Music and emotion

I've always been curious about why certain musical entities "feel" a certain way to most people -- for example, why does a major chord feed nicer than a minor, or just a bunch of other random notes?

For elements of feeling which seem to be built in for most people, it makes sense to look for an evolutionary cause. Here are two possibilities that might relate our backgrounds with our current musical interpretations:

(1) Rising vs. dropping pitch. Usually when you hear an increasing pitch, it's associated with increasing anticipation, while decreasing notes often reflect a release or departure. Simple examples include the half-note rising motive of the Jaws theme, or the ultimate farewell denoted by Chopin's falling funeral march. Perhaps these connotations could have arisen from the natural Doppler effect -- when a predator is approaching, or we are approaching prey, any noises from the nearing object will be increasing in pitch, while if our prey is escaping or we from a predator, the pitch is decreasing.

(2) Major and similar "nice" chords vs. dissonance. It seems to me that many "nice-sounding" chords are composed of pitch spectrums which could occur naturally as the overtones of a single sound-producer, while more dissonant progressions have less natural frequency ratios. In nature, we could associate something like a major chord with a single producer -- be it friend or foe, it is easy to understand and work with. Dissonance could only arise from a large group (who are not inclined to speak in pitch). So our sense of harmony among chords might have some foundation in the mere plurality vs unity of a sound-producing unit.

I'd be interested to see more work along these lines. Perhaps if we can physiologically or evolutionarily understand why some things sound good to us, and others less so, we could be more conscious of how to make good music :) with more awareness than the current traditional teachings -- which I think are more historically based than scientifically discovered -- would otherwise encourage.

puzzle answer: avg(sums(a set))

This is the answer to the last math puzzle. You should check out the puzzle before you read the answer!

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It turns out that

avg(sums(S)) = ½ ∑S.

Why?

The main observation is that, for any T ⊂ S, avg({∑ T, ∑(S-T)}) = ½ ∑ S. We should also note that if ∑ T₁ = ∑ T₂, then ∑ S-T₁ = ∑ S-T₂; this means that whenever the sums of T₁ and T₂ overlap in sums(S), so do the sums of their complements. If we think of adding each pair {∑ T, ∑(S-T)} one at a time to build sums(S), then either both elements of the pair are already in the set, or both are not -- either way, the average remains the same.

math puzzle: average of the set of sums

My friend Chris Altomare once asked me a math question which inspired this one:

Given a set S of real numbers, define the set

sums(S) := {t: t = ∑ T for some T ⊂ S}

and the set function

avg(S) := ∑ S / #S,

where #S denotes the size of the set S (we can leave avg(empty set) undefined).

As an example, let S = {2, 5, 7}. Then

sums(S) = {0, 2, 5, 7, 9, 12, 14};

the 0 is always included in sums(S) as the sum of the empty set. In this example, we also have avg(sums(S)) = 7.

Can you find a general formula for avg(sums(S)), for any set S?

This is actually not super-hard, but the trick is that the "obvious proof" is wrong -- in other words, if we thought of the sums function as giving a multiset, then it becomes a linear function, which would make everything straightforward (since avg is based on linear functions, so avg(sums) is still linearly analyzable and thus easy to find and prove). But this is not the case!

Good luck! I'll post the answer soon.