The music of pi
will convert the first 10,000 digits of pi (3.14159...) into music, based on starting notes of your choosing.
From fuyugare's comment here.
I just wish I had headphones so I could try it out...I'll just have to wait until I get home :(
Danke schön!
Ditto that. No audio here, either.
Here's a question, in case there are mathematical monkeys about: pi is an irrational number, yes? Yet, given the formula for a circle's area, pi could be expressed as the area divided by the square of the radius.
Now this means one of two things: either pi is an irrational number that can magically be expressed by a ratio (a logical contradiction), or somehow the area and/or radius of a circle must be irrational, which seems so counterintuitive that it makes me wish I was drunk right now. Surely the area and radius of a circle would be a definite value expressed by a rational number?
Please hope me.
Oh, cool link.
A rational number is not just any 'ratio', but a ratio of integers.* Every real number can be expressed as the ratio of two real numbers (the number itself and 1, for instance), but not every real number is rational. Furthermore, there is nothing particular about areas that require them to be rational.
* more precisely, a ratio of an integer over a non-zero integer.
[IHBT, probably.]
I find it amazing that we can actually calculate pi. One would think it could only be physically measured and then held as a standard.
Irrational music.
Specifying two octaves of pentatonic yields a sound like a gamelan played on a cheap Casio.
[on prev, fuyu already defined irrationality]
This reminds me of a neighbor's art studio. The previous occupant used spraypaint templates and covered all of the walls with the digits of pi. Started in the upper left and worked around the room. Very, very cool link (thanks fuyugare and rocket88).
roly: the infinite series 1 - 1/3 + 1/5 - 1/7... converges on pi/4. The more terms you calculate, the more digits of pi you get.
But as long as both the area and radius are rational, then the ratio of the area to radius must be rational, by definition. So the only way pi can be irrational is if one or both of the values is irrational. It seems very counterintuitive to me to say that the area or radius or circumference of a circle cannot be measured exactly.
Sorry. I'll stop with the derail now.
OH MY GOD.
I knew this kid in high school who had pi memorized to over 500 digits (actually over 1000 last I heard of him) and this is exactly how he did it.
He had perfect pitch so he just made "the pi song." It sounded AWFUL but it was an effective mnemonic-type device for him.
MCT, pi is not the ratio of the area to the radius. It's the ratio of the diameter to the circumference.
You will never find a circle with a simultaneously rational diameter and circumference.
I mean, the circumference to the diameter.
Uh, middleclasstool, the radius and area can't both be rational.
For instance, if the radius is 3, the area is 9*pi, an irrational number. They won't both be rational, EVER.
And, uh, what goetter said.
It's the same with 45/45/90 triangles. The ratio is 1:1:sqrt2. Which means that it's also an irrational ratio because one of the sides must always have an irrational length.
It seems very counterintuitive to me to say that the area or radius or circumference of a circle cannot be measured exactly.
What do you mean by exact? The radius of a unit circle is 1 unit. Its area is also 1 unit (in base-π). These are pretty exact figures!
Also, pi seems a boring number to make into music, as its decimal expansion is too random. Intuitively, I conject that a more biased sequence would map to a more interesting sound.
Hey, lookit. MCTool's going nerd-hunting.
"Shhhh... ah think thar's a whole thicket of 'em in that thar filter. Lemme flush 'em.
PI MUST BE RATIONAL!"
[SFX: confused squawking]
[SFX: multiple shotgun reports]
"Hoo-ee. Them's good eatin'."
goetter, I always thought they would make better trophies than vittles.
(man, I am dying to record that piece)
goetter wrote:
Also, pi seems a boring number to make into music, as its decimal expansion is too random. Intuitively, I conject that a more biased sequence would map to a more interesting sound.
I think so, too.
If you draw a curve that goes through all the notes, you get a complex sine-wave. If you so a fourier analysis of this wave, you can get a frequency distribution of how the music is changing. Low frequencies correspond to longer, slower changes, and high frequencies to quicker, more abrupt changes.
If I remember correctly from studying fractally generated music, random sequences that make the "most music-like" music have a frequency distribution of 1/f, or so-called "pink" noise. This implies that the lower frequencies are more highly represented, which is to say the transitions between the notes flow farily smoothly, but not too smoothly since you also have some small amount of high-frequency quick changes to shake things up. However, these quick changes are low-amplitude, so you don't get many big jumps between notes.
Human-composed music if analyzed will usually have a frequency distribution pretty close to 1/f.
If you smooth it out too much, say with a 1/f^2 frequency distribution, you lose most of the rapid variations between nearby notes, and it sounds too boring because there aren't enough changes.
