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Personal Ranking of Pokemon Regions

Posted by Ul-Pahrak in Mt. Coronet, Oct 19 2017 · 7 views
pokemon

Because why not. Subject to change as time goes on, surely, but as of right now, here’s how I feel about the seven main series regions.

1. Sinnoh, to the surprise of no one. You’re probably saying this is solely based on nostalgic bias, and that is a huge part of it, I’ll admit that. Pokémon Diamond was what brought me back into Pokémon, and it and Platinum were the first times I started to really consume all the data I could about the games, which set the trend going forward. Sinnoh’s the reason I’m playing Pokémon today, so it will always have a very special place in my heart. Of course, there is more to love about Sinnoh: personally I very much like the general design aesthetic of Gen IV Pokémon, I adore that the region is steeped in legend and myth, Cynthia, you can play your badges like a xylophone, and how could you not enjoy the Underground? I know Sinnoh has faults, but it’s very special to me, and I dearly hope the eventual remake is on a system I own.

2. Alola. Probably not surprising either. It’s a beautiful and extremely unique region, and it introduces many new twists on the Pokémon formula without rendering it unrecognizable. Many things I liked about other games/generations/regions were present in Alola, and while they weren’t all handled perfectly, I do think they were handled well, and it helps give the region a more “complete” feel. The plot and mythology are both engaging, though I do wish they’d intertwined a bit more. The Pokédex features favorites from every generation, and as for the new Pokémon…well, I’m a bit 50/50 when it comes to Gen VII’s design aesthetic, but that doesn’t weigh it down too much. And Alola forms are inspired (even if I can’t seem to get behind any that end up Dark-type). There are a few miscellaneous things from ORAS that I wish had stuck around, but Sun and Moon are improvements over X and Y in every conceivable way, and that helps me just really love this region. And judging from what we’ve seen of USUM, I’m beginning to wonder if it could even give Sinnoh a run for its money.

3. Johto. Kind of bumpy getting here, but…nostalgia played a big part again. Gold version was the first Pokémon game that was really mine and I didn’t have to share, so it’s the focal point of nostalgia as far as I’m concerned. Yeah, some of my memories of Gold are hazy, I don’t feel like I excelled when I played HeartGold, and now that I’ve got VC Gold I’m just baffled by some of the Pokémon distribution choices, but the feel of the region is something that just works. Difficult to explain, but this is all subjectivity anyway. And there’s something interesting about this graphical style, something notably old but still in vibrant color, that helps it retain this feeling of being a refreshing update to Gen I. There are some great Pokémon here too, though it can be difficult to find some of them—Typhlosion will always be a favorite, and darn it, I love Unown, I know they’re utterly useless in a battle but I’m too fascinated by their concept to feel anything but love towards these letters with eyes.

4. Hoenn. Before ORAS, if you heard me mentioning Hoenn, it was only to disparage it. I lost interest in Pokémon during Gen III, a large part of which I still blame on not being able to send Pokémon from Gen II up to the new Generation, and even now a large number of the Pokémon designs introduced here just feel a bit odd to me. Again, this hatred was all before ORAS. ORAS, when they came out, were the utmost pinnacle of Pokémon games on a mechanical level, and are still pretty darn close. They took a great deal of the good things in X and Y and really drove them home, capitalized on everything people already liked about the region (except for trumpets), and introduced plenty of new things that all worked very well. I utterly adore soaring, I can’t speak too highly about that feature. Hoenn’s personality also shone really well in those games, and it’s helped me find a much greater respect for the region. Though, since nearly all I like about it is strictly from ORAS, Hoenn itself falls smack dab in the middle.

