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Math, Music, and Pentatonic Scales

No one knows how long humans have been creating music, but the earliest attempts at understanding the theory seem have been made by that strange society called the Pythagoreans. Its founder, Pythagoras, lived on the Greek island of Ionia, sometime between 600 and 500 B.C.E. He is famous mostly for the statement of the Pythagorean Theorem — a fact known to the Babyloneans for at least a millenium before him. There are lots of interesting legends — some of which may actually be facts — about the Pythagoreans, but I don’t have time to go into them now. As Casey Stengel used to say “You could look it up.”

Anyway, the Pythagoreans seemed to have experimented with the sounds produced by strings and hollow flute-like tubes of differing lengths. They noticed that (all other things being equal, like tension for strings or diameter for tubes) that the most pleasant intervals between notes are produced when the lengths were in simple ratios. For example, if the length of one string is double the length of another, then the shorter one produces a tone that, in our current terminology, is called one octave higher than the longer one. This is the interval between, say, a C on the piano, and the next C above it. It’s the interval from Do to Do in the scale Do Re Mi Fa Sol La Ti Do.

The next most interesting ratio is 3 to 2. If one string is 3 units long, and the other 2 units, the shorter string will be a perfect fifth above the lower one. This is the distance from C to G on the piano, or from Do to Sol, as in Do Re Mi Fa Sol. Other ratios are: 4 to 3: a perfect fourth (C to F or Do to Fa) and 81 to 64: a major third (C to E or Do to Mi). The third was considered a dissonance in early music.

We now know that sound is simply a vibration of the air; the faster the vibration, the “higher” the pitch. A modern “middle C” is 256 cycles or vibrations per second. The vibrations per unit time — say second — is called the frequency. If two similar strings or tubes have lengths in a certain ratio, then the frequencies they produce will be in the opposite ratio. In other words, the shorter the length, the higher the frequency. If C has a frequency of 256, then G should have a frequency of 256 times 3/2 or 384.

It turns out that the fifth, or 3:2 ratio, can be used to get all the other “named” notes in the scale that we use. Starting with C, say, we keep going up by fifths. This gives C – G – D – A – E – B – F# – C# etc. You eventually get back to C, so this is called the “circle (or cycle) of fifths.” If you take the first five of these, C, G, D, A, E and arrange them in order: C, D, E, G, A, you get what’s sometimes called the “major pentatonic scale”. Pentatonic means, simply, “five-toned.” There are other forms of pentatonic scales, but this is what is usually meant by the term. Play these on the piano or whatever instrument you can find, or go to

The Pentatonic Scale.

to hear them on your computer.

It turns out that there are serious flaws in trying to create all of the notes of the modern “12 tone” scale using the circle of fifths: you don’t exactly get back to where you started, and some of the intermediate notes which should be, say, a fifth apart themselves, aren’t! This created problems for centuries in playing music in various keys and on different instruments. It wasn’t solved until about the (early) 18 century with the development of the “well-tempered” musical tuning. The idea in well-tempering, basically, is to divide the ratio 2 to 1 (the octave) into 12 equal parts or ratios; 12 because there are 12 “half tones” in an octave: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C. The frequency ratios between successive notes in the “chromatic” scale (all 12 tones) are set equal. This means that the product of all 12 of these ratios must equal the ratio 2:1 ( = 2 numerically) of the octave. Thus, each ratio is the twelfth root of 2. Now the twelfth root of 2 (written 21/12) is about 1.059463094 (by my calculator). Thus, to get from C to G, a space of seven of these half tones, requires multiplying the frequency of C by 21/12 seven times. Mathematically this is 27/12 which is approximately 1.4983: very close to 3/2 ( = 1.5), but not exactly. By making this ever so slight mathematical adjustment, we can build and tune instruments which sound beautiful together in all keys and in all combinations.

Thus we use math in our daily musical lives.

May 18, 2010 Posted by | Off Topig, Pentatonic Scales | , , , | Leave a comment

Tree pruning, accounting and mathematics

Well OK, I had a chance to look at a tape of Friday’s episode. There were several items that I’d like to comment on.

