Mathematical structures in biology and art

So far in this class we have focused on tools having to do with the specifics of computer graphic displays and how they may be used for music production and performance.   This will allow you to open windows, see and manipulate on icons, like when you things with a mouse or pointer.  These are very practical tool, but they do not themselves show you how to make good or better music.

To make better music, we must acquire additional tools and skills.  This post will focus on a few sets of conceptual tools that are especially easy to use in the context of computer music.  Three concepts we will explore today are the Golden ratio, the Fibonacci sequence and self-similarity or fractals.   The examples provided will be graphic in nature, but its the numbers that are interesting, as we can use numbers in our DAWs with ease.  We can use these concepts to relate our creations to those of great artists and even creations in the natural world.

The Golden Ratio (also Golden mean, Golden section, Divine proportion) is a relationship between two sections where the ratio of the larger of the two sections relative to the smaller of the two sections is the same as the ratio of the sum of the two sections to the larger of the two.    While that description may be confusing on first read, it may be easier to check out the golden rectangle.

     lets say the length of the sides of square B  is 144 pixels, and the total length of A and B on the long side is 233 pixels.  This would make the sort side of rectangle a 89 pixels.  Running some simple calculations, you would see that the ratio between 144/89 (1.6179) is about the same as the ratio 233/144 (1.6180).  If we round these to 3 significant digits you get 1.618.    This number is the golden number.   Now that you have seen it in its simplest form.  Take a quick look at how often you see it in life and art: (google images)

We can apply this to music, by structuring the durations of the sections of our productions and compositions to have durations in golden ratio relationships.   For example we could have a section that is 8 bars long, followed by a section that is 5 bars long.   We will explore some of these options in labs and projects

You can read much more about the golden ratio and its application in arts at

The fibonacci sequence is a series of numbers where the next number in the sequence is the sum of the previous two.  It starts as follows:  1, 1, 2, 3, 5, 8, 13, ect.      An interesting feature of the fibonacci sequence is that as the numbers get bigger,  the ratios between adjacent numbers gets closer and closer to the golden ratio.      21/13 =  1.6153,     34/21 = 1.6190,  …  337/233 = 1.6180

Since music often works in chunks that must remain whole to remain in a genre or style (like a bar of 4 beats in dance music).  These fibonacci numbers are convenient counts to use when arranging sections to approximate golden ratio durations.

Self similar structures or Fractals, are shapes or objects where the parts have similar characteristics to the whole.  Some of the clearest examples are Sierpinski triangles.     You may have accidentally created such triangles by doodling in your not book.    Start by drawing a large equal sided triangle.  Then, inside this triangle draw another equal lateral triangle, upside down, by connecting the middle points of the larger triangle.  You will then see the large triangle, a smaller upside down triangle, and three even smaller triangles.  You can continue drawing triangles within triangles until they are too small to draw.

This process can be done from the top down, in the case of starting with a large triangle and adding smaller ones inside, or from the bottom up.   For example draw a small square on a piece of paper, now draw a second square off the right side of the first, this time twice the side of the first.  Next draw a square on bottom of the second square 3x the size of the first.   Then a square on the left side of the previous square 5x the size of the first.  This is easier with graph-paper…  If you continue this patter (clockwise rotation around the edges of the previous square with a square that has a sides thats lengths equal to that of the two previous squares,  you may find you have made something like this:

So far all of these examples have been graphic in nature.   But these graphics are produced with simple number patters.   The main controls we use in our computer applications are numerical in nature.  We sequence notes (numbers) we repeat patterns (a number of times) the patterns themselves are numerical in nature (whole, half, quarter, eight and sixteenths notes for example) and even faders and effects are all controlled with numbers.

By exploring our numerical options with these ideas,  we may connect  the work we do with complex calculators (computers) with something more profound and hopefully more musically interesting.   If you explore more reading, videos or explanation of the golden section, fibonacci and fractals,  you will quickly see that these patterns are in almost every human creation that we identify as beautiful as well as all sorts of naturally occurring complex structures (animals and vegetables).   After seeing these patterns you will notice the most attractive human faces are those that best fit the golden ratio proportions.

In your next independent lab or project, explore how you can use these patterns to make music.

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