5. Unova. Really, I’ve always had a bit of a love-hate relationship with Unova, largely stemming from the fact that during Gen V I was unfortunately going through a bit of a genwunner phase. (Thank God that’s over.) Make no mistake, I still stand by some of my criticism: Black and White try far too hard to distance themselves from the other games, worst of all by expressly forbidding us from catching any Pokémon from the first four Generations until we beat their main story, and that issue is exacerbated tremendously by the general design aesthetic of Gen V Pokémon, which I find to be, well…pretty terrible. (With notable exceptions—Chandelure, Gigalith, and Zekrom are some of my all-time favorites.) But there’s also a lot that Gen V does right. It was an enormous step forward in story and characterization—the Gym Leaders never felt more like an actual ensemble of people, to say nothing of Team Plasma, and it’s always impressed me that the plot actually interrupts your initial League challenge. There’s also a wealth of post-game areas to explore, and we’ve still only covered Black and White. Black 2 and White 2 (as opposed to the expected Pokémon Gray) were a unique choice in and of themselves, and retained the best of their predecessors while strengthening the mechanics, adding loads more content, and no longer restricting you to Unovan-original Pokémon, thank God. (Though now that I think about it, I did only use Gen V mons when I played B2 and W2…huh…) Taking a fairer look at Unova, it really is a wonderful place, and as that teenage inclination towards arbitrary bad decisions abates it might actually move up this list. Still…Gen V Pokémon designs…I haven’t gotten over most of you.

6. Kanto. Kanto is just, well…basic. And I mean, it’s the first Gen, it kind of has to be since it’s the template. But still, I just kind of…I don’t know, I guess I’ve used up all the nostalgia I had for it and feel like there isn’t really much else going for it? I’ve trekked through this region probably more than any other, but I don’t know that I really have anything in particular to say about it. The last two slots on this list were hard to decide between, and part of why Kanto’s here and not the bottom might just be extra slack for having the burden of getting the franchise off the ground. Again, I don’t know that I really have anything to say.

7. Kalos. And like, I like Kalos. I do. But X and Y left me wanting more from it in the worst possible way, and while I was focusing on the positives and giving it the benefit of the doubt when I was expecting them to do a third version or a second pair to fix that, they dropped this region like it was hot and moved right along to Gen VII. So ultimately, Kalos feels like a half-baked trial run for the rest of the 3DS titles and even GameFreak doesn’t really care about it all that much. And that sucks. Because there are a lot of things I like about Kalos: the Gen VI Pokémon design aesthetic is marvelous, its Pokédex being a collection of favorites across all Gens was a wonderful decision that thankfully seems to be the new norm, and I love Mega Evolution and have honestly docked points from Sun and Moon for being so quick to discard it. It’s controversial, but I’m very glad for the changes made to experience—I’m in my 20s, I don’t have the time or patience to grind and now I don’t have to. But for everything Kalos did well, there’s a lot that is just so sloppy. Characterization is a huge step back from Gen V. Plot is about as bare as Gen I. Can’t say there’s much of a post-game. X and Y, in hindsight, just feel like they could have been so much more, and while we can forgive that at the start of a generation, not following up with a correction just makes it look so much worse. Because now X and Y are all we have to judge Kalos with, and it doesn’t compare well. Again, there’s a lot that I like about X and Y and Kalos—I enjoyed playing these games. I just wish I could have had an even better Kalos game after that. Sun and Moon feel like far more complete games, and remembering how abrupt the transition from Gen VI to Gen VII felt at the time, that’s somewhat strange to me. So I guess my relationship with Kalos now is a bit like what I once had with Unova, and maybe as time goes on I’ll be less bitter about the lack of a third game and I’ll feel better about this region. Chances are there will in the distant future come a day when X and Y are remade, and I dearly hope those take better advantage of Kalos.

