dissonance curves: experiment or theory?

Bram wrote:

>I've been reading over Tuning, Timbre, Spectrum, Scale, and doing some
examples, and have a question. In appendix E it gives a formula for dissonance
of two pure sine waves, but no particular explanation of why that formula
should be the way it is. Are the exponents used based on some abstract
idea, or are they simply the result of measurement?

The curves come from the experiments of Plomp and Levelt.
The form of the equations (as a difference
between two exponentials), is unique to TTSS. By choosing the parameters,
(using a least squares fit) it is possible to make this formula match the data
quite accurately. So the formula is derived from experiment,
which has been confirmed several times in different settings
(e.g., Kameoka and Kuriygawa)

>Also, I seem to recall it saying in Roederer that
the region of dissonance is farther away and wider in bass notes.

Yes, this is true. Sensory dissonance is tied roughly to the critical band.
This is incorporated into the equations in the "s" and "f" parameters,
which essentially make the basic curve shape wider at lower frequencies
and narrower at higher frequencies.

With this said, observe that most of the "theorems" in the appendix
use a simpler model in which the "s" and "f" parameters are jetisoned,
and a single curve is assumed for all frequencies. Thus the corresponding
reults must be considered only approximatley true over a broader
frequency range.