5 ms·

Nineteenth century machine that performs Fourier analysis

86 points by Phithagoras 12y ago

graycat 12y ago
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.
- appleflaxen 12y ago
  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.
  - graycat 12y ago
    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.
- yzzxy 12y ago
  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?
  - graycat 12y ago
    > 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!
amelius 12y ago
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! :)
- jacobolus 12y ago
  http://www.youtube.com/watch?v=s1i-dnAH9Y4 http://www.youtube.com/watch?v=s1i-dnAH9Y4
saretired 12y ago
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.
- cbd1984 12y ago
  Probably the publication bias against negative results. /s
- InclinedPlane 12y ago
  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.
- maaku 12y ago
  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 ;)
spitfire 12y ago
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_1.html http://www.eugeneleeslover.com/VIDEOS/fire_control_computer_...
gaze 12y ago
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.
timothya 12y ago
Direct link to the eBook PDF: http://www.engineerguy.com/fourier/pdfs/albert-michelsons-harmonic-analyzer.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=PL0INsTTU1k2UYO9Mck-i5HNqGNW5AeEwq https://www.youtube.com/watch?v=NAsM30MAHLg&list=PL0INsTTU1k...
mdturnerphys 12y ago
An even older "machine" that performs Fourier analysis: a lens [1]. [1] http://en.wikipedia.org/wiki/Fourier_optics http://en.wikipedia.org/wiki/Fourier_optics
- skierscott 12y ago
  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-transforms-and-optical-lenses/ http://scottsievert.github.io/blog/2014/05/27/fourier-transf...
amathstudent 12y ago
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
- SilasX 12y ago
  >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.
- engineerguy 12y ago
  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!

86 points by Phithagoras 12y ago

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.

appleflaxen 12y ago
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.
- graycat 12y ago
  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.
yzzxy 12y ago
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?
- graycat 12y ago
  > 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!

amelius 12y ago

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! :)

jacobolus 12y ago
http://www.youtube.com/watch?v=s1i-dnAH9Y4 http://www.youtube.com/watch?v=s1i-dnAH9Y4

saretired 12y ago

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.

cbd1984 12y ago
Probably the publication bias against negative results. /s
InclinedPlane 12y ago
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.
maaku 12y ago
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 ;)

spitfire 12y ago

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_1.html http://www.eugeneleeslover.com/VIDEOS/fire_control_computer_...

gaze 12y ago

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.

timothya 12y ago

Direct link to the eBook PDF: http://www.engineerguy.com/fourier/pdfs/albert-michelsons-harmonic-analyzer.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=PL0INsTTU1k2UYO9Mck-i5HNqGNW5AeEwq https://www.youtube.com/watch?v=NAsM30MAHLg&list=PL0INsTTU1k...

mdturnerphys 12y ago

An even older "machine" that performs Fourier analysis: a lens [1]. [1] http://en.wikipedia.org/wiki/Fourier_optics http://en.wikipedia.org/wiki/Fourier_optics

skierscott 12y ago
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-transforms-and-optical-lenses/ http://scottsievert.github.io/blog/2014/05/27/fourier-transf...

amathstudent 12y ago

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

SilasX 12y ago
>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.
engineerguy 12y ago
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!