#logarithms specifically

[effervescent-experience]

you know you've hit academic rock bottom when the most effective study method is writing scenes where your comfort character teaches you the material instead of just doing math like a normal person

[lucyandthepen]

get you alone | ljn ( m )

ideally, jeno should have his hands full with teaching. (un)fortunately, he only seems to have his head full of you.

pairing: tutor!jeno x reader verse: college au rating: r ( minors, do not interact! ) warnings & tags: jeno is a college algebra math tutor & reader is failing, written in lapslock, not beta’d in any shape or form so please excuse mistakes, unprotected sex (wrap it before you tap it, folks), piv, oral (f!receiving), use of pet names (kitten, angel, sweetheart), praise, reader calls jeno ‘sunbae’ until she doesn’t, size kink i guess if u squint! word count: 8.5k

a/n : actually this was written for a different fandom but i’ve decided to make it a jeno fic bc idk why not! first time writing in a different perspective so it’s a bit odd for me & i can't say i fw with this style nor am i particularly proud of this fic but she is ... sumn! also i fear i have a thing for the math tutor trope but that’s neither here nor there AHA enjoy !! 

if you liked it, please consider reblogging to support (especially because this may get flagged for mature content)!

there wasn’t anything special about your case; at least, that’s what jeno had thought when he picked up your request before he met you. before he met you, you were just another student trying to demystify the painfully enigmatic art of getting through college algebra. before he met you, he had already tagged this case as another charity stint — a good way to get brownie points with the dean’s office and the mathematics and natural sciences department. in fact, thinking of all his tutoring cases as community service made them somewhat palatable, if not a little forgettable. he was quite sure, at the time, that you’d be in and out — both of the tutoring center and his memory. such was the case with most of his other tutees, anyway. 

he hadn’t expected you to be… well, you — a pretty little thing, with your sweet smile and your wide doe eyes. on the first day, you’d stood out; you’d arrived at the tutoring center’s lobby in a short dress, knit cardigan, and coquettish makeup, as if every fiber of your being were bidding the spring a solid farewell. multiple heads had turned, including his, as you came up to the front desk and asked for one lee jeno for college algebra. you were eager for summer, jeno had learned as you broke the ice little by little, in part because you looked forward to visiting okinawa with your family, but also because you were eager to get your first semester out of the way. that much, you had in common with most of his other students — almost all of the ones seeking help in college algebra only took it as a depressing core requirement of whatever degree they were doing. you, specifically, were focusing on fashion design; that very vividly explained your attention to your looks. this mathematics class was a thorn in your side, a mandatory thing that was simply supposed to get you through later business-oriented classes in your degree program. for jeno, however, college algebra had become the perfect excuse from the moment he’d laid eyes on you. 

the more time he spends with you, the more he thinks you’re exactly his taste. it starts off with little things he finds attractive, things he picks up while he’s watching you fill out the practice sheets he’s prepared for you on quadratic equations or while trying to get you to understand logarithms — your neat, tiny handwriting, almost like print; your habit of boxing your final answers in firm strokes, even if they’re hopelessly wrong; your colored tabs, cascading down the page side of your textbook. but as the weeks wear on, he sees all the little things in between — the way your long eyelashes quiver when you stop and close your eyes as you think for the answer, the upturn of your plush lips when you have the same answer on the practice sheet as he does, the deepening of your artificial blush with a natural hue when you realize you don’t know the answers to his gentle questions. he notices that you refuse to wear anything longer than a knee-length skirt despite the still-strong winds, notices that your tiny palms are always smooth and pink, that your hair always smells of coconut milk. these are things he can’t help but jot down in his memory — that was exactly what you were, after all: memorable. 

and the more he remembers about you, the more jeno wants you. yet he’s never made a move, never given so much as a hint of his interest, not only because there are prying eyes all around the building but also because you have never so much as shown a smidge of desire back. in fact, he has to wonder if you’ve ever thought of him in a different capacity — not as a tutor, but as a man. if you have, you’ve never made that obvious; you always talk to him respectfully, the little wall you’ve erected between the both of you remaining steady, and you never let your eyes linger on his face for longer than it takes for him to explain what you don’t know. jeno has had his fair share of female students, and in all of them, he’s seen the same kind of hunger — to few, he’s catered to their whims, if only to pass the time, if only for his own benefit. but you, with your ribbons in your hair and your sweet, sweet mouth, have never once shown that same kind of desire. 

he doesn’t know if it frustrates him, but he does know one thing — it makes him want you all the more. 

he wants you even now, as you sit across from him, dolled up as usual. even now, as your eyes take on a glassy sheen of defeat, your cheeks puffing out in the way that tells him you’re admonishing yourself once again, he craves you — maddeningly so. and he realizes that it doesn’t really matter if you're not the one to fall first, as long as he can still have you. 

“time out,” you beg, your fingers meeting the palm of your hand to signal a break. “my brain feels like it’s going to explode.”

“you just had a break ten minutes ago,” jeno reminds you, though there’s a lighthearted amusement to his voice that makes you smile sheepishly. “at this rate, you’ll be on more breaks than you’ll be taking the time to actually learn.”

“i’m trying,” you groan, your fingers curling against your forehead as you bump your head against your fist. “i just don’t think i’m cut out for this polynomial whatever — trial and error bullshit.” 

“you’ll hate me for saying this — but you’ll never know unless you keep trying.” 

“funny.” your sigh rustles the papers in front of you gently. “how do you do it, sunbae?”

“hm?” 

“you’re not only good at this stuff, but you’re so good you’re able to take the time to teach people like me.” 

“strengths and weaknesses — it’s the natural way of the world.” jeno smiles gently at you, and he notes how his chest feels tighter when you return the sentiment shyly. “i could never do what you’re doing in your own degree, try as i might. anyway, you’ll get there. i won’t let you become my first ever failed project, you know.”

“i wouldn’t want to let you down either, sunbae, but—” the back end of your pencil taps lightly against the surface of the table. “it just feels hopeless. i can’t focus on anything. it’s so… so abstract, and everyone here is talking all at once, and i don’t even know what i’m ever going to get out of this class in the long run.” 

even when you’re dejected, you look pretty; your bottom lip juts out naturally when you whine like this, and for a moment, jeno can’t say anything in response. he’s too busy wondering what your mouth would feel like on his — on him. when he snaps himself out of his brief reverie, he notices you’re looking around at everyone else — and he has to agree that with the noise level in this whole building, it isn’t the most conducive site for learning, especially when the learner is already so averse to the subject matter.

“i can’t help much in the way of it being too abstract,” he says kindly. “but it’s not a requirement for us to have our sessions here. i know it can be quite distracting, all these voices flying around, so why don’t you look for a place that better suits you, and we can start meeting there instead? the more comfortable you are in your environment, the better you’ll be able to absorb the material, i’m sure.” 

“you think?” your pencil comes to a slow halt as you refocus on him, a thoughtful light glimmering behind your gaze. “yeah — yeah, i actually wouldn’t mind that. then, i’ll look for a different place for us to meet, and we can start there next week. how does that sound?”

“whatever suits you suits me,” he responds easily. 

he lowers his gaze immediately after you flash him a blinding grin; there are far too many people here, as you both very well know, and if he keeps looking at you and your pretty little expressions any longer, he might just give them something to actually look at. 

it had been your idea, not his, so why did jeno feel like he’d dragged you into a compromising situation?

you’d texted him over the weekend that your search for a new venue had been absolutely fruitless; every cafe and study space you’d been to was either too expensive or equally as packed with people, if not both. jeno had seen the preview to your message, but he hadn’t been prepared for what it read out in full when he’d actually opened it. 

sunbae, would it be too difficult to just meet at my apartment? i attached a map, so let me know!

it wouldn’t be too difficult; logistics-wise, it was walking distance from campus and almost directly across the train station he takes home. it also definitely promised an environment you were comfortable in, and you wouldn’t have to worry about excess noise from any other tutoring groups. no, the difficulty really only lied in himself — you two, all alone, would certainly mean his mind would be up to no good for the two hours every monday, wednesday, and thursday you would be together. 

but for your sake, he’d try to rein it in, with the operative word being try. 

your place is as neat and as pretty as you are; he doesn’t know if you’ve cleaned up for him, or if you’re naturally this organized, but he likes it all the same. it smells of toasted marshmallow and expensive perfume, and all your furniture matches. jeno supposes he likes that in a woman — someone able to care for herself, someone who cares about herself. and you’re always just as neat and pretty to match, with your hair always styled sweetly, your makeup always enhancing your features. 

the problem is that now that he’s in here, where you live, and where you spend most of your time, jeno’s mind seems to wander too much towards thoughts about what you do in private. he rejects studying on the couch, not just because it’s bad for posture and concentration but also because he can’t help but imagine you pressed into the cushions by his hand. he suggests the small dining table you have, but on the second meeting at your place, he starts thinking about what you might look like seated on the table, your ass hanging over the edge and his face buried between your thighs. whenever you look up to ask him something, he drinks in your lovely, made-up face again, and starts wondering what your makeup would look like ruined before he interrupts that trainwreck of a thought with the answer to your question. 

by the end of the week, jeno’s defenses are all but shot, and he realizes that this situation might be optimal for you, but it definitely isn’t doing him and his now constantly straining cock any great favors. 

he supposes that your performance has somewhat improved; you’re less likely to trail off when you’re thinking and can actually do practice sets for a lot longer without all the noise and hubbub around you. your only real hindrance is yourself and your frustration; you have a habit of giving into your carelessness that sends you spiraling into despair, and it doesn’t help that when you press your cheek against the surface of your dining table and whine, the comfort jeno offers is noticeably delayed because he’s too busy thinking about his cock between your lips. 

