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.