RE: [tuning] "N" again(??)

Joseph wrote,

>I'm having a little trouble with the value "N" of the Farey series...
>Is it kind of like a "resolution limit" of the ratios being
>considered??

Good morning, Joseph!

It's a limit on the ratios being considered (in the Farey case, it means
that neither numerator nor denominator can exceed N), but doesn't really
have anything to do with "resolution".

>And, if so, how do you derive the "N number" limit from
>the ratios?? Some easy, hopefully, examples might help...

N is an input, not an output.

Let me know if this makes sense now.

Anyway, Manuel just sent me all the harmonic entropy curves in the first
octave from N=80 up to N=405, using s=1% and s=1.2% (thanks, Manuel!). For
the s=1% case, the N=80 curve shows local minima at 264, 316, 386, 438, 498,
582, 622, 702, 814, 884, 970, 1020, and 1052 (7:6, 6:5, 5:4, 9:7, 4:3, 7:5,
10:7, 3:2, 8:5, 5:3, 7:4, 9:5, and 11:6), while the N=405 curve shows local
minima at 276, 320, 388, 500, 589, 634, 822, 886, 702, and 972 (7:6, 6:5,
5:4, 9:7, 4:3, 7:5, 10:7, 3:2, 8:5, 5:3, and 7:4). A look at the curves
gives you a better sense of what's going on. I uploaded a graph that
includes all of them to
http://www.egroups.com/files/tuning/perlich/manuel.jpg -- the lowest curve
is N=80, and the highest is N=405. They have basically the same shape,
sloping downward overall, but the N=405 curve has considerably shallower
dips than the N=80 curve. Because of the slope, some of the dips, though
visible, don't turn up as local minima at all in their shallower form (9:5
falls into this category), and (again because of the slope) for all the
dips, a point noticeably to the right of the center of the dip turns up as
the local minimum when the dips are in their shallower form (hence the
slight "stretching" of the cents vales for the N=400 minima -- even the
shallower dips for N=80, such as 10:7, show this right-shifted effect).

One valuable goal would be to use a series other than the Farey series in
order to decrease this overall downward slope, and hence the right-shifting
and skipping of some shallow dips when looking only at local minima. I've
proposed a numerator times denominator limit for other reasons, and it would
help in this area as well. Unfortunately, it's harder to compute such a
series compared with computing a Farey series, but perhaps between Manuel
and myself we can tackle this.

The shallowing of the dips as N increases is not something I anticipated in
my original posts, which you've read. I would now conjecture that as N
approaches infinity, the depth of the dips approaches zero. This means that
for harmonic entropy to yield something useful, we need to set a value for
N. This unfortunately means the harmonic entropy function has two free
parameters, N and s (not to mention the choice of whether N defines a limit
according to a Farey series or some other kind of series). Choosing an N
means, how high up the harmonic series could we recognize intervals if we
had perfect hearing resolution? That's a very difficult question to settle
experimentally, since we don't have perfect hearing resolution. Given a
realistic estimate of the hearing resolution, such as 1%, the fact that the
curves for N=80 and N=405 are so qualitatively similar that is seems highly
unlikely that any psychological experiment could determine which curve is
closer to the truth and which N is closer to the "true" N.

Although the arbitrary parameters are ugly, what is nice is that, for a
given s, the dips in the curve are pretty much the same for such a wide
range of N values. For s=1%, you pretty much see dips for all ratios with a
denominator (i.e., smaller number) no more than 6, maybe 7, but certainly
not 8 or more. That's a pretty powerful result. It's saying, no matter how
high up the harmonic series our brain might be able to recognize
conceptually, a hearing resolution of 1% limits the set of "special" ratios
to those whose smaller number is less than 8 (where special means something
like "easier to recognize than its immediate neighbors"). Now certain ideal
listeners have shown s=0.6% in ideal registers, and having rich timbres and
more than two notes at a time might effectively further decrease s, but
whatever s is, we can find an inherent limit to the complexity of the
intervals that could be recognized as "special".

As a denominator limit like this implies an odd limit if you require all the
octave inversions and extensions of a consonant interval to be "special" in
this sense, this kind of calculation could determine how far the odd limit
concept can realistically be taken. I believe that in the context of otonal
chords, Partch's extension of the odd limit from 5 (or 9, as Partch says) to
11 is not unrealistic; however, for utonal chords, the dyadic harmonic
entropy probably says it all, so the denominator limit of 7 found above
might be the farthest one can realistically go if one expects most listeners
to be able to "follow along". . . .

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> I uploaded a graph that
> includes all of them to
>http://www.egroups.com/files/tuning/perlich/manuel.jpg -- the
>lowestcurve is N=80, and the highest is N=405. They have basically
>the same shape, sloping downward overall, but the N=405 curve has
>considerably shallower dips than the N=80 curve. Because of the
>slope, some of the dips, though visible, don't turn up as local
>minima at all in their shallower form (9:5 falls into this
>category), and (again because of the slope) for all the dips, a point
>noticeably to the right of the center of the dip turns up as the
>local minimum when the dips are in their shallower form (hence the
>slight "stretching" of the cents vales for the N=400 minima -- even
>the shallower dips for N=80, such as 10:7, show this right-shifted
>effect).
>

Well, this is certainly a beautiful graph. It looks a little like a
tapestry gone mad... Clearly though, as you mention, all the
different values for "N" still give much the same overall picture.

If I'm understanding the concept of harmonic entropy at all, when
higher values of N are used... i.e. larger numbers for the ratios and
more included pitches, there is not so much "entropy," since there is
more choice and the smaller values don't "stand out" so much as when
there is less choice and N is smaller (??)

And, characteristically for this list, we didn't have to look too
far... John deLaubenfels wrote:

>OK, I'll bite, Paul. I'm a programmer, and potentially clever enough,
>and I have a native C++ compiler with no arbitrary limitations such
>as you seem to have to put up with. Could you send me some
>quasi-readable source code for the binary calculations you do, that
>I could port over to C++...

This is really great news! I hope this works out, since I personally
believe this experiment is one of the most exciting things I have
seen on the tuning list in the year I have been here!! It's like
creating a theory of concordance out of nothing! What a realm of
sound to explore!!

But, it will be complete if I could kindly post some of the results
as AUDIBLE .mp3 files on the little "Tuning Lab" site... I'm anxious
to do that...

I'm following this with great interest!
_________ ______ ___ __ __ _
Joseph Pehrson