The Lord of The Rings is at once both one of my favorite books and one of my favorite film trilogies. And I don't really feel the need to write another sentence justifying that.
In any case, I reacted with some consternation upon finding out the Amazon was, having attained the rights to Tolkien’s world, developing a new series set in Middle-earth. On the one hand, we get to return to that world. On the other, it's hard to top Peter Jackson’s interpretation of that world – how else could Minas Tirith look if not like that?
But then, revisiting Middle-earth means a chance to do some things differently. Like maybe making the world look a little more inclusive.
The Lord of The Rings is very white. That's not so much a judgement as it is a fact. It doesn't make it any worse as a work, it's just how it is. So if we're telling new stories, let's ask why not and mix things up and cast some people of color as these characters.
Now, my own knee jerk response is “hey, let's make all the elves Asian!” because that way you'll be forced to have an Asian actor on screen anytime an elvish character is in play (and also we’ll get Elrond, half-Asian). But equating fictional races with real life ones becomes real hairy real quick. It runs the risk of feeling like stereotyping and, in the case of my own “make all elves Asian” orientalism and exoticism. Because if they don't look like the normal, clearly they must be other, so let's make them not-human. That line of thinking falls back on to the white-as-default mindset, where if you need a normal Everyman you make him a white guy. And let's not do that.
Because if we're diversifying Middle-earth, let's let everyone be everyone. Let's have black elves and surly Asian dwarves, let's have Latino hobbits and an Indian shieldmaiden of Rohan.
Because why not.
The Lord of The Rings, and a lot of high fantasy with it, falls into the trap of looking a lot like Western Europe in the Middle Ages. Which, I suppose, is fair, given that Rings is the forerunner of modern fantasy and that in writing it Tolkien wanted to give England its own myths to rival those of Greece. So of course it's gonna portray a very (white) England-inspired place. But that’s done, and it doesn't excuse modern fantasy works (and the upcoming Amazon show would indeed count as a modern fantasy work) from being very white and European.
Cuz there's nothing in The Lord of The Rings’ mythology that precludes a more diverse cast. Sure, you'd have to ignore Tolkien’s descriptions of characters as fair and golden-haired, but that's not a loss. Heck, even adding more women makes sense; we've already got characters like Lúthien and Galadriel who've kicked butt in their time. Eowyn’s given the title shieldmaiden so she’s probably not the first. There’s no reason not to.
This is a fantasy world with magic rings and enchanted swords (and, y'know, elves and dwarves and stuff), there is literally no good reason why everyone has to be white. The only reason a black elf or Asian dwarf sounds so odd is because it's outside what we've internalized as normal for the genre. We're simply used to seeing these archetypes as white. And that's s gotta change.
And where better for that change to happen than in the world of The Lord of The Rings? This is the book that elevated fantasy from children’s books to something taken seriously. It's what inspired the world of Dungeons & Dragons, it's the basis for just about every modern work of high fantasy. This is a chance to shift the framework, to redefine how fantasy usually looks.
I love The Lord of The Rings (and The Hobbit and The Silmarillion). Why can't I, someone who's reread the books countless times, quoted the movies in the opening to his thesis, and dominated Lord of The Rings bar trivia, get to see people in those stories who look more like me?
Also, Route 1 has Munchlax now. And sometimes they have Leftovers. I caught a Munchlax with Leftovers only five hours into the game. Rad.
Hm, what else? The Photo Club is fun, I'll probably be spending some time there, and the Roto Loto has proved quite useful already. Most of the changes I'm noticing right now are fairly minor, but I imagine that'll snowball as I go forward. Also, Mantine Surf is pretty darn fun. While you were required to do it, you were not required to be good at it, which was an immense relief.
Next thing I need to do is take on Lana's trial, which I'm...a bit nervous about. Pikachu's going to need to pretty much solo it and the only electric move he knows is Thunder Shock. Hopefully it won't take too long, I need that fishing rod to catch a Wishiwashi for my team.
(Also, Pokemon Bank will apparently be getting patched to work with USUM in late November. Trading with Sun and Moon works fine, though.)