So, there you have it. Who’s offended? :P

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Thursday Tangents

Posted by Nahaki in I Have No Idea, Oct 19 2017 · 21 views
what a thursday

what a thursday today is. Ever wondered what it's like to experience a thursday?
  • Do you remember when there were theories about Hero Factory having a Villain Factory final wave with Von Nebula
  • 2011 was a pretty good introduction for CCBS, the first wave, Ordeal of Fire, was about characters, the second wave, Sava Planet, was about creatures
  • I have to wonder if Savage Planet/Ordeal of Fire were based on potential set designs for G1 BIONICLE - Jungle environment in the second half of 2011, and Bota Magna was meant to be in 2011. I doubt that is a coincidence.
  • Bota Magna: It has several regions. There are three major settlements. I'll tell you now: there was a Vorox settlement in canon. Here though, it got wiped out by a volcano's eruption. Vorox still exist. I'm just simply not letting you have stinger tails because that would be an unfair advantage over other players. Should I reconsider?


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Review 70620: Ninjago City - Level 1

Posted by xccj in Team J.A.F., Oct 19 2017 · 11 views

This model is so big and packed with detail, that a single blog entry won't suffice. (But not a full on front page review, that would be exhausting to finish.) I haven't even finished building it; partially because I was to spread out the build for as long as possible, partially because it already takes a really long time. (This would make for a great group speed build set, IMO, but I'm not rushing through it myself.) Anyway, I wanted to talk about some of my favorite bits, and because the levels are almost independent of each other, I thought I'd split it up that way.

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The first level is build using six numbered bags... although that's slightly off, since most numbers have multiple bags. It's a LOT of pieces. Not a lot of vibrant colors on this level; the buildings are mostly brown and white, with gray, dark tan, and olive green built into the base. The water uses a TON of trans blue tiles, which itself was a mighty task to put down properly, but the instructions weren't too bad. The water itself is a cool design, because they have different colors underneath the tiles, giving a vague idea that there's a waterscape below the city. (An interesting bit is that trans neon green was used as the base beneath the sewer... so take that for what you will.) There have been a few unusual parts here and there, so this does make for a good parts pack, but hopefully some of the newer, rarer elements will start showing up in cheaper sets.

The first thing that surprised me was how small the set's footprint is. Others have commented that it's only on a 32x32 baseplate, but at least a third of that is already taken up by water tiles, so the entire model is build up on three corners of a baseplate. :o There's already a lot going on the first level, as you have the clean up robot's storage area, the fish market, a small bedroom, and a tea house, as well as a cool bridge and the start of the elevator. The individual rooms are rather small, but have distinctive looks to them. It does remind me why I decided to avoid module buildings; there's a lot of detail packed into them, but they're hard to look in when the set's build, and even harder to access due to their size. (The cleaner robot's storage room is particularly irksome to access.) There are some nice details, and I especially liked the green Sabine hairpiece used as a plant. Some of them also have distinctive roof designs, but my favorites so far is the design using the crowbars. (Did that crowbar piece always snap onto a bar like that? Was I seriously missing their potential?)

The water is mostly tiles, but there are still a few studs popping out, either as plant life (the paint palette as a lily pad is great) or as 1x1 plates (to fit in the corners that the 1x2 tiles couldn't finish.) On one hand, that means the boat can't be slid along effortlessly through the water, but then there are also some connection points for it. The boat is an awesome little design, and I really like how they built the cover. (I mean, it doesn't feel like it's minifig scale, but I can let that slide.) They also included a neat dock on the back using brown technic connectors. The bridge is also a great design, and I'm impressed at how they designed the angles rails. Previous sets have used the clips, but they leave exposed gaps and sometimes aren't secured in. Well, they cleverly used the neck brackets to hide the gap, and the balljoints secure the tile at the bottom. The elevator is also a simple but effective and modular design. I skipped ahead and build the elevator stand, and it works great. There's also the small tower on the far corner, which is okay but the cooler part is the design they used to anchor it at the 45 degree angle, which is something we don't usually see in system sets.

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The figs are... well, they're not exactly the highlight of the set. Three figs have traditional Asian robe designs... that are seen in a lot of the Ninjago sets, wore by various civilians. Civilian Kai is great, mostly because he completes my civilian ninja collection, but the actual design is still pretty meh. I guess the best here is Ivy Walker, with her colorful torso design. (I thought she might be Jay's sister, since his last name is Walker on the TV show, although he doesn't have siblings. Instead, maybe she's his Aunt, and little Sally is his cousin?)