I suspect that pi produces a "Brownian" frequency distribution which is even across the frequency spectrum, or so-called "white" noise. This makes music which changes too much and too fast to sound musical.
Interestingly enough, 1/f noise is the most common type of randomness found in nature. I suspect that our brains are structured to "like" this kind of music the best because we're tuned to recognize this pattern.
Neat link. A long time ago, I tried to convert pi to music in just this way and quickly realized that it would never cease to sound, well, like it does. I messed around a little with decoding schemes other than a digit=8th note, e.g. inserting rests and/or decoding sequences into note-length pairs. The result wasn't a whole lot better, but maybe I just chose the wrong decoding scheme.
I thought about but never implemented some kind of genetic algorithm to evolve a good decoding scheme. I suspect that wouldn't have helped much either, for fatoudust's reasons. If a pleasing decoding method were found, that would mean that the digits of pi may not be as random (in the uniform-distribution sense) as we thought. Not likely. Still, not a bad experiment to try.
Randomness is a funky concept. If you talk of randomness in the compressibility sense (can the sequence be represented by a smaller string), then pi is, in that sense, not terribly random.
Next, think of the (uncountable) infinitude of numbers like pi and what meanings they might have.
MM: That's an impressive method to memorize pi. I tried to do it the old fashioned way and only got around 100 permanent.
don't shoot me *hides*
All those complaining of the randomness of the music should note that you don't have to pick ten *different* notes for the digits. Try picking within a chord, and repeating some notes (like the root and fifth) several times within the ten notes.
Still not musical, but better than the major scale.
What do you mean by exact?
Be vewwy vewwy quiet. I hunting newds.
I think what gets me here is the concept that the measurement of the circumference (or area -- I meant the ratio of the area to the *square* of the radius, sorry) could be irrational.
It's counterintuitive to me, I suppose, because it seems to me that when you measure, say, the length of a line, there comes a point where your measurement stops. Maybe this is faulty reasoning on my part, but it strikes me as bizarre that something which clearly has a fixed length (e.g., the circumference of a circle), could be represented by a decimal number that never stops going and never repeats. I'm aware that there's a fallacy here in conflating the length of the line with the "length" of the number, but it just strikes me as bizarre that something of fixed, finite length cannot be represented by a fixed, finite number.
Great link, btw.
Irrational numbers do bend the brain!
I remember when I was studying the topology of the real numbers, and having something in me simply refuse to understand a key fact about how rational and irrational numbers relate. That even though between any two rational numbers there is an irrational, and between any two irrational numbers there is a rational, there still are "infinitely" more irrationals than rationals!
Infinitely big and infinitely small are pieces of cake compared to how weird things can get when you talk about infinitely interwoven.
(I can bring up math-y words like "cardinality" and go into more detail if anyone wants, but that isn't really on-topic. I guess I'm a frustrated math teacher at heart :) Too bad we don't have a topic-driven "monkeytalk" section... )
Actually, middleclasstool, you're having problems with something that is at the heart of a rather profound philosophical split in the mathematical community. Don't feel bad, because some very intelligent people have problems with that very same idea!
There are people who follow Brouwer's philosophpy of mathematical intuitionism who would also have a problem with the idea of an irrational number corresponding to a real length.
It's really the old saw, "infinity, does it exist?" from the ancient Greeks recast into mathspeak.
Uh, that's actually garbage, fatoudust. Intuitionism and irrational numbers are not at odds at all. See, for instance, any construction of real numbers as converging infinite computations. Modern intuitionism can be essentially boiled down to a rejection of non-constructive axioms like excluded-middle, or equally of rejecting truth without proof. (See eg. Per Martin-Löf's Siena lectures.)
(If you're talking about the so-called Ultrafinitism, then you'll have to do a lot more than simply cite the work of Volpin. I haven't seen any satisfactory presentation of it.)
Gah, converging
Fuygare, I know I read somewhere about a faction of intuitionism and the concept of irrational lengths. It's been over a decade for me, but I'll try to dig around and post a reference before this thread drops off the face of the monkey!
Thanks for calling me on it, though.
mct, hunt me if you like. I think you've hit it on the head. The fact is that any measurement of a real-life circle's dimensions will have to end at some decimal place, simply because we don't have measuring instruments with infinite precision. It's equally impossible to measure a real-life line that is exactly 0.333.... cm. So for any real-life circle you get an approximation of the radius and the circumference and from those get only an approximation of pi. So, as an added consequence to our lack of infinite precision, we can say that there is no such thing as a perfect circle (nor, for that matter, a perfect 45/45/90 triangle) outside the realm of abstraction. But if you don't believe me you can check that "Things That Don't Exist" song.
