Nineteenth century machine that performs Fourier analysis | Hacker News Reader

Nineteenth century machine that performs Fourier analysis

engineerguy.com

86 pointsPhithagoras28 comments

The designer of this machine is the Michelson of Michelson-Morley, Nobel prize 1907; his interferometer was a brilliant apparatus that disproved the aether hypothesis. I'm surprised the blurb for the book doesn't mention who he was.

Probably the publication bias against negative results.

The fascinating thing about the Michelson-Morley interferometer experiment is that they were attempting to validate the aether hypothesis, and measure the Earth's absolute velocity relative to the aether. Instead they failed to do so, and the experiment showed that the speed of light is the same in all directions relative to all observers regardless of motion. A quite shocking and non-intuitive result that served as a foundation for the theory of relativity.

Even today most non-scientists do not understand or appreciate this result, they still operate within a mental model of the universe involving absolute frames of reference.

Explaining the Michelson-Morley experiment required assuming a constant speed of light regardless of relative velocity, combined with old-school frame agnosticism. Of course making this intuitive leap is what led Einstein to special relativity (Lorentz had already provided the math). To experimental physicists, these unsung heros deserve just as much credit ;)

It would be nice if there was a resource that summarized the mechanical counterparts of these operations:

- summing displacements

- subtracting displacements

- turning displacements into rotation

- multiplying displacements by a constant factor

- multiplying two displacements together

etc.

Using these primitives, any computation could be transformed into a mechanical device. One could even build a compiler for it! This compiler could take a program as input and a 3d-printable model as output. Happy coding! :)

http://www.youtube.com/watch?v=s1i-dnAH9Y4

How about an 18th century machine that performs Fourier analysis? A violin!

Also it has a really good numerical approximation of 2^(7/12).

Why? A violin has 4 strings. The lowest string in frequency is the G, the first G below middle C on a piano. The other three strings are, left to right to the player, D, A, and E. Each adjacent pair of strings differ in frequency by a musical perfect fifth so that, three times the frequency of the lower string in the pair is the same as two times the frequency of the higher string in the pair. So, as move across the strings from G to D, A, and E, at each step are multiplying frequency by 3/2 which, then, must be, and is, a nicely good approximation to 2^(7/12) which is essentially what a piano is forced to do for those notes.

A lot in violin playing is from listening to harmonics, that is, overtones, that is Fourier analysis.

Also that each string has a natural frequency of vibration is essentially a Fourier point.

But, wait, there's more, and much earlier! Newton ran a beam of white light through a prism which is a Fourier analysis -- essentially a power spectrum via Fourier transform. Then Newton was smart enough (uh, he was a bright guy) to use a second prism and show that the individual colors could not be further decomposed. So, somehow the first prism did some fundamental decomposition.

Also, some armies knew that soldiers walking across a bridge should not march in step. Why? Because a Fourier analysis of the bridge might show that frequency of the marching was a natural frequency of the bridge and could cause large amplitude motions of the bridge, failure of the bridge, and the army to fall into the water below.

The main parts of the human ear do a Fourier analysis.

X-ray diffraction is essentially 3D Fourier analysis.

I don't understand how that would work. The video gives a great explanation of analysis, which is that you are given an arbitrary waveform, and need to decompose it into it's component sines or cosines. Even with your description, I can't see how a violin would accomplish that.

The sound from one string on a violin is essentially periodic. So, can apply Fourier series. When do that, get the amplitudes of the overtones.

The periodic sound from a violin string is not a sine or cosine wave, but, as in Fourier theory, each of the overtones is.

And it is just those overtones a violinist uses to tune the instrument. That is, use a tuning fork to tune the A string to, say, 440 Hz. Then bow the A string and the E string together, listen to the third overtone from the A string and the second overtone from the E string and adjust the E string until the beats (for each x Hz difference in frequency, x beats a second) disappear. The overtones being used are the sine-cosine components in the Fourier series decomposition of the periodic sounds.

Wouldn't the human then be performing frequency analysis, not the violin itself?

You seem to be conflating decomposition of a signal with generation of the same, and Fourier analysis with a lot of things. How does one take a "Fourier analysis" of a bridge? Isn't that just resonant frequency?