Honestly, I've never hunted, because I never had the time. Back when I was a kid, I took a hunting class that included a lot of details about safety. That was the main thin, safety. A lot about safety. I do think that that's the most important thing to keep in mind when hunting. If you nothing else, you must know how to never shoot yourself in the foot, or worse, accidentally shoot someone else. One of the #1 rules of hunting is to always know where you're aiming your rifle and always know what you're shooting at before you pull the trigger. Always be aware of whether or not a shot is safe to take before shooting. Always take extra precautions to know with absolute certainty that you're the only person in the area. There are also many rules for handing a gun safely from one hunter to the other, how to handle guns in groups, how to cross fences with a gun, and even how to pass axes between two people (upside-down, with the blade perpendicular with your arms).
In case you were wondering, I learned all of this in a Boy Scouts-y type organization.
Anyway, that's safety. When it comes to ethics and other traditions, there are certain things to bear in mind. In a nutshell, always respect the law and the customs of the land. This includes:
- Knowing if you're required to have a permit to hunt.
- Knowing if you're the right age to hunt.
- Knowing what hunting equipment is legal.
- Knowing if you're allowed to hunt in an area.
- Knowing if a particular animal is legal to hunt and in-season.
- Knowing what you're allowed to do with an animal once you've hunted it.
- Respecting people who are uncomfortable with hunting.
There are certain things that aren't illegal, but it is still the responsibility of the hunter to act in an ethical manner. A hunter should always make sure that his hunting does not hurt the ecosystem. The law does not always reflect what is healthy for the environment. If you're on a hunting trip in another country where a reckless behavior is legal, that legality does not make the hunting ethical. You must use your judgment. Some countries do not have proper protection laws against endangered species. Make sure you recognize an endangered species and don't take advantage of their availability. A hunter must be a hundred times better educated than the average person on what impact their activities leave on the animal kingdom, since they bear a great responsibility.
Another thing that isn't illegal everywhere, but one should strive to avoid, is any unnecessary animal cruelty. Never allow an animal to suffer. My father, who likes to fish, enjoys catching fish and tossing them back into the water. My sister objects to this, because it's pointless and doesn't serve any point. She does not object to fishing, so long as the fish's body is used in some way and to some end, but returning them to their place worse than how they were found is not only cruel, but completely pointless. I happen to agree with her. I wish that my father would stop that habit if only for her sake. Unfortunately, my father doesn't really care much about hunting ethics, or any kind of ethics, or respect for people, or human decency, or boundaries, or love, or kindness, or any notion that any sort of living people holds any sort of dignity worth his acknowledging, or any sense of obligation to treat people with respect, or comprehension that he has any obligation to follow any sort of reality other than his own, or any ability to show empathy, or any inkling that sadism is actually the opposite of integrity.
[I had to delete this paragraph after I realized just how angry this subject makes me.]
So basically, there are a lot of rules, written in civil laws and in natural laws, for what one cannot do as a hunter. Once you respect those, you have freedom to determine your own hunter's ethics and traditions with what's left over. Different people have their own perspectives on hunting, and hunters are not a homogeneous group. In some places, hunting is seen as a rite of passage or as a means of proving one's worth. it can be a strong tradition, or a weak tradition. Many people have religious perspectives on hunting; for some people, their religion forbids it altogether, and among those religions that permit hunting, there's no consensus of what it means.
How I was brought up, I was taught that a hunter must contribute to the land, or at the very least, do no harm. It's sort of like the Hippocratic Oath. I see hunting as a spiritual experience, one where you bond with nature, one where you take part in the circle of life. I see humankind as holding a special place as both the caretakers and the masters over nature. A friend of mine, a farmer and actress (uncommon combination, I know), takes care of pigs, truly loves them, and holds similar views as me. As it happens, she still ends up slaughtering her livestock. The way she sees it, in her role as master, she determines the course of her pigs' lives; how they're conceived, how they're raised, what sort of health and happiness they will know while they're on this Earth, when and how they will conclude their lives, and to what ends they will die. The deaths are painless, and they are meaningful in her eyes. When I apply similar principles as hunting, I think of my grandmother's garden. She trims her bushes, uproots weeds, and sometimes even uproots beautiful flowers as she alters and shapes her garden into a plot of land that she deems desirable. I see hunting in a similar way; you're in a garden, and the death of a game can be like the trimming of a bush. Hunting shapes nature in small ways. Hunting allows someone to become one with nature, to appreciate the small details of nature's garden. It's more interactive than merely hiking. It has an intimacy to it.
