--- In harmonic_entropy@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In harmonic_entropy@y..., "gdsecor" <gdsecor@y...> wrote:
> >
> > Well, hello there! When will dinner be served?
> >
> > --George
>
> i take it you found the "margo" or vos-based curve more appealing?
> that was your "appetizer" -- the chef would like to know how you
> enjoyed it. take some time to examine it and to reflect on how
> well/badly its particulars suit you -- this will help our kitchen
> staff select the right tools for the next course . . .

Ah, yes! I was so busy reading the menu that I hadn't noticed that
it had arrived on my table:

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote [main
tuning list, #38302]:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > > > I calculated numerical values for the peaks of consonance as
1/(m*n)
> > > > for each ratio m/n.
> > >
> > > harmonic entropy, for the simplest ratios, turns out to go very
much
> > > as tenney harmonic distance = log(n*d), so we have very similar
> > > measures in some respects.

While different quantitatively, it does have the same qualitative
result in that ratios will occur in the same order from simplest to
more complex, or most-to-least consonant.

> > > > Where I found specific ratios to conincide with
> > > > what I perceived as points of minimum local consonance, I
used the
> > > > same formula.
> > >
> > > harmonic entropy predicts that the formula breaks down, most
likely
> > > already when m*n = 100.
> >
> > I did observe a neutral third (11/9, m*n=99) to be a local
maximum,
> > however.
>
> do you mean a local maximum of consonance?

Yes. (Sorry, but I keep forgetting that we're looking at this from
opposite directions.)

> i can tune 17:13 by ear, but i don't think that makes it consonant,
> or even necessarily locally consonant.

If you can hear the partials and eliminate the beats, then I would
consider that a reason for calling 17:13 a local consonance -- not
very consonant, mind you, but slightly more consonant than for its
mistunings over a very, very small distance.

However, in my own experiments involving tones less rich in harmonics
than unfiltered sawtooth waves, I didn't identify this as a local
maximum consonance, but rather as a point of local maximum
dissonance. So perhaps we are in agreement about this particular one.

> i do have an alternate set of
> curves (equally simple to compute) that show many, if not all,
> rational numbers at a local minimum of dissonance, for example:
>
>
http://groups.yahoo.com/group/harmonic_entropy/files/dyadic/margo2.gif

That looks more like what I had in mind. (More comments below.)

> > Still, this would indicate that it is not quite as
> > dissonant as the product of the integers would lead us to believe.
>
> exactly. back to my original, default, "smooth" harmonic entropy
> curve formulation, these graphs point that fact out in vivid detail:
>
> http://groups.yahoo.com/group/tuning/files/perlich/stearns.jpg
> http://groups.yahoo.com/group/tuning/files/perlich/stearns2.jpg
> http://groups.yahoo.com/group/tuning/files/perlich/stearns4.jpg
>
>
http://groups.yahoo.com/group/harmonic_entropy/files/dyadic/default.gi
f
>
> > Something that I am suspicious about in your graph is that it
doesn't
> > shown enough local points of maximum consonance. Between 5:4 and
> > 4:3, for example, I see a local minimum consonance at 448 cents
> > without any indication that 9:7 is more consonant than a
mistuning,
> > something about which I had no doubt in my observations. (The
same
> > could be said for 10:7.) Now when it comes to an interval such
as
> > 11:9 (or even 11:8), the difference between it and a mistuning is
> > very subtle -- my graph exaggerates the difference (reflecting
the
> > fact that you can hear a lot more under laboratory conditions
than
> > you can in a musical performance) -- and I was completely unable
to
> > hear 16:13 or 17:14 as such -- but rather as points of local
minimum
> > consonance.
>
> > Perhaps there is not enough resolution in the vertical direction
in
> > your graph -- 7:6 and 9:5 barely show up as consonances -- and a
> > higher resolution might make a difference for 8:7 and 9:7. (A
bump
> > in the road isn't going to show up on a topographical map, but I
sure
> > feel it when I drive my car over it.)
>
> it sounds like you'd probably prefer the "pointy" formulation as in
> margo2.gif above -- let me know.

Yes, this is very much what I was looking for.

> there's also the s (resolution)
> parameter which can always be tweaked to give more or less
importance
> to more complex ratios (the better your hearing resolution, the
more
> easily you can identify the complex ratios "as such", because the
> complex ratios are in more "crowded" areas amongst all the ratios).

For that we can refer to my graph:

http://groups.yahoo.com/group/tuning-math/files/secor/consonce.gif

It appears that the point at which I no longer heard n:d as a local
consonance is when n*d reached a value around 150 (with 16:9 and
17:10). So margo2.gif would have a little more sensitivity than what
I observed.

> > > > I recently did some experiments with chord progressions
involving
> > > > suspensions in which two tones a small "semitone" apart
resolve to a
> > > > minor or subminor third. After trying different sizes of
semitones
> > > > in various keys, I came to the conclusion that 70 cents
(~24:25) is
> > > > very close (closer than, say, 63 cents, ~27:28) to the most
> > > > harmonically dissonant semitone.
> > >
> > > isn't this *highly* dependent on register and timbre, though?
maybe
> > > on loudness as well?
>
> > With my most recent experiments involving suspensions (to find
the
> > most dissonant semitone) I varied the timbres and registers
somewhat
> > in reaching my conclusion. I found that there is more to
dissonance
> > that just the beat rate -- there is also a psychological
perception
> > conveyed by a semitone of a given size that conveys an impression
of
> > dissonance that seems to be less influenced by (if not
independent
> > of) the register in which it is heard. In connection with this,
> > consider our perception in comparing two major thirds, for
example,
> > mistuned by the same number of cents, one wide and the other
narrow.
> > There is a somewhat different psychological perception associated
> > with each that is a function of size (but independent of beating)
> > that will influence our perception of dissonance. (Might this be
> > that an interval's melodic properties also come into play when
the
> > interval is heard harmonically?) I believe that something of
this
> > sort played a part in my conclusion that the 70-cent semitone is
the
> > most dissonant.
>
> well, can you give a sense of *how much*, if at all, this
conclusion
> is dependent on timbre and register?

In running my experiments I avoided the unusual and the extremes of
timbre and register, inasmuch as I thought that these would be of
less value than the usual and customary.

My timbres contained both odd and even harmonics, and I varied the
amount of filtering of the upper partials. This didn't seem to make
much of a difference in judging the most dissonant semitone, since
the source of the most obvious beating is between the fundamentals of
the two tones.

I avoided extremes in register, staying within the two octaves or so
centered around the F above middle C. This is where we customarily
hear chords in closed position.

As I said in another place, my experimentation was not so exhaustive
as to try to pin down an exact value, say 70 as opposed to 68 cents,
but it was enough to establish that value in preference to 63 cents.

I have recently had an extensive discussion with Margo Schulter (for
whom I would presume margo2.gif was made) about what is the most
effective size of semitone for melody, in addition to the most
dissonant (and therefore effective) for harmony. She is in general
agreement with me in advocating a value of 70+-10 cents for both of
these. (And she insisted that this optimal range should include
28:27, ~63 cents.)

I trust that this should give the chef enough information to work
with to prepare the main course.

--George