@PrinceNorthLæraðr Okay, I just read what partials are. (Apparently they're alternative sounds played with the same fingering on the holes but with different adjustment of the valves.) Why does that mean you can play trumpet? (I'd think that, if anything, it would mean you can't, since you need a hand to adjust the valves and a hand to handle the holes.)
(Not to mention a third hand for the -- I think it's called -- mute.)
@msh210 So on trumpet (and brass instruments) partials are determined more or less by the pressure of your lips
Partials mean that I can play different notes with a single type of fingering
(Individually. Can't play them all at once, obviously)
Basically, because the sound is produced at the lips, "buzzing" (lip vibrating) at a faster rate means that the frequency coming out of the trumpet results in a higher pitch
The valves are added in to modify and help focus the pitch. But technically valves aren't even necessary on trumpets
Once you start playing really, really high, the partials become close enough that you can adjust them just by slightly modifying the embouchure (fancy word for how we form our lips to create sound)
Pitch is also determined by your tongue position and the rate at which air is being pushed outwards
@Sciborg, I edited (well, suggested an edit) out code blocks that were being used for emphasizing quotations on English Language & Usage - COMPLETELY UNECESSARY, SERIOUSLY PEOPLE REALLY? - and I thought of your screen reader.
still trying to find some way to use some anagrams of WHAT CAN UP GIRLS for WALPURGISNACHT, but also &lit, after my failed attempt in the CCCC, but it's not quite working out
shame
anywany, at least puzzling is up
and North might've knocked out that musical connect wall - nice!
Also, this financial aid application wants the exact value of my parents' retirement accounts (all of them. added together. they don't actually know this value.)
a few more words about the algorithms I gestured vaguely at above. Mediants: if a/b,c/d are rational numbers then so is their "mediant" e/f=(a+c)/(b+d)=1. If further ad-bc=1 -- i.e., a/b and c/d are "as close as possible" -- then the same is true for a/b,e/f and e/f,c/d. So, if you have any positive number, bracket it with 0/1 and 1/0, and then repeatedly find the mediant and replace one of the endpoints with it. This gives you an ever-decreasing sequence of intervals containing your number.
Continued fractions: suppose x is a positive number. Then its integer part [x] is an approximation to x, obviously, and then the remainder x-[x] is between 0 and 1. Now replace x with 1/(x-[x]) and repeat the process. If x is rational, it will eventually stop with a remainder of 0; if x is irrational, it will go on for ever. At any point you can stop and just ignore the remainder (pretending it's 0). Why would you bother doing any of this?, you ask. I'm glad you asked.
Suppose you stop at some point. Then you've got a rational approximation to x. I should explain how; the easiest way is with a concrete example. Let's take pi. We have pi = 3 + x1 where 0 <= x1 < 1, and then 1/x1 = 7 + x2 where 0 <= x2 < 1, and then 1/x2 = 15 + x3 where 0 <= x3 < 1. (Those integers 3, 7, 15 are the integer parts above and the x1, x2, x3 are the remainders.) So we have the following approximations:
And each of those (except possibly the first) is the best approximation whose denominator is smaller than the next one's denominator. So, e.g., to do better than 22/7 you need a denominator of at least 106. To do better than 333/106 you need a denominator of at least 113, because the next one is 355/113.
And to do better than 355/113 you need a denominator of at least 33102, because the next one is 103993/33102.
These things looking like 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + 1/...)))) are called "continued fractions", and there are some nice algorithms to make computing them and using them less painful than the explanation above probably makes it seem.
And now I'll shut up about continued fractions, but they are pretty cool and if you're interested in rational approximations you absolutely need to know about them.
It isn't quite true that each continued fraction "convergent" is the best approx up to the next one's denominator. They are always the best up to their own and often somewhat further, but not always up to the next one. Sorry about that.
@HTM Oh, very "saucy" at times. In Chrom and Robin's friendship level up interactions, Chrom walks in on Robin taking a shower, and Robin accidentally does the same to Chrom
Nothing explicit is shown obviously, but it's SOOOO funny
One of my favorite vg themes of all times, btw is this
The answer is:
