Physics: Beating the musical uncertainty principle?

Edited: June 8, 2018, 11:26 AM · The musical uncertainty principle states that the pitch of a tone cannot be determined more accurately than (1/(2*pi))/duration. But I did an experiment in Audacity that appears to allow me to determine pitch more accurately than the uncertainty principle allows. I recorded a bowed open string as an .mp3, uploaded it into Audacity, then generated a sine wave close to the frequency of the open string on a new audio track, merged the tracks, and then counted the beats and the measured the time during which they occurred on the merged track. The number of beats divided by the time during which they occurred equals the difference in frequency between the recorded tone and the sine wave. When I did this with a violin A string tuned to 440 hertz, and merged the .mp3 individually with sine waves of 440, 441, and 442 hertz, I got calculated values of pitch that agreed more closely than the uncertainty principle allows (0.024 hertz [0.1 cents!] versus 0.04 hertz). And why not? The uncertainty principle assumes that you can't do the procedure I described. Is there a flaw in my reasoning? If not, why has the uncertainty principle been accepted?

Replies (10)

Edited: June 8, 2018, 12:02 PM · I am not sure if I understood everything in what you have written, yet I guess the major flaw is the following: You are performing different measurements (by comparing the signal with 441Hz, 442Hz, etc.) and each will give you a different result. Now you compare these results with the real value of 440Hz, which you would not know in an actual experiment. Here, you determined it upfront by playing the tone much longer. If you did not know upfront it was 440Hz, and you would do your measurement, you could not be more certain about the result than the uncertainty relation.

That shall say, uncertaintity relations does not "forbit" that by chance (or actually by a good chance) a single measurement is better. Just you will get different results and you cannot be certain what is the right result because of your accuracy limit.

June 8, 2018, 12:07 PM ·

"Since researchers have known for a long time about the cochlea's nonlinearities, the current results are not quite as surprising as they would otherwise be."

June 8, 2018, 12:07 PM · I'm not familiar with this "uncertainty principle". Whose idea was it and by whom has it been accepted? What do you mean by "determined" - subjectively distinguished from adjacent tones, or objectively discriminated by methods other than "sounds a bit higher"? "Pitch" of course is a purely subjective phenomenon, largely but by no means fully determined by frequency. How steady is the pitch of a bowed open string?

I think you're right to ask "why not?" and your reasoning seems fine. If it's simply down to counting beats the only limit would seem to be that there would have to be at least two beats within the duration of the tone so you can measure their separation.

June 8, 2018, 12:14 PM · Uncertainty principle doesn’t tell you anything about what the pitch actually is, it just tells you the absolute minimum error you could have when measuring. In other words, the uncertainty principle is telling you that your .024Hz difference is meaningless because you can’t determine the pitch without an error of at least .04Hz. The fact that you calculated a pitch .024Hz different from 440hz is nothing but an artifact of the computer estimating. The computer, nor anything else, can say the pitch is closer than .04Hz from 440Hz. Of course this is assuming you calculated the correct value of .04hz. I think you don’t need the 2pi either because the relation is ft>1 and you would only need the 2pi if you wanted wavelength instead of frequency.
June 8, 2018, 7:09 PM · The absolute pitch of a note doesn't matter. The amount of resonance that a note stimulates from the instrument in the form of partials is what makes us sound "in tune" or "out of tune."

What is "uncertain" to me is what you are trying to accomplish.

Edited: June 9, 2018, 11:41 PM · To be annoyingly literal; the uncertainty principle is from theoretical physics; the Heisenberg-Schroedinger uncertainty principle is that at the atomic level we can measure the position or momentum of a particle , but not both; aka the observation effect; to locate an electron you have to pump so much energy into the system that it jumps orbitals. The clearest photo that we can ever get of an atom will be a fuzzy sphere. It has analogies to normal life, like; you can be in the parade or watch the parade, but not both. Or, you can buy a house, or have enough money to buy a house, but not both. In my previous day job as a bio-medical lab tech. I frequently ran into this problem; I can only study the bio-chemistry of the creature if I kill it, which changes it. The music problem that you might be thinking of; two A-strings pitched 440 and 441 might sound the same to most humans, but when played simultainiously they will generate an interference/beat frequency of 1/sec. The limit of human pitch discernment is reported to be about 5 "cents" = 1/20 of a half-step. (Not the same as Hz.) Related topic; the mental limit for thinking or hearing separate notes/events is about 16/sec. Runs faster than that are heard as a blur. The orchestral player doing a Tchaikovsky symphony goes home wondering "did I really play those runs correctly? I have no idea..
Or: if your vibrato is 5/sec and the notes are going by at 8/sec., PLEASE do not use vibrato. The value of pi has little to do with this. A rolling circle will generate a sine wave, but sound waves are not like waves on the surface of the ocean, sound is varying areas of partial air compressions. ~jq (I dropped out of the physics/math major the second year when they started talking about the square root of negative one, and calculus in more than three dimensions.-- I switched to music, with only 12 notes to wrestle with.)
June 8, 2018, 11:24 PM · I'm also a literalist when it comes to scientific terminology, since nothing can be proven or agreed upon unless all terms are precisely defined. We all tend to confuse "pitch" with "frequency". In fact the subjective pitch (tautological - all pitch is subjective, as is colour, taste, texture) of a pure sine-wave tone is slightly lower than that of a complex harmonic tone with the same fundamental. Complex harmonic tones with the fundamental and lower harmonics filtered out retain the subjective pitch of the original, which is how we can still hear the sound of a cello over the phone.
Edited: June 9, 2018, 4:22 AM · Pitch vs frequency. Pressure vs weight. Acceleration in stationary objects.
Scientists often take everyday words and give them a more precise, restricted meaning. This is essential, sorry, vital for effective dialogue, but non-scientists find this petty and irritating and prefer a miasma of half-truths.

Mathematics is a desperate but fascinating attempt to to put a neat, simplified face on a chaotic universe which has no obligation to comply!

Edited: June 9, 2018, 4:13 AM ·

If the root frequency is lower in volume than one of its harmonics, you would think this would throw off the experiment. Add compression to this (mp3) and now you have a lot of the harmonics that are equal in volume to the root note.

What does your spectrum analyzer indicates about your violin's 440, before and after compression?

Edited: June 9, 2018, 5:14 PM · Eureka! Scott Bailey provided the missing puzzle piece, which is that the ear CAN beat the classical uncertainty principle (due to nonlinearities of the cochlea). What confused me is that Heisenberg's uncertainty principle is considered absolute, so I assumed that there is also no way to get around the classical (musical) uncertainty principle. But sound is a physical pressure wave, whereas the quantum wave function is not a physical wave, and therefore cannot be recorded and measured. So the musical uncertainty principle does not prevent measurement of frequency to any desired degree of accuracy. The musical uncertainty principle is based on the idea that if you judge frequency by beats between the test tone and a reference tone, then you really can't resolve fractions of a beat--with your ear! But we have better equipment than that (electronic).

Another thing that convinces me that there is no such limit to the ability to accurately measure frequency is that I can image a touch-sensitive fingerboard that records the exact position of the violinist's fingers. Obviously, the accuracy of such measurements is not limited by the musical uncertainty principle.

If we can measure frequency to such extreme accuracy, imagine making a recording of a violinist playing a very fast run, and determining how much each note is out of tune! I'm sure it can be done.

Thanks, Steve, for the distinction between pitch and frequency.

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