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....
The real parts of the complex solutions are within half a cent of ( 2^{-2.5} ), the note 3 octaves below the equally tempered tritone (\sqrt{2}) ! I'm not sure what this means yet, and the New Year's party is about to start, but that's something to think about when the others have gone to sleep tonigh ;) It has to mean something.
@edvin - Happy New Year to you too! This is an interesting puzzle to ponder after I party. Could it be related to how the positive 4th root of 5 is very close to 3/2? Hmm...
@johncarlosbaez
I've been pondering the interpretation of complex valued musical intervals, and I can't make interpreting the imaginary part as damping make sense.
My main issue is that what we are talking about are musical intervals, rather than tones. The difference might be subtle, but the thing about intervals is that they can be stacked together, as in added and subtracted from each other.
When two intervals are added, their frequencies are multiplied, so the whole thing is logarithmic, and this is why it's so hard to make sense of anything but intervals represented by anything else than strictly positive reals.
What COULD make sense is to think about complex numbers using their POLAR representation, and interpret the argument as PHASE.
If we stack two tritones, represented by (\sqrt{2}), we want to get an octave, and (\sqrt{2}\sqrt{2} = 2). We want 4 equally tempered minor thirds to add up to an octave, and find that they must have frequencies (\sqrt[4]{2}).
In the same way, if everything is to make sense, stacking 4 of whatever interval is represented by (i) should result in a perfect prime, i.e. the interval represented by the number 1. (i^4=1) after all. An "interval" that shifts the phase by one quarter period checks that box!
An interval (X=1+i) would then correspond to scaling the frequency by (|X| = \sqrt{2}) and shifting the phase by one eighth of a period, and (X^4=-4) would be the note two octaves up with inverted phase.
@johncarlosbaez
What I have NOT managed to make sense of is addition of complex numbers representing intervals. Or, more urgently, subtraction. The whole point of the equation in the main toot was to match the beat of two intervals, and the beat is determined by the ADDITIVE difference of two notes, not the multiplicative difference.
If we want the complex intervals to conform to this it's enough to just consider numbers with norm 1, and see that the beat of the note at (i) and the note at 1 should have frequency (\sqrt{2}), and I don't know what to make of that...
which give a sound of frequency (ω+ω')/2 pulsing at a slow frequency (ω-ω')/2.
But these identities hold for complex frequencies too, so my approach will be to figure out what they imply. As long as the frequency difference ω-ω' is real we will get that pulsation expected from beats, even if ω and ω' are not real.
@johncarlosbaez By the way, there are two possible fifths and two possible thirds that all have the same beat frequency. Correct me if I'm wrong, but it seems we are going for the lower of the two fifths and the higher of the two thirds. Also, we seem to aim for the third one sixth above the fifth.
[
F^4 + 2F - 8 = 0 \Leftrightarrow \frac{F^4}{2} - 2\frac{5}{4} = \frac{3}{2} - F
]
Is there some reason for this or is this just some historic accident?
@johncarlosbaez Another possible equation, if we instead want notes to be on the same side of their just values, would be
[
\frac{F^4}{2} - 2\frac{5}{4} = -\frac{3}{2} + F.
]
The equation can be visualised as two curves, one line represented by the right hand side and a fourth degree curve that's symmetric around the y-axis.
Since both curves have a zero close to 1.5, it's easy to guess there should be an intersection close to there. Flipping the sign of right hand side doesn't move the zero of the right hand side, and thus should not move the intersection much either. Indeed, the equation above has a root at (\approx 1.4945), which is slightly worse than
the 1.496 that results from the original one.
On the other hand, the equation above has roots -0.7976 and (-0.3485 \pm 1.2475i). -0.7976 is very close to (-\frac{4}{5}) ((-\frac{4}{5}) is a major third down from 1). The other roots both contain an imaginary major third, and their real part is close to two octaves down from (-\frac{7}{5}). The latter isn't among the "traditional" intervals, but a reasonably small fraction.