“my dad’s going to kill me if i fail this midterm,” you grumble, stabbing the practice sheet with your pencil; it skids sideways, and jeno robotically fixes it back into proper alignment for you, careful not to brush against the arm that’s folded inwards, supporting your chin. “he only agreed to let me take this degree because of the business aspect of it. as if i’ll need to know about—” you check the header of the worksheet. “domain and range when i’m doing actual design work.”

“you’ll never know what might be useful later on in life. i definitely thought this was nonsense back in high school — and then i got this job.” 

“and now you’re rolling in dough?” you smile slightly. jeno chuckles. 

“i’m a long way away from having myself a scrooge mcduck golden pool, but i make enough to get by very comfortably, thanks to this.” 

“thanks to me, you mean.”

“you’re not my only student,” he snorts, pinching your elbow; you cry out exaggeratedly. “focus up. the hour’s almost over, and you should have finished with this much earlier.”

“can you leave it as homework?”

“not a chance.”

you blow out a sharp puff of air. “my mom used to do this thing where she’d give me rewards if i did well with my homework. i wish i’d still get something out of this.” 

“what kind of rewards did she give you?” 

“chocolates — candy, or sometimes we’d go out for milk tea together, if i did a particularly good job.”

“this is math tutoring, not a trip to the dentist,” jeno says, amused. 

“a trip to the dentist would be more enjoyable,” you mutter under your breath, picking up your pencil and doodling an angry face next to the number you’re only halfway through solving. “this totally blows.” 

“try to finish this before the hour’s up, and i’ll see if i can get you something nice. out of my own paycheck,” he stresses, prodding at your cheek to shift your attention back to the paper. he doesn’t miss the fact that your eyes light up, childish as the promise is. 

he doesn’t know if that’s really what motivates you, but you do manage to finish the worksheet with a few minutes to spare before the clock hits seven, and that earns you some light, solo applause. it isn’t much by way of true praise, but you flush with pride all the same. jeno packs his things in silence as you get yourself a glass of water, and you see him to the door. only there does he notice your eager eyes, your expectant smile. 

“what’s going through that pretty little head of yours?”

“are you really going to give me a reward? i did great today, you know,” you respond bluntly. 

“you were serious about that?” he laughs. 

“absolutely. i earned it.” you raise a slim finger, wagging it in his face. he trails it with his gaze, no shortage of amusement in his eyes. “next monday, i want something sweet.”

jeno takes in the sight of you, keeping your door open with your hip; he wonders if you know what you’re doing to him, what you’re asking of him — if you even know there’s nothing that could possibly be sweeter than you at this very moment. he drinks in the sight of your feigned haughty expression on your pretty features, the unnervingly low dip of your tank top, the tempting hemline of your shorts, and feels like you must be aware of what he’s going to do next. 

“if it’s something sweet you want, you don’t have to wait until next week.” 

he does it before he can think it through — surely, there’s nothing too harmful about a quick kiss? he angles your chin upward with his thumb and forefinger before you can even react to his words, and he tastes you like that for the first time. you’re just as soft and as sweet as he’d imagined, if not more so. 

when jeno pulls away, you step back; there’s shock written all over your face, your mouth still hanging open slightly. your voice is gentle, shaky when you start speaking. 

“sunbae, wha—”

“see you next week. rest up over the weekend, or there’ll be consequences.” 

he finds it easy to joke with you now, even after what he’s done — finds it easy to wave goodbye with nonchalance as he walks to the elevator, now that he’s gotten one thing out of his system. the look on your face, the growing blush across the bridge of your nose and your temples is indication enough for jeno to feel confident — if you hadn’t thought about him that way before, you were sure to spend the next few days doing exactly that. 

it’s exactly a week before your midterm exam, and jeno notices you’re less than focused. 

he’d let you stew over the weekend, not expecting much by way of communication; indeed, his phone hadn’t once been jostled by your texts. he’d taken that silence to assume that you’d been wrapped up in thoughts of the kiss he’d left you with, and you did not disappoint on that front; the next monday saw you fidgety, flushed, and constantly faltering in your words. you asked less questions, which normally indicated a problem, but today, he’d let it slide; you definitely had a little too much on that pretty little brain of yours. 

he notices you’re still dolled up — your eyelids are shimmery, and your lips are glossy; you’re wearing a tennis skirt that hits all the right buttons for him, too. it’s true that you’re always pretty well-dressed and put together, but today somehow feels different. if before, jeno had always seen you dressed up simply to look good, today it feels a little more like you’re dressed up to look good for him. he knows it’s a little bit egotistical to assume as much, but he also doesn’t miss the side glances you throw at him when you think he’s not looking at you answering your textbook or the way your cheeks glow when you make the slightest bit of eye contact. 

still, you try to focus as much as you can; it’s adorable, in fact, to see all your valiant efforts to appear unperturbed. he figures he’ll play along for as long as you will — what matters to him, after all, is that you’re in the game to begin with. you complain less today, focus on your worksheets, and jeno even manages to witness the sight of your forehead creasing up as you concentrate on a particularly difficult item. you’re adorable, in the kind of way that makes him want to pin you down and have his way with you. 

you finish your work without a fuss today; you only actually asked for his help twice, which was a feat in and of itself. and again, when the session is over, you walk him to the door.

this time, when you linger, he waits; you’re clearly not good at hiding your true intentions, as it’s become clear you have something you want to say. as you try to piece your thoughts together, jeno reaches into his backpack’s front pocket and extracts today’s gift — an actual chocolate bar, albeit a rather run of the mill one. 

“what’s this?” you ask, your thought process clearly derailed as confusion takes over your features. 

“your reward. for a good job last week and today — you said you wanted one, didn’t you?” 

“but i thought—” you stop yourself, your mouth opening and closing, suddenly wordless. jeno grins. 

“not good enough? i picked that up from a convenience store on my way here, so it definitely isn’t anything special, but i thought it would at least be a good motivator.”

you’re turning red, and there’s turmoil in your eyes — he enjoys this, he realizes, the way he flusters you. if he had known this would be the result, he would have made a move much sooner. you shift your weight from one foot to the other, back and forth, obviously weighing out your options too. finally, you say, “alright.”

“you seem disappointed.”

“i’m not.”

“i’ll get you a better brand next time, if you really don’t like it.” 

“it’s not that.”

“so what is it?” he doesn’t expect you to say it, and you don’t defy expectations; your bottom lip just quivers, and jeno chuckles low under his breath, stepping forward just past your doorway, just a little bit closer to you. “don’t tell me you wanted something completely different?”

you don’t say so, but he knows; he can tell by the way you tilt your head back, the way your lips part slightly, the gloss still trailing along the seam. he can tell by the way your torso arches just a little bit closer, almost like an accident. he can tell by the way your eyes bore into his, almost pleading. 

“what you did last week…” you start, but your voice trails off into nothing soon after. he chuckles again.

“ah, that. i might have gotten ahead of myself.” 

“was that all?” you press.

“and what would you do, if it wasn’t?”

“well — do you always like to play games?”

“i have a penchant for playing with my food before i eat it, if that answers your question.” he smiles down at your still-reddening face. “i was giving you a reward, as you wanted. i came up short on options then and there. you’ll let it slide this once, won’t you?”

“you did that just because i did well last week?”

“of course.”

“well, i did well today, too.” 

“you did, and that’s why you have this.” he gestures to the chocolate bar in your hand. 

“i don’t want this.” your voice is stubborn now, heated and frustrated, and you stuff the chocolate back into his hand. you must not like having to ask for something so blatantly — it’s too bad jeno wants to hear it in those exact words. 

“tell me what you really want, then.” 

you’re still unable to find the words, but your hands do the talking for you; they press into his shoulders and give you leverage to tiptoe until you’re just close enough to his lips. but you don’t close that gap, your mouth quivering only inches away from his, and oh, jeno wants to toy with you, but you’re just too irresistible this close to him. his warm palms press against your jaw, keeping your face steady as he closes the gap, and this time, he doesn’t just get a brief taste of you — jeno claims your lips with the thirst of a man who’s stumbled upon an oasis in the desert. 

you must have thought about this moment long and hard over the weekend, because the nonchalant side of you that’s turned a blind eye to him is completely gone; he drinks in your soft noises and short, breathless gasps — all signs of your eagerness — until he’s drunk on the taste of you. the deeper the kiss gets, the less you can keep up, but you try, and jeno always likes rewarding your efforts, his wide tongue taut and flush against your tiny one in the sweet, warm cavern of your mouth. he licks every inch of it, leaves the mild nicotine taste of himself there, before he pulls away slowly. your eyes are still closed when he creates distance, fluttering open in a happy haze a few seconds later. 

“good enough for you?” he murmurs, tucking a soft lock of hair behind your ear. you hum in assent through your dazed smile, and jeno knows he won’t be the only one looking forward to this coming wednesday.

you’d done really well today.

jeno’s proud of you — prouder than he’s been of most of his students in his career here at the university, actually. you’d finally answered a worksheet almost perfectly, save for a couple of numbers where you’d forgotten to round up, and those things are absolutely negligible at this point (by his books, anyway). you’ve been on your best behavior yet, avoiding all forms of complaint, and he knows fully well why, but he won’t criticize you for your hard work all the same, no matter the motivation behind it. 

in fact, you’ve done so good that he doesn’t wait until he’s about to leave to give you your sweet reward — which is why, twenty minutes before he’s meant to go, he’s got you on your couch, your legs spread, each one hooked over his shoulders. 

truth be told, you’d been good way before the lesson had started; you’d answered the door in a crop top and the tiniest pair of shorts you’ve dared to wear yet — all clothes that you couldn’t yet wear outside yet, given the weather. selfishly, jeno is thankful for this fact, and if he had to list down other things he’s thankful for, just off the top of his head, it’s that you no longer meet in the tutoring center and that your apartment’s walls seem thick and well-reinforced. 

