Wednesday, June 21, 2017

Musical Notes and the Hilbert Curve

3blue1brown logo
seriously go watch the videos
Grant Sanderson makes a great series of YouTube videos, called 3blue1brown, that explore math concepts with animations.  This video talks about Hilbert curves, which are continuous fractal curves that completely fill a two-dimensional area.  The video uses them for a hypothetical system of encoding a two-dimensional image (as a grid of pixels) into a one-dimensional sound (as a range of frequencies).  I'm not sure such a system would work very well as described (to me, this system called "vOICe" seems a bit more practical) but that's not really the point of the video.  It's just a setting for talking about the curves, and how a concept like "approaching infinity" can have practical applications.

Fourth order Hilbert curve
would not make a good maze
Here, the Hilbert curve is used as a mapping from a 1D space into a 2D space.  A useful property of these curves is that when you increase the resolution (or the number of iterations) the mapped 2D location of a given 1D value is not drastically changed.  So someone can get used to 0.083 being at the top-center of the image, and it doesn't much matter what resolution is chosen for the curve - it's still basically at the top-center.

After watching the video, I wondered about the reverse procedure.  That is, what kind of images would it take to produce a given sound.  In particular, where would the standard musical pitches lie on the curve, and what would chords look like?

So I made a little web experiment to find out.  I call it Hilbert Chords.  Here's how it works:  I mapped the 1D space onto a single octave, the range of frequencies between middle-C and high-C.  It's a logarithmic scale, so the 12 pitches in the chromatic scale (plus one for the repeated C) are spaced evenly apart.  You can move your mouse over either the 2D or 1D regions to hear the corresponding pitch and also see where each point maps to in the other space.  Also you can click a handful of buttons that play preselected chords and see what the chord would look like.

locations of pitches on the chromatic scale
in color.. because, you know..
I was surprised, at first, to see that the notes all lie on corners of the four quadrants.  After thinking about it a bit, it makes sense.  Each quadrant connects to the next one at one of its corners.  The pair of corners connecting two quadrants are basically the same pitch.  So that leaves three distinctly-pitched corners per quadrant, for a total of twelve (and again there's the final corner for the repeated C).

When we increase the number of iterations, the curve gets denser, but the points representing the pitches move less and less.  As the number of iterations approach infinity, the pitch-points approach specific points on the square.  I don't have a complete understanding of the math, but it seems to me that four of them (E, G#, and the two Cs) approach the corners, six of them (C#, F, G, A, and B) approach the centers of the sides, and three of them (D, F#, and A#) approach the center.  At sufficiently high resolution, you wouldn't be able to see the difference between some very different chords, and the D augmented chord would look like a single point in the middle.

Ok, not a good visualization of the chromatic scale.  A plain circle is much better.  Or even just a keyboard.

Tuesday, June 13, 2017

Do not Discharge Stink Bombs in Andover

I'm not a lawyer.  Let's start with that.

(Los Angeles, not Andover)
credit: Steve Isaacs
But I was, you know, browsing through the code of bylaws of Andover, Massachusetts, and a few things jumped out at me.  First, the prohibition on certain items of amusement in Article XII §39(c):
The sale, distribution and discharge of stink bombs, smoke bombs and aerosol silly string products within the Town of Andover are prohibited.
This is apparently not an uncommon kind of ban.  Silly String is also banned in Middleborough, among other towns, and notably in Los Angeles, though only on Halloween.  Stink bombs are banned in Oklahoma City.  And there's New Hampshire 2017 House Bill 100, which would provide an exemption for "toy smoke devices" on the apparently more general prohibition of smoke bombs.  I guess serious smoke bombs would still be banned.

I don't know of any other town that has banned all three.  But I didn't do the most thorough of searches.  Nor did I, in ten minutes of googling, see any codes that provided an exception like that of §39(d):
This by-law shall not apply to public safety personnel in the conduct of their duties.
(Don't worry, this was photoshopped)
credit: What_No_Cookie
So it's fine for a police officer to use silly string, as long as it's part of their job.  I don't know how often that comes up.  Maybe if they're demonstrating things you're not allowed to do?  "Hey kids, see this stuff I'm spraying everywhere?  This is Silly String.  Looks like fun, huh?  Well, you can't do this!  Also, drugs."

§39 was enacted in 1998, after the town "experienced severe problems" with these sorts of devices.  It's hard for me to imagine how tough it must have been back then; I wasn't living in Andover at the time, so who am I to judge?  One thing's for sure though, I haven't had any severe silly string problems since I've moved in.  So I guess the ban is working!