> How does one take a "Fourier analysis" of a bridge?

Give the bridge an 'impulse. Then it's Fourier transform is has all frequencies. Then see what the bridge does: Get out the transfer function of the bridge. It might have a peak -- then don't march the troops at that frequency. Or the output motion of the bridge is the inverse Fourier transform of the Fourier transform of the input to the bridge from the soldiers times the transfer function of the bridge. If the signal from the soldiers puts too much power at a frequency that is high in the transfer function, then that is a "resonate frequency" and the bridge is threatened.

For the violin, yes, the human ear does a lot of Fourier analysis with the little hairs inside that coiled up thing, whatever it is called, but for the violin I was considering the beats when bow two adjacent strings together. So, bow the A string at 440 Hz together with the E string at about (3/2)440 Hz, and listen to the third harmonic of the A string and the second harmonic of the E string -- they should be the same frequency. If their frequencies differ by x Hz, then the sound will have amplitude modulations, beats, of x per second. That's how a violinist tunes the instrument.

The Fourier part? The sound from the A string is roughly periodic but not a sine wave. Similarly for the sound from the E string. Do Fourier analysis, that is, Fourier series on each of the two periodic signals. Take the third term from the Fourier series of the A string and the second term from the Fourier series for the E string, and listen to the beats. While the signals from each string are periodic, they are not sine waves, but the terms from the Fourier analysis are sine waves which is partly why the beats are so easy to hear. So, in effect, are using, say, the E string to find the Fourier coefficient of the third harmonic of the A string.

The A and E strings are separated in frequency by an interval of a perfect fifth which is 7 semi-tones or, from a piano, about 2^(7/12) which is close to 3/2. Well, that's the story for a perfect fifth. But there is also a perfect fourth, 5 semi-tones, a major third, 4 semi-tones, and a minor third, 3 semi-tones, a perfect 6th, 9 semi-tones, and, of course, an octave, 12 semi-tones. In perfect tuning, each of these intervals has, for the two notes, overtones with the same frequency (for two small whole numbers, one of them times the frequency of the lower note is the same as the other times the frequency of the higher note) so that can tune the interval by listening to beats. Of course, the easiest one to hear is the octave.

So, when using two strings this way, really are doing a Fourier analysis using one string to take the Hilbert space inner product of the two signals and using a sine wave from one of the overtones and its inner product with the other signal to scan that signal for beats and, thus, where the overtones are and, thus, really doing what Fourier analysis does, a projection via Hilbert space inner products onto orthogonal axes (sine waves at the overtone frequencies, that is, at frequencies that are whole number multiples, whole numbers again, of the frequency of the original periodic signal). So, adjust one of the two strings and in effect scan the sound from the other string for it's overtones -- that's essentially Fourier analysis. I will resist writing out all the math, but, really, a lot of Fourier theory is fairly intuitive stuff where a lot of the main ideas can be done with just pictures.

Or, yes, it would be a little simpler to play just, say, the A string and have an audio oscillator that puts out a sine wave and, then, slowly sweep the frequency of the oscillator sine wave and listen for beats with the sound of the A string -- that's essentially just Fourier series analysis of the periodic signal of the A string. But if don't have an audio oscillator handy, and use another string, say, the D or E string, and its overtones, each of which is a sine wave, and adjust the frequency of the D or E string and, thus, sweep a selected (listen carefully!) overtone of the D or E string past selected overtones of the A string and do much the same as with the audio oscillator.

Gee, I knew there was some Fourier theory in there somewhere!

Ok, I see your point of how a violin can be used to "search" a solution space for the transform. I still think it would be a bit of a stretch to say the violin performs the Fourier analysis here, as the human brain is required to do computation as part of that process.

As for the bridge, I doubt that soldiers were informed by anything other than trial and error, so I don't think they are an example of an 18th-century Fourier transform. Even when engineering on bridge resonance began it was probably focused on simpler classical standing wave calculations.

In the violin tuning, a human just listens to the beats. In principle, a tone deaf person could do that. So, the beats just need sensitivity to volume, amplitude modulation, not really frequency. So, the frequency stuff is in the violin, not the human.