We look in on Charlie “pruning a tree.” What is that and what does it have to do with mathematics? A tree is a way of organizing data in the memory of a computer. Think of this memory as a bunch of mailboxes, each with an address. A mailbox contains data of some sort. This data can consist of various facts: name, age, social security number, preference in cars etc. for a person, or cash flow on a given date, debt, etc. for a company. A mailbox can also have a pointer — i.e. address — to one or more other mailboxes. For example, a mailbox could have the name, income and other information for the CEO of a company, and also pointers to the mailboxes of, say, the various vice-presidents reporting to the CEO. Each of these VP’s mailboxes could contain personal or financial information about them, together with the addresses (pointers) of the mailboxes for employees reporting to them etc. If the CEO’s mailbox is drawn at the top of a sheet of paper, and the arrows are drawn downward toward the VPs working under the CEO, and more arrows are drawn downward from each of them to the employees working for that VP, then you get a picture that looks a bit like a pine tree (you’ll have to imagine the arrows since they are too hard to type):

CEO

VP1 VP2 VP3 VP4

E1 E2 E3 E4 E5 E6 E7 E8 E9

etc.

There will be arrows from CEO to each of the 4 VPs. There will be arrows from each VP to some of the 9 employees. For example, VP1 may point to E1, E6 and E7, while VP2 may point to E1, E6 and E10. And so on.

This kind of tree is easily constructed in the memory of computer by programers who are trained in setting them up. In practice they can be very big and complicated, with many mailboxes (sometimes called nodes), each with many pointers and lots of data. There can be many more layers than the ones shown here. The “chains of command” can be followed by following the pointers downward.

So, a lot of interconnected data can be efficiently stored in this way, and accessed by following the arrows or pointers from node to node.

The same concept of a tree can be used to store many other kinds of things: relations among people (“family trees”), the relation of words or phrases in a sentence, and the interconnection of suspects in a criminal investigation.

Charlie has set up some sort of tree — we don’t know exactly what the links are — but it has gotten too complicated. There are pointers to nodes that are not really relevent. He is more interested in motives for murder in the corporation than in corporate chains of command. He is interested in financial links, not necessarily in corporate hierarchy. So, to deal with his tree more efficiently, he has to get rid of mailboxes containing irrelevent information. When a mailbox is removed, pointers (arrows) to it also have to be removed so that when the computer follows a path it doesn’t end up with a pointer to … nothing. The process of removing unnecessary nodes and the pointers to them is called “pruning”, for obvious reasons. Setting up, maintaining and analyzing trees is generally considered the work of computer scientists. However, like scientists in general, computer scientists study mathematics to help them analyze the patterns found in constructions like trees. This analysis can involve the creation of efficient algorithms for building and traversing (following the pointers of) trees or for estimating how long it will take to find certain kinds of data in a tree. Charlie is an expert in this kind of math, sometimes called combinatorics.

I don’t really know how Charlie’s tree was constructed. If it requires a supercomputer to analyze it, it must be very big, with many and complex paths. It is unlikely that Charlie could have typed this all in, even though we know he had been working on this case in the past — for the Securities and Exchange Commission (SEC).

Whatever Charlie is looking for, the computer must search all nodes by all paths to find it. For example, if each node has 10 pointers, then a 4 level tree of the type shown above could have as many as 1111 nodes: the top node pointing to 10 below it, with each of these pointing to 10 below it (10×10 = 100) and each of these pointing to 10 below it (10x10x10 = 1000). Of course, each level might not contain 10 different nodes, but to make up for it, there might be a more complex array of pointers.

This is beginning to get a bit fuzzy here. Why is a theoretical mathematician, who publishes research articles on pure and applied mathematics, doing the job of an accountant and a computer programmer? I know hundreds of professional mathematicians and none of them is involved in accounting (other than doing their taxes). It seems to me very unlikely that Charlie would be moonlighting analyzing financial data for the SEC — I’m sure he’s well-paid and busy in his job at Cal-Sci. But I could be wrong. Any mathematicians out there who do this kind of work?

Is an accountant a mathematician? Depends on what you mean by a mathematician. My dictionary says that mathematics is “the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically,” and a mathematician is someone who studies or practices mathematics. Simply “using numbers” is not really enough to qualify for being a mathematician: you have to be studying the theoretical patterns in these numbers and expressing them symbolically; or at least using general formulas and equations to solve problem. So, accountants use numbers, but even though the work they do is complicated and hard, it is not using general formulas and equations, so is not usually considered mathematics; thus, accountants are generally not classified as mathematicians.

