The question is simple; I rephrased it a little from the original:
A musician plays the A at 440 Hertz (Hz), then the nearest half-tone, G#, which is close to 415 Hz. He plays with the speed ‘prestissimo’, which means 200 beats per minute. “Every note has four beats.” The musician plays “notes of length 1/32” (so these notes have only 4/32 beats – musicians’ terminology is often confusing). Can anybody actually hear the difference and distinguish A and G# at this speed? [1]
Quite obviously, the author wants to make the point that the fundamental relation between the uncertainty in frequency and the uncertainty in time makes it that we cannot hear the difference.
Now first of all, this is really important. Many people think that the famous Heisenberg uncertainty relation is the fundament of quantum mechanics; that quantum uncertainty is very fundamental and leads to many other quantum phenomena. Actually, the Heisenberg uncertainty relation is no more but Fourier’s uncertainty, pure mathematics known long before modern physics, except for the assumption that a frequency implies some energy, but that has nothing to do with uncertainty. All the quantum mystery is in the assumption of that mass (energy) comes with a frequency, but the uncertainty relation is Fourier’s and has nothing to do with quantum mechanics. Uncertainty is not the essence of quantum mechanics and much of the quantum uncertainty can easily arise in warm classical ethers, where the temperature of the underlying ether wiggles the emergent phenomena about so that there is a minimum uncertainty.
Since many, including too many so called experts such as academic physicists, still carry also the misconception that quantum mechanics is supposedly all about small little tiny microscopic things, what could be more enlightening than having the uncertainty impact a trusty old grand piano? So the question is clearly a great homework exercise, or better, it could be made into one, however …
The established textbook answer is totally wrong: The maximum uncertainty in time (say if you were to ask when precisely each related phonon has actually been emitted), written “Δt”, is 4/32 beats divided by 200 beats per 60 seconds, that makes 3/80 seconds, or 38 milliseconds (ms). This much is correct. The uncertainty in frequency, Δf, is at least roughly one divided by the uncertainty in time, and so Δf is at least as large as 80/(3s) = 26.6 Hz. The difference between 440 Hz and 415 Hz is only 25 Hz, so they can supposedly not be distinguished.
Already as a school boy, I did not believe it! Look mate, the oscillation period at 440 Hz is 2.3 ms, so there are about 16 full wavelengths coming at us over the allowed 38 ms! You telling me that I cannot know the difference in frequencies if I am given 16 wavelengths? Color me doubting.
Now the first obvious mistake is that the proper uncertainty relation involves the angular frequency, which is 2 Pi times the frequency, and that the product of the uncertainties is bigger than one half, not one. These make together a whopping factor of 4 Pi, more than ten, a whole order of magnitude. So the uncertainty in frequency is only about two Hertz. But again, this is in some sense not really the main mistake, otherwise it could perhaps be correct at lower frequencies where a half-tone step is less than 25 Hz of difference. But it won’t work, and not only because we have great difficulties to hear a difference of only two Hz anyway, also at low frequencies and with slowly played notes, or because a half-tone is only two Hz away at frequencies that humans cannot hear. Before I get to the final reasons for why it won’t work with real instruments however, …
Two remarks on human nature: 1) Giving this task to university science students in the context of a lecture that is specifically on critical thinking and scientific argumentation, being repeatedly told that the homework is all about actually understanding instead of taking science as religion, having been mislead by the evil lecturer, yours truly, several times before in order to teach that message, makes all no damn difference. The students perceive the desired answer from the framing of the question, and although they were given the correct equation and use it, they nevertheless manage to provide the wrong answer, whatever the authority seemingly desires to hear, not caring a moist rat’s behind about truth, no urge for confidently understanding, but instead quite some sophistication in making up nonsense on demand – great future scientists indeed, totally fit for excellent participation in today’s scientific community and academia.
2) Moreover, it takes about one minute playing around with a guitar or some such instrument in order to recognize that the desired answer cannot be correct. There is no way in hell anybody could play A and G# in such a way that I would not hear the difference quite clearly. And I am not even any good at hearing frequencies – in fact, I discovered that I played my guitar for at least two years a whole note lower than properly tuned. It started with the damn thin e-string always snapping and me getting pissed about certain industries and stores apparently surviving entirely on people buying e-strings. Being lazy and despicably stingy, I therefore put less and less tension on the strings – call it “Master Feng’s Jewish Drop-D tuning” if you like – if you knew about guitar tunings and weren’t a PC liberal with your head firmly up your dark hole, you could have laughed about this joke – yeah man, just a joke, don’t be calling the litigious anti free speech slime bags from the ADL or SPLC on me now. Anyway, nevertheless, although theoretically as well as experimentally obviously mistaken, the question made it into textbooks and hundreds if not thousands of teachers have taught it, and many thousands of pupils and students have been graded according to how well they provide the wrong answer - there is natural selection in the evolution of functional social structure for you.
The mistake is not just the factor of 4 Pi. If that were true, it should be getting obviously more difficult to hear a difference at lower frequencies. Playing around with any real instrument, you quickly realize that even the lowest available notes can still be clearly distinguished no matter how quickly they are played as long as they are produced at all. Of course, if the sound of the scratching of the finger along the string takes as long as the time of the note, then the note cannot be said to have been played in the first place, so this is not a fundamental limit of uncertainty, however, it in some sense actually is, and here is the reason why:
Again, at the stated breakneck speed, 440 Hz provide still 16 wavelengths. Two octaves lower, the lowest A on the guitar, we are at 110 Hz, and there are therefore still four full periods. Another octave lower, we get into the range where many humans cannot hear the frequency at all, but there are still two full periods. If you have several periods, it implies that the excitations on a string went forth and back along the string, and so resonant frequencies such as the higher harmonics at two, three, four and more times the fundamental frequency survive the damping. It is those harmonics that make the sound of an instrument! In fact, the resonance frequencies of the whole instrument make the sound, and so we will never have the situation that the frequency is so low and pure that we cannot hear the difference on grounds of the fundamental Fourier uncertainty.
And this is the hint that tells us that once you start having less than a wavelength in the signal, the problem is the generation of the signal in the first place, not the detection – for this we do not even need to consider that also the Fourier uncertainty quite obviously and unsurprisingly starts to matter around this “one wave length limit”. A string being distorted into some shape by a finger or piano hammer does not result in any particular frequency. The desired frequency comes about from that the system swings around for a while, damping away everything but the harmonics. Yes, that holds equally for the speaker of a computer, so computers instead of musicians with wooden instruments are not the solution. By the way, generating 440 Hz with my computer needs me to tell mathematica to please produce 426 Hz, so much on how greatly reliable it is to do stuff with computers without carefully checking ev-er-y-thing. For-f-n-get-it!
In order to do this experiment, one would need to first produce pure sinusoidal signals at for example 100 and 104 Hz, something that somebody actually can distinguish at all. The well isolated standing waves should be kept behind beam choppers that only open up for 19 ms, first one, then the other. Then ask the participants which one was higher, number one or two. I don’t think I would then hear the difference, but likely not because of fundamental uncertainty but because I doubt that I can hear the difference between 100 and 104 Hz even if you allow 50 ms – maybe I am wrong here.
So, what to do? The homework exercise would still be great for hammering down that most of what is widely known to be quantum is nothing to do with it except for that it sounds super smart if one talks about nano quantum hybrid synergistic shit. Ha – here is how. I keep the homework question as it is and let the answer teach something about the human condition as well, that’s after all the purpose of the course and truly fundamental to science.
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[1] S. Vongehr: “Fundamental Science for Applied and Social Science Students.” Lecture Notes / Draft (2017)
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