consonance + dissonance

There are 2 ways to think about consonance and dissonance. One way is to think about the issue purely physically, e.g. dissonant might mean the beats that are produced between two notes, perhaps two notes that should be the same unison tone but are a little off. The second way is to think about what consonance and dissonance mean with respect to the Western tradition of harmony (or any tradition of music practice), e.g. a perfect 5th is consonant and a minor 2nd is dissonant.

The second way of thinking about consonance and dissonance is largely conventional: while no one would probably regard the octave as dissonant there was a time when 3rds were not consonant enough, say, for inclusion in final cadences (in Medieval music) -- and even when they did become a part of final cadences the preference was for the major 3rd. Schoenberg -- again in his Theory of Harmony -- makes mention of this aspect of consonance and dissonance and states that there really is no difference between the two. His analogy is that the numbers 2 and 10 are not opposites, and that all intervals are, well, equally intervals.

Regarding the first way of thinking about this issue here's a fabulous post by a physicist. In it the idea of physical (or acoustic) consonance and dissonance has to do with the overtones produced by the tones involved. Consonant intervals are explained as those intervals which have many overtones in common, e.g. the octave of a tone's overtones are all contained in the fundamental's overtones, hence not only does the interval sound consonant in this case but they sound like the same note.

There is an issue raised and addressed in the comments. Take for example a sine wave. There are no overtones (the sine wave is a pure tone), hence according to the above mentioned theory of consonance and dissonance there souldn't be any difference between, say, and octave and a major 7th. They do the experiment and verify that in fact you do still hear the 7th as more dissonant than the octave.

Anyway, great food for thought. Also mentioned at the beginning of the post is this site -- check it out, too, it's fab.