One thing I don't usually look forward to in sets is the stickers, but this one is an exception. Only, the ground level doesn't exactly have a ton. The two tiles along the bridge do reference Bionicle (they were the first stickers applied too) but sadly they're partially hidden by the plant life. My favorite here would probably be the kimono lady design on a door panel, but the tiger poster in the tea house is also well done.

There aren't really any fun play features on the ground level, although I know we get some as we move up in the set. I guess there's the elevator, as well as a few sliding door panels.

In conclusion, the bottom level is a solid design, and is sturdy enough to add more rooms to the top of this. The room designs are okay... some would be nice as their own mini set. The highlights so far have been the boat design, the crowbar roof, and the stone bridge design. I'm eager to build the next level and see what cool designs are waiting there.

:music:

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kaput goes computer

Posted by Finch in Alrest, Oct 18 2017 · 24 views

so my laptop went kaput

good news is i got a new laptop. which isn't a garbage one from 2013. hallelujah.

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Larry Nassar

Posted by Jean Valjean in Kraggh's Works ♫♪, Oct 18 2017 · 26 views
MeToo, Olympics, McKayla Maroney and 1 more...

:kaukau: Quick do a Google search, and you'll know what I'm talking about. The point is, when it comes to people abusing people of power, it's incredibly important to name names, even if it's humiliating to do so. I want to see people like this behind bars, as expediently as possible, and I think that in order to end the culture of abuse, people need to start taking pragmatic steps to combating it. Reporting people who abuse their power when they have, or have had, power over you takes quite a bit of courage, but those who show such courage have my admiration.

Yesterday I said that I admires Olympic athlete Gracia Gold. Today I voice my appreciation of McKayla Maroney. She is an inspiration, and has done much to contribute to a culture that has provided me a great deal of inspiration over the years, to be the best person I could be. It is good to see that she is inspiring others who need inspiration far more than I ever did.

24601



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Rrrrrrebels

Posted by Bfahome in You're attacked by a Repair Nektann, Oct 16 2017 · 43 views

Rrrrrrebels Well we're in the home stretch now. Final season and it's starting off with a bang.

Spoiler


also pic related

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Progress on Fighting CA Fires

Posted by Vorahk1Panrahk2 in V1P2's Blog, Oct 16 2017 · 32 views
General Life Stuff

It was really hard to be optimistic on Wednesday when I made that blog entry. The fire was huge, the area had doubled overnight, strong winds were expected to pick up again. And after posting that entry, the area where my girlfriend lives was advised to evacuate. But hey decided to stay until until it was mandatory, which fortunately never came, and as of now even the recommended evac has been lifted.

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.

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92% of teens have moved on to memes,

Posted by Pomegranate in Hundreds of Crimson Seeds, Oct 14 2017 · 73 views

if your one of the 8% that still use old message board fads, put this in your sig

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The Illusion of Choice

Posted by Ta-metru_defender in TMD's Creatively Named Blog, Oct 14 2017 · 51 views

Essays, Not Rants! 290: The Illusion of Choice

When not raiding Soviet bases to 80s hits in Metal Gear Solid V, I've been playing Until Dawn with my roommate. Now, I don't really do horror, like, at all. But Until Dawn features a supposedly robust choices and consequences system, which I am, of course, a sucker for.

We’ve finished the game and there's been a good deal of payoff to some of the choices we've made. The big thing we're looking forward to, though, is playing it again and making different choices to see what would happen.

Because right now a lot of what happened feels like a direct result of the choices we've made and I wanna know how much of that is really because of what we did. Every little plot turn can’t be the result of our decisions, even though it can feel like it.

A lot of the time, when we play a game with multiple choices, we want everything we do to be impactful and for it to create a tailored set of consequence that are entirely dependent on what we did.

