Mathmagic Land: A Second Small Step into Mathematics

Mathematics is the alphabet with which God has written the universe.

In our second small step into mathematics, I wish to explore that wonderful world that many generations of students will hopefully be familiar with: Mathmagic Land. This was an Academy Award nominated short film featuring Donald Duck, adventuring through a place entirely unfamiliar to him. After first asserting that math is for “eggheads”, a friendly narrator guides him through the math involved in music, architecture, art and games. In my first post I said I wished to go beyond the utility of math, and I believe this movie transcends utility to explore the natural wonder of math and where it appears in the wild.

This video is just under half an hour and is freely available on YouTube (linked above) so I recommend that you watch it as a prerequisite to this post. I hope to extend some of the concepts that are so nicely presented in the video, while also adding some of the places where I love seeing math show up in organic ways. So without further delay, let us begin!

    One of the first places we see math show up in Mathmagic Land is music, where perfect intervals create our western notion of the major scale. This discovery was fundamental to developing music the way that we did, allowing for a common way to teach and play music, though notation would lag behind drastically. The ability to split a string into certain ratios — 4 to 3 and 3 to 2 for our perfect fourth and fifth, and the 2 to 1 ratio for the octave directly explained in the video — to produce new sounds that seem to blend so harmoniously together is amazing. This order that seems to be present in many areas of nature that we explore is rather profound, and generally makes me smile uncontrollably, and I hope that the video captured your attention and interest as you watched this part.

    Now I want to make my first detour from Mathmagic Land specifically, and talk more about where math involves itself with music. For more information on what follows (as I will not do it the justice it deserves), definitely look into Howard Goodall’s documentary series Big Bangs, but for the math, particularly the episode about equal temperament. Now, it is important to know that the discovery of the ratios involving the major scale created what is now known as just temperament. This sort of tuning depended on a particular note, and the other notes in the scale did not sound very good except for when they were played in the context of the base-note’s scale.

    Let me try and give you a concrete example. The most basic major scale (to write out and play on a piano) is the C-major scale: C-D-E-F-G-A-B-C. This is the only major scale on the piano that consists entirely of “white” keys. However, let us say we wanted to play a G-major scale: G-A-B-C-D-E-F#-G. Well, the C that sounds good in the C-major scale will not be the same exact pitch in the G-major scale when we are in just temperament, since the ratios do not line up perfectly. For a very long time, until the time when Bach was popular, western music was developed under just tuning. Thus if a pianist (or more era-specific, a harpsichordist or organist) wanted to play a piece of music in the key of C, they would need a specific instrument tuned to C. They would never attempt to play in the key of G on this instrument, since the turnings would not align in a pleasant way.

    As years went on, people started to realize this inconvenient and rather unwieldy way to perform music needed to change. And so equal temperament was developed.  This allowed every note on your piano to sound almost right no matter what key you played in, and to normal human ears there was no real difference. However, this is why choirs rarely perform with pianists. Choirs tend to perform much more in just temperament since a tiny off-key sound tends to be exacerbated in the human voice, especially en masse. So the note a choir wants to sing to make a beautiful sound disagrees with how the piano is physically made to play the note.

    Now that you have a (slightly more than necessary) background, (but hey, I love music and it is fun to talk about), let me tell you where the math comes in. As was mentioned before, just temperament was based on whole-number ratios, dividing a note cleanly in a harmonious, natural way. On the other hand, in order to allow every note to be played in an acceptable fashion despite the key, equal temperament requires equal ratios between notes the same distance apart. In western music, the fundamental size is the half-step, which is what you get moving from one white key on a piano to the next black key (or the adjacent white key if going from B to C or E to F). In western music, using this half step, you end up finding that the ratio necessary between successive half-steps is the twelfth root of 2. That’s right, 2^{1/12} \approx 1.059. This, though not a pretty number, is an unequivocally mathematical solution to a problem in our world. I think this is really cool. A lot of math and physics needed to be developed in order to understand how to get this ratio correct. An understanding of the logarithmic nature of sound was necessary, as well as the mathematical framework to get equal temperament close enough to just temperament to be reasonable moving forward.

    As tempting as it is to provide a more in depth explanation of equal temperament, I think this would go beyond the scope of what I am trying to accomplish in these posts. So well will press on to the golden ratio, the golden spiral, and Fibonacci numbers.

