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overtone series and uncertainty principle

Anonymous Sun 20 Dec 18:32:43 2020 No.12488504 View Reply Original Report

Quoted By: >>12489045 >>12489302 >>12491077 >>12491212

am i correct in assuming that the overtone series in music and the uncertainty principle are directly correlated? if the overtone series concerns adding up a bunch of sin waves to create different types of sounds and timbre, and a fourier series, given a short amount of time, has trouble separating those waves compare to a longer period of time, then you arrive at the uncertainty principle

curious because it would be nice to have a fundamental correlation between two major hobbies of mine

Anonymous

Anonymous Sun 20 Dec 2020 20:37:57 No.12489045 Report

Quoted By:

>>12488504
>am i correct in assuming that the overtone series in music and the uncertainty principle are directly correlated?
That depends on what it is that's "waving".

Anonymous

Anonymous Sun 20 Dec 2020 21:41:21 No.12489302 Report

Quoted By: >>12490148 >>12491052

>>12488504
nope.

the issue of a FT applied to a short period signal being unable to measure low frequencies is related to the Nyquist sampling theorem.

the uncertainty principle is a statement about non-communing operators. a quantum state is a vector. each type of operator/measurement is associated with a vectorspace basis that can be used to expand the quantum state in. when a measurement is made, the quantum state "collapses" to one of the basis vectors associated with the measurement operator. uncertainty principles happen when two different operators have different basis vectors, such that measuring the quantum state with one operator puts the quantum state in a configuration that is a mix of states of the other operator.

a geometric example is a 2d rectangular basis, and a copy of that basis rotated 45 degrees. being bases, the basis vectors of each basis can be expanded in the other basis. however, the rotated basis vectors are linear combinations of the non-rotated basis vectors. if we had a 2d quantum state of some sort and operators corresponding to the bases under consideration, when we measure the quantum state with one operator, the quantum state becomes one of the basis vectors in one basis, but a mix of basis vectors in the other, and vice versa.

operators that are commuting turn out to be associated with bases that are made up of (more or less) the same basis vectors. in mathy-speak, the operators are "simultaneously diagonalizable".

Anonymous

Anonymous Sun 20 Dec 2020 21:56:29 No.12489383 Report

Quoted By:

i'll add that in Fourier analysis, the period $T$ is inversely proportional to the frequency resolution $\Delta f$ , and the sampling time $\Delta t$ is inversely proportional to the bandwidth $B$ .

Anonymous

Anonymous Mon 21 Dec 2020 02:22:09 No.12490148 Report

Quoted By: >>12490864

>>12489302
wow great response anon, so the idea is that if two operators aren't commuting they don't share any common basis vectors, and that in turn leads to uncertainty when you make a measurement because looking at one operator's basis vectors makes the other's a superposition of the basis we're choosing to look at?

Anonymous

Anonymous Mon 21 Dec 2020 07:26:45 No.12490864 Report

Quoted By:

>>12490148
that's the gist of it, yeah

Anonymous

Anonymous Mon 21 Dec 2020 08:34:54 No.12491052 Report

Quoted By: >>12491087

>>12489302
this is a pretty narrow matrix-mechanics interpretation of the uncertainty principle. You're not wrong, but the concepts are a lot more related as you are representing them to be, especially when you consider the definitions of the position and momentum operators, (or I guess, eigenstates)

Anonymous

Anonymous Mon 21 Dec 2020 08:45:39 No.12491077 Report

Quoted By: >>12491090 >>12491200

>>12488504
Sort of, but not really. While it is indeed true that long windows resolve frequencies better and shorter windows have better temporal resolution, nothing prevents you from using multiple windows in the signal at once.

Anonymous

Anonymous Mon 21 Dec 2020 08:50:48 No.12491087 Report

Quoted By:

>>12491052
The way I kinda see it is the two distributions of vector space and we can alter how well we sample either one aspect of the other. Either we have a good maximum likelihood estimator for one of the vectors in space but the sampling on the other vector is much less defined as it wasn't sampled as well.

Anonymous

Anonymous Mon 21 Dec 2020 08:52:50 No.12491090 Report

Quoted By: >>12491114 >>12491200

>>12491077
im curious, what do you mean by temporal resolution, in comparison to resolving frequencies better? it seems to me you're saying both long and short windows resolve are better for resolving frequencies, but im not sure im catching the difference in resolving frequency and resolving time

in addition, would using multiple windows get rid of uncertainty entirely? apologies for the questions, this topic interests me

Anonymous

Anonymous Mon 21 Dec 2020 09:01:42 No.12491114 Report

Quoted By:

>>12491090
Think it's a corollary of
https://en.m.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem

Basically the frequency of the observation can influence what sort of information can be easily gleaned from the experimental signal

Anonymous

Anonymous Mon 21 Dec 2020 09:48:30 No.12491200 Report

Quoted By:

>>12491077
>>12491090
In music, most sounds come in the form of wavelets, in the sense that notes have a start and an end. The shorter the note, the harder it is to pin down its exact frequency due to harmonic distortion when measured in the fourier basis (what engineers like to think of as the fourier transform, and what musicians might think of as the spectrogram) But the shorter the note, the more precisely one can define its exact start/end times (sharper in the standard basis, what engineers think of as time domain).

The same mathematical mechanisms are what lead to the result of the quantum uncertainty principle, except for the extra (important) complication of what constitutes a measurement in QM, where the state is a variable that gets projected (or entangled with, depending on your metaphysical preference) to the measurement basis.

That's the extent of my poorly-remembered knowledge on the subject anyways. I'm sure I got at least one thing horribly wrong and other anons can correct me on it.

Anonymous

Anonymous Mon 21 Dec 2020 09:55:34 No.12491212 Report

Quoted By: >>12491278

>>12488504
the uncertainty principle stems from x, p and E, t to be fourier transforms, so if you want the analogue in music, find a pair of fourier transform variables.

Anonymous

Anonymous Mon 21 Dec 2020 10:24:34 No.12491278 Report

Quoted By:

>>12491212
the waveform view operator and the spectrogram operator

or if we're talking about the DAW itself, the single-payment-license operator and the monthly-subscription-license operator :^)

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