On preview: Yes, infinity is highly abstracted and poorly understood concept (by many including myself). The first barrier for me was to understand that infinity probably shouldn't be considered a number in most usages of the term, but an idea of endlessness or unboundedness. As it becomes more abstracted, wackier things develop, like sets of infinitely many points that add up to no "length" or "area" (e.g. fatou dust!). Or the interweaving of rationals, irrationals, and numbers like pi.
You've brought back memories of the involved process of constructing the real numbers from rational numbers. I am compelled to mention one of my favorite-sounding terms: Dedekind Cuts!
This thread is really pissing me off. Take a measuring tape graduated on one end in units of 1cm, and on the other in units of πcm. Then you can measure the radius of the circle and the circumference of the unit circle exactly. Don't think πcm is a valid unit? What exactly makes 1cm so special? Recall that the definition of a metre is defined as the distance travelled by light in some fraction of a second (a second defined as some number of state-transitions in a Cesium atom). Everything is a mutiple of some unit or other.
If you're a certain kind of Physicist, everything is some multiple of Plank-length or Plank-time anyway. All measurements are then integral!
err, Planck. *hangs head in shame*
regarding the measurability of an irrational number.
perhaps things that we think are whole, solid, and incapable of flux, really do change throughout, offering infintisimal and immeasurable changes that our perception is not able to register.
thus, whatever numbers we may use to measure things we perceive to be real, are in fact, not real, and the reality is that cosmos is indeed change, and only entertains our perception for our rational moments of discovery.
I like that. It's good to get some poetic philosophy in when one starts talking of infinity and the real world. Such a statement, of course, would apply to the measurability of any number, not just irrationals. This, on top of the lack of infinite precision in the empirical world, is why, in theory, you can make a tape measure graduated in both integer units and units of pi, and measure your formal unit circle with it, but outside the perfect abstraction of mathematics, such a tape measure (and unit circle) can only be approximated. So it isn't unreasonable that someone might have some cognitive dissonance with the infinite precision of pi if they take as their guide the world around them.
So, no, pi isn't particularly special here, nor is 1 (though both have interesting properties). We could be talking about a challenge to duplicate an arbitrary length of string and the same arguments would apply.
Perhaps someone could direct me to a perfect circle in the natural world?
in the natural world
Well, you know, so to speak. I find it useful shorthand.
Wolof - what about the ripples formed round a pebble landing in a still millpond?
So it isn't unreasonable that someone might have some cognitive dissonance with the infinite precision of pi if they take as their guide the world around them.
I would forgive them if they are similarly uncomfortable with any number.
I think this discussion got off on the wrong foot. There is no need at all to define π in terms of some measurement when there are any number of abstract definitions; for example: (a) smallest positive solution of sin(x) = 0, (b) square of the error function, (c) ln(-1)/i, etc. Why fetishize the circle?
I am scared by infinity.
The concept to understand is limits. A perfect circle can be thought of as the limit of the n-sided polygon as n approaches infinity. Infinity is never reached, just approached. Many infinite series and infinite sums have non-infinite solutions.
A perfect circle can be thought of as the limit of the n-sided polygon as n approaches infinity.
Or it could be thought of as a circle, which is easier to define and need not involve limits.
What fuyugare said.
Sorry if that sounded rude btw :(
I believe the ancient Greeks used this method (n-sided polygon) to calculate pi using known linear geometric principles.
what about the ripples formed round a pebble landing in a still millpond?
Wouldn't these ripples be formed by some finite number of water molecules?
Yes, but as you go further down, to atoms, then to (theoretically) quarks and strings, you begin to approach an infinite number. I think.
To me, perfection is as elusive a concept as infinity. Those ripples are probably as close as you can get in the natural world. I tend to believe that they still aren't perfect due to tiny perturbations. Or maybe by some chance they are perfect, but we will never be able to verify it.
Why fetishize the circle?
Initially because it's the simplest definition for non-mathematicians. It's a good place to start without diving head first into abstract land. Almost everyone has an idea of what a circle is. Unfortunately by defining it in terms of something we think we can measure, we back ourselves into a corner, and in order to get the "true" value of π we have to make the leap into abstraction anyway. Now in abstraction there's no way to measure the circle, so we need some way to get at the circumference, which is why Archimedes came up with the limiting procedure rocket88 mentioned.
The value of π shows up in so many places now that we could define it in many ways. Who knows, maybe it is best to start with some other definition, thereby avoiding the infinite precision quandary altogether, which isn't related at all to π but to the application of the abstract to the natural world. So we don't so much avoid the problem as delay it, unless we never leave abstract land.
converging