"Resonance underlies aspects of the world as diverse as music, nuclear fusion in dying stars, and even the very existence of subatomic particles."
I learned recently that the harmonics of consonant musical notes have aligned frequencies. Meaning: the most consonant notes produce the most resonance between the strings. For example, two strings at a consonant interval of a fifth (3:2) have harmonic resonance every 3rd and every 2nd harmonic band. This is much easier to show with a visual of the spectrograms: https://docs.google.com/presentation/d/1cdRfvPtek44rH8k2GpMV...
As oscillations of neurons in the brain will also show this harmonic structure (due to the frequency following responses [1]), consonant notes may also produce the most neural resonance! But, we don't have the ability to measure and test this hypothesis yet.
[1] Bidelman, G. M., & Momtaz, S. (2021). Subcortical rather than cortical sources of the frequency-following response (FFR) relate to speech-in-noise perception in normal-hearing listeners. Neuroscience Letters, 746, 135664.
This gets a little more complicated when you discover that not all fifths are the same interval, as any string player will attest. Violinists playing double stops have to use ever-so-slightly different finger positions than they do when playing single notes. And instruments with fixed tuning like pianos use something called "equal temperament," which isn't quite the 3:2 perfect fifth.
Perfect fifths are a bitch to play on the violin indeed. You need to cover two strings with one finger, which needs to be placed at just the right angle. And it's very obvious when you are slightly out of tune, unlike single notes.
But pianos don't exactly use equal temperament, either. There's a curve applied to equal temperament, which makes everything sounds more in tune.
At the end of the day, musicians follow their ears. Whatever sounds the best is the most important.
And thank god for equal temperament, because the last thing you want is perfect intervals that don't leave any space for proper resonance.
«In general, the interference equation can be used to measure resonant amplitudes for any musical interval under any temperament or octave division. This equation tells us that minimum resonance occurs at the fourth root of an octave (or square root of twelve) while maximum resonance occurs at the cube root of half an octave. Taken together, these results offer clear evidence that harmonic interference balances naturally around 12 as the most rational and harmonic number possible.»
«We find here the most amazing thing. The arithmetic mean converges toward PI, or mathematical constant π ≈ 3.14159, located in the middle of the curve. We further find this point in the distribution curve to be equal to Unity (or 1) when the domain value X = 12. This is significant because twelve is the square root of 144, the value shared by both harmonic and Fibonacci series in a 12-step octave. Squaring each of the table values and dividing by twelve confirms that 12.02383 ≈ 12 is the point of balance between foreground and background.
The significance of twelve as a point of balance in the octave interference pattern is proven further by plugging it into the equation, confirming the curve height equal to Unity at the octave. But even more significant than this is the fact that plugging the square root of twelve into the equation results in the amplitude y = 5.0666. Care to guess what this number represents?
It is none other than the y-axis amplitude for the golden ratio in an octave. Yes, the square root of twelve in the Gaussian interference pattern occurs precisely at Φ, right in the “cracks between the keys” of a major 3rd and minor 3rd in an octave. Just like the dense lattice region between a major 6th and minor 6th, the infinite golden ratio also provides an anti-harmonic proportion in the lower half of an octave. This occurs naturally at the square root of 12 (or fourth root of 144) in a 12-step octave.
No matter how you do the math, both harmonic and Fibonacci series reach a harmonic balance with one another at n=12 and an anti-harmonic dead zone at n=√12. Division of the octave by twelve (not eleven, nineteen or any other number) is revealed here as a completely natural pattern produced by linear harmonics that are curved in pitch space by Fibonacci proportions as they converge to Φ. Could Gioseffo Zarlino’s decision to divide the octave into twelve steps have involved some knowledge of this simple relation between harmonics and the Fibonacci series?»
