Monday, January 5, 2015

An Appendix: Dissonance and Consonance

Consonance and dissonance are traits we ascribe to sounds. Thus, our perception of sounds is somewhat important to this classification. Clearly, it's not a fully objective quality.

In the main post, I note how Pythagoras contributed to the understanding of harmony.  Although Pythagoras did not understand what sound was, he did understand that relating the sizes of the objects on which you play (say, different lengths of otherwise identical strings) by simple ratios produced appealing sounds. There have been a number of hypotheses as to why this would be the case. The most widely accepted one these days relates to the harmonics (overtones) described previously.

If we add together two sine waves, a and b, of rather similar frequencies,  the resulting wave will have a complication - its amplitude changes periodically in a wave-like way:

Sum of sin(tx) and sin(ytx), where y is a constant relatively close to 1, and t is an arbitrary constant.
I have no idea what causes the graphical misshap close to the second through.
Picture from, try out sin(x) + sin(yx), with y≃1 there for a variety of y.

 The frequency of this "metawave" is the same as the difference between the two frequencies a and b. If the difference is small, it does not sound bad - just like a slight wavering volume, somewhat similar to a vibrato in sound, and if it is large we don't perceive it as dissonant either. The range in which we perceive dissonance is called the critical bandwidth. Empirical research has shown that it covers a range from about a handful hertz to 6/5 of the frequency. However, this might seem to fail to explain dissonances over wider ranges, such as the major seventh (which is roughly 15/8, which clearly is wider than 6/5) or the very dissonant tritone (which is sqrt(2), which is a bit less than (6/5)^2, and thus clearly wider than 6/5).

I previously mentioned overtones. These provide us with the actual explanation! To calculate the dissonance for an interval, we should actually look at the amount of overtones for each of the tones in the interval that come within each others' critical bandwidths. The relative amplitudes of course also contribute, but in ways that is less easy to investigate by just eyeballing a graph.

Let us compare two intervals. A440 and E660 vs. A440 and D#622. (Note: for ease of calculation, I have reduced the usual D# by an ever so slight bit.) These have the following overtone series:

We can see that the A column (the one starting out with 440), and the E column (660) often coincide. Even if they didn't perfectly coincide (say, we replaced 660 with 659 or 661), the numbers would be close, and thus not reach the requisite width to enter into the critical bandwidth until several overtones down the line. However, 622 quickly enters it - 1320 is within the critical bandwidth of 1244 (or rather, they're within each other's range), 1866 is within the critical bandwidth of 1760, etc. Sure, 1980 is within the critical bandwidth of 2200 too, so E will cause some slight dissonance. However, the further up the overtone series we have to go to find critical bandwidth issues, the less dissonant an interval is. Of course, timbre may also affect the dissonance - clarinets and many woodwinds lack even-integer overtones (so, a tone sounding at a 100hz will only have overtones at 300hz, 500hz, etc), and for a good enough analysis, we would have to look into them as well.

Relative amplitude of the overtones is relevant, but we're not going to look at that now. More detailed models for understanding dissonance exist (e.g. 'harmonic entropy'), but this post is mainly meant as an appendix to an upcoming piece of reasoning about scale construction (that is part of a greater piece of reasoning regarding claims made by A432hz enthusiasts). Harmonic entropy has been used by people interested in scale construction, and various predictions made by it seem to have been accurate.

Anyways, this is a very short introduction to the issues of consonance and dissonance, and one where further complications can ensue - instruments where the overtones are not integer multiples of the fundamental, for instance, have their own complications with regards to what intervals are consonant and what intervals are dissonant.

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