“sunbae, don’t tease me.” your silly little whining voice makes its first appearance of the day, but all jeno does is smile — it’s an almost wicked expression, set firmly between your thighs. “you said i did really well today. don’t tell me you’re backing out on rewarding me?”

“not at all, sweetheart,” he hums, pressing a small kiss to your inner thigh. he likes seeing you shiver at the contact, likes the way you’re chewing on your lip in what appears to be slight agitation. “just thinking of how much of a reward you deserve.” 

in all honesty, jeno would like to take every bit of you now; you’re already so ready for him, anyway. he can smell the faint perfume of your arousal, can see the way you’re anticipating the most from him, and a part of him doesn’t want to deny you of that. the larger part of him has dreamed of burying his cock into you, anyway, and why wouldn’t he do that? but something also tells him to wait — or, rather, to make you wait, to make you want him just a little more. 

and so, he decides.

his mouth finds your skin again, pressing kisses up your thigh; they get wetter, hotter as his mouth moves up, until his nose and lips are buried against your clothed core. you squirm in response, but his grip on your thighs keeps you relatively steady, even as his tongue presses against thin fabric. the wet muscle pushes sharp against your tiny entrance, the tip meeting slight resistance against your shorts and panties, but he finds a way, burying half his tongue in alongside damp cloth. 

you’re already wet like this, and so needy that it might be possible for jeno to get you off just like this, still clothed, but the hunger in him spikes once you call out to him. 

“sunbae, please…”

with a groan, his fingers yank the fabric aside, exposing your pussy to the warmth of his breathing. it’s as pink, as pretty, as tiny as the rest of you, as fuckable as he’d imagined it would be, and he wastes no time in pressing his tongue flat against your folds, dragging it up in a wide, messy stripe; the muscle only tenses when it bumps against your clit, his tongue flicking upwards to tease it. 

you’re so reactive, even at the slightest things — you whimper, you squeeze your eyes shut, you squirm. you’re begging to be fucked, and jeno’s cock is strained tight against his jeans, but your taste is so addicting that he can’t help but dive back in. his tongue eases between your folds now, spreading them apart until they’re lewd and sticky with his saliva, and the nub of your clit has grown so pronounced now — so pert and lovely that he can’t help but purse his lips around it and suck with excess force. 

“sunbae — f—fuck,” you mewl; you almost sound tearful. “f—feels so good…”

jeno wants to tell you how fucking good you taste, how beautiful the sounds you’re making are, but his mouth is too busy; his teeth rake down your cunt lightly, earning him a jerk of your hips, and he has to place pressure down on your thighs again to make sure you’re still enough for him to slip his tongue into your cunt. 

he can tell even just by that how tight you’d be around him; your walls are warm around his tongue, and there’s a pressure against the muscle that tells him how good it’d feel for his cock to take its place. as if to simulate his desires, he presses his tongue deeper in, fucks you shallowly with its wetness until your whimpers become little sobs, broken and choked back. his thumb drags across your slit then settles against your clit, and he can feel the thrum of your pulse against the pad of his finger, beckoning him. he complies, easily, thumb tracing circles around the nub that start off slow, only for him to ramp up the pace alongside his tongue. 

you’re easily at fault for that; the way you whine for him, call him sunbae, tell him how good it feels over and over — why wouldn’t he want more of you? 

he’s not sure which of you really earns the sweet reward today; you cum on his tongue, your cunt trembling against his mouth and your fingers threaded into his hair, but he’s the one who comes out licking his lips like he’s had the best treat of his damn life.

come the middle of next week, jeno finds himself face to face with a test paper — one already clearly marked, with a number circled on the top-right corner. ninety. a stellar grade for anyone, and especially for you. 

you know it, and you look absolutely triumphant; you’re practically shining as you perch on your little dining table, your perfectly manicured finger jabbing at the score in emphasis. 

“flying colors, wouldn’t you say?” 

“color me impressed,” jeno replies smoothly, a genuine smile of pride tugging at his lips; he turns the page over, scanning your responses. you still draw your parabolas a little on the small side, making them a bit difficult to discern, and you’ve still got the habit of not rounding your answers up, but this is tremendous work, and he’ll be the first to praise you for it. “your dad must be filled to the brim with joy now, right?”

“i haven’t told him yet. you were the first.”

“well, i’m proud of you, sweetheart.” 

“proud enough to give me a reward?” 

he looks down at you in feigned thoughtfulness. here you sit, back in your little tennis skirt, looking up at him with hopeful eyes under those long, curled lashes. for someone who spent the first half of this semester acting ostensibly nonchalant, you’d very easily shown your true colors soon after — not that he really minds. in fact, he’s taken a decided kind of liking to how eager and willing you’ve come to be. 

“we’ve only just started our session, though,” he hums out, an idle thumb grazing his chin as he watches your expression turn from bright to cloudy, the beginnings of strategy darkening your gaze. it’s not like he wants to say no; he has no real intention to. but seeing you squirm in want makes him feel good about his decision to hold out a little longer — never mind the ache in his cock even then. “don’t we usually leave the rewards for a later time?” 

“i was thinking — since it’s the start of a new lesson —” 

“we wouldn’t want you falling behind from the start, would we?”

“i promise i won’t,” you pout. “i promise i’ll put in my best effort next time.” 

“next time? sweetheart, don’t tell me you’re thinking to get off scot-free today…” jeno trails off, his hand falling to the nearest surface it can reach — which, logic seems to dictate, is your soft, milky thigh. he feels you tense under his palm, and he bites back a smile, keeping his expression level. “i just don’t know.”

your small hands grip at the front of his shirt, and he hears you, for the first time, doing something he’s always wanted to hear you do. 

“please, sunbae?”

how could he say no to you? he hadn’t really planned on it, had only wanted to see you do this, but it’s still too much and beyond his expectation — your misty gaze, your quivering lip. it’s almost laughable that you don’t think he’d notice the way you shift yourself so that his hand, still warm against your thigh, slides up your skin, the hem of your skirt bunched up in the junction between his thumb and forefinger.

jeno chuckles — isn’t this exactly where and how he’s always wanted you? “how could you ask me like that and expect me to refuse, angel? in that case, i have no real choice but to dedicate all our time today to your reward.” 

your breathing hitches — in anticipation, in desire, in excitement — as his hand continues its trail upward, deliberately now, fingers hooking into the waistband of your panties. his head dips down, rests into the crook of your neck, and he inhales the thick, sweet scent of your perfume, your shampoo, of you and all that he’ll take from you. 

“just remember, you asked for this,” he murmurs against your skin. “so i’m going to take every bit of you until there’s nothing left for anyone else.” 

you’re so willing, so ready even before he can get his full bearings; your hips are rising slightly off the table, and jeno feels like it’s you that’s telling him to move faster. he tugs down your panties, letting gravity take its course until they’re a tiny puddle of fabric on the floor, and he slots himself between your legs. like this, you have no choice but to spread, and you do so without hesitation, your knees locking against his sides as he pulls you in for a tight, hungry kiss. there’s that taste of you he loves, that clean, sweet buzz that draws him in, and his hands are bruisingly tight on your waist as he reclaims your lips. 

you already look dazed when he pulls away, which is always cute, but a little unfair — jeno wants you to be aware still when he takes you, and damn, if he doesn’t want to take you right fucking now. he kisses you again, harder and more demanding, as if willing your attention back to him, while his hands explore you — run up your thighs, fingers brushing against the plush curve of your ass. it’s not enough, not by a long shot, and he’s pushing the waistline of your skirt up your stomach with his hands, letting his warmth transfer onto your skin; he chuckles as your stomach sucks inward at his touch, just as you let out a gasp against his lips.

and he wants desperately to hear that noise again; in fact, he wants to know what you sound like in every capacity. his mouth works down your neck, pleased to find that suckling wet and languid on a spot just above your collarbone has you writhing and whimpering. are you sensitive or touch-starved? whatever the reason, he wants to draw all of that out of you, his hands drawing back down to hook under your thighs. jeno drags you to the edge of the table, until your bare cunt is flush against the front of his jeans, and he lets you feel him — a brief tease of what’s to come. 

“i’m s—so wet already,” you whisper, as if he doesn’t know — as if you know it’s exactly what he wants to hear anyway. “sunbae, please, i need you.”

“not that,” he murmurs, his teeth grazing your collarbone as he speaks. “not sunbae. jeno. call me jeno, angel.”

“jeno,” you exhale shakily, and it’s music to his ears — as if the last thing holding him back from you had shattered. 

“that’s it — what a good girl,” he purrs, his hips rocking forward against your pussy before they retract, leaving just enough space for his hand to slip between. slender fingers trail down your folds, sticky and slick. “you are all wet for me, aren’t you? ready to take me deep inside?” 

even the way you nod, a tiny movement of assent, drives him wild, yet a part of him still wants to test the limit of your patience, his middle finger stretching to circle your entrance. 

“wouldn’t want to shock your tiny little pussy, though, would i? will you let me stretch you out first, kitten?”

“yes,” you mewl, sounding almost tearful. “anything— anything, please.”

jeno drinks in the long, drawn-out keen you set free when his digit sinks into you; he’s already felt your walls against his tongue, but a small part of him is still surprised at just how tight you are. that same part nags that he might not fit easily into you, but whatever that voice is is easily drowned out by a more assertive promise — he’ll make it fit. 