Others simply love the primal aspect of hunting. Remember how I compared hunting to nudism? The comparison holds up. It's natural. It's a means of rediscovering our core identity. Some might say that we've grown past this, that we're better and more sophisticated, but others have a mind that technology will never change our core identity. We are hunter-gatherers. We go out and we tame nature. We celebrate being at the top of the food chain. So long as no one is hunting anything endangered or off-season, I have no problem with this mentality.
There are those who do it because it's a sign of independence. It makes a personal statement. There are those who do it because it's better than playing video games. Some do it to be tough, which I find a weak reason, but if they respect the laws of man and nature, then there's no reason to stop them from hunting. Then there's people like my father, who talk about hunting all of the time and never do it, and demean people who don't hunt, and will do thinks that deliberately make people feel uncomfortable for the sake of being macho, and
[Another paragraph and a half deleted.]
Believe it or not, I do want to go hunting with my father sometime. It's an experience that I feel that I need to have, if we can agree on what it ethical and what the meaning of the hunt is. It would probably be one of my better experiences with him. He never developed beyond parallel play; that is, he doesn't actually interact with people all that much, but he still desires for people to do the same things that he's doing as he's doing them. If he's interested in something, he wants to do it, and he wants everyone else that he knows to do it with him. Without copying and pasting a definition of parallel play, that's how I'll describe it. Basically, hunting seems like the ideal situation for this, since you don't have to talk much while you're doing it, but it's definitely something that he reveres and considers constructive, so it just might form a bonding experience. So long as he doesn't revel in his capacity to cause pain and demean everything in this universe that isn't him, it just might work out.
[Fighting the temptation to fit in another rant.]
Outside of my father, I sometimes wonder what people I would want to hunt with. I wonder what sort of father I'd be. When I took my hunting class, I remember a picture in my textbook of a father hunting with a daughter. That image stuck with me, and to this day I see that as something that I very much want. I don't give much thought to these days on romance and marriage and other intimate mushy stuff, and I like the idea of staying single for my entire life because of how hardcore and na-na-na-na-can't-touch-this that sounds, but then I think of these things and remember what I'm missing out on. I think of what things that I can offer someone that I've never witnessed in my lifetime, and hopefully in ways that are far more loving and prosperous than they ever would have been if I had had the opportunity. I only just realized as I was writing this that if this ever happened, there's a good chance that I wouldn't walk with my children in my own homeland of the North American Great Planes, but in a far-off country like China, and this image of hunting with my children that I always had might be completely different from the one that happens in reality. I'm very much interested in international travel.
On the note of China, hunting has been suspended there altogether since 2006. It isn't a permanent ban, but rather the country has been trying to figure out its laws for the last decade. Most people, when they think of hunting, think of the United States, Canada, Australia, Argentina, and the continent of Africa, but China has a rich history of hunting. Eventually, when they figure out their hunting laws, I would be very interested in going to the sparsely populated regions of the country to experience nature in ways that few Americans have thought of. Before hunting was put on hold, they had various species of deer, goats, gazelles, and argali. The argali are a group of wild mountain sheep with twisted horns, with several subspecies, and are the largest type of wild sheep in the world. They are unique to the Himalayan region. There were a few other species, such as the Tibetan Antelope and wild yaks that required special permits. Presumably, most of these will still be legal when hunting is allowed again, with updated quotas. China is also trying to figure out what hunting laws will apply to citizens, since oddly enough it was only legal for tourists to hunt before they started on their revisions.