Computation is not defined by difficulty for a human. It is closer to being defined by the ability to follow a (possibly arbitrary) decision procedure. In this violin method, a human is required to follow a decision procedure, unlike in a prism or the "Harmonic Analyzer" above.

Fine, but the Fourier part is really in the violin and not in the human. Listening for beats is very old stuff in signals, and no doubt circuits and/or software are sometimes used to detect them; if so, then we can do the stuff with the beats without humans.

Hello, I just have to ask, are you suffering from mental illness? If you aren't, it may be a good idea to get to a GP/Doctor and get checked out. Your comments on this thread read EXACTLY like some of the writing I have read by a family member with Schizophrenia. Feel free the flag / down vote me, but if you aren't aware then it could potentially help a lot...

Your comments on this thread read EXACTLY like some of the writing...

Care to elaborate on why that is the case?

Let's see: (1) Make some progress learning to play violin. I did. E.g., I made it through not all of but over half of the Bach "Chaconne", regarded as great music and challenging by nearly all violinists. (2) Learn some Fourier theory, pure and applied. I did that, for work with the fast Fourier transform on sonar problems for the US Navy and other problems. Also I took some grad math courses that covered Fourier theory carefully, right, based on measure theory.

I wrote the material here quickly, and better explanations could be possible:

For a violin, when tuning, and really also for much of the playing, to get the frequency ratios correct, which is most of what playing a violin with good innotation is about, use overtones, that is, the terms of a Fourier series expansion of a periodic (not necessarily sine or cosine) signal. In particular, when bow two strings together, i.e., at the same time, say, the A and the E, with the A already at 440 Hz from, say, a tuning fork, and slowly adjust the frequency of the E string, then are, in part, adding an overtone of the A string with the signal of the E string and, really, as adjust the E string, sweeping in frequency, as in the terms of a Fourier series, a sine wave overtone of the E string the terms of the Fourier series of the A string. When that overtone of the E string gets close to the frequency of a term in the Fourier series of the A string, get beats, that is, an amplitude modulation which violin students learn to listen for and hear. When the beats go from a few a second down to less than one a second and basically go away, then have found the frequency of the desired overtone of the Fourier series of the A string, that is, have essentially part of the Fourier series of the A string.

As do other cases of bowing two strings together, get to find more overtones: E.g., want to use a finger of the left hand on the A string to play B, C, C#, D and E. E.g., Beethoven's 9th Symphony has "Ode to Joy" and can play that in A Major with C# C# D E E D C#, .... Well, to get the B correct, bow it with the E string and look for a perfect 4th. For the C, look for a perfect major third. For the C#, look for a perfect minor third. For the D, bow with the open D string an look for an octave. For the E, bow with the E string and look for unison. In eadh case, as adjust finger on the A string, will be doing a sweep in frequency looking for a term in the Fourier series of the other string.

For the bridge, treat it as a linear system. Then given and input signal, to get the output, take the Fourier transform of the input, multiply it by the impulse response of the bridge, and then take the inverse transform. The impulse response is what get when hit the bridge with an impulse, that is, a signal with all frequencies with equal power. If the bridge has a resonant frequency and the troops march with that frequency, then the product of the two Fourier transforms and the inverse transform will be large and the bridge might fail. Fourier transforms win again.

My comments on Fourier theory are fine and should be entertaining for the HN audience.

I wrote the remarks quickly and kept the content intuitive. If I wrote it all out in terms of measure theory, then I'd be still more difficult to read. That you found something objectionable with what I wrote is absurd.

Your remarks are ignorant about Fourier theory and/or just hostile to me. A guess is that I wrote something you didn't understand and, thus, you got hostile. Such hostility is not appropriate on HN.

Put the two together and the criticize what I wrote about where essentially Fourier theory pops up playing a violin. There's more, e.g., the image through a lens of a point source and, then, much of antenna theory, right, also for sonar, especially the phased array case. And there's the issued of power spectral estimation -- did quite a lot of that via Blackman and Tukey.