“can’t tell you how much i’ve wanted to feel your pretty little hole around my cock,” he presses on, his finger pushing deeper in; he feels you tense a delicious kind of tightness, as if it’s almost too much for you. is it? “ever since that first day you came into the tutoring center, dressed up all cute — did you do that on purpose, sweetheart?”

“yes,” you admit, breathless; the syllable is lengthened into a weak moan as jeno pumps his finger into you, slow, deep strokes that tease your tacky walls open. “wanted — wanted to make a good impression…”

“and you did, didn’t you? kept looking so sweet for me, so pretty every single time — got me thinking about all the ways i wanted to have you. got me so fucking hard every time we’d meet — is that what you wanted?”

jeno doesn’t give you much room to respond, but he can make his own answers to appease himself anyway; he reclaims your lips, already eager for another taste of you, and you comply with the same amount of desire, your soft whimpers melting against his teeth. in the space of pseudo silence, wet, messy noises, he manages to tease another digit into you, and you cry out against his lips as it pushes in, joining the first in how deep it reaches. he absorbs that too, takes in every minute sound you make, relishes the way you pulse around his fingers. even without the noises, he can tell your pleasure’s heightening, with the way you clench around him, your hips rocking pitifully as you’re eager to rut against his palm. 

“look at you now.” he’s selfish, but he doesn’t care — he wants to ruin you, and if the telltale squelch of your cunt as he fucks his fingers into it isn’t indication enough, then the way your mouth hangs open as he pulls away, letting his name fall freely from your lips, definitely is. “legs spread, all desperate to feel good for me. what a needy little kitten you are. this good enough for you, angel?”

you shake your head, only to squeal as he pulls you closer, his fingers shoving deeper into you; your hips are re-angled, allowing him to brush the pads of his digits against the rough, sweet spot, and he feels triumph bloom in his chest as you throw your head back, teary eyes squeezed shut.

“no, no, no,” you babble, and he can see the bob of your throat as you swallow hard, clutching at sense to make words. “want — need your cock, want to cum on your cock so badly, jeno — want you to fuck me, stretch me open, please —”

“greedy, aren’t you?” he murmurs, leaning in to nip at the spot he’d left reddened above your collarbone. “go on then — show me how much you want it. show me what a good girl you are, and cum on my fingers.” 

“but—” 

“come on, angel,” he urges above the squelching noises, increasing surely in volume. his fingers meet resistance when they spread apart inside you, but all it does is create a delicious friction that has you squirming in his hold. “don’t hold back. let me see you fall apart.” 

and you do, so prettily, your eyes rolling back and your voice unrestrained. jeno’s fingers ride you through your orgasm, pumping deep and steady despite how slick you’ve gotten, your juices coating his hand and wrist. he watches the flush rise to your neck, stopping at your cheeks, watches the heaving of your chest, the shine of your skin from a thin sheen of sweat, and he doesn’t want to let you come down from this high, but his cock is aching — practically bursting from his jeans — and all he can do is make the silent vow that the next time you look like this, he’ll be balls deep in you. 

“that’s my girl,” he coos gently, watching the tension slip from your shoulders; his free hand is at the small of your back quickly, easing you down as your torso falls back, and you’re laying on the table. “pretty little thing, aren’t you? cumming so sweetly for me.” 

“jeno,” you groan out weakly, your tiny hand clasping around his wrist. “cock — i want your cock, please—” 

“can’t wait?” he’s indecent for sounding amused, but even that does nothing to stay his arousal; how eager you are simply makes him want you all the more. “okay, angel — since you asked so nicely.” 

a slight twinge of disappointment runs through him as he pulls his fingers out, but it’s quickly buried by the feeling he gets once he gives you a clear sweep of a once-over; how slutty you look, still half-dressed but already half-ruined, your thighs shaking in an effort to keep them open for him, the remnants of your last climax still leaking out of your hole. the sight of you has him so distracted that unbuttoning and unzipping his pants feels like a fever dream of an act; he barely notices what he’s doing until he’s already bare in front of you, and alertness has crawled halfway back into your consciousness as you push yourself up on your elbows to look at him.

“it’s so—” you have the decency to blush, though there’s a pleased look on your face that tells him you’re not really embarrassed. “i didn’t think you’d be this big.” 

“does that worry you?”

“i’ve never had anyone… this big.” pride blooms in his chest — good, he thinks, because if he can’t be as memorable as your first, then he’ll take being the most in something as a prize. “i don’t think — will it fit?”

“does it matter?” he chuckles, and your blush deepens. “no matter what — you’ll take all of me in, won’t you?”

you chew on your bottom lip, as if considering your options, but to jeno, there’s really only one choice — the correct one, and you make it when you nod your head. 

“it’ll feel good, though, you know,” he muses. his hand wrapped around his base, he lines himself up with you, the tip grazing against your folds. “even better than just now.”

with just a little more pressure, he has his shaft flush against you; his girth sits against your slit, the tip pressed against your clit, and he starts to rock his hips — into his fist, against your cunt. your hips quiver, and a shiver runs through you as your pleasure spikes again, but he can tell it isn’t enough. your bottom lip is back between your teeth, and your eyes are flitting between his face and his cock. jeno reaches out, eases your lip out from between your teeth, strokes it gently, almost tenderly. 

“say it,” he commands in a soft, silky voice. 

“fuck me, jeno,” you breathe out, barely missing a beat. “fuck me, fuck my pussy, please.”

and if you ask that desperately, he’ll waste no time; he draws his hips back, dragging his cock down until he’s aligned with your entrance. his eyes are trained on your face, even when he pushes in, so that he can take in your expression — the widening of your eyes as his tip breaches the first wave of resistance, the way your mouth falls agape as his fingers dig hard into your flesh. he’s never seen a prettier sight in his life.

“stretched you out already, but you’re still so fucking tight,” his voice is a soft, melodious croon, a stark contrast to the way he’s forcing past your tightness. “tight and wet, like a good girl.” 

“so big,” you whimper, your fingers stretched far enough to tickle the front of his shirt. “can’t — can’t take it.” 

“of course you can, angel.” jeno doesn’t give you the time to brace yourself fully before he’s rocking his hips in a little more sharply, his cock now halfway into you. your fingers curl into a little fist, immediately flying back to block the noise from your mouth. “ah ah. don’t get shy on me now; you’ve been so noisy for me all this time.”

but he doesn’t really mind the way you clap your palm over your mouth to muffle your high-pitched squeal as he thrusts in fully, the adjustment period after the last movement close to nothing; he’s too busy focusing on how good you feel around him, how warm and wet your insides are. this is heaven, easily, and jeno wants to stay here for as long as he can. 

“god, you’re fucking tight,” he repeats, an appreciatory gaze running over where you’re joined. his thumb stretches over your folds, rubbing them — something of an apology, perhaps, although all it does is stimulate you more, and you shiver at the extra contact. “how deep is it, baby?”

“can feel you here,” you mumble out, your small hand pressing just above your pelvis. he feels the tightness multiply as you place pressure, even just for a moment. “your cock’s so much deeper than anyone else.” 

your hand falls away, limp, as he draws his hips back; you inhale, long and deep, before letting it out as a broken moan when he pushes back in. it drives him crazy, to start off this slow, when all he wants is to find a pace that has you sobbing, but the resistance of your pussy against his length isn’t easy to ignore. jeno works you open, his jaw set and his grip tight against your frame, and it isn’t long before he’s picking up speed, the slap of his flesh against yours fueling him exponentially, mingling with your cries, steadily increasing in volume. 

“that’s it. let everyone hear you,” he eggs on, his thumb now circling tight around your clit; your legs are quivering, threatening to close, but he keeps you steady, one arm wrapped around your thigh. his thrusts grow rougher, more deliberate, and when he looks up from where you’re joined back to your face, he sees your expression as a mixture of incredulity and ecstasy. a thin line of drool hangs from the corner of your mouth, your pretty pink lip gloss smeared, and fuck if he doesn’t want to make sure you look like this every single time he comes over. “let them know who’s fucking you good, angel.”

“j— jeno!” your voice hitches, lilts up as he presses in at a different, deeper angle, and he almost cums right then and there from the way your walls pulse around him. “your cock feels so good, fucking me just right— more, god, more—” 

he complies without hesitation, gathering both your thighs and pushing them closer to your chest; you look even lewder like this, folded in half with your sopping cunt presented to him like it’s all his to take, and it is, isn’t it? there’s an increase in the intensity, the vigor in which he pumps his cock into you, and he knows he’s brushing repeatedly against your spot by the way you’re blubbering his name out in a way that suggests you sincerely think no one else in this building can hear you. 

“that’s my girl,” he hums approvingly, though there’s a thickness in his voice that has him sounding a little more strained. “such a good girl, with your cunt all nice and sloppy for me. do you like it when i go this deep? does it feel good when i fuck you where no one else can?” 

“yes!” you sob out, your hands crumpling the end of your skirt up into tight fists. “jeno, i— cum, i need to cum again, please—”

“i’ve got you, kitten,” his tone is reassuring, a stark contrast to the rigor of his hips. “don’t have to hang on for me, you know; always love seeing you fall apart.” 

“m’close, so close —” 

“let go, then,” he urges, his blunt nails digging into your flesh. “let me feel that sweet cunt cum on my cock.” 

you comply without hesitation, though if you’d done it willingly, he can’t really tell; he has to pin your hips down to stop you from bucking up and causing him to slip out, and you writhe against him as you sob in ecstasy, your walls fluttering before they clench. stray tears leak from your eyes, squeezed shut, and jeno wants nothing more than to eat you up like this — broken, fucked out. 

you’re not even fully down from your high when he feels it — that sudden wrenching in his gut that tells him he’s about to follow suit. with a low groan, he peels your thighs apart again, lets you watch him as he bullies straight into your leaking hole. your voice is a staccato, punctuating every deep, sharp thrust into you, and it’s exactly to that melody that he wants to get off. 

