April 2007 - mathfun@mailman.xmission.com

[math-fun] pi to 500 digits on piano
by Thane Plambeck 01 May '07

01 May '07

I can't decide if it's the F major drone in the accompaniment or the (human determined?) rests that makes this so musical: http://www.tomdukich.com/math%20pi%20piano%20keyboard.html -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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[math-fun] sunday funnies
by Schroeppel, Richard 01 May '07

01 May '07

A factoring cartoon. --Rich http://xkcd.com/c247.html

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Re: [math-fun] Fourier transforms
by Daniel Asimov 30 Apr '07

30 Apr '07

Henry writes: << Fourier analysis can easily separate out the various sinusoidal frequencies in a waveform. Is there an analogous analysis that can separate out various real exponentials in a real waveform? I.e., if a signal is the sum of various real exponentials (i.e., no sinusoidal components), is there a simple analysis that will pull out the coefficients & exponents? Is there a "fast" version analogous to the FFT for this procedure? I recall studying Laplace transforms, but can't recall whether they solve this particular problem. >> It would indeed seem that the Laplace transform is what you're looking for; see < http://en.wikipedia.org/wiki/Laplace_transform >. Since the (bilateral) Laplace transform is equivalent to the Fourier transform under a change of variables (real <-> imaginary), there by rights should be a corresponding fast [or finite] Laplace transform. (Cf. V. Rokhlin, "A fast algorithm for the discrete Laplace transform", J. Complexity, v. 4, 1988.) In simple discrete cases -- say where you already know f(x) == sum_{n=-oo to oo} c_n exp(nx) then integrating f(x)*exp(-n_0 x) over the imaginary interval [0, 2pi*i] will equal 2pi*i c_n (modulo technical details). --Dan

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Re: [math-fun] pi to 500 digits on piano
by Daniel Asimov 29 Apr '07

29 Apr '07

Thane wrote: << I can't decide if it's the F major drone in the accompaniment or the (human determined?) rests that makes this so musical: http://www.tomdukich.com/math%20pi%20piano%20keyboard.html >> Now I'm curious to hear a musical pi transcribed more naturally. Say, the digits of pi in base 2, expressed mod 12 . . . or better, pi expressed in "factorial base", also mod 12. (With Z_12, of course, corresponding to the notes of an octave. Or perhaps better, to the simultaneous playing of a coset of all tones whose frequency ratios are powers of 2 (i.e, with the same chroma) and within the range of human hearing.) --Dan

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Re: [math-fun] sunday funnies
by Daniel Asimov 29 Apr '07

29 Apr '07

Michael wrote: << Rich wrote: > A factoring cartoon. --Rich > > http://xkcd.com/c247.html 1) In general, xkcd rocks. He's talking at MIT on 5/14. 2) Someone implemented the factoring clock, of course. http://hacks.atrus.org/factor_clock/ >> I don't know the factoring algorithm used here, but if you try to count seconds along with the clock, it's clear that harder numbers to factor correspond to, um, longer seconds. --Dan

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Re: [math-fun] pi to 500 digits on piano
by David Gale 29 Apr '07

29 Apr '07

Surely the rests are not human determined. I suspect they correspond to the digit zero. The transcribers chose to use the scale of F. I guess mathematicians are in some ways very much like people. We have our own myths, rituals and superstitions with our pi-dolatry. Surely pi- music will be musically indistinguishable from sqrt[2]-music or the music of any random sequence of digits. If this should turn out to be false maybe I too will become a pi worshipper. dg At 10:41 PM 4/28/2007, you wrote: >I can't decide if it's the F major drone in the accompaniment or the (human >determined?) rests >that makes this so musical: > >http://www.tomdukich.com/math%20pi%20piano%20keyboard.html > >-- >Thane Plambeck >tplambeck(a)gmail.com >http://www.plambeck.org/ehome.htm >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] pi to 500 digits on piano
by N. J. A. Sloane 29 Apr '07

29 Apr '07

Re: listening to pi: Try the new improved (thanks to David Applegate) "listen" button on the OEIS. A000796 is dec. exp. of pi, A001203 is the cont. fraction and sounds a lot better. You can choose any of the 128 musical instruments that MIDI can synthesize (try #102!). 10,000 digits or 20,000 terms of the cont. frac. can be played By the way, > I'm giving a talk in the Applied Math Dept at MIT on Monday Apr 30 > about the OEIS. Tea at 4:00 in Room 4-174 (Math Majors Lounge), > talk 4.30-5.30 in Room 2-105. See http://www-math.mit.edu/amc/spring07/ > for more information and a link to a nice poster. I'm hoping this > will be an opportunity to meet OEIS fans that I've never met as well > as to see old friends. Neil