On the other hand, the creation of modern notions of accounting and bookkeeping sometime in the early renaissance went hand in hand with the gradual acceptance of negative numbers, a concept resisted by even the most creative mathematicians of antiquity. This resistance eventually yielded to the very real (and painful) concept of a loss, or deficit, most conveniently represented by a negative entry. This was concealed at first by having a separate page in the “books” for amounts one owed; but, as early as the 12th century, the mathematician Fibonacci was using negative numbers to denote (financial) losses. It was Fibonacci who, among other things, introduced modern number notation — invented in India and used by the medieval Arabs — to Europe.

Well, it’s getting late. In another blog — maybe Wednesday — I’ll tell you about the “vector fields” that Larry uses to describe the water flow (leak) in Charlie’s office, and that seem to inspire Charlie to punningly analyze cash flow.

By for now.

May 18, 2010 Posted by | mathematics, Off Topig | , , | Leave a comment

Shadows, position and sundials

If you stick a vertical pole in the ground, its shadow will move as the day progresses. This is because the rotation of the earth makes the sun appear to travel across the sky. As the sun “moves”, the shadow that a fixed object casts moves with it. This suggests that you could make a clock: simply draw a line from the pole along the shadow at noon: mark it with a “12”; then draw another line along the shadow at 1 PM and put a “1” on it; the line at 11AM is marked with an “11”, etc. This gives a series of radial lines which should enable you to tell the hours on other days. Unfortunately this is not the case. If you lived at the North Pole it would work, but the problem is that when the sun travels through the sky for one hour, the shadow of a vertical pole moves through an angle which is not the same in every season; except at the north pole, the relation between time elapsed and the angle a pole’s shadow moves varies from day to day. You can not calibrate a vertical pole clock so as to tell time all year round! (More on this later.)

In Friday’s episode, the shadow pole is a basketball net post, presumably vertical. Two photographs are available of it, with local time stamps. Unfortunately, these just tell us the time it is everywhere within a given time zone. This is significant as we shall see.

What do the pictures tell us? If we know the height of the pole and the length of the shadow (which we do since a basketball rim is exactly 10 feet off the ground, and the shadow’s length is computed by counting bricks on the ground), we can use simple trigonometry to find the angle the sun makes with the ground, called its elevation.

Now the path of the sun through the sky depends only on the time of year (date) and your latitude. For example, in northern latitudes, the sun stays low on the horizon in the winter and higher in the summer. This is because the earth’s axis of rotation, which always points in the same direction (or very nearly) is tilted with respect to the plane of its orbit around the sun. So as the earth travels around the sun, the south-north axis may point toward or away from the sun. When it points toward the sun, the sun rises higher in the sky in the northern hemisphere, and so warms the Earth more. It is summer. You get theidea.

So let me repeat: The path of the sun through the sky depends only on the time of year and your latitude. We know the time of year and we would like to find the latitude, so we must determine the path. If we knew the sun’s exact elevation at a particular time, we could determine the path. This can be done by looking through all paths and finding the unique one with the sun at that position at that time; even better, there are mathematical ways of finding the path exactly, using what’s called spherical trigonometry. But we don’t know the exact time, just the time-zone time. Time zones are 1000 miles wide, approximately. So here is why we need two pictures at a definite time interval apart. Only one path will have the sun changing in altitude by exactly that amount in that time. Thus, two pictures taken a known time interval apart determine a unique path, which in turn gives a unique latitude.

Even better, once you know the path, you can predict at exactly what time the sun will be highest. Say this is at 12:20 local time. (Remember that the sun can’t be highest at noon everywhere within a time zone.) That means that you are 20 minutes west of noon in your time zone. There are tables which showexactly where (in terms of longitude) in a time zone noon occurs, so you can calculate your east-west coordinate — that is, your longitude — from this.