There's also Australia. I have a friend who was raised in Australia. When we ran in to each other on our first day of college, we went on a nature walk, and we did it regularly. On one occasion, we saw a couple of deer roam by our dorms, and we pulled out knives and chased them through a construction site. We've also done other things in nature, such as camping on the Omaha Tribe Reservation, which we visited regularly (and really ought to again sometime, if we can get in contact with our friends there), playing broomball on a frozen lake, sledding and skiing down the hills of our river valley neighborhood back when winters were cold. He moved back to Australia for a year, and upon his return often asked me if I wanted to move there with him. There we can explore the slot canyons, roam the desert, and skedaddle through the natural parks. He also mentioned legal means of hunting. There's an open season on all non-native wildlife, and it doesn't stop there. As sacrilegious as it sounds to Americans who would never dream of shooting the national symbol of their country, Australians don't hesitate to shoot their signature animal. Kangaroos are considered by many to be pests, and furthermore, though exotic they aren't endangered, so Australian hunters commercially hunt over one-and-a-half million kangaroos per year. Traditional, pre-colonial means of hunting kangaroos included throwing sticks, where people would throw heavy sticks at kangaroos and break their necks. These throwing sticks come in a few shapes, the most recognizable being a boomerang shape. An actual boomerang is considered to be lighter, and it thrown in to trees in order to startle birds into flying up into traps set in the higher branches. Contrary to popular believe, boomerangs were not designed to cross-breed yo-yo's and Frisbee's. I really like my friend's idea, and I am seriously considering it, since our lives are advancing similarly and we will both be in an ideal place to move to Australia at about the same time.
Hunting with throwing sticks fits my idea of hunting. I do not consider hunting with guns to be unethical, and as an American I consider firearms to be a quintessential inclusion in my nation's mythology. However, older tools such as the bow and arrow have always grabbed my attention. They're less precise and more frustrating, but I wanted that challenge. Furthermore, firearms scare me on an innate level, in part because of negative experiences with you-know-who. [Okay, I didn't have to delete a paragraph this time, but I removed a particularly unpleasant sentence right here.] I like that it forces me to conserve my ammunition, and to go search for it when I miss my target. I like the ability to select my arrows and even decorate them, and that arrows can be retrieved from their targets, and that arrows marking special achievements can be preserved. I like that everything about a bow and arrow is completely manual and dependent upon my strength. I like the physical exertion that it takes to create the torque. I love the technical design and the engineering that goes in to a bow.
What I didn't appreciate was being asked by a physics teacher to bring my bow to class back in high school without providing a means for me to do it without frightening other students in the hallways. Again, this comes back to the ethics. Hunting wasn't even involved here, but I still wish I had found a way to respect other people's comfort and not to startle anyone. A mature and responsible owner of such a tool ought to know how to never cause disharmony with it, because even freedom from fear isn't a legal right, it is still right to never let them know fear. For those wondering, the bow was brought to class for an experiment in measuring different types of force. In hindsight, I should have proposed meeting him outside of school so that he could have safely and inconspicuously brought it to the lab himself.
As cool and as challenging as it is to hunt with a bow and arrow or just a simple throwing stick, hunting with firearms isn't a walk in the park, either. Many people believe that hunting defenseless animals with guns at a safe distance is cowardly and isn't an accomplishment. Naturally, this has a point, and it's obvious that I agree with it to some extent seeing as I chose to train in archery instead. However, the name of the sport is hunting, not shooting. The weapon is only a part of the hunting experience, and anyone who has taken a hunting class knows this. The reason why I have not yet gone on a serious hunting trip is that the actual act of hunting is time-consuming, and takes a lot of knowledge. A good hunter must know where to find game, how to track game, how to wait for game, and if necessary, how to lure game. Hunting isn't like an arcade game where the deer are right there in front of you and all you have to do is aim and shoot. The vast majority of the time, you aren't aiming at anything. The hunter must use a backdrop of technical know-how to read the environment and find animals. I never quite mastered that level of outdoorsmanship. In addition to knowing the art of hunting in and of itself, avid hunters ought to know how to deal with the elements when things go wrong. If someone goes on a hunting trip in the Canadian shield, he needs to know how to take care of himself if he gets lost, how to cope with hypothermia if he falls into a freezing river, and how to start fires and create shelter. Having a gun makes things a bit easier, but it's still a hardy experience. People can have different opinions on ethics, which all hunters must respect, but I also believe that non-hunters must all understand the challenge that hunters undertake. Hunters can't simply be dismissed as lazy, insecure people who claim a cheap sense of accomplishment.
