
Nineteenth century machine that performs Fourier analysis - Phithagoras
http://www.engineerguy.com/fourier/
======
saretired
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.

~~~
InclinedPlane
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.

------
amelius
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
[http://www.youtube.com/watch?v=s1i-dnAH9Y4](http://www.youtube.com/watch?v=s1i-dnAH9Y4)

------
graycat
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.

~~~
yzzxy
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
> 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!

~~~
robinduckett
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...

~~~
graycat
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.

~~~
eric_bullington
> 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.

------
spitfire
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_...](http://www.eugeneleeslover.com/VIDEOS/fire_control_computer_1.html)

------
amathstudent
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

~~~
engineerguy
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!

------
timothya
Direct link to the eBook PDF: [http://www.engineerguy.com/fourier/pdfs/albert-
michelsons-ha...](http://www.engineerguy.com/fourier/pdfs/albert-michelsons-
harmonic-analyzer.pdf)

Direct link the the accompanying video playlist:
[https://www.youtube.com/watch?v=NAsM30MAHLg&list=PL0INsTTU1k...](https://www.youtube.com/watch?v=NAsM30MAHLg&list=PL0INsTTU1k2UYO9Mck-i5HNqGNW5AeEwq)

------
mdturnerphys
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
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...](http://scottsievert.github.io/blog/2014/05/27/fourier-transforms-
and-optical-lenses/)

------
gaze
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.

