There is a very simple high school homework exercise question that I have kept with me for 30 or so years because of its deep relevance for the understanding of fundamental physics. It teaches about the nature of quantum uncertainty, but sadly also about how terribly wrong textbooks can be, how nonsense makes it into print and is taught to millions as the wisdom of science, although about one minute playing with a guitar, or ten minutes of critical thought, should have told the author, or some teacher, or somebody for crying out loud. Now as I am teaching and writing a book on fundamental physics for lay people as well as applied and social scientists, it shall finally be resolved properly.

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.