One thing that most people can agree on, though, is that laser-sighted rifles are unfair. Even in America, these are illegal in every state but the great state of Texas. At least, last I checked. It might be illegal there now, too. I wouldn't argue against its legality if it was legal in my state, but I would strongly discourage hunters from using them and I wouldn't want to hunt with such a person as my partner. This isn't surprising from the guy who enjoys archery. Riflery is more precise, but people still often miss. Having laser sightings is like using cheat codes in a video game. Can you imagine Jumanji if you had cheat codes? Good luck impressing Karen Gillan with that sort of sportsmanship.
Overall, I do still expect to hunt with a rifle someday, especially if I ever decide to get sentimental and have kids, and especially if those kids are raised in America. Rifles are a part of American iconography, just as swords are a part of England's. It's a part of the culture, and I do desire to hunt in part for cultural reasons. They say you haven't truly experienced another country if you haven't tried their food, and in some ways that goes for hunting. Really, any old traditions having to do with subsistence. As I said earlier, if I was in Australia, I would want to hunt a kangaroo with a boomerang-shaped bludgeon, not only because of the sportmanship but also because of culture. I could, after all, hunt with a throwing stick anywhere, but I would specifically do it in Australia because of its connection to the heritage of the land. The same goes with America, which had a history of riflery since its very inception. One simply has not had the "full" American experience without understanding riflery.
This leads me to one last thing with relation to hunting methods ending with the "ry" suffix. I mentioned riflery, archery, boomerangery (that isn't an actual word), and there's one more. It's called falconry. This is, hands down, the single coolest method of hunting ever devised by man. For those not in the know, it's when you capture a raptor and train it to fetch wild quarry, like having a dog catch a . People who practice falconry are called falconers, and come to think of it, that would make for an awesome name for a baseball team. But I should get back to the circle of life before I get off on a tangent. While the game that Thorondor brings home might not make for as impressive of a photo-op as the triceratops that Steven Spielberg shot, you get to pose with a bird of prey on your wrist as your loyal companion. You can be that person. Throw in a steed, and I'm pretty sure that you'll be the coolest person in the room no matter where you go. You win in life. Game over.
Of course, it isn't just awesome because it looks cool, but because it truly is a great achievement and puts you in a rare tier of hunter. Falconers are the SEAL Team Six of hunters. You don't buy a raptor for this. You capture an actual wild animal and forge a bond with it. You need to spend a great deal of time every single day with it. You most likely need to be single. Many places don't hand out permits for it unless you take a written test on it. You usually end up spending a small fortune on books because of the sheer amount of knowledge required for both training and caretaking. It can take two years to finish an apprenticeship, and it takes over seven years to become a Master. Most falconers will refuse to teach you anything unless you provide proof of the seriousness of your commitment. You have to spend great deals of time in the middle of nowhere, far away from roads, rifle hunters, power lines, barbed-wire fences, and all other things that could put all the time that you invested with Hedwig at risk of amounting to nothing. You never feed them food from the pet store, but raw meat that you expect them to later hunt for you. You have to be emotionally prepared for the possibility that Hawkeye might pull a Richard Parker and dump you at any given moment, after so much effort was put in to trying to share your life with nature's most graceful predator?
Got that? Now throw in ethics. The law mandates that you provide your raptor proper housing, and that you have the proper equipment. The American federal government has no laws saying that falconry impacts the environment, but falconers have to hold themselves to a higher standard than what the law permits. A falconer owes it to other falconers not to damage the reputation of the craft, and must never harm any birds. If you or another falconer looses a bird, then custom dictates that you put serious effort into finding it. If you run out of money to support this lifestyle, you must pass your raptor on to another qualified falconer or safely release it into the wild. The future of the sport rests on your shoulders to sponsor apprentices. You shouldn't let your friends touch it. You shouldn't do anything commercial with it. And remember what I said earlier about photo-ops and being the coolest person in the room? The falconry community actually discourages publicity, and much of what you do will go without recognition. At this point, we're not just talking about ethics, but the mettle of one's character.
