Happy Holidays Margo!

To Margo Schulter, who has brought me many gifts including a
knowledge of Vicentino and Benedetti, I'd like to offer this fractal
depiction of intervals within an octave:

/tuning/files/dyadic/vos.gif

This was inspired by the research of Joos Vos, one of whose articles
I summarized here.

It is a harmonic entropy curve.

For the probability discribution, I used the kind of curve Vos
obtained, exp(-abs(d/t)), instead of the usual normal standard one
based on exp(-d^2/t).

One may note a striking resemblance to curves easily obtained from
Sethares's dissonance algorithm (though the resemblance would sharply
drop if one were to use triads instead of intervals).

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

I think my ears like 9/7 and 12/7 a little better than this curve does.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> I think my ears like 9/7 and 12/7 a little better than this curve
does.

The curve is actually incorrect -- but while I work on a correction,
what do you think of the other harmonic entropy curves I've produced?
Please reply at

harmonic_entropy@yahoogroups.com

and you'll find some good examples in the files section of both this
list and that one.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> I think my ears like 9/7 and 12/7 a little better than this curve >
does.

You might like smaller values of t (the rate at which the
exponential "fuzz" function decays) better. I'll post an example to
harmonic_entropy@yahoogroups.com . . .

By the way the corrected curve looks almost exactly the same.

Hello, there, Paul, and thank you for a really nice "harmonic entropy"
map of one possible view of the intervals.

Here I might add that I often seem to like a range of points on the
curve, with lots of timbre-influenced effects I'm not sure how to
explain.

To give this some accentuated relevance to the focus of another forum also
on topic here, "Make Micro Music!" I want to draw the very practical
moral: Try it, and see if you like it in a given timbre.

I'd describe harmonic entropy as a kind of topographical model for
which intervals seem more "simple" or "complex" in certain usual
harmonic timbres. Just as a map doesn't attempt to predict whether a
given traveller might prefer valleys or plateaux, but simply indicates
where different elevations might be found, so I would take harmonic
entropy as a model which people can use to describe their own tastes.

For example, in a harmonic timbre suggested to me by a soundcrafter of
recognized expertise, I find that 12:14:18:21 (0-267-702-969 cents) or
14:18:21:24 (0-435-702-933 cents) is rather "smooth," while 11:14
(~417.51 cents) in a sonority close to 22:28:33 (0-418-702 cents --
actually about 0-418-704 cents) has an effect at once pleasantly
"active" and somehow also "smooth" or "resonant" in a certain way.

These are musical impressions: harmonic entropy theories I've heard
often tend to take the first kind of sonority as presenting simple
intervals or "valleys," with 11:14 regarded as more of a point on a
"plateau" or more complex area rather than a distinct "valley" or
minimum in itself.

In this timbre, a bit like a reed organ (to which I've added some
plucked string sounds, maybe a bit harpsichord-like or harp-like), the
tempered 22:28:33 has a feeling at once "exciting" and "rather sweet,"
the kind of quality I like for a place to pause in a 14th-century
style.

This is also the kind of effect I get in some nice timbres with a
usual Pythagorean 64:81 (about 408 cents), the "historically
authentic" choice.

Maybe I should explain my own view of "harmonic entropy" from a
certain musical outlook: it's simply a kind of map of how simple or
complex intervals might be expected to sound in various regions of the
spectrum. Whether something is pleasing or otherwise depends on the
style, and on the listener.

One prediction of harmonic entropy theory that does seem more or less
to fit my experience is that a "large major third" type of interval
may be at maximum complexity -- which can translate into "dissonance"
in some harmonic textures -- at around 423 cents. This is right around
17-tET, where a regular major third has a size of 6/17 octave or
423.53 cents.

I've noticed that these 17-tET major thirds may call for a bit of
sensitivity to timbre, so that they sound _relatively_ blending in a
neo-medieval kind of style where they should be unstable and at the
same time _somewhat_ "concordant." A bit of adjustment can make them
very pleasing indeed: rich and complex, and yet "compatible." This may
mean simply picking a more "pastelized" kind of texture, with gentler
definition of "peaks" and "valleys."

Anyway, Paul, I enjoy your gift both as a piece of visual art and as a
map encouraging musical exploration -- one theme of this forum.