In summary, then, having two photos giving elevations of the sun at two local times enables you to find both your latitude and longitude. This can all be done using spherical trigonometry, a subject that started with the Greek mathematician Ptolemy nearly 2000 years ago, and was brought to fruition by the medieval Arab astronomer/mathematicians over 1000 years ago. Spherical trig applies the functions of trigonometry — sine, cosine, tangent — to studying circular arcs in 3-dimensional space. These arcs are either on the surface of the earth, or in the heavens directly above the earth. Such arcs are completely described by the angles which lines drawn from their endpoints make at the center of the earth; so, both the angles and the sides of a “spherical triangle” are actually given as angles. It’s a complex and beautiful theory, but I don’t have time to get into details here. In any case, it is exactly the math needed to describe the apparent paths of heavenly bodies through the sky, or ships moving across the seas.

How accurate are the calculations made on the show? Charlie claims that they are accurate to within 1/100 of a degree, which amounts to a little more than a kilometer. This is probably optimistic, since who knows if the basketball pole is exactly vertical, or how accurately we can count bricks in the photo to find the length of its shadow. Be that as it may, one might be able to find an aerial photo of a square kilometer block in California and have a computer do a pattern search with other features of the photos to pinpoint the exact location. Presumably that’s what happened.

By the way, the Arab mathematicians of about 600 years ago were able to fix the problem of the shadow pole not being accurate with the changing seasons. The idea was that the earth rotates around its axis, which is a line running between the North and South Poles. If you want equal hours to correspond to equal shadow angles, you need to have your shadow pole parallel to this axis. So, at least in the northern hemisphere, simply point the shadow pole in the direction of the North Star (Polaris). The angle that such a pole makes with the vertical turns out to be 90 degrees minus your latitude (so at the North Pole it is vertical). The Arabs produced very accurately calibrated sundials; the tilted “shadow pole” became a triangle with hypotenuse pointing north and angled so that it points to the North Star at night; it became known as a gnomon. They were able to calculate, using spherical trig, the exact calibrations so that it would tell accurate time all year long!

May 18, 2010 Posted by | Off Topig | , , , , , , , , | Leave a comment

What are vector fields?

While looking at the leak in Charlie’s kitchen, Larry drops some ink from a pen into the puddle and talks a bit about “vector fields”. What are they?

Well, for some things we want to measure, we need two kinds of information: their size or magnitude and their direction. One example is a force. The magnitude of the force is how strong it is; maybe this would be measured in pounds (or newtons in the metric system). For example, someone may shove you, or a wind might push you, with a certain number of pounds of force. But for a force, you have to know more than the magnitude: you have to know the direction. So, a force may push you sideways, or forward, or upward, or some direction in between. So a force is described by two things: its magnitude and its direction.

In physics we talk about another quantity like this: velocity. To know a velocity you need to know its magnitude; for example, 130 meters per second. However, you usually need to know what direction you’re travelling in. So a velocity might be described as: 130 meters per second in the direction 30 degrees north of east.

Quantities that need a magnitude and direction to describe them completely are called vectors. There is a nice way to picture vectors: as arrows. The length of the arrow gives its magnitude (when we know the scale) and the direction the arrow points gives the … well…direction. Thus, the velocity described above might be represented as an arrow 130 centimeters long, pointing 30 degrees north of the due east direction.

So what is a vector field? A vector field is a bunch of vectors drawn at a lot of different points. They represent forces or velocities at those points. Here’s an example.

Suppose you have a fish tank and start pumping water into it. (To make sure that it doesn’t overflow and make a mess, you may want to drain water out at the same rate). Now as the water flows in, it swirls around, creating currents in the tank. At different places in the tank the water is moving in different directions and at different speeds. If you had a lot of highly trained fish, you could get them to swim to different positions in the tank and measure the speed and direction of the water at some instant at all these different positions. Each fish could then hold an arrow at its position, with the length of the arrow representing the magnitude of the velocity of the water there, and with the arrow pointing in the direction of the flow at that point. This collection of arrows is a vector field: one vector for each point. Depending on how many fish you had, you could get a better (finer) or worse (coarser) visual representation of the actual flow of the water. Since the water itself is transparent, these arrows would give you a better idea of what’s actually taking place in the flow within the tank. Although it is physically impossible to achieve it, a vector field with vectors at each of the infinitely many points in the tank would give you complete information about the flow. Even though we can’t do this practically, mathematicians nevertheless can envisage and represent such vector fields. How do they do this?