Right, the Michelson-Morley interferometer, like Young's double slit, is basically antenna theory and, thus, also Fourier theory. I omit the details of the math.

What I wrote was supposed to be fun reading.

There's nothing wrong with what I wrote. Maybe you don't like it; and of course it was not a full course in Fourier theory; and I omitted the math; but for much of a STEM technical audience it should have been easy to read.

Your medical diagnosis is totally wacko nonsense, incompetent, irresponsible, erroneous, inappropriate, insulting, and provocative.

Here's your logic: You know some sick people who write. You observe that I write. So, you conclude that I must be sick. Erroneous. Nonsense.

> What I wrote was supposed to be fun reading.

It was! As a (very) amateur-level musician and programmer, I greatly enjoyed reading your comment. It took a couple of times (because of my shaking understanding of Fourier transforms, not your writing), but I understood your point in the end.

So thanks for sharing. I'm glad you're enthusiastic about this stuff, it'd make a great blog post.

Just because someone is incredibly passionate about something technical that you don't understand doesn't mean that person has a mental illness. It's pretty clear from reading what he's written that it isn't technobabble or nonsense in the least. Schizonphrenic individuals do not generally write comprehensible and logically sound ideas down. The worst you could characterize graycat's comments as are "quirky". Your comment is both hurtful, since it was made publicly, and completely wrong.

This is apropos for the topic. Before the era of digital computers, ships used to use analog fire control computers.

It is fascinating to watch these. It reminds me to pay attention to the fundamentals, the particulars are only temporary.

http://www.eugeneleeslover.com/VIDEOS/fire_control_computer_...

It's hardly 'long-forgotten'...

From Körner's book 'Fourier Analysis' (CUP, 1988):

"[...] Kelvin ... designed and built a ... machine (the harmonic analyser) to perform the task 'which seemed to the Astronomer Royal so complicated and difficult that no machine could master it' of computing the coefficients from the record of the past height [of tides].

Kelvin's harmonic analyser has a good claim to be the grandfather of today's computers not only because he obtained government money to build it but also because it represents the first major victory in the struggle 'to substitute brass for brain' in calculation. It is pleasant to record that Kelvin's instruments were so well adapted to their purpose that it took electronic computers 20 years to replace them.' - p.30-1

and

'We have seen ... how Kelvin invented machines which could compute periodic functions from their Fourier series and conversely obtain the Fourier series of a given periodic function. One such machine was constructed by Michelson to work to a higher accuracy and to involve many more terms that previous models. (Michelson's ability to build and operate equipment to new standards of accuracy was legendary. Of his interferometer which he invented and used in the Michelson Morley experiments it was said that it was a remarkable instrument - provided you had Michelson to operate it. His experiments to measure the diameter of the nearest stars using an interferometer were not reproduced for 30 years.)" - p.62

>Of his interferometer which he invented and used in the Michelson Morley experiments it was said that it was a remarkable instrument - provided you had Michelson to operate it. His experiments to measure the diameter of the nearest stars using an interferometer were not reproduced for 30 years.

From the perspective of the scientific method ("on the words of no one" etc), that's ... not actually a good thing.

I love Körner's book: it is amazing, interesting, accurate, his personality shines through . . . . I spent quite a bit of time with it when we had debates about some of the technical issues underlying the machine. This book likely isn't the place to start your study of fourier methods, but it is THE place to end!

Direct link to the eBook PDF: http://www.engineerguy.com/fourier/pdfs/albert-michelsons-ha...

Direct link the the accompanying video playlist: https://www.youtube.com/watch?v=NAsM30MAHLg&list=PL0INsTTU1k...

An even older "machine" that performs Fourier analysis: a lens [1].

[1] http://en.wikipedia.org/wiki/Fourier_optics

I wrote a blog post[1] about the Fourier transform and lenses. Lenses taking Fourier transforms has surprising evolutionary effects.

[1]:http://scottsievert.github.io/blog/2014/05/27/fourier-transf...

If I remember right, this machine inspired the discovery of gibbs phenomena. One of the few times a mathematical result was seen in practice before writing it down on paper.

Reply on news.ycombinator.com