With peace, love, and best holiday wishes,

Margo

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> For example, in a harmonic timbre suggested to me by a soundcrafter
of
> recognized expertise, I find that 12:14:18:21 (0-267-702-969 cents)
or
> 14:18:21:24 (0-435-702-933 cents) is rather "smooth,"

All the intervals in these chords are within the 9-limit, so no
surprise there . . .

> while 11:14
> (~417.51 cents) in a sonority close to 22:28:33 (0-418-702 cents --
> actually about 0-418-704 cents) has an effect at once pleasantly
> "active" and somehow also "smooth" or "resonant" in a certain way.

Well, you may note that on the new graph I just produced for you,
11:14 is a visible "valley" in the curve.

> These are musical impressions: harmonic entropy theories I've heard
> often tend to take the first kind of sonority as presenting simple
> intervals or "valleys," with 11:14 regarded as more of a point on a
> "plateau" or more complex area rather than a distinct "valley" or
> minimum in itself.

Well, this new model may be more useful to you if you find a just
11:14 to be distinct from slight mistunings of 11:14.
>
> I've noticed that these 17-tET major thirds may call for a bit of
> sensitivity to timbre, so that they sound _relatively_ blending in a
> neo-medieval kind of style where they should be unstable and at the
> same time _somewhat_ "concordant." A bit of adjustment can make them
> very pleasing indeed: rich and complex, and yet "compatible." This
may
> mean simply picking a more "pastelized" kind of texture, with
gentler
> definition of "peaks" and "valleys."

Yes, and Easley Blackwood's 17-tET etude demonstrates this very, very
nicely.

> Anyway, Paul, I enjoy your gift both as a piece of visual art

I'm working on a prettier version, which will have even more ratios
labelled. I found that the choice of the "fuzz" parameter, which I
usually set to 0.6% or 1%, seems to have no effect on the curve,
which seems to be a fractal and may, if calculated finely enough,
prove to have a local minimum at _each and every rational number_ --
one of those bizarre mathematical constructs that is a favorite theme
of recreational fractal mathematics.

I hope to post this prettier version later today.

Happy Holidays!

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> I'm working on a prettier version, which will have even more ratios
> labelled. I found that the choice of the "fuzz" parameter, which I
> usually set to 0.6% or 1%, seems to have no effect on the curve,
> which seems to be a fractal and may, if calculated finely enough,
> prove to have a local minimum at _each and every rational number_ --
> one of those bizarre mathematical constructs that is a favorite theme
> of recreational fractal mathematics.

It should if the calculation is pushed to the limit. The whole business probably requires a Stieltjes integral or something like it to make the maximum mathematical sense of it.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> I think my ears like 9/7 and 12/7 a little better than this curve
does.

Let me know if you like this curve any better:

/tuning/files/dyadic/margo.gif

This time I used an exponential decay rate t=0.6% instead of t=1%.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Let me know if you like this curve any better:

I did like it better; this time, I'll raise another issue--in my estimation, the effect of a nearby octave or unison is a little stronger; 8/7 does not strike me as nearly as consonant as 9/7.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Let me know if you like this curve any better:
>
> I did like it better; this time, I'll raise another issue--in my
>estimation, the effect of a nearby octave or unison is a little
>stronger; 8/7 does not strike me as nearly as consonant as 9/7.

First question: does 9/7 sound like a local minimum of dissonance to
you? How far out can you go before the next local minimum?

If yes to the first question, how do you explain that lots of people
think 11/9 is more consonant than 9/7? Dan Stearns for one, Vicentino
it seems for another (is this correct, Margo?)

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Let me know if you like this curve any better:
>
> I did like it better; this time, I'll raise another issue--in my
estimation, the effect of a nearby octave or unison is a little
stronger; 8/7 does not strike me as nearly as consonant as 9/7.

Do you feel that the graph should have an overall downward slope?
Have you seen the graphs that result from a Farey-limited, rather
than Tenney-limit, set of seed ratios? Check the Files section of
harmonic_entropy@yahoogroups.com -- they replace the general
horizontal trend with a downward-sloping one, but are otherwise
nearly identical.

Assuming you feel that the graph should go straight across and not
slope downward: A t value of 1.5% makes 9/7 more consonant than 8/7,
even though the latter has a much deeper valley. I'll post the graph
as an "Everyman Vos" example to

harmonic_entropy@yahoogroups.com

It would be good to continue the discussion there.

