The Plomp-Levelt dissonance curve and more

Hi all! I would like to contribute to this discussion, because it happen to
arise at just the right time . . .

Dante Rosati wrote,

>Theres no doubt that, with two sine tones, the dissonance action around
>1/1 is also heard in an attenuated form around 2/1. The same effect is also
>happening around 3/2. Theres more going on, but it needs more listens. I
>also noticed that my ear can hear/conceptualize (with the pure sine tones)
>stable intervals that "change quantum state" to the next important
>ratio/scale step. I have to listen to it a few hundred more times... (one
>of the quintessential "tone mandalas", I'd say)

In another one of those crazy coincidences, I just e-mailed Dante to tell
him he should look into Sethares' _TTSS_ and the Plomp-Levelt consonance
curves, and the next thing I know other people start talking about them on
the list!

I have two comments:

1. There are a wide variety of phenomena that are not fully understood in
connection with these phenomena. There appear to be varieties of beating and
of combination tones that happen not in the ear (the way normal beating and
non-equipment-based combination tones arise) but instead in the brain. These
effects interact with the ear-based effects in ways that are not accounted
for by quantitative models. Read Reiner Plomp, _Aspects of Tone Sensation_
for the nitty-gritty dirt. A lot of these phenomena are covered in Juan
Roederer's _Physics and Psychophysics of Music_ with excellent illustrations
(including the waveform diagrams one poster described, along with a
discussion of their relevance and irrelevance to the perceived sound), but
the material is presented in too organized a fashion to address the all the
confusing results Plomp found (by the time Plomp wrote his book; the early
studies for which he is famous are but a small piece of the puzzle).

2. Either of these books will tell you about the virtual pitch phenomemon.
Quite distinct from the difference-tone phenomenon (as experiments have
demonstrated -- see the books or, better yet, the original papers to which
they refer such as Goldstein's), virtual pitch represents the brain's
attempt to fit a fundamental to a set of sine waves. One actually hears the
virtual pitch -- or even several if the set of sine waves does not closely
approximate the proportions of lower members of harmonic series. If we
interpret the "salience" of these several possibilities as probabilities,
one can calculate an entropy measure to describe how simple the virtual
pitch sensation is. This is what I call "harmonic entropy". Harmonic entropy
is a component of dissonance missed by Sethares, but musically very
important. It is there for sine waves, but complex tones provide the
auditory system with more clues (such as upper partials in the frequency
range where the ear is most accurate at calculating virtual pitch) as to
what the frequency ratios might be. Sethares would predict that otonal and
utonal chords (harmonic-series-based and "subharmonic series"-based chords
have about the same dissonance (whether you use just sine waves or full
complex tones), but in a certain sense the otonal chords (especially the
higher-limit ones) are much more consonant, and the harmonic entropy concept
explains this (think about it). Joe Monzo has collected my thoughts on
harmonic entropy and a few graphs into his commentary-rich page at
http://dkeenan.com/Erlich/index.htm.