complexity formulae, "fuzzy" notes

[Mulkers:]
> What do you exactly mean with an harmonic
> entropy minima?.

Refer to Paul Erlich's postings on Harmonic Entropy,
archived at my website**:
http://www.ixpres.com/interval/td/entropy.htm

It seeks to mathematically model a component of
dissonance in dyads which is based on the periodicity
aspect of our pitch-perception, showing that
simple-integer ratios are the ones most likely
to be implied or perceived in hearing intervals
(which may actually be different, and not rational
at all), and confirming Partch's Observation One
regarding Field of Attraction. Erlich derived it
from the work of van Eck.

Minima refer to areas along the "octave"
pitch-space where harmonic entropy is at a low
point, meaning that there is one interval most
likely to be perceived as the heard interval.
The area around 3:2 is a good example.

The maxima, conversely, refer to areas along
the "octave" pitch-space where many ratios of
roughly the same numerical complexity all have
about the same probability of being perceived
as the heard interval. The area around 16:15
is a good example, because all the superparticular
ratios between approximately 21:10 and 15:14
(I'm totally guestimating here, but you get the idea)
sound roughly the same and could be mistaken for
one another (disregarding any other cues given
by any musical context).

These high and low points occur because the complexity
of the numbers in ratios varies within the "octave"
in an uneven way, with smallest numbers having the
largest Field of Attraction or probability. Because
larger numbers both have smaller fields and are closer
together, we are less able to pinpoint one ratio from
among several of them.

Carl Lumma, John Chalmers, and myself all think
that Erlich has formulated an important aspect of
musical harmony and tuning here.

[Mulkers:]
> I have a strange feeling you're
> touching here some ideas that are mine as well.

If you're right - sorry,
didn't mean to touch without asking permission first 8-)

[Mulkers:]
> Actually, two tones don't fit because they "behave" like
> overtones, It's because of the fitting overtones themselves.

It's both, and more. Helmholtz's beating, which
is what the "fitting overtones" avoid doing, is one
part of the sonance model, but it may not be as
important as either harmonic entropy, tonalness,
or integer/odd/prime-complexity. They all play a part.
The one thing that seems to tie them all together
is the *model* of the overtone series.

In the post you quoted, I said:

[Monzo:]
> Would the minima describe a subset of the
> harmonic series that is the determinant of
> the most likely implied fundamental of
> "tonalness"?

The fact I am assuming is that the harmonic series
is a kind of psycho-acoustical template that we
use to understand musical harmony, and that we
try to fit what we hear to it, even if the match
is not perfect [= just].

I purposely worded it the way I did to remain non-specific,
so that my proposition could also be expanded to include
notes that are possibly acting as xenharmonic bridges
among the prime-factors.

Erlich's harmonic entropy works for dyadic intervals,
but no one has yet figured out how to apply the idea to
chords, to describe the consonance/dissonance of
triads, tetrads, and higher-ads.

I believe that finding some mathematical way of
combining dyadic harmonic entropy with the ideas of:

1) "tonalness" (where simultaneous notes tend to
imply some harmonic series, and thus some fundamental)

and

2) the integer/odd/prime-complexity idea

is the solution to calculating both the sonance
of chords, and the finity/periodicity of harmonic
systems. Lattice diagrams are a tremendous aid
in visualizing these processes.

I also believe that xenharmonic bridging is going
on all the time while we listen to music - that
intervals and pitches of one prime-factorization
are able to substitute, and in varying degrees
(depending on composer, performer, listener, etc.)
*are* substituting for those of other prime-factors.
(lattices model this well too)

But this is a very complex, fluid, and subjective
process. I think it possibly involves the way
different areas of the brain process different
information.

-Monzo

** I have a new version of the harmonic entropy
webpage ready to go on the web, complete with
graphs and a commentary thru-out, based on an
explanation Erlich gave me in person. I've been
having trouble with my FTP.

|\=/|.-"""-. Joseph L. Monzo...................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\ c/o Sonic Arts, PO Box 620027, San Diego, CA, USA
_U//_/-/__/// | "...I broke thru the lattice barrier..." |
/monz\ ((jgs; | - Erv Wilson |

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Dave Keenan has kindly made my first four graphs of harmonic entropy
available on the Web. Thank you Dave! They all assume a 1% frequency
resolution and show intervals from 1-1200 cents.

http://dkeenan.com/Erlich/hentropy3.GIF

uses as the set of possible interpretations the ratios in the Farey
series of order 80, i.e., improper fractions whose numerator does not
exceed 80. This is based on assuming that the upper note is being held
constant at a frequency 80 times the lowest perceivable fundamental, and
all maxima and minima are given in cents.

http://dkeenan.com/Erlich/hentropy2.GIF

is the same for 81 instead of 80.

http://dkeenan.com/Erlich/hentropy1.GIF

compares all such curves for 80-84 (the curves get higher as this
"limit" gets higher)

http://dkeenan.com/Erlich/HENTROPY.GIF

instead of a Farey series, uses a series where the sum of the numerator
and denominator does not exceed 112. All maxima and minima are given in
cents.

Sorry Paul,

In between telling you where your harmonic entropy charts were and getting
digest 118 I arranged to also put up Monzo's commentary linking your
charts. It disagreed on the filenames so I renamed the files. 1 and 3
swapped numbers and unnumbered became 4. This is a more logical order. It
corresponds to the order in which you described them to the list. I don't
suppose it matters since each chart is well labelled internally.

Anyway, you can now just go to http://dkeenan.com/Erlich

Regards,

P.S. Do you intend to address this conflict between Dan Wolf's experimental
result (dissonance increases with d when *lowest* tone fixed) and your model?

-- Dave Keenan
http://dkeenan.com

>P.S. Do you intend to address this conflict between Dan Wolf's
experimental
>result (dissonance increases with d when *lowest* tone fixed) and your
model?

I wonder if Dan would be so kind as to repeat the experiment with the
highest tone fixed.