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Doesn’t that sound cool? Every choice you make has consequences! Siding with Miranda or Jack when they argue aboard the Normandy in Mass Effect 2 could spell disaster down the line! If Walker doesn’t spare that guy in Spec Ops: The Line what will it mean for the future?

The problem is, games are a finite medium. What’s done, is done, and has to have been doable. There’s a limit to your free will, a limit set by the game developers and their bother and/or budget. It turns out that choosing Kaiden or Ashley has no real choice on the rest of Mass Effect, as the survivor fulfills basically the same role in the sequels. Picking Udina or Anderson doesn’t have much bearing on Citadel politics, because Mass Effect 2 doesn’t have much of it, and by the time 3 rolls around, Anderson (if you chose him) has stepped down so that Udina represents the humans and the intrigue on the Citadel proceeds accordingly.

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Now, I am kinda picking and choosing some examples, Mass Effect does have some brilliant moments of consequence (whether or not you saved Mealon’s research in the second game has a massive impact on the third – it’s that it’s one of the few choices of that nature that make it stand out so), but a few different playthroughs, the cracks in the game’s design start to show. No matter what, Udina will end up on the council. The Rachni will return whether or not you kill their Queen. Whether or not you sacrifice the Council in the Battle of The Citadel doesn’t mean much ultimately. To quote Eloise Hawking in LOST: the universe has a way of course correcting.

Which is a bummer, because what if, to beat a dead horse, picking Anderson or Udina made for totally different plot lines in Mass Effect 3. Maybe Anderson as Councilor meant that Cerberus never managed to attack the Citadel, but in exchange made the mission to Earth that much harder without him in your corner. It does mean a lot of resources, but it also means a more personalized experience.

I think that might be why I’m hesitant to jump back into Until Dawn. Right now everything happened as a result of my choices. Little tweaks to the game’s horror were because of my answers to questions posed to me (Snake-Clowns with Needles, though the snakes never showed up). Playing the game again (which I absolutely want to do to, why else, see what would happen) will probably show where the seams are and reveal how little impact my decisions had. That it doesn’t on the first play through speaks to good writing.

Because choice in games are an illusion, and will continue to be until you have an infinite number of monkeys typing up an infinite number of outcomes to an infinite number of players’ decisions. But until then, players can be tricked into thinking we have a decision. If the game’s narrative makes the causality feel like it had to happen, like that your choice led you here no matter what, then the illusion isn’t broken. Just spackle those cracks with good writing and we’re onboard.

For the first playthrough or two, anyway. After that it boils down to just gaming the system as much as you can (how can I make sure everyone dies in the most gruesome way in Until Dawn?).

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My birthday

Posted by Angel Beat in ~ Canterlot's Sparkling Jewel ~, Oct 10 2017 · 36 views

It was pretty nice yesterday, because this arrived in the mail:

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new star wars trailer

Posted by Elittra in Great Games, Oct 09 2017 · 55 views

yes please

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Stress and misery

Posted by ToaDally Spooked in Terrible Things for a ToaDally Low Cost, Oct 07 2017 · 52 views

When stuff really starts getting to you and you can't even fall asleep until nearly 4 am, I'm such a mess lately and anything I try to do about it makes me worry more about it all

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Equation of the Day #20: The Fourier Transform

Posted by Akano in Akano's Blog, Oct 03 2017 · 80 views
Autobots, roll out and, Fourier and 2 more...

Today I wanted to talk about one of my favorite equations in all of mathematics. However, I won’t do it justice without building up some framework that puts it into perspective. To start out, let’s talk about waves.

A wave, in general, is any function that obeys the wave equation. To simplify things, though, let’s look at repeating wave patterns.


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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),

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or rectangles (a square wave),

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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.


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Image: Wikipedia


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:

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Animation: LucasVB on Wikipedia


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,

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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,

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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,

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or, using Euler’s identity,

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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,

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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.


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One thing we could do is start with a square wave, but make the valleys larger to space out the peaks.