Clearly, it isn't for everyone. However, I hold the ethical expectations for falconers as the classic standard which all hunters should take inspiration from. The future of the sport depends on the nobility of its participants.
Anywho I changed my profile pic, which gives me a grand total of one pic change and one name change in the seven years I've been here. All ya hooligans with your name contests every month give me vertigo.
'finds story about a live action Dora The Explorer movie being made with Michael Bay as a producer'
Me: Yeah, that's something to blog about.
So Michael Bay is helping to make a Dora The Explorer movie. I didn't see that coming, TBH. Apparently the movie will age Dora up to a teenager and will be about her moving to the city to live with her cousin, Diego.
So congrats to all the Dora The Explorer fans for getting a live action movie for your show. Let's just hope that Michael Bay doesn't blow it up.
Since then, things have been optimistic. As of now most of the larger fires seem to be at least 50% contained, which does not mean half way out. They just have preventative lines drawn around them using whatever techniques fire fighters use (or at least, that's how I am interpreting this). If the winds pick up strongly again the fires could get over the containment lines, but so far it doesn't look like those will come.
Though many people have been allowed to return to their homes, and some shelters have closed due to lack of necessity, no one is out of the woods yet. I don't know what the plan is to put out the fires, but it might be a while before they are 100% out. That said, the amount of help California is receiving is incredible. Apparently there are over 10,000 fire fighters working, and as of yesterday (I think) there were fire fighters from 17 additional states (Oregon, Washington, Idaho, Montana, Nevada, Utah, Colorado, Arizona, New Mexico, New York, Minnesota, Florida, South Caroline, Alaska, South Dakota, Wyoming and Indiana) and Australia helping out. Australia! Together they contributed 266 fire engines, 79 fire crews, and 56 other personnel. I am very grateful for the support that they have shown and for the work that they all are doing. If I weren't on the other side of the world I would sit outside the local fire station and cook pancakes all morning for them. My girlfriend works right next to it and will be dropping off a few batches of homemade brownies.
So while the battle is far from over, things are looking up, and I'm hoping they keep looking that way.
A wave, in general, is any function that obeys the wave equation. To simplify things, though, let’s look at repeating wave patterns.
The image above depicts a sine wave. This is the shape of string and air vibration at a pure frequency; as such, sinusoidal waveforms are also known as “pure tones.” If you want to hear what a pure tone sounds like, YouTube is happy to oblige. But sine waves are not the only shapes that a vibrating string could make. For instance, I could make a repeating pattern of triangles (a triangle wave),
or rectangles (a square wave),
Now, making a string take on these shapes may seem rather difficult, but synthesizing these shapes to be played on speakers is not. In fact, old computers and video game systems had synthesizers that could produce these waveforms, among others. But let’s say you only know how to produce pure tones. How would you go about making a square wave? It seems ridiculous; pure tones are curvy sine waves, and square waves are choppy with sharp corners. And yet a square wave does produce a tone when synthesized, and that tone has a pitch that corresponds to how tightly its pattern repeats — its frequency — just like sine waves.
As it turns out, you can produce a complex waveform by adding only pure tones. This was discovered by Jean-Baptiste Joseph Fourier, an 18th century scientist. What he discovered was that sine waves form a complete basis of functions, or a set of functions that can be used to construct other well-behaved, arbitrary functions. However, these sine waves are special. The frequencies of these sine waves must be harmonics of the lowest frequency sine wave.