Inspired by @johncarlosbaez's post on complex frequencies, thinking about complex numbers:
If you are learning complex analysis, and want to really understand branch cuts, and live in the US, I recommend moving to east Asia.
No, really: I lived in South Korea for a few years, working at KAIST, and having to deal with the international date line -- the branch cut for the Earth -- forced me to understand that little bit of complex analysis.
This is especially relevant today, December 31st, because as I write this in the morning (US central time), it's already tomorrow -- January 1st, 2024 (or, apropos of my recent post, January 1st, 45² - 1) in parts of Australia. Happy #newyear2024.
@ddrake - I went to the North Pole to withdraw some money from my bank, because I heard they had a branch nearby. But when I got there I saw nothing but endless snow! Turns out there was a branch cut near the pole.
@RefurioAnachro - temperature is not proportional to average speed, even for ideal gases. For an ideal monatomic gas in 3 dimensions we have
E = 3kT/2
where E is (kinetic) energy, T is temperature and k is Boltzmann's constant. This shows that for such a gas temperature is proportional to the square of the average speed. More importantly, it shows that kT has the same units of E. If we work in units where k = 1, then temperature has units of energy. If we also use units where ℏ = c = 1, then temperature has units of inverse time. And that is the deep way of thinking about it that Hawking took, with great success!
Arrr. That was my first hunch. No Idea why I didn't stick with it. Thanks, @johncarlosbaez!
h is in joules per hertz. Hm... why wouldn't we simply say joules times second instead? I remember how that came about historically, but are there other reasons?
Boltzmann's constant k has units joules per kelvin. That is obvious from E=3kT/2. Why is 3/2 not part of k?
A joule is one kg m² / s² and c is of course in meters per second. I don't quite see how c plays a role here. It's probably obvious and I'm just too dense to c it.
@RefurioAnachro - "h is in joules per hertz. Hm... why wouldn't we simply say joules times second instead? I remember how that came about historically, but are there other reasons?"
You must have read "joules per hertz" somewhere, I always say "energy × time", e.g. joule-seconds. But I can make up a reason why joules per hertz is nice. Say you have a photon with frequency N hertz. Then just multiply N by ℏ to get its energy!
"Boltzmann's constant k has units joules per kelvin. That is obvious from E=3kT/2. Why is 3/2 not part of k?"
The 3 is just because a quantum harmonic oscillator with n degrees of freedom in thermal equilibrium at temperature T has expected energy NkT/2. It would be silly to make such a big deal about the fact that a free particle in 3d space acts like a harmonic oscillator with 3 degrees of freedom. That's just an application.
You could argue that the 1/2 should be in there, but really it shouldn't: the much more fundamental fact is that in thermal equilibrium at temperature T, the probability that a system will be in a state of energy E is proportional to exp(-kT). All the crap about quantum harmonic oscillators, particles, etc. just follows from this!
" I don't quite see how c plays a role here. It's probably obvious and I'm just too dense to c it."
You're right, c is not necessary to convert from energy to inverse time!
@RefurioAnachro@johncarlosbaez
"Why is 3/2 not part of k?" This one at least I feel mildly qualified to answer: the 3 is just the number of degrees of freedom (namely three translational degrees). For multiatomic gasses you may get 5/2 or other numbers depending on the vibrational and/or rotational degrees of freedom of the molecule. So the "more fundamental" statement would be "1/2kT per degree of freedom".
(Why the 1/2 isn't simply absorbed into k, I wouldn't know)
@SvenGeier - the kT/2 per degree of freedom law only holds for certain special systems: those where energy is a quadratic function. This follows from a more general rule called the 'Boltzmann distribution', which has a k in it but no 1/2.
@johncarlosbaez said:
> tuning systems where the frequency ratios are complex.
If I remember correctly, modular synths usually model notes (envelope curves) using three numbers: attack a, sustain s, and decay d. You can't simply start a note at 100% volume, that sounds really bad. This is typically implemented using three knobs with linear scale, and the result feels right.
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