“tell me where you want it, angel.” he doesn’t trust his voice, sharp and short as it is now. “should i mark your pretty face? your stomach?”

“want it against my pussy,” you whisper out, and jeno almost loses his mind as he watches you spread your folds apart with your forefinger and middle finger, inviting him. “make a mess of it, sunbae.”

he’s barely able to pull out before he’s spilling against you; he ruts against your slit, coating your folds and the insides of your thighs in thick, creamy white. you hold your legs apart for as long as you can until they start to tremble, and he catches them and gently eases them down. 

when you sit up to kiss him, you’re still demanding; he feels your hips rock closer, your sticky cunt pressing against the underside of his cock.

“not enough,” you murmur against his lips, and jeno chuckles as you bind your hands around his neck. 

“don’t worry, kitten,” he hums back. “we’ve got all afternoon.”

[olenvasynyt]

@kateprincessofbluewhales made this post about high fae aging and how the older high fae should look like they’re 40s-50s instead of 20-30 and I completely agree and I wanted it show this insane headcanon that I made charts for because I’m crazy lmao. I thought the high fae aging was so confusing and badly written but I really needed to understand it!

So how I think the High Fae age is this: the child grows at a steady rate that appears the same as how a human ages. So when they are one year old they look like they’re one, when they’re 5 they look like they’re 5, etc until we get to approx. when a high fae puberty starts. Mor got her bleeding late, she got it when she was 16, (as seen by the red dot I wrote her age wrong sorry). I headcanon the average age of puberty starts around 12-13 which is about the same as a human’s.

But when the high fae begins their development, this is when the age in appearance starts a logarithmic growth rate. It comes quickly in the beginning but begins to rapidly decrease as they age. Mor is 16 when she got her first bleeding so she might have appeared a little younger; Rhys (purple mark), was in his thirties when he was fighting in the Human War so he might have appeared 19 or 20.

This time in a high fae’s life is when their magic is developing, and it stops developing when they reach maturity, which is 80 years old. I put a teal mark for Tarquin, which works out because I always saw him to be 20 or in his early twenties. And at 80, their age in appearance evens out to a rate that is almost impossible to notice until you expand it out.

So that purple mark is Rhys’ age at ~500 years old. And I always pictured him to look like he’s in his late 20’s early 30’s so this steady growth matches up perfectly.

And this also works well with the idea that High fae children are rare. So they are rare because they’re hard to conceive and because they age rapidly compared to the immortal high fae life.

And if we expand this out further, this growth rate matches up with how I always see Beron, so I always pictured him middle-aged so approximately 50. And with this rate, he would be about 1,500 years old which I think is perfect because I headcanon that Beron was near the same age as Rhys’ father.

And if we expand this further, we can see Madja and how old I think she could be.

Madja is confusing to me and we know almost nothing about her. But Feyre describes her to be ancient looking: she’s got white hair, she has wrinkles, her hands are wrinkled and knobbly. So in my head, I pictured her looking like she’s about 80 or 90. And with this growth rate, she would be about 4,000 years old. Which makes sense to me.

We don’t know when High fae usually die. They’re immortal, but even this immortality has a limit. Amren said that she was immortal before in her previous form, and she says not immortal like the high fae but Immortal. So Madja might be in a high fae’s elderly stage. And also she’s pretty rare too we haven’t seen another eldery high fae because a high fae usually doesn’t have a safe life that allows them to live this long. They had the war, they have these very competitive lives, they love fighting in battles to the death.

So yeah, that was my very autistic explanation of my very specific headcanon for how the high fae age in ACOTAR lmao

[diz-eaze]

having albedo and scaramouche as both your roommates—ohh…damn. double whammy…

like yeah, if you think scaramouche’s picture taking habits are bad, take a look at this guy *thumb pointing at albedo* this guy is NUTS!!! i know he’s probably broken a camera or two because of that. and on a freak-off scale between the two, i’d wager money on albedo breaking that scale in full. sure, both are weird and creepy, but it’s always the calm and stoic ones that got the craziest unimaginable tricks up their sleeves. and tbf scaramouche isn’t completely overt about his interests and motives, but he’s a little more pathetic about it bc he’s prone to accidentally exposing himself if he ever fucks up. not albedo though. he’s too perfect. too pristine. and he has that same cleanliness and smile as to idk hannibal. albedo is perfect premeditated. scaramouche is just a hot mess with a wetter dick.

also kabukimono scaramouche 💆‍♀️💆‍♀️ i like to imagine that it’s some kind of childhood friends/crushes (possibility one sided), who got separated because one or the other moves somewhere, then reunited only to figure out that little boy you used to hang out with is now all grown up and completely different than what you remember. but he still likes you. but he reallllyyy likes you this time around. and might have been following you around all these years oh well

or kabukimono > scaramouche pipeline might have had happened after an intense breakup and he crashed out and did an entire personality 180. scaramouche has phoenixed into a bitter ex-boyfriend. his mom thinks it’s a phase like most kids this days, but oh boy this little guy truly means it this time

running laps around this ask because you get me so bad,,, you get me !!

yandere albedo is so delicious because it's impossible for him to get outed as a freak !! his countermeasures against being found out are too high; too elaborate. and also... it's speculated by the traveler in the paralogism quest that albedo would never be put on trial for murder because if he did commit the crime, he'd never be found out because he's just too good, thus never being suspected in the first place LMFAOOOO. his reputation and personality in general causes people to think of him being a freak to be like,,,, an impossible concept. fanfiction levels of OOC for them, if you will. but if they had mind-reading powers, then they'd see the degeneracy that plagues albedo's thoughts 24/7 :(((

random npc: wow, mister albedo looks so serious! :0 i wonder what he's thinking about! albedo, thinking about (y/n): no lube, no protection, all night, all day, from the kitchen floor to the toilet seat, from the dining table to the bedroom, from the bathroom sink to the shower, from the front porch to the balcony, vertically, horizontally, quadratic, exponent al, logarithmic, while i gasp for air, scream and see the light, missionary, cowgirl, reverse cow girl, doggy, backwards, forwards, sideways, upside down, on the floor, in the bed, on the couch, on a chair, being carried against the wall, outsid-

whereas scara, specifically modern au roomie, is the ultimate boyfailure who's just too unhinged and much of a gooner to keep all his freakyness to himself. it's quite impossible for him to function as a normal human being whenever your in the vicinity, in fact, just a simple whiff of your scent is enough to have his pants tightening :(( what more if you willingly speak to him, or accidentally... touch him !? he's so obvious to the point where even you get the hint a month or two in him being your roommate. though you mistake his infatuation as a simple crush instead of what it actually is: a maddening obsession.

and childhood friend scara is such a cute concept,,, he's literally been in love with you since the age where you could only waddle around, and he's never stopped being in love with you even after all this time... despite the distance and unfamiliarity. and while his version of love is a tad bit too intense for normal human comprehension, it's very much real and genuine :((((

[ell-arts]

PMATGA Headcanons: Surnames

This thing has been a headscratcher for as long as I have been a fan of the show. Or it might be just me-

But I've always struggled to imagine what some of the characters' surnames/last names could be. Especially the main trio. Mostly because it's tricky to come up with surnames that would fit them, and something in line with the show's running theme of naming things after geometry or character-specific quirks.

We know that Stratos' last name is Spheros (derived from stratosphere) and thus it's also Betrayus' surname. We know of Mr Strictler, who only revealed his surname and never his first name, so we can deduce that Sherry's last name is Strictler (she's his daughter). Then there's Mr Dome, Ms Globular, Sir Cumference, etc. But other than these characters, we never really heard of any other surnames mentioned in the show.

There is Spheria Suprema, and hey it could be probable that that is her surname, but I'm leaning a bit more towards the idea that it's her celebrity name/Pac-Pong champion title/nickname rather than her actual given surname. The chances of Spheria being given a name that rhymes with a descriptive title such as 'Suprema' at birth is a little slim if you ask me (this is just my rationalisation, don't come at me xD)

So that leaves us with the rest of the cast. I've seen some really creative attempts from the fandom to give the characters their surnames. So much so that I can't really come up with any good ones in comparison xD

But hey, these are the best I could come up with, along with some headcanons or reasons for why I chose them. Subject to change!

-

Pac Orbon (Yellow Orbs, duh...)

Cylindria Eden (based on the Garden of Eden, and sounds hippy-ish)

Spiral Logarith (based on the logarithmic spiral in math)

Elliptica Spheros (her mother kept her last name)

Sir Perimet Cumference (his first name is based on perimeter (of a circle), which can be shortened to 'Peri' but he still prefers 'Sir C')

Trayus Rotundin Spheros (he changed it to Betrayus Sneakerous Spheros as part of his rebellion-turned-revolt)

Zac Orbon (again, yellow orbs...)

Sunny Orbon (maiden name was Sunny Solari - a play on sol and solar, from the sun - which is also round and yellow)

Spheria Solari (tends to go by Spheria Suprema, her celebrity name, since that's what many people know her as)

Specter Eidolon (greek for 'idol' or 'phantom')

Blinky "Shadow" Scarlos (based on 'scarlet')

Inky "Bashful" Whimson (based on 'whimsical')

Pinky "Speedy" Chacier (Old French for "chase/hunt")

Clyde "Pokey" Oren (just based on orange, but can also have different meanings depending on the language)

[The Ghost Gang's names are a combination of lore that I've found relating to the original Pac-Man games. For example, Inky's original name in the retro games was Bashful, and his Japanese name is kimagure, which means 'whimsical', hence Whimson.]

