Why is the sound of a flute different than the
sound of a clarinet? How does your brain recognize different
instruments by their timbre? Today I will show you how to
decompose a musical sound into constituents that reflect the way we
recognize different instrument sounds.
I recently bought a Sansa Clip+ music player. It is like the
older generation iPod Nano but much less expensive since it is not
made by Apple. I use it primarily to listen to podcasts -
specifically, recorded lectures of university classes. There are a
wide array of fascinating topics to listen to, from high-quality
places like Yale and MIT where physically sitting in the class as a
student would cost, quite literally, thousands of dollars. I
particularly like listening to Humanities classes, since you can
generally follow along quite well with just the audio. This past
winter I listened to an excellent Philosophy class by Shelly Kagan
at Yale. I am currently enjoying a seminar on Tolkien's The
Silmarillion by "The Tolkien Professor".
Numbers I presented one way of thinking about complex
numbers - numbers like the square root of negative one, which do
not "exist" in the "real" world, but which are nevertheless quite
useful in many scientific applications.
That article motivated imaginary numbers as the solution to word problems like "find a number whose square
is -1", just as negative numbers solve word problems like "find a
number which when added to 5 gives 2". Today I'd like to show you a different way of thinking about them.
I recently came across a fascinating
article that shows how to produce music using a one-line C program.
Granted, the musical result is not in the same league as a Bach fugue,
but the fact that a 57 character C program can produce sound patterns
with interesting dynamic textural variation is really
neat. Particularly so because the formula only involves arithmetic and
bitwise operations on whole numbers between 0 and 255: no trig
functions, lookup tables, Fourier transforms, or any of the many other
modern tricks for sound synthesis.
The original inventor of this technique is a Finnish
artist/programmer called viznut. He posted several articles and
video/sound-recordings about the idea in 2011, which then spawned a
whole community of people doing experiments. You can find the original
articles and listen to sound-clips at
symphonies from one line of code -- how and why?.
How can we measure complexity?
Whether we examine the pattern of notes of a piece of music, the pattern of
light and color in a painting, or the pattern of architectural
features in a building, it is intuitively clear that some patterns are more
complex than others. Is there a way to quantify this idea
mathematically? Yes; in fact, there are several ways, with interesting
connections to Information Theory and Entropy.
Today we begin with a short illustration of one approach
(known as Kolmogorov-Chaitin Complexity) from Guest
A. Salingaros, Professor of Mathematics, Urbanist &
Architectural Theorist at the University of Texas at San Antonio.
I will follow his contribution with a few additional remarks of my own.
Nikos A. Salingaros, The University of Texas at San Antonio.
Music has many connections with
mathematics. Approximately 2500 years ago, the ancient Greek
mathematician Pythagoras and his students observed that the sounds of
two plucked strings blend nicely if the ratio of their lengths is a
ratio of small whole numbers. Today I want to explore the way our
modern Western 12-note musical scale relates to that observation.
Suppose you have $100 and you decide to go to a casino to try to
double your money. Are you better off putting the whole $100 on a
single bet, or should you make a lot of smaller bets? Maybe you should
adjust the size of your bet as the evening progresses? Should you stop
as soon as you reach $200 (if you ever do), or keep going?
Lots of people have opinions about questions like these. Today, I will
show you how to calculate the correct answers yourself with
just a few lines of code in R. Even better, you will understand
the approach, which means you can do your own analysis of whatever
strategy you want to test. And best of all, it's free: no need to
spend real money at an actual casino to find out.