Hello, there, Paul, and after looking again at your most pleasant
holiday gift vos.gif which also marks a birthday (December 25) for me,
I can maybe explain my previous statements about 11:14 in relation to
your graphic as labeled in the first version I downloaded. The new
version, of course, makes everything quite explicit -- a wonderful
gift indeed in both versions.

Looking at the first version, I could see a couple of small "valleys"
between 4:5 and 7:9, but on the basis of the diagram alone, I would be
cautious about placing those at any specific point in the spectrum.
Once I know that one of them is at 11:14, I have no problem at all
locating one that seems to fit that location (the second and deeper of
two minor minima in the region). Your very clever refinement in the
second version of labelling the ratios for "shallower" minima or
valeys using an arrangement of the type neatly fitting these numbers
into the diagram looks like an ideal solution.

Smiling, I reflect that at least I erred in the right direction in my
response to the first version: given my enthusiasm for 11:14, it's at
least better that I "underinterpreted" a local minimum actually
indicated in your diagram rather than "overinterpreted" something that
might have been several cents off in either direction, for example. Of
course, thank you for quickly enlightening me on the actual minimum
that's there in the curve, and is now expressly identified by ratio.

What I did quickly note in that version was a local minimum for what I
was pretty sure was 11:13 (~289.21 cents), very easy to confirm in the
new version with its optimized spacing of type for the ratios.

If both 11:13 and 11:14 turn out to be distinct minima, this might fit
the classic intuition that mediants often tend to identify points of
relative simplicity between deeper "valleys."

Once I did have the experience in a harmonic timbre of seeming to hear
11:14 a bit "smoother" than the regular major third of 29-tET at about
413.79 cents -- but maybe there were some timbre-specific factors (not
necessarily common to other harmonic timbres), since at other times I
have found a 29-tET major third somewhat "milder" than a third around
11:14, or at least that was my casual impression.

As you might guess, I also got a kick out of seeing 14:17 as a local
minimum on the second version. Would harmonic entropy theory suggest
that 14:17:21 is maybe analogous to a Partchian "otonal" version, like
6:7:9, with the smaller third placed below in the more "natural" or
"harmonic" arrangement?

Also, might one make a fine distinction between "rootedness" where the
lowest note of a sonority is equivalent to an octave of the
"fundamental"; and "anchoring" which might be brought about by an
interval such as 2:3, 4:5, or 4:7 above the lowest note, as in 6:7:9
or 12:14:18:21, although the lowest note is not, strictly speaking,
"rooted" -- giving it a "rootlike" quality?

By the way, I'll maybe share a little "present" here, something less
artistically impressive than your curve and diagram, however -- but an
item regarding a certain "pet topic" of mine.

Most appreciatively, and Happy Holidays to you and all,

Margo Schulter
mschulter@value.net

P.S. I'll comment on your newest remarks and graphics today in another
message -- and that margo2.gif does look like a nice representation of
my own approach to these things.

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> As you might guess, I also got a kick out of seeing 14:17 as a local
> minimum on the second version. Would harmonic entropy theory suggest
> that 14:17:21 is maybe analogous to a Partchian "otonal" version,
like
> 6:7:9, with the smaller third placed below in the more "natural" or
> "harmonic" arrangement?

I'm not sure I understand your question . . .

> Also, might one make a fine distinction between "rootedness" where
the
> lowest note of a sonority is equivalent to an octave of the
> "fundamental"; and "anchoring" which might be brought about by an
> interval such as 2:3, 4:5, or 4:7 above the lowest note, as in 6:7:9
> or 12:14:18:21, although the lowest note is not, strictly speaking,
> "rooted" -- giving it a "rootlike" quality?

Yes, I think you asked me this once before and I replied that
anchoring is a good term for a distinct quality that is important
especially in the minor triad. We can discuss further on the
appropriate list, and you might want to look at Parncutt's book, if
you haven't already.

> By the way, I'll maybe share a little "present" here, something less
> artistically impressive than your curve and diagram, however -- but
an
> item regarding a certain "pet topic" of mine.

Looking forward to it!

Did you see margo3.jpg?

Merry Christmas!

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_31851.html#31875

>
> I'm working on a prettier version, which will have even more ratios
> labelled. I found that the choice of the "fuzz" parameter, which I
> usually set to 0.6% or 1%, seems to have no effect on the curve,
> which seems to be a fractal and may, if calculated finely enough,
> prove to have a local minimum at _each and every rational number_ --

It doesn't do anything particularly different for *PRIMES??* :)

JP