[canmom]

Music Theory notes (for science bitches) part 3: what if. there were more notes. what if they were friends.

Hello again, welcome back to this series where I try and teach myself music from first principles! I've been making lots of progress on zhonghu in the meantime, but a lot of it is mechanical/technical stuff about like... how you hold the instrument, recognising pitches

In the first part I broke down the basic ideas of tonal music and ways you might go about tuning it in the 12-tone system, particularly its 'equal temperament' variant [12TET]. The second part was a brief survey of the scales and tuning systems used in a selection of music systems around the world, from klezmer to gamelan - many of them compatible with 12TET, but not all.

So, as we said in the first article, a scale might be your 'palette' - the set of notes you use to build music. But a palette is not a picture. And hell, in painting, colour implies structure: relationships of value, saturation, hue, texture and so on which create contrast and therefore meaning.

So let's start trying to understand how notes can sit side by side and create meaning - sequentially in time, or simultaneously as chords! But there are still many foundations to lay. Still, I have a go at composing something at the end of this post! Something very basic, but something.

Anatomy of a chord

I discussed this very briefly in the first post, but a chord is when you play two or more notes at the same time. A lot of types of tonal musical will create a progression of chords over the course of a song, either on a single instrument or by harmonising multiple instruments in an ensemble. Since any or all of the individual notes in a chord can change, there's an enormous variety of possible ways to go from one chord to another.

But we're getting ahead of ourselves. First of all I wanna take a look at what a chord actually is. Look, pretty picture! Read on to see what it means ;)

So here is a C Minor chord, consisting of C, D# and G, played by a simulated string quartet:

(In this post there's gonna be sound clips. These are generated using Ableton, but nothing I talk about should be specific to any one DAW [Digital Audio Workstation]. Ardour appears to be the most popular open source DAW, though I've not used it. Audacity is an excellent open source audio editor.)

Above, I've plotted the frequency spectrum of this chord (the Fourier transform) calculated by Audacity. The volume is in decibels, which is a logarithmic scale of energy in a wave. So this is essentially a linear-log plot.

OK, hard to tell what's going on in there right? The left three tall spikes are the fundamental frequencies of C4 (262Hz), D♯4 (310Hz), and G4 (393Hz). Then, we have a series of overtones of each note, layered on top of each other. It's obviously hard to tell which overtone 'belongs to' which note. Some of the voices may in fact share overtones! But we can look at the spectra of the indivudal notes to compare. Here's the C4 on its own. (Oddly, Ableton considered this a C3, not a C4. as far as I can tell the usual convention is that C4 is 261.626Hz, so I think C4 is 'correct'.)

Here, the strongest peaks are all at integer multiples of the fundamental frequency, so they look evenly spaced in linear frequency-space. These are not all C. The first overtone is an octave above (C5), then we have three times the frequency of C4 - which means it's 1.5 times the frequency of C5, i.e. a perfect fifth above it! This makes it a G. So our first two overtones are in fact the octave and the fifth (plus an octave). Then we get another C (C6), then in the next octave we have frequencies pretty close to E6, G6 and A♯6 - respectively, intervals of a major third, a perfect fifth, and minor 7th relative to the root (modulo octaves).

However, there are also some weaker peaks. Notably, in between the first and second octave is a cluster of peaks around 397-404Hz, which is close to G4 - another perfect fifth! However, it's much much weaker than the overtones we discussed previously.

The extra frequencies and phase relationships give the timbre of the note, its particular sound - in this case you could say the sense of 'softness' in the sound compared to, for example, a sine wave, or a perfect triangle wave which would also have harmonics at all integer frequencies.

Perhaps in seeing all these overtones, we can get an intuitive impression of why chords sound 'consonant'. If the frequencies of a given note are already present in the overtones, they will reinforce each other, and (in extremely vague and unscientific terms) the brain gets really tickled by things happening in sync. However, it's not nearly that simple. Even in this case, we can see that frequencies do not have to be present in the overtone spectrum to create a pleasing sense of consonance.

Incidentally, this may help explain why we consider two notes whose frequencies differ by a factor of 2 to be 'equivalent'. The lower note contains all of the frequencies of the higher note as overtones, plus a bunch of extra 'inbetween' frequencies. e.g. if I have a note with fundamental frequency f, and a note with frequency 2f, then f's overtones are 2f, 3f, 4f, 5f, 6f... while 2f's overtones are 2f, 4f, 6f, 8f. There's so much overlap! So if I play a C, you're also hearing a little bit of the next C up from that, the G above that, the C above that and so on.

For comparison, if we have a note with frequency 3f, i.e. going up by a perfect fifth from the second note, the frequencies we get are 3f, 6f, 9f, 12f. Still fully contained in the overtones of the first note, but not quite as many hits.

Of course, the difference between each of these spectra is the amplitudes. The spectrum of the lower octave may contain the frequencies of the higher octave, but much quieter than when we play that note, and falling off in a different way.

(Note that a difference of ten decibels is very large: it's a logarithmic scale, so 10 decibels means 10 times the energy. A straight line in this linear-log plot indicates a power-law relationship between frequency and energy, similar to the inverse-square relationship of a triangle wave, where the first overtone has a quarter of the power, the second has a ninth of the power, and so on.)

So, here is the frequency spectrum of the single C note overlaid onto the spectrum of the C minor chord:

Some of the overtones of C line up with the overtones of the other notes (the D# and G), but a great many do not. Each note is contributing a bunch of new overtones to the pile. Still, because all these frequencies relate back to the base note, they feel 'related' - we are drawn to interpret the sounds together as a group rather than individually.

Our ears and aural system respond to these frequencies at a speed faster than thought. With a little effort, you can pick out individual voices in a layered composition - but we don't usually pick up on individual overtones, rather the texture created by all of them together.

I'm not gonna take the Fourier analysis much further, but I wanted to have a look at what happens when you crack open a chord and poke around inside.

However...

In Western music theory terms, we don't really think about all these different frequency spikes, just the fundamental notes. (The rest provides timbre). We give chords names based on the notes of the voices that comprise them. Chord notation can get... quite complicated; there are also multiple ways to write a given chord, so you have a degree of choice, especially once you factor in octave equivalence! Here's a rapid-fire video breakdown:

Because you have all these different notes interacting with each other, you further get multiple interactions of consonance and dissonance happening simultaneously. This means there's a huge amount of nuance. To repeat my rough working model, we can speak of chords being 'stable' (meaning they contain mostly 'consonant' relations like fifths and thirds) or 'unstable' (featuring 'dissonant' relations like semitones or tritones), with the latter setting up 'tension' and the former resolving it.

However, that's so far from being useful. To get a bit closer to composing music, it would likely help to go a bit deeper, build up more foundations and so on.

In this post and subsequent ones, I'm going to be taking things a little slower, trying to understand a bit more explicitly how chords are deployed.

An apology to Western music notation

In my first post in this series, I was a bit dismissive of 'goofy' Western music notation. What I was missing is that the purpose of Western music notation is not to clearly show the mathematical relationships between notes (something that's useful for learning!)... but to act as a reference to use while performing music. So it's optimising for two things: compactness, and legibility of musical constructs like phrasing. Pedagogy is secondary.

Youtuber Tantacrul, lead developer of the MuseScore software, recently made a video running over a brief history of music notation and various proposed alternative notation schemes - some reasonable, others very goofy. Having seen his arguments, he makes a pretty good case for why the current notation system is actually a reasonable compromise... for representing tonal music on the 12TET system, which is what it's designed for.

So with that in mind, let me try and give a better explanation of the why of Western music notation.

In contrast to 'piano roll' style notation where you represent every possible note in an absolute way, here each line of the stave (staff if you're American) represents a scale degree of a diatonic scale. The key signature locates you in a particular scale, and all the notes that aren't on that scale are omitted for compactness (since space is at an absolute premium when you have to turn pages during a performance!). If you're doing something funky and including a note outside the scale, well that's a special case and you give it a special-case symbol.

It's a similar principle to file compression: if things are as-expected, you omit them. If things are surprising, you have to put something there.

However, unlike the 简谱 jiǎnpǔ system which I've been learning in my erhu lessons, it's not a free-floating system which can attach to any scale. Instead, with a given clef, each line and space of the stave has one of three possible notes it could represent. This works, because - as we'll discuss momentarily - the diatonic scales can all be related to each other by shifting certain scale degrees up or down in semitones. So by indicating which scale degrees need to be shifted, you can lock in to any diatonic scale. Naisuu.

This approach, which lightly links positions to specific notes, keeps things reasonably simple for performers to remember. In theory, the system of key signatures helps keep things organised, without requiring significant thought while performing.

That is why have to arbitrarily pick a certain scale to be the 'default'; in this case, history has chosen C major/A minor. From that point, we can construct the rest of the diatonic scales as key signatures using a cute mathematical construct called the 'circle of fifths'.

How key signatures work (that damn circle)

So, let's say you have a diatonic major scale. In piano roll style notation, this looks like (taking C as our base note)...

And on the big sheet of scales, like this:

Now, let's write another diatonic major scale, a fifth up from the first. This is called transposition. For example, we could transpose from C major to G major.

Thanks to octave equivalency, we can wrap these notes back into the same octave as our original scale. (In other words, we've added 7 semitones to every note in our original scale, and then taken each one modulo 12 semitones.) Here, I duplicate the pattern down an octave.

Now, if we look at what notes we have in both scales, over the range of the original major scale.

Well, they're almost exactly the same... but the fourth note (scale degree) is shifted up by one semitone.