«As a surprising correspondence between music and math, this little trick reveals the Pythagorean comma accurate to 3 decimal places. More amazing still, if we recalculate using the un-rounded arithmetic mean 12.02383 found earlier in place of 12, we obtain a slightly better estimate for the Pythagorean comma good to 4 decimal places. This bizarre associative property in the interference equation using the anti-harmonic golden ratio location of n=√12 proves the golden ratio is a physical property in the natural harmonic series and not some kind of error or “evil” in nature as portrayed by the Church. Vibration needs room to resonate in space and the Pythagorean comma created by the golden ratio appears to be just the right amount of room needed.»
I skimmed a bit and it's pretty weird. It reads like pseudoscience and mysticism mixed with a fair amount of true (but oddly interpreted) ideas. Kind of reminds me of https://en.wikipedia.org/wiki/Systemantics
It doesn't read like that at all; your unfamiliarity with the subject matter doesn't mean it's "pseudoscience and mysticism", that's a classic argument from incredulity. It's rigorously based in the mathematics of music and physics of sound.
Amazingly, brainwave bands have octave relationships -- they are doublings of frequency. E.g., alpha at 10hz, beta around 20hz, gamma around 40hz. This allows for neural resonances between bands. Further, the width of brainwave bands have a golden ratio relationship, where a high and low theta can independently operate without interference. The irrationality of the golden mean results in wave peaks that never meet/synchronize. This allows for multiplexing or non-interference between brainwave bands. Described in this paper, cited over 1000 times:
Klimesch, W. (2012). Alpha-band oscillations, attention, and controlled access to stored information. Trends in cognitive sciences, 16(12), 606-617.
This rabbit hole is part of why I got into eurorack/modular. It was to apply my musical ear to intuitions to help learn about waves and harmonics. There is a lot of woo around that crossover, but when I can hear the difference in a transform and then see it reflected on an oscilloscope or in a spectrogram, its easier to get a feel for what the constants and coefficients are supposed to do. (especially with the Maths module) A generation of kids playing with their parents' rigs and getting that feel may yield a new renaissance in physics in a couple of short decades. I'm just a hacker and a writer, and beneath what might be called an artist, but I have the sense that an emerging trend of popular quantitative culture is building momentum.
If particles can be understood as resonances in a field then it’s hard for me to understand “stable” particles. What keeps them around? My mental model for resonance and waves requires some kind of constant input energy to maintain the oscillations in the field which produce the particles. What am I missing?
Resonances are like vibrations on a trampoline; stable particles are more like lumps in a carpet.
Usually what keeps particles stable is that they carry some quantum number(s) that is/are conserved, and no lighter combination of particles can be made with that combination.
Superconductors can theoretically produce perpetual motion: once you set up a supercurrent, it will keep going forever. The caveat being that the electrons that are to move perpetually cannot do work of any kind*. That is, the field generated by the supercurrent cannot be interacted with.
Likewise, an ideal LC circuit made of superconducting material can theoretically ring forever, provided "nobody hears the sound". So, given my understanding of the physics, nature does not forbid stable perpetual resonances (though engineering constraints may forbid even microscopic perpetual resonators).
* When people talk about "perpetual motion machines" they tend to want to extract energy from them. That's folly.
Is that assuming that the superconductor itself does not undergo particle decay, or is there something special about superconductors that make them unusually stable compared to other matter in the universe?
Part of the difficulty in that analogy is that we normally think of things as damped by some surrounding media. For particles (modeled as field resonances storing quantized spatially concentrated energy) the only damping is coupling to other resonances (e.g. other particles). The particles were created by those interactions and they be annihilated (or decay) into other particles depending on how those resonances couple. Often due to quantization, this requires multiple particles including photons to "transmute" and that combination tends to reduce the likelihood of the interaction. That's my intuition from 30 years ago, anyway.
> My mental model for resonance and waves requires some kind of constant input energy to maintain the oscillations in the field which produce the particles.
Your intuition is inverted from reality.
Quantum mechanics dictates that a harmonic oscillator in a state will remain oscillating in that state indefinitely unless perturbed.
