Cartwright's nonlinear model

Paul wrote:

>Also, there are many musicians around here devoted to exploring non-
>JI harmony and/or inharmonic timbres, so these types of studies make
>a "dang" difference to them. Especially when the dominant ideology in
>the music theory of inharmonic timbres, which is that of Sethares,
>completely, totally ignores these phenomena.

I agree with Paul on several counts here - understanding pitch shifts
when dealing with inharmonic timbres is important,
and it would be great to be able to take this into account
when dealing with scale/timbre interactions.

With this said, however, Im not sure that this particular paper
is likely to be useful towards this goal.
First, there is no new psychoacoustic data in the paper -
rather they have done a (quite admirable) modelling of the
data from the 1962 paper by Schouten, and related it to
dynamical system theory.

The data that they model is limited (only looking at small
pitch shifts with a small number of partials). I do not think
they provide a way to generalize to other situations than
a collection of sequential partials (k, k+1, k+2).
Accordingly, they miss a larger issue that is, I think,
far more important.

In simplest form, the larger question is this: given a collection
of (inharmonically related) partials, *how many* notes will be heard.
Because of their focus on collections of sequential partials,
the answer in their case is always "one". Yet if you play around
with inharmonic tones (bells provide a good non-electronic example)
then you realize right away that sometimes you "hear" one tone,
sometimes two, and sometimes more.
(No doubt Paul will be happy to
point out that current theories also ignore this!)

My reservation here is that until you know how many "notes"
are sounding, it is not particularly meaningful to talk about
what their pitches are.

Bill Sethares

--- In tuning@y..., SETHARES@E... wrote:
>
>
> Paul wrote:
>
> >Also, there are many musicians around here devoted to exploring non-
> >JI harmony and/or inharmonic timbres, so these types of studies make
> >a "dang" difference to them. Especially when the dominant ideology in
> >the music theory of inharmonic timbres, which is that of Sethares,
> >completely, totally ignores these phenomena.
>
> I agree with Paul on several counts here - understanding pitch shifts
> when dealing with inharmonic timbres is important,
> and it would be great to be able to take this into account
> when dealing with scale/timbre interactions.

I'm not sure if pitch shifts have much to do with what I was thinking of here -- rather, I was thinking
of the fact that the ear will create virtual pitches wherever it "sees" an approximate harmonic
series, and not otherwise -- both of which considerations should be very important when trying
to make music with highly inharmonic timbres. Also, the propensity of a chord to create a single
clear virtual pitch represents an extra component of consonance, often more important than that
due to the lack of beating/roughness between partials.
>
> In simplest form, the larger question is this: given a collection
> of (inharmonically related) partials, *how many* notes will be heard.

My harmonic entropy model provides a way of attacking this problem, though currently
calculations have been performed only for dyads. Conceptually, though, I think I'm on the right
track.

> (No doubt Paul will be happy to
> point out that current theories also ignore this!)

Hmm . . . have you read Parncutt's book, _Harmony: A Psychoacoustical Approach_? One of
the parameters he discusses is "multiplicity", which is the apparant number of pitches in a chord.
Parncutt was one of the two main inspirations for harmonic entropy theory, along with Van Eck.