Re: Entrobreed

>>Maybe the answer is that consonance and dissonance aren't mutually
>>exclusive opposites. For example: in the case of large dyads, maybe we
>>should say that the dissonance decreases, and so does the consonance!
>
>Exactly the thoughts I had to on reading Graham Breed's post. You could
>post this to the list.

I posted it right after I sent it to you. I also posted my conclusions on
the lattice metric stuff, which I am feeling pretty good about.

>Except I don't remember Paul Erlich saying that consonance decreased, only
>that dissonance did.

I don't remember what Paul said about this. But Graham's post seemed to
indicate that they consider dissonance to _increase_ with span.

I proposed that maybe they meant that consonance decreases with span, as
dissonance is obviously not increasing. Follow?

>Problem is, this complicates things, perhaps needlessly. I *can* convince
>myself to think of wide dyads as highly consonant. Maybe Graham can too.
>Anyway this confirms my resolve to only talk about DISSONANCE (for now),
>since we at least agree that wide dyads have low dissonance. Perhaps we
>could add another dimension at right-angles and call it DEGREE OF
>INTERACTION.

Hmm. The way I read it, Graham and Erlich were saying wide dyads had high
dissonance. This does not fit the definition of dissonance that we've been
using. So I suggested that maybe they meant wide dyads had low consonance.

I'm not sure what you mean by DEGREE OF INTERACTION, and I'm not sure my
distinction between con and dis is a needless complication. I could see
consonance as a measure of ease of tuning by ear and dissonance as the
subjective "restlessness" of an interval. Something like...

---------------------------------------------------------------------
name width (cents) consonance dissonance
---------------------------------------------------------------------
small dyads 15-200 mostly low mostly high
mediam dyads 270-1900 varies varies
large dyads over 2000 mostly low mostly low
---------------------------------------------------------------------

For small and medium sized dyads, they ought to be reciprocal. But they
both ought to get small for wide intervals.

>I said it, and Paul E. accepted it. As a COMPLEXITY measure it goes on
>indefinitely (but no-one can directly hear COMPLEXITY), but as a DISSONANCE
>measure it fails after that. It's purely because when the numbers get that
>high you are too close to other lower numbered ratios and TOLERANCE comes
>into play. All COMPLEXITY measures run into TOLERANCE eventually, it's just
>that n+d seems to have a particularly simple rule for *when*.

Thanks! I do recognize the need for tolerance. Harmonic entropy seems to
be the ideal tolerance metric, and maybe the ideal measure of what I call
consonance above. Sethares' roughness stuff ought to take care of what I
call dissonance.

I usually recognize the 19-limit as a safe and effective tolerance cutoff,
and I am surprised that (n+d) would fail around the 9-limit.

>This is another reason why 15/4 vs 12/5 is not a clearcut counterexample to
>n*d COMPLEXITY. Pauls original objection was to an octave-equivalent
>version of n*d (which no-one was proposing) where 15/8 vs 6/5 was a
>clearcut counterexample. But I'm not interested in octave-equivalent
>complexity. Except perhaps partial octave equivalence via the prime
>weighting scheme I proposed, however that may be unnecessary.

Thanks. I'm down with this.

Just so we're cool, what exactly do we mean by complexity? I'd like to
define it as a do-in-your-head measure that represents both roughness
(dissonance) and tonalness (consonance) vectors. What do you think?

On top of this basic definition, I think it ought to be octave-specific and
as timbre-invariant as possible (working well enough on all generic
harmonic timbres). I'd also like to have it as frequency-independent as
possible. Lastly, I don't think it should include tolerance. I think
tolerance is better treated afterwards with cents detuning or harmonic
entropy. I do like the 19-limit cutoff very much, and so I'm not worried
if the thing goes nuts with bigger numbers than 19 (that is, why would
anybody feed it such a number?).

Carl

[Carl Lumma, in response to a message of mine that didn't make it to the
list]:
>I'm not sure what you mean by DEGREE OF INTERACTION, and I'm not sure my
>distinction between con and dis is a needless complication. I could see
>consonance as a measure of ease of tuning by ear and dissonance as the
>subjective "restlessness" of an interval. Something like...
>
>---------------------------------------------------------------------
> name width (cents) consonance dissonance
>---------------------------------------------------------------------
>small dyads 15-200 mostly low mostly high
>mediam dyads 270-1900 varies varies
>large dyads over 2000 mostly low mostly low
>---------------------------------------------------------------------
>
>For small and medium sized dyads, they ought to be reciprocal. But they
>both ought to get small for wide intervals.

Yes. I could accept those definitions. What consonance and dissonance would
you give to silence and to white-noise.

>Thanks! I do recognize the need for tolerance. Harmonic entropy seems to
>be the ideal tolerance metric, and maybe the ideal measure of what I call
>consonance above. Sethares' roughness stuff ought to take care of what I
>call dissonance.

Don't get too carried away with the "ideal" bit. How many such measures
have you compared? I'm working on a simpler way of imposing TOLERANCE on
COMPLEXITY.

>I usually recognize the 19-limit as a safe and effective tolerance cutoff,
>and I am surprised that (n+d) would fail around the 9-limit.

You mean "...would start to fail after the 9-limit"? Remember this is dyads
only, not chords. How many 19 limit dyads can you recognise as distinct
from nearby simpler ratios?

>Just so we're cool, what exactly do we mean by complexity? I'd like to
>define it as a do-in-your-head measure that represents both roughness
>(dissonance) and tonalness (consonance) vectors. What do you think?

That sounds ok, but I'd say it is any function that is defined only for
ratios that attempts to represent their relative dissonance when the
numbers are small enough to avoid TOLERANCE effects and close enough to
avoid SPAN effects. It doesn't have to be do-in-your-head to be called
COMPLEXITY, but of course we'd like it to be.

Paul seems to me to be saying that dissonance/consonance has two
components: roughness/non-roughness and non-tonalness/tonalness. I don't
think it is safe to use consonance as a synonym for tonalness etc.

>On top of this basic definition, I think it ought to be octave-specific and
>as timbre-invariant as possible (working well enough on all generic
>harmonic timbres).

True.

>I'd also like to have it as frequency-independent as
>possible.

This is the one I'm stuck on at the moment. Waiting for Dan and Paul to
sort it out.

>Lastly, I don't think it should include tolerance. I think
>tolerance is better treated afterwards with cents detuning or harmonic
>entropy.

True.

>I do like the 19-limit cutoff very much, and so I'm not worried
>if the thing goes nuts with bigger numbers than 19 (that is, why would
>anybody feed it such a number?).

Well, it should preferably have the property that no matter where you stop
with the ratios (as long as it's high enough), it gives the same shaped
curve after tolerance is applied.

Regards,
-- Dave Keenan
http://dkeenan.com