In fact, we've seen this set of notes before - it is after all nothing more than a cyclic permutation of the major scale. We've landed on the 'Lydian mode', one of the seven 'modes' of the diatonic major scale we discussed in previous posts. We've just found out that the Lydian mode has the same notes as a major scale starting a fifth higher. In general, whether we think of it as a 'mode' or as a 'different major scale' is a matter of where we start (the base note). I'm going to have more to say about modes in a little bit.

With this trick in mind, we produce a series of major scales starting a fifth higher each time. It just so happens that, since the fifth is 7 semitones, which is coprime with the 12 semitones of 12TET, this procedure will lead us through every single possible starting note in 12TET (up to octave equivalency).

So, each time we go up a fifth, we add a sharp on the fourth degree of the previous scale. This means that every single major scale in 12TET can be identified by a unique set of sharps. Once you have gone up 12 fifths, you end up with the original set of notes.

This leads us to a cute diagram called the "circle of fifths".

Because going up a fifth is octave-equivalent to going down a fourth, we can also look back one step on the circle to find out which note needs to be made sharper. So, from C major to G major, we have to sharpen F - the previous note on the circle from C. From G major to D major, we have to sharpen C. From D major to A major, we have to sharpen G. And so on.

By convention, when we write a key signature to define the particular scale we're using, we write the sharps out in circle-of-fifths order like this. The point of this is to make it easy to tell at a glance what scale you're in... assuming you know the scales already, anyway. This is another place where the aim of the notation scheme is for a compact representation for performers rather than something that makes the logical structure evident to beginners.

Also by convention, key signatures don't include the other octaves of each note. So if F is sharp in your key signature, then every F is sharp, not just the one we've written on the stave.

This makes it less noisy, but it does mean you don't have a convenient visual reminder that the other Fs are also sharp. We could imagine an alternative approach where we include the sharps for every visible note, e.g. if we duplicate every sharp down an octave for C♯ major...

...but maybe it's evident why this would probably be more confusing than helpful!

So, our procedure returns the major scales in order of increasing sharps. Eventually you have added seven sharps, meaning every scale degree of the original starting scale (in this case, C Major) is sharpened.

What would it mean to keep going past this point? Let's hop in after F♯ Major, at the bottom of the Circle of Fifths; next you would go to C♯ Major by sharpening B. So far so good. At this point we have sharps everywhere, so the notes in your scale go... C♯ D♯ E♯ F♯ G♯ A♯ B♯ C♯ ...except that E♯ is the same as F, and B♯ is the same as C, so we could write that as C♯ D♯ F F♯ G♯ A♯ C C♯

But then to get to 'G♯ Major', you would need to sharpen... F♯? That's not on the original C-major scale we started with ! You could say, well, essentially this adds up to two sharps on F, so it's like F♯♯, taking you to G. So now you have...

G♯ A♯ C C♯ D♯ F G G♯

...and the line of the stave that you would normally use for F now represents a G. You could carry on in this way, eventually landing all the way back at the original set of notes in C Major (bold showing the note that just got sharpened in each case):

D♯ F G G♯ A♯ C D D♯ A♯ C D D♯ F G A A♯ F G A A♯ C D E F C D E F G A B C

But that sounds super confusing - how would you even represent the double sharps on the key signature? It would break the convention that each line of the stave can only represent three possible notes. Luckily there's a way out. We can work backwards, going around the circle the other way and flattening notes. This will hit the exact same scales in the opposite order, but we think of their relation to the 'base' scale differently.

So, let's try starting with the major scale and going down a fifth. We could reason about this algebraically to work out that sharpening the fourth while you go up means flattening the seventh when you go down... but I can also just put another animation. I like animations.

So: you flatten the seventh scale degree in order to go down a fifth in major scales. By iterating this process, we can go back around the circle of fifths. For whatever reason, going down this way we use flats instead of sharps in the names of the scale. So instead of A♯ major we call it B♭ major. Same notes in the same order, but we think of it as down a rung from F major.

In terms of modes, this shows that the major scale a fifth down from a given root note has the same set of notes as the "mixolydian mode" on the original root note. ...don't worry, you don't gotta memorise this, there is not a test! Rather, the point of mentioning these modes is to underline that whether you're in a major key, minor key, or one of the various other modes is all relative to the note you start on. We'll see in a moment a way to think about modes other than 'cyclic permutation'.

Let's try the same trick on the minor key.

Looks like this time, to go up we need to sharpen the sixth degree, and to go down we need to flatten the second degree. As algebra demands, this gives us the exact same sequence of sharps and flats as the sequence of major scales we derived above. After all, every major scale has a 'relative minor' which can be achieved by cyclically permuting its notes.

Going up a fifth shares the same notes with the 'Dorian mode' of the original base note, and going down a fifth shares the same notes with the 'Phrygian mode'.

Here's a summary of movement around the circle of fifths. The black background indicates the root note of the new scale.

Another angle on modes

In my first two articles, I discussed the modes of the diatonic scale. Leaping straight for the mathematically simplest definition (hi Kolmogorov), I defined the seven 'church modes' as simply being cyclic permutations of the intervals of the major scale. Which they are... but I'm told that's not really how musicians think of them.

Let's grab the chart of modes again. (Here's the link to the spreadsheet).

We can get to these modes by cyclically permuting the others, but we can also get to them by making a small adjustment of one to a few particular scale degrees. When you listen to a piece of music, you're not really doing cyclic permutations - you're building up a feeling for the pattern of notes based on your lifelong experience of hearing music that's composed in this system. So the modes will feel something like 'major until, owo what's this, the seventh is not where I thought it would be'.

Since the majority of music is composed using major and minor modes, it's useful therefore to look at the 'deltas' relative to these particular modes.

To begin with, what's the difference between major and minor? To go from major to natural minor, you shift the third, sixth and seventh scale degrees down by one semitone.

So those are our two starting points. For the others, I'm going to be consulting the most reliable music theory source (some guy on youtube) to give suggestions of the emotional connotations these can bring. The Greek names are not important, but I am trying to build a toolbox of elements here, so we can try our hand at composition. So!

The "Dorian mode" is like the natural minor, but the sixth is back up a semitone. It's described as a versatile mode which can be mysterious, heroic or playful. I guess that kinda makes sense, it's like in between the major and minor?

The "Phrygian mode" is natural minor but you also lower the 2nd - basically put everything as low as you can go within the diatonic modes. It is described as bestowing an ominous, threatening feeling.

The "Lydian mode" is like the major scale, but you shift the fourth up a semitone, landing on the infamous tritone. It is described as... uh well actually the guy doesn't really give a nice soundbitey description of what this mode sounds like, besides 'the brighest' of the seven, this video's kinda more generally about composition, whatever. But generally it's pretty big and upbeat I think.

The "Mixolydian" mode is the major, but with the seventh down a semitone. So it's like... a teeny little bit minor. It's described as goofy and lighthearted.

We've already covered the Aeolian/Natural Minor, so that leaves only the "Locrian". This one's kinda the opposite of the Lydian: just about everything in the major scale is flattened a bit. Even from the minor it flattens two things, and gives you lots of dissonance. This one is described as stereotypically spooky, but not necessarily. "One of the least useful", oof.

Having run along the catalogue, we may notice something interesting. In each case, we always either only sharpen notes, or only flatten notes relative to the major and minor scales. All those little lines are parallel.

Indeed, it turns out that each scale degree has one of two positions it can occupy. We can sort the diatonic modes according to whether those degrees are in the 'sharp' or 'flat' position.

This is I believe the 'brightness' mentioned above, and I suppose it's sort of like 'majorness'. So perhaps we can think of modes as sliding gradually from the ultra-minor to the infra-major? I need to experiment and find out.

What have we learned...?

Scale degrees are a big deal! The focus of all this has been looking at how different collections of notes relate to each other. We sort our notes into little sets and sequences, and we compare the sets by looking at 'equivalent' positions in some other set.

Which actually leads really naturally into the subject of chord progressions.

So, musical structure. A piece of tonal music as a whole has a "palette" which is the scale - but within that, specific sections of that piece of music will pick a smaller subset of the scale, or something related to the scale, to harmonise.

The way this goes is typically like this: you have some instruments that are playing chords, which gives the overall sort of harmonic 'context', and you have a single-voiced melody or lead line, which stands out from the rest, often with more complex rhythms. This latter part is typically what you would hum or sing if you're asked 'how a song goes'. Within that melody, the notes at any given point are chosen to harmonise with the chords being played at the same time.

The way this is often notated is to write the melody line on the stave, and to write the names of chords above the stave. This may indicate that another hand or another instrument should play those chords - or it may just be an indication for someone analysing the piece which chord is providing the notes for a given section.

So, you typically have a sequence of chords for a piece of music. This is known as a chord progression. There are various analytical tools for cracking open chord progressions, and while I can't hope to carry out a full survey, let me see if I can at least figure out my basic waypoints.

Firstly, there are the chords constructed directly from scales - the 'triad' chords, on top of which can be piled yet more bonus intervals like sevenths and ninths. Starting from a scale, and taking any given scale degree as the root note, you can construct a chord by taking every other subsequent note.

So, the major scale interval pattern goes 2 2 1 2 2 2 1. We can add these up two at a time, starting from each position, to get the chords. For each scale degree we therefore get the following intervals relative to the base note of the chord...

I. 0 4 7 - major

ii. 0 3 7 - minor

iii. 0 3 7 - minor

IV. 0 4 7 - major

V. 0 4 7 - major

vi. 0 3 7 - minor

viiᵒ. 0 3 6 - diminished

Now hold on a minute, where'd those fuckin Roman numerals come from? I mentioned this briefly in the first post, but this is Roman numeral analysis, which is used to talk about chord progressions in a scale-independent way.