The rest of Andover's bylaws are, for the most part, as interesting as you would expect bylaws to be.

Awww yeah, candlepin.
Article XI §4, which allows the Selectmen to "grant licenses for the operation of bowling alleys on the Lord's Day between the hours of 1:00 and 11:00 p.m.".  I think that bit refers to "[Massachusetts] General Laws, Chapter 136, Section 4B", which doesn't seem to exist at the moment.  But it's probably talking about something in the Common Day of Rest Law, which goes on and on about things you're not generally allowed to do on Sunday in Massachusetts, at least not without a license: such as dancing in exchange for money, offering to watch people dance in exchange for money, and just generally conducting business.  I'm not a lawyer, and that whole law seems ridiculous, so I'm probably not reading it right.  Anyway, it's good to know that if there were any bowling alleys in Andover (there aren't) then they could operate on Sunday if they got the appropriate license.

What you can't get a license for, according to Article XI §8(a), is an "automatic amusement device [kept at a business and operated for hire] which presents a risk of misuse as a gaming device".  I'm not sure about "misuse as a gaming device".  Is this indicating that gaming devices in general are bad, and we wouldn't want anyone using a non-gaming amusement device for gaming?  Maybe the misuse is bad, but only the kind of misuse that can happen with gaming devices.  MGL Chapter 140 Section 177A seems to indicate that pinball machines are "automatic amusement devices", but I would assume using it as a gaming device wouldn't be misuse.

Somehow I think they had slot machines in mind here, but they also helpfully list characteristics associated with risky amusement devices.  If it has any of those characteristics, then no license for you.  For instance, any device that "involves matching, random numbers, patterns or cards".  Or one that "accumulates more than 26 plays" (26 seems like a strange threshold).  Look out for any device that's "equipped with a "knock off" switch, button or similar device", those also carry a risk of misuse, somehow.

Probably needs a rabies shot, though.
There is the bit where it says one may keep or board pet cows (part of Article VII §, which I guess is cool, and let me to discover in the Board of Health regulations that the "Keeping of Goats Permit" is only $30.

So in summary, I'm not a lawyer, but I think you can ride your goat to the bowling alley on Sunday.  And you can bring your Stink Bombs but you can't use them.  And, please, play pinball only in moderation.

Thursday, June 1, 2017

BlockFractal: generator of blocky fractal-like shapes

The frabjous isle of North Osia
BlockFractal is an algorithm that generates blocky fractal-like shapes.  The idea is to make something that would be suitable for a continent, island, or lake, in a procedurally generated 2d tile-based game.  The demo page shows the generated shape with a Google-Maps-like interface, where you can zoom and pan around.

First it generates a random place name, like New Gharnia, Superior Iukauton, or Twuq'eros.  That name is used as a random seed to generate a random fractal.  You can enter your own place name, and you can control two properties of the fractal generation.

One is "iterations".  This sets how many times the BlockFractal algorithm is run on the initial seed shape.  The initial shape is a simple square, but each iteration adds finer and finer random detail to it.  Drag the iterations slider around to see how the shape was built.

The other is "variation".  This basically affects the craziness of the shape.  Low variation produces a simpler but blockier shape.  High variation produces a shape with many more twists and turns.  Somewhere in the middle seems to be pretty good for continent-looking shapes.

Seed square
Iteration in progress
New shape after first iteration

There's a thorough description of the algorithm at the Algorithm Overview page on the github site, but basically it goes like this:  Take the initial seed shape (a 2x2 square in the first of the three example images above), double it, and then for each unit edge make a random decision to leave it be or to move it "inward" or "outward".  That produces a new shape which is more complicated, and around twice as big, as the last one.

The frumious land of Yixonia
Repeat this algorithm as many times as you like, though it does take twice as much memory with each iteration, so to avoid memory issues I stopped the demo at 9 iterations.

There are a few edge cases to consider, for instance I don't want the path to intersect itself.  If you're curious the Algorithm Overview page has more details.

The resulting shapes make me think of Koch snowflakes and quadric fractals.  Also, check out Brainfilling Curves, for some similar and amazing shapes.

As you play with the demo, imagine if this were the high-level map of a top-down RPG or roguelike game.  I like the idea of scattering some different biomes and rivers around, randomly placing some cities and towns, and maybe using some Delaunay triangulation to connect them with roads.