1. |
A fugue is ...
03:40
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A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards. However, the notion of fugue is much less rigid than that of canon, and consequently it allows for more emotional and artistic expression.
The telltale sign of a fugue is the way it begins: with a single voice singing its theme. When it is done, then a second voice enters, either five scale-notes up or four-down. Meanwhile the first voice goes on, singing the “countersubject”: A secondary theme, chosen to provide rhythmic, harmonic, and melodic contrasts to the subject.
Each of the voices enters in turn, singing the theme, often to the accompaniment of the countersubject in some other voice, with the remaining voices doing whatever fanciful things entered the composer’s mind.
When all the voices have “arrived”, then there are no rules. There are, to be sure, standard kinds of things to do—but not so standard that one can merely compose a fugue by formula.
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2. |
Fishes and Scales
05:54
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Escher took the idea of an object’s parts being copies of the object itself and made it into a print: his woodcut Fishes and Scales.
Of course these fishes and scales are the same only when seen on a sufficiently abstract plane. Now everyone knows that a fish’s scales aren’t really small copies of the fish; and a fish’s cells aren’t small copies of the fish; however, a fish’s DNA, sitting inside each and every one of the fish’s cells, is a very convoluted “copy” of the entire fish— And so there is more than a grain of truth to the Escher picture.
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3. |
Bach and Cage
02:40
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On the other hand, to appreciate Bach requires far less cultural knowledge. This may seem like high irony, for Bach is so much more complex and organized, and Cage is so devoid of intellectuality.
But there is a strange reversal here: Intelligence loves patterns and balks at randomness. For most people, the randomness in Cage’s music requires much explanation; and even after explanations, they may feel they are missing the message—
Whereas with much of Bach, words are superfluous. In that sense, Bach’s music is more self-contained than Cage’s music. Still, it is not clear how much of the human condition is presumed by Bach.
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4. |
The Carroll Dialogue
03:03
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A system of reasoning can be compared to an egg. An egg has a shell which protects its insides. If you want to ship an egg somewhere, though, you don’t rely on the shell. You pack the egg in some sort of container, chosen according to how rough you expect the egg’s voyage to be.
To be extra careful, you may put the egg inside several nested boxes. However, no matter how many layers of boxes you pack your egg in, you can imagine some cataclysm which could break the egg. But that doesn’t mean that you’ll never risk transporting your egg.
Similarly, one can never give an ultimate, absolute proof that a proof in some system is correct. Of course, one can give a proof of a proof, or a proof of a proof of a proof— But the validity of the outermost system always remains an unproven assumption, accepted on faith.
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5. |
Crab Canon
03:08
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Hallo! Hulloo! What’s up? What’s new?
You see this bump, this lump?
Given me by a grump. Ho! And such a fine day!
You see, I was idly loafing about the park when up lumbers this giant fellow from Warsaw— A colossal bear of a man—playing a lute.
He was three meters tall, if I’m a day.
I mosey up to the chap, reach skyward and managed to tap him on the knee, saying, “Pardon me, sir, but you are Pole-luting our park with your mazurkas.”
But WOW! He had no sense of humor—not a bit, not a wit—
and POW!— he lets loose and belts me one, smack in the eye!
Were it in my nature, I would crab up a storm,
but in the time-honored tradition of my species, I backed off.
After all, when we walk forwards, we move backwards.
It’s in our genes, you know, turning round and round.
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6. |
The Brain's Ants
03:44
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The type of decisions which a neuron faces—and this can take place up to a thousand times per second—is this: Whether or not to fire—that is, to release ions down its axon, which eventually will cross over into the entry ports of one or more other neurons, thus causing them to make the same sort of decision. The decision is made in a very simple manner:
If the sum of all inputs exceeds a certain threshold, yes; otherwise, no. Some of the inputs can be negative inputs, which cancel out the positive inputs coming from somewhere else. In any case, it is simple addition which rules the lowest level of the mind.
To paraphrase Descartes’ famous remark, “I think, therefore I sum.”
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7. |
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15 is odd, so I make 3n + 1: 46
46 is even, so I take half: 23
23 is odd, so I make 3n + 1: 70
70 is even, so I take half: 35
35 is odd, so I make 3n + 1: 106
106 is even, so I take half: 53
53 is odd, so I make 3n + 1: 160
160 is even, so I take half: 80
80 is even, so I take half: 40
40 is even, so I take half: 20
20 is even, so I take half: 10
10 is even, so I take half: 5
5 is odd, so I make 3n + 1: 16
16 is even, so I take half: 8
8 is even, so I take half: 4
4 is even, so I take half: 2
2 is even, so I take half: 1
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8. |
The Origin of Life
03:30
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There are various theories on the origin of life.
They all run aground on this most central of all central questions: “How did the Genetic Code, along with the mechanisms for its translation, originate?”
For the moment, we will have to content ourselves with a sense of wonder and awe, rather than with an answer. And perhaps experiencing that sense of wonder and awe is more satisfying than having an answer—at least for a while.
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Thomas Oboe Lee Cambridge, Massachusetts
Thomas Oboe Lee was born in China in 1945. He lived in São Paulo, Brazil, for six years before coming to the United States in 1966. After graduating from the University of Pittsburgh, he studied composition at the New England Conservatory and Harvard University. He has been a member of the music faculty at Boston College since 1990. ... more
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