:9 and 2:3 is
fixed. You can fix the error of 2:1 as well to get the pure octaves
version.

-Mike