Model 3: Linear HE

The last model I posted made use of Minkowski's ?(x) function in order
to assign each generator a scalar value representing how distinct, on
average, its intervals are. This came from the basic result, which is
in accordance with the initial desiderata I posed, that the best
rank-2 systems for this criteria are those which avoid the generator
being close to a simple rational division of the period. Scales which
fail to do so inevitably end up generating a ton of very small
comma-sized intervals, and hence will also generate a lot of MOS's
with nearly-equal intervals that differ only by these comma-sized
intervals (e.g. in the role of c). To reiterate, I suggest that
intervals which differ only by a very small amount are more difficult
to tell apart than intervals which differ by a larger amount, and
hence scales like this will generate a lot of very close, ambiguous
intervals.

As a second pass in modeling this same feature, we might resurrect our
old friend, the Harmonic Entropy curve: much like ?(x), this also
allows rationals to compete with one another, weighted by their
complexity, and allows us to find the spots that are maximally close
and far from them. However, this time, there's a twist: since we're
trying to avoid simple fractions of the period, the ratios we're going
to use will now be in -linear- space, not logarithmic space.

Note that in this case, we're not saying that there's some process in
the brain that matches intervals to equal divisions of the period.
However, we are saying that a listener won't be able to distinguish
between intervals that are really close in size, such as 399 and 402
cents, and so will tend to perceive those intervals as being the same
interval, or perhaps different intonations of the same interval. And,
we're saying that as a side effect of this, a scale generated by a 399
cent interval will tend to produce lots of these ambiguous intervals,
and that this effect simply happens more as the generators get close
to simple rational divisions of the period. The "fields of attraction"
here could perhaps instead to be thought of as "fields of repulsion" -
things we want to avoid - and hence we'll look for maxima in the
resulting curve.

Let's make sure that HE is at least modeling the cursory aspects of
the behavior of MOS scales and ambiguous intervals correctly in ways
that agree with the other plots I've posted.

Firstly, we want to make sure that simple rationals are worse to be
near than complex ones. For instance, a generator that's 585 cents
will tend to generate a bunch of small 15-cent intervals early in its
MOS series, but a generator that's 285 cents will generate small
intervals of 60-cents early in its MOS series, which is four times
better. HE does this by assigning simpler rationals a larger field of
attraction.

Secondly, for the simple rationals, we want to make sure that
generators get a worse rating if they get closer to a rational number,
because the ambiguous comma-sized intervals produced, which will be
chromata for the MOS's preceding their appearance, will be even
smaller. HE obviously does that.

Thirdly, it's probably likely that complex rationals don't really
matter at all, because you'll have to use really large MOS's before
you start seeing any sign of ambiguity whatsoever. For instance, we
probably don't care at all if the generator is tuned to something like
29/50ths of a period, because we won't start seeing any c-ambiguity
until we get to the 50-note MOS, nor any s-ambiguity until we get to
the MOS that's one step larger (either 69 or 81 notes), nor any
d-ambiguity until we get to the MOS that's one step smaller (31
notes). If we note that the denominator thus reflects the largest
effective MOS size we care about - or the largest equal division of
the period we care about avoiding, which is more or less equivalent -
HE allows us to model this by seeding the rationals with a Farey
series, with the N cutoff bounding the complexity of the things we
care about.

Fourthly, in order to make any claim that we're representing some
MOS-agnostic property of generators themselves, it would be good to
make sure that the curve converges as N goes up. HE mostly does this,
but keeps increasing in height; if we can normalize this by dividing
by the Harmonic Renyi Entropy of order 0 it can be shown empirically
to converge.

Fifthly, we want to model the variation that likely exists in
listeners' abilities to distinguish between similarly-sized melodic
intervals. For instance, one person may find intervals that differ by
~30 cents to be very ambiguous, particularly in a melodic context, and
tend to perhaps perceive them as different intonations of the same
interval, whereas another person might find it pretty easy to
distinguish between them as unique and totally different things. We
also might want to model any contribution to this variation that
timbre might provide. HE attempts to do all of this by smearing
everything out with a Gaussian that takes its standard deviation as a
free parameter. (You might argue that smearing things out cents-wise
is a bad idea, but it's a start for now.)

So for now, it all seems decently solid. I'll be posting some plots in
a minute to show how it looks.

-Mike

Rather than post plots, I just updated my online HE app to let you
play around with linear HE. You can find it here:

http://www.mikebattagliamusic.com/HE-JS/HE.html

Here's a rather crammed plot showing the settings I used:

/tuning/files/MikeBattaglia/HElins=0.6%.png

I set a=2 for no other reason than that it very closely resembles
exp(HE) with a=1, and hence makes it easier to see maxima. I also set
N=100, but if you want you can try bringing it up to N=2000 if you
want to verify that it appears to converge. The choice of s is
non-trivial, and as I basically have no real data to see what value
might be best, I just set it to 0.6% as a starting point in the
picture above. This is about 7 or so cents.

Using the above settings, maxima pop out at (in no order)

737 cents - father, in about 13-EDO
702 cents - pythagorean
672 cents - mavila, in around 25-EDO
422 cents - squares
378 cents - magic, in about 16-EDO or 19-EDO
320 cents - keemun/orgone, in about 15-EDO
280 cents - a very sharp orwell, in about 17-EDO
257 cents - semaphore, in about 14-EDO
220 cents - machine, in about 11-EDO
185 cents - glacial, in about 13-EDO
159 cents - porcupine, in about 15-EDO

Although the generators have been shifted by a few cents, which has
hence led to some different labels (like meantone being replaced by
pythagorean), many of the same sorts of generators are popping up as
before: something which creates the diatonic scale, something which
creates the superdiatonic scale, something which creates father, etc.

A sensi-ish and dicot-ish generator didn't pop up this time, but if
you change s to 0.4% to make it finer, the peak at 422 cents splits
into two peaks at 415 and 441 cents, and the peak at 378 cents splits
into two peaks at 384 cents and 353 cents. Additionally, the 702 cent
peak splits into two at 707 and 697 cents, and the peak at 320 cents
splits into 333 and 315 cent peaks.

It's hard to make any real predictions from this without any idea on
how to set s, so these results must be treated as preliminary and are
just to investigate the concept. While it's hard to use this model to
make any claim about the superiority of any generator, it's noteworthy
that many of the same sorts of generators keep turning up, although
the generator sizes may change by a few cents and the relative
ordering may change: we get things generating the ubiquitous diatonic
scale, something that's close to father, sometimes sensi, and orgone,
orwell, semaphore, machine, glacial, porcupine, etc. This shouldn't be
too much of a shock, because if we set up models of the sort I've
proposed here, we'll typically see phi-based generators returning,
especially the ones in which the L/s ratios of their MOS's hit phi
quickly on in the series.

Play around and have fun with it. Next time we'll try to get past some
of this confusion by tightening things up a bit more.

-Mike

On Thu, Mar 29, 2012 at 4:50 PM, Mike Battaglia <battaglia01@...> wrote:
>
> So for now, it all seems decently solid. I'll be posting some plots in
> a minute to show how it looks.