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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 1 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,

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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

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where

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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, . Putting this together, we arrive at the equations

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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

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Or, graphically,

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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,


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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,


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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:

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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.


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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. :psychotwitch: Let me know what you think of the layout/formatting/whatever!

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Rust Ghoul

Posted by Lord Oblivion in Oblivion's Blog, Sep 29 2017 · 65 views

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(links to topic)

Flickr


hi



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Someone managed to decode Matoran

Posted by JMSOG in Odd Thoughts by JMSOG, Sep 28 2017 · 155 views

On my dorm door, I put a whiteboard, and wrote a message in Matoran saying "If you can read this, I want to talk to you". It's a longshot, but the idea was that if someone was able to recognize what the alphabet was (or, even better, was a dedicated enough fan of Bionicle to read it fluently), that was a person I wanted to get to know.

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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A Thirty Fifth Random Question

Posted by Toa Smoke Monster in The Island., Sep 18 2017 · 78 views
35th, Random Questions

What is your favorite song covered by a band and/or singer who didn't originally sing it? (AKA a cover song)

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Ch-Ch-Ch-Ch-Changes

Posted by SwagtronYOLO in Designated Tekulo Crying Corner ♡, Sep 17 2017 · 127 views

Turn and face the strAAAAAAAAAAAAAAAAAAAAAAAAAA

I GOT A JOB IN WASHINGTON AS A PASTRY CHEF AND I ACCEPTED IT AND NOW I HAVE TO APARTMENT HUNT AND FIND A PLACE TO LIVE WITHIN A MONTH AND THIS IS REAL ACTUAL ADULT STUFF THAT MY EDUCATION AND LIFE EXPERIENCES DID NOT PREPARE ME FOR.

WHAT IS? HOW DO?

But, like, everyone at work keeps telling me finding a place is the easy part and that I'll be fine and that is reassuring.

@w@

This is gonna be fuuuuuuuuuuuuuun.

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I'm Getting There

Posted by Kagome in Wind's Weblog, Sep 11 2017 · 218 views

There will be a day where I will not succumb to self-effacing urges during a depressive breakdown, promise.

So let's see:

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I'm cute, so bi, so trans, still part of Thunderfury, all that.
Dang, I can't believe it will have been two years since I started HRT in November.

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Final Boss

Posted by Primus in Prime Time, Sep 11 2017 · 63 views

Built some competition for an old MOC of mine. Topic to come later today.

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Ideas for a Broken Lhikan Greatsword Axle?

Posted by Illuminatus in The Minimalist, Sep 10 2017 · 77 views

So just a few hours ago I got what I thought was a great deal on Toa Lhikan and Kikanalo: about $25 in local currency for a seemingly complete set; no box or instructions included. When disassembling the set and doing a piece count, I found a few were missing and several others were in the wrong color. All in all, nothing I wasn't prepared for (these are typical issues when buying from "normies"); I've already ordered the instructions from a US seller and all the necessary pieces from a UK seller. By my current estimations, the extra total shouldn't be more than about $15.

Now the bit that I wasn't prepared for was the fact that Lhikan's Greatswords apparently have very fragile and breakable axles. Upon close inspection of the pieces, I realized one of the axles was glued to the actual sword and when I poked at it it came off with barely any effort; needless to say I'm rather disappointed with the seller, but I've decided to let it go. Here's the issue: Lhikan Greatswords cost a lot on BrickLink, precisely because of their breakability. So I've decided that I'll go ahead and try to mend broken fences, i.e. axles.

My idea right now is to basically even the surface out of both the sword and the axle and then glue them together at as perfect an angle as I can manage. First I'll need to fill in the small crater on the sword with glue or plastic, then file it to produce an even surface. Then I'll file the broken surface of the axle. Then I'll glue the two.

What do you guys think? Yea or nay? Any better ideas? Have any of you gotten a broken piece like that and how did you deal with it?

Perhaps I should get a cheap piece in the same pearl light gray (I think?) color, melt it somehow and use that instead of glue?



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