This has the counterintuitive effect that an atom with an electron in an excited state will remain there indefinitely if you can cut off the atom from external interactions.
This was, in fact, one of the arguments that Bohr used against Einstein. Experimentally--atoms decay--there is no denying that. However, Einstein's formulations said that atoms should not decay without external influence.
It turns out--Einstein was right. However, those kinds experiments require things like laser traps that were far beyond the capabilities of the time.
"Such a bump is the unmistakable signature of “resonance,” one of the most ubiquitous phenomena in nature."
If so then, physicists call "resonance" a "particle". Or what physicists call particles are resonances. But why? If you observe resonance call it resonance. I don't get it.
It's still called a particle because that's a useful abstraction, and also keep in mind that correctly identifying it as resonance happened much later than people first started to use said abstraction.
In reality, we know by now that there's no such thing as a particle:
At first, resonance was used as a programmatic analogy to guide understanding of spectral lines. Then, it turned out that resonance wasn't a metaphor, it was a mechanism.
as for why, probably so that literature written now still makes sense following on from older material from the 20th century that isn't wrong but does include some antiquated terminology.
Say, perhaps, an audience waving their hands to the beat of the music? The muscle movements would then require pressure from the heart, which begins to beat synchronously as well.
Also, what about very small respiratory particles; could they be resonated and "dance themselves to pieces" by using light or sound at the right wavelength?
I had a weird idea related to this about 3D “shaping” massive structures in outer space. So, say you go and mine an asteroid, then put a bunch of metal into a giant “box” for lack of a better word. Then say you could tap into a large energy source from a star. Could you emit energy waves into this box that causes the metal to form itself into a certain shape? I’m thinking of those videos where they put grains of sand on a surface and vibrate it at different frequencies to produce varying shapes. With the right interference patterns, could you 3D form a spaceship hull inside the giant box full of metal floating in space?
At the root of reality there's some "wavy" equation that propagates changes at a certain speed (of light). The exact form of the equation is of little significance, only its "waviness" matters. This waviness is often called causality. Such equation in a confined area, any area, always and necessarily has many repetitive modes where the original state repeats itself after a certain interval, but all thosr modes exist in abstract, i.e. potentially. An external driving force allows to select one of those modes. This is probably a tautology, but maybe someone will find it interesting.
34 comments
[ 4.2 ms ] story [ 77.1 ms ] threadI learned recently that the harmonics of consonant musical notes have aligned frequencies. Meaning: the most consonant notes produce the most resonance between the strings. For example, two strings at a consonant interval of a fifth (3:2) have harmonic resonance every 3rd and every 2nd harmonic band. This is much easier to show with a visual of the spectrograms: https://docs.google.com/presentation/d/1cdRfvPtek44rH8k2GpMV...
As oscillations of neurons in the brain will also show this harmonic structure (due to the frequency following responses [1]), consonant notes may also produce the most neural resonance! But, we don't have the ability to measure and test this hypothesis yet.
[1] Bidelman, G. M., & Momtaz, S. (2021). Subcortical rather than cortical sources of the frequency-following response (FFR) relate to speech-in-noise perception in normal-hearing listeners. Neuroscience Letters, 746, 135664.
But pianos don't exactly use equal temperament, either. There's a curve applied to equal temperament, which makes everything sounds more in tune.
At the end of the day, musicians follow their ears. Whatever sounds the best is the most important.
«In general, the interference equation can be used to measure resonant amplitudes for any musical interval under any temperament or octave division. This equation tells us that minimum resonance occurs at the fourth root of an octave (or square root of twelve) while maximum resonance occurs at the cube root of half an octave. Taken together, these results offer clear evidence that harmonic interference balances naturally around 12 as the most rational and harmonic number possible.»
«We find here the most amazing thing. The arithmetic mean converges toward PI, or mathematical constant π ≈ 3.14159, located in the middle of the curve. We further find this point in the distribution curve to be equal to Unity (or 1) when the domain value X = 12. This is significant because twelve is the square root of 144, the value shared by both harmonic and Fibonacci series in a 12-step octave. Squaring each of the table values and dividing by twelve confirms that 12.02383 ≈ 12 is the point of balance between foreground and background.
