You may have seen this Jedi math trick: take something that makes sense with real numbers and try it with complex numbers. Weird yet useful things happen.
For example it turns out that imaginary time is 1/temperature. Hawking used this to compute the temperature of black holes.
An easier example is that the exponential function applied to an imaginary number gives you the trig functions sine and cosine.
Indeed, if ω is any complex number, exp(iωt) is a function of time that oscillates at a frequency equal to the real part of ω, and decays exponentially at a rate equal to the imaginary part of ω. So we can think of ω as a complex frequency! Its real part is an ordinary frequency, while its imaginary part is a decay rate.
Thus, in music it makes sense to consider tuning systems where the frequency ratios are complex. I haven't yet found anything interesting to do with this thought. But it makes sense to have notes that oscillate but also decay.
Here's a dumb idea. Nobody knows Bach's original well-tempered scale. In 1977, Herbert Anton Kellner had a wacky suggestion: the beats in the major third (which is close to a frequency ratio of 5/4, but not quite) should have the same frequency as those of the perfect fifth (which is close to 3/2, but not quite).
This led him - the derivation is too long to fit in the margin of this post - to the 'Bach equation':
F⁴ + 2F - 8 = 0
where F is the frequency ratio of the perfect fifth. He got a solution
F ≈ 1.495953506
for the perfect fifth. But it also has a negative solution, and two complex solutions that aren't real. Do these mean anything?
Maybe you can come up with a better idea about complex tuning systems....