Well, it is standard to represent a point in space by its coordinates. To keep things simple, let’s say we are dealing just with a flat space, as we do in plane geometry. Think of it as a piece of the floor. Each point, as you learn in coordinate geometry in high school, has two coordinates, and is represented as (x-coordinate, y-coordinate). Similarly, a vector or arrow on the floor is represented by two coordinates: the first tells how far to the right (positive) or left (negative) the arrow points; the second tells how far up (positive) or down (negative) it points. So, the vector (1,1) points one unit to the right and one unit up, so it makes an upward 45 degree angle with the horizontal. So does the vector (2,2), but it’s twice as long. Thus, two coordinates are enough to describe a vector on the floor. So a vector field on the floor would draw, at each point P = (x,y), an arrow or vector at P. The vector’s coordinates (which give its magnitude and direction) depend on the point at which it is to be drawn, so depend on the coordinates of that point. Thus, the first coordinate of the vector (giving its left/right direction), which we’ll call it X, depends on both x and y; we express this by writing X = f(x,y) or X is a function of x and y. Similarly, the vector’s second coordinate (giving its up/down direction) also depends on x and y, so is also a function of x and y; we write: Y = g(x,y). So a vector field for the points on the floor is represented by two functions X = f(x,y) and Y = g(x,y). Mathematicians put them together into a single pair which they write:

(X,Y) = (f(x,y), g(x,y))For example: (X,Y) = (x2 + y, 3x + 4y)This vector field draws the vector (x2 + y, 3x + 4y) at the point (x,y). So, for example, it draws the vector (X,Y) = (7,18) at the point (x,y) = (2,3), since 22 + 3 = 7 and 3(2) + 4(3) = 18.

When Larry drops some ink into the water on Charlie’s floor, he is trying to visualize the vector field of the flow. Each droplet spreads out along the flowing water. The faster the water is flowing at the point where the droplet is placed, the more elongated the drop becomes. It moves in the direction of the vector at that point, and at a rate roughly proportional to the length of that vector, at least during a fixed time interval, say of about a second or so. The more drops of ink Larry squeezes out of his pen, the more vectors get drawn. He see that some of this flow is not flat along the floor, but is moving in a pattern suggesting that some vectors are pointing up: the water is oozing upward, so the leak is coming from below.

Mathematicians use computers to visualize vector fields given by equations like the one I wrote above. Here is an example (I’ll post some more when I get a chance):

May 18, 2010 Posted by | Off Topig | , , , | Leave a comment

“Race in American Trials Collection” research guide goes online

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A brief research guide, Researching Race in the American Trials Collection, is now online. A link to the guide is in the Law Library’s Research page, under the heading “Legal Research Guides. While the guide’s focus is on trials involving slavery, segregation, and related issues, it’s also helpful for researching other topics in the American Trials Collection. Additional slavery resources are available via the Yale Slavery and Abolition Portal.

We’ve added about 100 titles to the American Trials Collection in the last year, including Trial of Thomas Sims, on an Issue of Personal Liberty (Boston, 1851), described by Paul Finkelman as the most complete record of “the first important and intensive investigation of the meaning of the 1850 [Fugitive Slave] Act” (Slavery in the Courtroom, p. 94). This pamphlet is also available online, courtesy of the American Memory website of the Library of Congress.

MIKE WIDENER
Rare Book Librarian

May 18, 2010 Posted by | Off Topig | , , , , , , , , , , , , | Leave a comment

“A Gross Injustice”

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African American soldiers in the United States Colored Troops originally did not receive equal pay. Some northerners demanded that Congress take action and change the policy. This editorial, which was published in Harper’s Weekly on February 13, 1864, asked readers to consider the issue:

  • “But the point for every honest man to ponder is this: We invited the colored man to fight for us: they have shown themselves brave, clever, and obedient, and we refuse to pay them what we pay other soldiers. Not to speak again of the sheer breach of faith and wanton injustice of such conduct, a distinction like this, even if it were honorably made, tends to maintain a feeling of caste which would be fatal to the army. All that we ask is fair play for every man who will risk his life for the country; and against foul play…we shall not fail to protest as earnestly and persistently as we can.”

Congress eventually instituted equal pay in June 1864.

May 18, 2010 Posted by | Off Topig | , , , , , | Leave a comment