The significance of twelve as a point of balance in the octave interference pattern is proven further by plugging it into the equation, confirming the curve height equal to Unity at the octave. But even more significant than this is the fact that plugging the square root of twelve into the equation results in the amplitude y = 5.0666. Care to guess what this number represents?
It is none other than the y-axis amplitude for the golden ratio in an octave. Yes, the square root of twelve in the Gaussian interference pattern occurs precisely at Φ, right in the “cracks between the keys” of a major 3rd and minor 3rd in an octave. Just like the dense lattice region between a major 6th and minor 6th, the infinite golden ratio also provides an anti-harmonic proportion in the lower half of an octave. This occurs naturally at the square root of 12 (or fourth root of 144) in a 12-step octave.
No matter how you do the math, both harmonic and Fibonacci series reach a harmonic balance with one another at n=12 and an anti-harmonic dead zone at n=√12. Division of the octave by twelve (not eleven, nineteen or any other number) is revealed here as a completely natural pattern produced by linear harmonics that are curved in pitch space by Fibonacci proportions as they converge to Φ. Could Gioseffo Zarlino’s decision to divide the octave into twelve steps have involved some knowledge of this simple relation between harmonics and the Fibonacci series?»
«As a surprising correspondence between music and math, this little trick reveals the Pythagorean comma accurate to 3 decimal places. More amazing still, if we recalculate using the un-rounded arithmetic mean 12.02383 found earlier in place of 12, we obtain a slightly better estimate for the Pythagorean comma good to 4 decimal places. This bizarre associative property in the interference equation using the anti-harmonic golden ratio location of n=√12 proves the golden ratio is a physical property in the natural harmonic series and not some kind of error or “evil” in nature as portrayed by the Church. Vibration needs room to resonate in space and the Pythagorean comma created by the golden ratio appears to be just the right amount of room needed.»
[1] https://en.wikipedia.org/wiki/Pythagorean_comma
Klimesch, W. (2012). Alpha-band oscillations, attention, and controlled access to stored information. Trends in cognitive sciences, 16(12), 606-617.
Usually what keeps particles stable is that they carry some quantum number(s) that is/are conserved, and no lighter combination of particles can be made with that combination.
Likewise, an ideal LC circuit made of superconducting material can theoretically ring forever, provided "nobody hears the sound". So, given my understanding of the physics, nature does not forbid stable perpetual resonances (though engineering constraints may forbid even microscopic perpetual resonators).
* When people talk about "perpetual motion machines" they tend to want to extract energy from them. That's folly.
Your intuition is inverted from reality.
Quantum mechanics dictates that a harmonic oscillator in a state will remain oscillating in that state indefinitely unless perturbed.
This has the counterintuitive effect that an atom with an electron in an excited state will remain there indefinitely if you can cut off the atom from external interactions.
This was, in fact, one of the arguments that Bohr used against Einstein. Experimentally--atoms decay--there is no denying that. However, Einstein's formulations said that atoms should not decay without external influence.
It turns out--Einstein was right. However, those kinds experiments require things like laser traps that were far beyond the capabilities of the time.
If so then, physicists call "resonance" a "particle". Or what physicists call particles are resonances. But why? If you observe resonance call it resonance. I don't get it.
In reality, we know by now that there's no such thing as a particle:
https://arxiv.org/abs/0807.3930
as for why, probably so that literature written now still makes sense following on from older material from the 20th century that isn't wrong but does include some antiquated terminology.
Pedagogically useful to use the word particle.
Say, perhaps, an audience waving their hands to the beat of the music? The muscle movements would then require pressure from the heart, which begins to beat synchronously as well.
Also, what about very small respiratory particles; could they be resonated and "dance themselves to pieces" by using light or sound at the right wavelength?