    The video makes much of the golden ratio, golden segment and golden rectangle, while also briefly mentioning the golden spiral that results. I think this is awesome. It is something that shows up in nature and provides something purely aesthetic that can be partially analyzed by mathematics. The forms of flowers, the natural proportions of people and many other objects, all have hints of this golden ratio in them. I suspect that many of you, unless you’ve had a cool math teacher that showed you this movie in school or taught you about it otherwise, may not have been aware of the golden ratio. However, in my experience many people have heard of the Fibonacci Sequence and Fibonacci numbers. If I remember correctly, they came into popularity with the book The DaVinci Code. Let me tell you how this sequence and the golden ratio relate to each other.

    To begin let us derive what the golden ratio is. We will do it first by looking at a golden rectangle:

    This image has two golden rectangles: the entire large rectangle, as well as the red rectangle. So the golden ratio is the ratio between the two sides of the rectangles. We have the ratio of (a+b) to a, and the ratio of a to b. We know these ratios are equal, since they are both the golden ratio, so we can set up the following equation:

    \frac{a+b}{a} = \frac{a}{b}

    Now, we can assume that a = 1, so we have that a + b must be equal to 1 times the golden ratio. We have a special symbol for the golden ratio, $latex\ phi$, which is written “phi” and pronounced “fie”, not “fee”. So we see that a = 1, a+b = \phi so we must have b = \phi - 1. Hence we can adjust our first equation to get:

    \phi= \frac{1}{\phi - 1}

    If we multiply the right side denominator over and subtract by 1 we get:

    \phi^2 - \phi - 1= 0

    If you’re savvy and remember your quadratic formula, it’s easy to solve this to get:

    \phi = \frac{1 + \sqrt{5}}{2} \approx 1.618

    So we have our value of \phi! This is exciting! But, how does this relate to the Fibonacci sequence? Let me first tell you what the Fibonacci sequence is if you are unaware. We begin with the numbers 1 and 2, though depending on who you ask you might start 1 and 1, but it ends up being the same. Then you take the two numbers and add them, getting 3. Then you take your number and the previous one, 3 and 2, and add them to get 5. Then add 3 and 5 to get 8. 5 and 8 get you 13. Formally, we can write:

    a_1 = 1, \;a_2 = 2, \;a_n = a_{n-1} + a_{n-2}

    This sequence is cool. Most flowers have a Fibonacci number of petals, their seeds are arranged by Fibonacci numbers, and there are many other places these numbers show up if you look around. But we want to find the golden ratio here. So, let us take ratios! Consider the ratio of some term and the previous term, a_n / a_{n-1}. If we start off, we get 2/1 = 2. Then 3/2 = 1/5. Then 5/3 = 1.67 or so. Then 8/5 = 1.4. Then 13/8 = 1.625. And we go on, and it seems that we are squeezing toward some value. Let us suggestively call this value \phi. Then we can say that as n becomes incredibly large, a_n/a_{n-1} tends toward \phi. But similarly we know that a_{n-1} / a_{n-2} would also tend towards this number. So let us set up an equation:

    \frac{a_n}{a_{n-1}} = \frac{a_{n-1}}{a_{n-2}}

    Well we have an equation for a_n, so let us use it to get the equation:

    \frac{a_{n-1} + a_{n-2}}{a_{n-1}} = \frac{a_{n-1}}{a_{n-2}}

    Now we can split up the left side to get:

    1 + \frac{a_{n-2}}{a_{n-1}} = \frac{a_{n-1}}{a_{n-2}}

    Now if we look at n getting very large, we have that the right hand side tends toward \phi by our original assumption. But then we can see that the second term on the left hand side is the reciprocal of the right hand side, so it must equal 1/\phi. Hence we can write:

    1 + \frac{1}{\phi} = \phi

    If we multiply both sides by \phi and set it equal to zero, we get a familiar equation from before:

    \phi^2 - \phi - 1=0

    We know the solution is the golden ratio! Hence if we take the ratio of Fibonacci numbers, we approach the golden ratio! This is why the spiral that shows up in the golden rectangle in the video is also associated with the Fibonacci sequence. It’s fun how we can get this natural ratio to show up elsewhere.

    I know for some of you this last part may have been hard to follow, and that’s okay. Math is about pushing ourselves and stretching our understanding of a topic to find new and interesting things in the world, and we just accomplished that! I don’t have much to say about the billiards section of the film, but I hope you enjoyed it. It is fun to rewatch it and try it out on a pool table by yourself, to see if you can play billiards the same way.

    There is a lot of wonder to mathematics, a lot of ways that it shows up in our everyday life, and I hope that reading through this stretched your mind a bit and that you enjoyed learning in Mathmagic Land. It’s a wonderful place, truly the land of adventure as we are told, and I look forward to continuing the journey with whoever wishes to come along with me.

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