The image above shows a harmonic series of a string with two ends fixed (like those of a guitar or violin). Each frequency is an integer multiple of the lowest frequency (that of the top string, which I will call ν1 = 1/T, where ν is the Greek letter "nu."), which means that the wavelength of each harmonic is an integer fraction of the longest wavelength. The lowest frequency sine wave, or the fundamental, is given by the frequency of the arbitrary wave that’s being synthesized, and all other sine waves that contribute to the model will have harmonic frequencies of the fundamental. So, the tone of a trumpet playing the note A4 (440 Hz frequency) will be composed of pure tones whose lowest frequency is 440 Hz, with all other pure tones being integer multiples of 440 Hz (880, 1320, 1760, 2200, etc.). As an example, here’s a cool animation showing the pure tones that make up a square wave:
As you can see in the animation, these sine waves will not add up equally; typically, instrument tones have louder low frequency contributions than high frequency ones, so the amplitude of each sine wave will be different. How do we determine the strengths of these individual frequencies? This is what Fourier was trying to determine, albeit for a slightly different problem. I mentioned earlier that sine waves form a complete basis of functions to describe any arbitrary function (in this case, periodic waveforms). This means that, when you integrate the product of two sine waves within a harmonic series over the period corresponding to the fundamental frequency (T = 1/ν1), the integral will be zero unless the two sine waves are the same. More specifically,
Because of this trick, we can extract the amplitudes of each sine wave contributing to an arbitrary waveform. Calling the arbitrary waveform f(t) and the fundamental frequency 1/T,
This is how we extract the amplitudes of each pure tone that makes up the tone we want to synthesize. The trick was subtle, so I’ll describe what happened there line by line. The first line shows that we’re breaking up the arbitrary periodic waveform f(t) into pure tones, a sum over sine waves with frequencies m/T, with m running over the natural numbers. The second line multiplies both sides of line one by a sine wave with frequency n/T, with n being a particular natural number, and integrating over one period of the fundamental frequency, T. It’s important to be clear that we’re only summing over m and not n; m is an index that takes on multiple values, but n is one specific value! The third line is just swapping the order of taking the sum vs. taking the integral, which is allowed since integration is a linear operator. The fourth line is where the magic happens; because we’ve integrated the product of two sine waves, we get a whole bunch of integrals on the right hand side of the equation that are zero, since m and n are different for all terms in the sum except when m = n. This integration trick has effectively selected out one term in the sum, in doing so giving us the formula to calculate the amplitude of a given harmonic in the pure tone sum resulting in f(t).
This formula that I’ve shown here is how synthesizers reproduce instrument sounds without having to record the instrument first. If you know all the amplitudes bn for a given instrument, you can store that information on the synthesizer and produce pure tones that, when combined, sound like that instrument. To be completely general, though, this sequence of pure tones, also known as a Fourier series, also includes cosine waves as well. This allows the function to be displaced by any arbitrary amount, or, to put it another way, accounts for phase shifts in the waveform. In general,
or, using Euler’s identity,
The collection of these coefficients is known as the waveform’s frequency spectrum. To show this in practice, here’s a waveform I recorded of me playing an A (440 Hz) on my trumpet and its Fourier series amplitudes,
Each bar in the cn graph is a harmonic of 440 Hz, and the amplitudes are on the same scale for the waveform and its frequency spectrum. For a trumpet, all harmonics are present (even if they’re really weak). I admittedly did clean up the Fourier spectrum to get rid of noise around the main peaks to simplify the image a little bit, but know that for real waveforms the Fourier spectrum does have “leakage” outside of the harmonics (though the contribution is much smaller than the main peaks). The first peak is the fundamental, or 440 Hz, followed by an 880 Hz peak, then a 1320 Hz peak, a 1760 Hz peak, and so on. The majority of the spectrum is concentrated in these four harmonics, with the higher harmonics barely contributing. I also made images of the Fourier series of a square wave and a triangle wave for the curious. Note the difference in these spectra from each other and from the trumpet series. The square wave and triangle wave only possess odd harmonics, which is why their spectra look more sparse.
One of the best analogies I’ve seen for the Fourier series is that it is a recipe, and the "meal" that it helps you cook up is the waveform you want to produce. The ingredients are pure tones — sine waves — and the instructions are to do the integrals shown above. More importantly, the Fourier coefficients give us a means to extract the recipe from the meal, something that, in the realm of food, is rather difficult to do, but in signal processing is quite elegant. This is one of the coolest mathematical operations I’ve ever learned about, and I keep revisiting it over and over again because it’s so enticing!
