The joy of the 11-limit

Joe Monzo wrote:

>I think placing the higher-prime/odd ratios
>closer to the chord harmonically makes more
>sense than placing it melodically where it would
>logically go. 7 isn't melodically close to
>anything in the 4:5:6, yet it appears there.
>So perhaps this is better:
>
> 5
> / \
> / \
> / 7 \
> / \
> 4---11----6

Oh no. This would imply that 11/8 is half of 3/2. Won't do at all. How do
you distinguish the 11/8 above 4 from the 16/11 above 6?

Paul Erlich wrote:

>I have a problem with defining 13-limit intervals in 31-equal beacuse
>31-equal is only consistent through the 11-limit. This means that if you
>try to construct a harmonic series through the 13th partial on a
>31-equal instrument, you won't be able to use the best approximations of
>all 13-limit intervals that are available in 31-equal.

Okay, no problem, don't use *all* 13-limit intervals. 12 and 19 steps are
the only "large" intervals in 31-equal that aren't 11-limit consonances, and
they look more 13-limit than anything else. My simple position is that if
you have the 11-limit, the 13-limit comes bounding in on its coat tails
because of the confusion between 18/11 and 13/8. Whether this interval is
really heard as a 13/8 is a question I find meaningless. When the 13-limit
intervals get very poor, that's no problem, because you'll get good 11-limit
intervals instead.

I suppose I'd better define "large". Obviously, as the smallest non-trivial
11-limit interval is 12/11, no interval smaller than 12/11 can be in the
11-limit until we get to the unison. So 1, 2 and 3 steps in 31-equal
obviously can't be 11-limit consonances. Because of octave invariance and
some other related concept, no interval between 11/6 and an octave can be an
11-limit interval either. So, that rules out 28, 29 and 30 steps. All
other intervals within an octave are the "large" ones.

I did try 11/8 and 13/8 last night. Listening around the middle of the
keyboard and with a clean brass sound, I preferred 11/8 to 13/8, but 18/11
sounded even smoother. 18/11 seems to be at a minimum of dissonance of some
kind. I may be imagining the last bit, though. It's difficult to do blind
trials with the TX81Z when I know really that I have to press the button 5
times to get to the "right" note.

Then, I tried at different pitches with different timbres. The 13/8 seems
to be more consonant than the 11/8 when the notes get higher, or the timbres
get richer. That is, less like a pure tone. This is still using harmonic
timbres. I didn't try 18/11 with this experiment.

So, those are my opinions. They may change were I to do the same
experiments on a different day. The effects are subtle.

Graham
http://www.cix.co.uk/~gbreed/