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.
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! :)
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.
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...
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.
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.
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.
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.
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
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!
>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.
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.