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Re: [math-fun] Fourier transforms
by Daniel Asimov 29 Apr '07

29 Apr '07

Henry writes: << The sinusoids are orthogonal to one another, allowing for the construction of excellent filters to detect a strong signal. The real exponentials aren't orthogonal, at least under the traditional defn of inner product. Perhaps the cleverness of Laplace was in coming up with a different inner product? >> Does this have a direct bearing on what I wrote below? [edited: c_n -> c_(n_0)] --Dan At 04:49 AM 4/28/2007, Daniel Asimov wrote: >Henry writes: > ><< >Fourier analysis can easily separate out the various sinusoidal frequencies in a >waveform. > >Is there an analogous analysis that can separate out various real exponentials in >a real waveform? > >I.e., if a signal is the sum of various real exponentials (i.e., no sinusoidal components), >is there a simple analysis that will pull out the coefficients & exponents? >Is there a "fast" version analogous to the FFT for this procedure? > >I recall studying Laplace transforms, but can't recall whether they solve this >particular problem. >>> > >It would indeed seem that the Laplace transform is what you're looking for; >see < http://en.wikipedia.org/wiki/Laplace_transform >. Since the (bilateral) >Laplace transform is equivalent to the Fourier transform under a change of >variables (real <-> imaginary), there by rights should be a corresponding >fast [or finite] Laplace transform. (Cf. V. Rokhlin, "A fast algorithm for the >discrete Laplace transform", J. Complexity, v. 4, 1988.) > >In simple discrete cases -- say where you already know > > f(x) == sum_{n=-oo to oo} c_n exp(nx) > >then integrating f(x)*exp(-n_0 x) over the imaginary interval [0, 2pi*i] >will equal 2pi*i c_(n_0) (modulo technical details). > >--Dan >>

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Re: [math-fun] Fourier transforms
by Henry Baker 28 Apr '07

28 Apr '07

The sinusoids are orthogonal to one another, allowing for the construction of excellent filters to detect a strong signal. The real exponentials aren't orthogonal, at least under the traditional defn of inner product. Perhaps the cleverness of Laplace was in coming up with a different inner product? At 04:49 AM 4/28/2007, Daniel Asimov wrote: >Henry writes: > ><< >Fourier analysis can easily separate out the various sinusoidal frequencies in a >waveform. > >Is there an analogous analysis that can separate out various real exponentials in >a real waveform? > >I.e., if a signal is the sum of various real exponentials (i.e., no sinusoidal components), >is there a simple analysis that will pull out the coefficients & exponents? >Is there a "fast" version analogous to the FFT for this procedure? > >I recall studying Laplace transforms, but can't recall whether they solve this >particular problem. >>> > >It would indeed seem that the Laplace transform is what you're looking for; >see < http://en.wikipedia.org/wiki/Laplace_transform >. Since the (bilateral) >Laplace transform is equivalent to the Fourier transform under a change of >variables (real <-> imaginary), there by rights should be a corresponding >fast [or finite] Laplace transform. (Cf. V. Rokhlin, "A fast algorithm for the >discrete Laplace transform", J. Complexity, v. 4, 1988.) > >In simple discrete cases -- say where you already know > > f(x) == sum_{n=-oo to oo} c_n exp(nx) > >then integrating f(x)*exp(-n_0 x) over the imaginary interval [0, 2pi*i] >will equal 2pi*i c_n (modulo technical details). > >--Dan

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[math-fun] Fourier transforms
by Henry Baker 27 Apr '07

27 Apr '07

I hope that this is a trivial question. Fourier analysis can easily separate out the various sinusoidal frequencies in a waveform. Is there an analogous analysis that can separate out various real exponentials in a real waveform? I.e., if a signal is the sum of various real exponentials (i.e., no sinusoidal components), is there a simple analysis that will pull out the coefficients & exponents? Is there a "fast" version analogous to the FFT for this procedure? I recall studying Laplace transforms, but can't recall whether they solve this particular problem. Thanks for any references on this problem.

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