Now, this is all awesome math that has wide applications to many areas of physics and engineering, but it has all been a setup for what I really wanted to showcase. Suppose I have a function that isn’t periodic. I want to produce that function, but I still can only produce pure tones. How do we achieve that goal?
Let’s say we’re trying to produce a square pulse.
One thing we could do is start with a square wave, but make the valleys larger to space out the peaks.
As we do this, the peaks become more isolated, but we still have a repeating waveform, so our Fourier series trick still works. Effectively, we’re lengthening the period T of the waveform without stretching it. Lengthening T causes the fundamental frequency ν1 to approach 0, which adds more harmonics to the Fourier series. We don’t want ν1 to be zero, though, because then nν1 will always be zero, and our Fourier series will no longer work. What we want is to take the limit as T approaches infinity and look at what happens to our Fourier series equations. To make things a bit less complicated, let’s look at what happens to the cn treatment. Let’s reassign some values,
Here, νn are the harmonic frequencies in our Fourier series, and Δν is the spacing between harmonics, which is equal for the whole series. Substituting the integral definition of cn into the sum for f(t) yields
The reason for the t' variable is to distinguish the dummy integration variable from the time variable in f(t). Now all that’s left to do is take the limit of the two expressions as T goes to infinity. In this limit, the νn smear into a continuum of frequencies rather than a discrete set of harmonics, the sum over frequencies becomes an integral, and Δν becomes an infinitesimal, dν . Putting this together, we arrive at the equations
These equations are the Fourier transform and its inverse. The first takes a waveform in the time domain and breaks it down into a continuum of frequencies, and the second returns us to the time domain from the frequency spectrum. Giving the square pulse a width equal to a, a height of unity, and plugging it into the Fourier transform, we find that
This is one of the first Fourier transform pairs that students encounter, since the integral is both doable and relatively straightforward (if you’re comfortable with complex functions). This pair is quite important in signal processing since, if you reverse the domains of each function, the square pulse represents a low pass frequency filter. Thus, you want an electrical component whose output voltage reflects the sinc function on the right. (I swapped them here for the purposes of doing the easier transform first, but the process is perfectly reversible).
Let’s look at the triangular pulse and its Fourier transform,
If you think the frequency domain looks similar to that of the square pulse, you’re on the right track! The frequency spectrum of the triangular pulse is actually the sinc function squared, but the integral is not so straightforward to do.
And now, for probably the most enlightening example, the Gaussian bell-shaped curve,
The Fourier transform of a Gaussian function is itself, albeit with a different width and height. In fact, the Gaussian function is part of a family of functions which have themselves as their Fourier transform. But that’s not the coolest thing here. What is shown above is that a broad Gaussian function has a narrow range of frequencies composing it. The inverse is also true; a narrow Gaussian peak is made up of a broad range of frequencies. This has applications to laser operation, the limit of Internet download speeds, and even instrument tuning, and is also true of the other Fourier transform pairs I’ve shown here. More importantly, though, this relationship is connected to a much deeper aspect of physics. That a localized signal has a broad frequency makeup and vice versa is at the heart of the Uncertainty Principle, which I’ve discussed previously. As I mentioned before, the Uncertainty Principle is, at its core, a consequence of wave physics, so it should be no surprise that it shows up here as well. However, this made the Uncertainty Principle visceral for me; it’s built into the Fourier transform relations! It also turns out that, in the same way that time and frequency are domains related by the Fourier transform, so too are position and momentum:
Here, ψ(x) is the spatial wavefunction, and ϕ(p) is the momentum-domain wavefunction.
Whew! That was a long one, but I hope I’ve done justice to one of the coolest — and my personal favorite — equations in mathematics.
P.S. I wanted to announce that Equation of the Day has its own website! Hop on over to eqnoftheday.com and check it out! All the entries over there are also over here on BZPower, but I figured I'd make a site where non-LEGO fans might more likely frequent. Let me know what you think of the layout/formatting/whatever!
I was just informed that two of my RA's had made it their personal challenge to figure out what the weird circles on my door mean. They have apparently spent the last three weeks trying to crack what it says letter by letter.
They managed to do it. They still don't know what alphabet that was.
I'm amazed and stunned.
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