Here, a capital Roman numeral represents a major triad; a lowercase Roman numeral represents a minor triad; a superscript 'o' represents a dimished triad (minor but you lower the fifth down to the tritone); a superscript '+' represents an augmented triad (major but you boost the fifth up to the major sixth).

So while regular chord notation starts with the pitch of the base note, the Roman numeral notation starts with a scale degree. This way you can recognise the 'same' chord progression in songs that are in quite different keys.

OK, let's do the same for the minor scale... 2 1 2 2 1 2 2. Again, adding them up two at a time...

i. 0 3 7 - minor

iiᵒ. 0 3 6 - diminished

III. 0 4 7 - major

iv. 0 3 7 - minor

v. 0 3 4 - minor

VI. 0 4 7 - major

VII. 0 4 7 - major

Would you look at that, it's a cyclic permutation of the major scale. Shocker.

So, both scales have three major chords, three minor chords and a diminished chord in them. The significance of each of these positions will have to be left to another day though.

What does it mean to progress?

So, you play a chord, and then you play another chord. One or more of the voices in the chord change. Repeat. That's all a chord progression is.

You can think of a chord progression as three (or more) melodies played as once. Only, there is an ambiguity here.

Let's say, idk, I threw together this series of chords, it ended up sounding like it would be something you'd hear in an old JRPG dungeon, though maybe that's just 'cos it's midi lmao...

I emphasise at this point that I have no idea what I'm doing, I'm just pushing notes around until they sound good to me. Maybe I would know how to make them sound better if I knew more music theory! But also at some point you gotta stop theorising and try writing music.

So this chord progression ended up consisting of...

Csus2 - Bm - D♯m - A♯m - Gm - Am - F♯m - Fm - D - Bm

Or, sorted into alphabetical order, I used...

Am, A♯m, Bm, Csus2, D, D♯m, Fm, F♯m, Gm

Is that too many minor chords? idk! Should all of these technically be counted as part of the 'progression' instead of transitional bits that don't count? I also dk! Maybe I'll find out soon.

I did not even try to stick to a scale on this, and accordingly I'm hitting just about every semitone at some point lmao. Since I end on a B minor chord, we might guess that the key ought to be B minor? In that case, we can consult the circle of fifths and determine that F and C would be sharp. This gives the following chords:

Bm, C♯dim, D, Em, F♯m, G, A

As an additional check, the notes in the scale:

B, C♯, D, E, F, G♯, A

Well, uh. I used. Some of those? Would it sound better if I stuck to the 'scale-derived' chords? Know the rules before you break them and all that. Well, we can try it actually. I can map each chord in the original to the corresponding chord in B minor.

This version definitely sounds 'cleaner', but it's also... less tense I feel like. The more dissonant choices in the first one made it 'spicier'. Still, it's interesting to hear the comparison! Maybe I could reintroduce the suspended chord and some other stuff and get a bit of 'best of both worlds'? But honestly I'm pretty happy with the first version. I suppose the real question would be which one would be easier to fit a lead over...

Anyway, for the sake of argument, suppose you wanted to divide this into three melodies. One way to do it would be to slice it into low, central and high parts. These would respectively go...

Since these chords mostly move around in parallel, they all have roughly the same shape. But equally you could pick out three totally different pathways through this. You could have a part that just jumps to the nearest note it can (until the end where there wasn't an obvious place to go so I decided to dive)...

Those successive relationships between notes also exist in this track. Indeed, when two successive chords share a note, it's a whole thing (read: it gets mentioned sometimes in music theory videos). You could draw all sorts of crazy lines through the notes here if you wanted.

Nevertheless, the effects of movements between chords come in part from these relationships between successive notes. This can give the feeling of chords going 'up' or 'down', depending on which parts go up and which parts go down.

I think at this point this post is long enough that trying to get into the nitty gritty of what possible movements can exist between chords would be a bit of a step too far, and also I'm yawning a lot but I want to get the post out the door, so I let's wrap things up here. Next time: we'll continue our chord research and try and figure out how to use that Roman numeral notation. Like, taking a particular Roman numeral chord progression and see what we can build with it.

Hope this has been interesting! I'm super grateful for the warm reception the last two articles got, and while I'm getting much further from the islands of 'stuff I can speak about with confidence', fingers crossed the process of learning is also interesting...

[heckinpupperino]

i always knew id yse english history and science in the real world but genuinely what was the point of learning about math after i learned my times tables. never use geometry. never use algebra. the amount of time i spent crying over logarithms in high school is the only time theyve ever come up in my life. useless pointless subject aimed to make specifically me miserable…

[skyes-messy-desk]

I'm halfway through a bio review session in which I'm teaching my classmates most of it and I honestly just feel bad for them because I am such a hot mess today. The bio final is tomorrow and I'm a little scared. All of my grades are on the borderline so I'm getting a wee bit concerned. But it's fine! I'll live!

Today's agenda:

Bio final review

Thermochemistry reaction problems

Calc practice problems (specifically logarithmic differentiation)

[transgenderer]

Re: universal science cooccurring w/capitalism

Invention/discovery of logarithms and books of them, and slide rules, etc. was important! I think. For computing things for navigation. I saw a whole YouTube video on this and everything. Almost 3D printed a slide rule about it

hmm see logarithms are important and all. but i feel like theres a very specific concept cluster, just barely emerging in newton and then gradually expanding until solidifying in the 19th century of behavior that looks like 20th century science. controlled experiments, mathematical models, all that jazz. and that particular cluster has been powerful in a very specific way. and it didnt exist before like, maybe the mid 18th century? which is around when you see the inklings of capitalism. which is weird

[lectronicproot]

I said this on Bluesky and I really should link these two accounts in a pinned post or some but eh, but basically I’m world building and I saw this video

I knew the relation between this three but it never came across my mind that it would be possible to make the relation between the three more transparent lol.

In my worldbuilding project the species I’m focusing on rn develops pretty complex math a while before most other civilizations and I wonder if it would seem natural to have their math notation have this relation be clearer

I wonder if it could be reflected in their language too, maybe the words for using powers is related to something like “folding”, since when you fold something the creases form an exponentially growing amount of segments.

Rooting could be unfolding, straightening, flattening, something like that. The whole paper analogy kinda depends on whether they discover paper though… they’re in a rainforest, and while it’s possible to make paper from trees, without modern technology they would specifically have to use soft wood, and I don’t know if rainforest trees are soft or hard lol, tho things like plant fibers are also possible materials

Log is harder, but something like just “finding the fold” would suffice, considering my conlang is kinda Chinese like with verb-noun compounds being grammaticalised into their own words. Maybe something like fold scale or something would work.

I feel like I’m focusing too much on the whole paper etymology though. My civilization is one that can naturally fly and are traders. Maybe exponents could be “reaching to the sky” by the exponent amount? Rooting is maybe descent then? Or maybe finding a root is finding what a number is “anchored” or “tied down” to. And logarithm is referred to as the “shadow” of these two?

Conlanging is always hard for more abstract concepts but I find it really interesting to try and do regardless lol. If anybody else has any ideas of other ways a culture could refer to these functions then I’d like to hear them!

[irlactualhuman]

favorite word?

I don't really have a favorite. It's like music. I like music that goes hard, not really bound to genre or specific song. There are too many of either.

Obstreperous comes to mind. Fun to say. Very much enjoy a good O word. Like obfuscate or obsequious.

Anagram is a very pleasing word in construction and mouthfeel.

Torpid (being afflicted with a torpor) is good.

I like when simile is bound within and exemplified by a single word. Like Dionysian. Euclidean. Words that convey a feeling, image, and vibe all at once. If you're at all familiar with the concept, you can grip the feeling it's selling pretty damn easy.

Kinetic is just fun to say.

Logos. Logarithmic. L and G make very good friends in my brain.

I hope my rambling about words is enlightening. They are one of my favorite things.

[c-official]

What is a data structure or tree

Well a data structure is a structure to store data in a way that optimizes some desired operations. An array can be thought as a very simple data structure designed to support the operations

Get i: get the data associated to the number i in constant time

Set i: set the data associated to the number i in constant time

With the restriction that i must be between 0 and some preset length.

Usually we want to support more advanced operations like a segment tree that is designed to

Set i: set the data at index i in O(log n)

Sum i to j: get the sum of values from index i to index j in O(log n)

We could like with an array have Set be in constant time, but that would be at the cost off getting the Sum operation be slower, so a segment tree compromises and finds a way to do both in logarithmic time.

A tree is a specific type of graph. So first lets get into a bit of graph theory. A graph is a collection of objects with edges/connections between them. Example: If you look at a group of t-girls you can then draw each of them as a dot. Now for each pair of girls who have kissed you can now draw a line between them to represent that. Congratulations you have now drawn a graph!

Now a tree is a graph without loops. Example: Lets look at the family tree of a species that reproduce asexually. A loop can now not be possible because that would require a being to be it's own grandparent or such. This is in particular a rooted tree where we have some structure that says that a node has a parent and some children. The root of the tree is now the one node only node who does not have a parent. (Lets say that we only draw the family tree nine generations back).

[erbiumspectrum]

Hi! Do you have any tips for studying chemistry? For some reason I cant seem to get all the formulas in my brain.

Hey!

My unhelpful but still favorite advice for shoving formulas into one's brain is to understand them 😅 A purely memorization-based approach is very bad for chemistry.

If the problem seems to be particularly understanding/ remembering formulas:

Ask yourself if this particular formula is just words turned into numbers and mathematical symbols. I think it may not work for everyone, but for example I found it easier to remember the literal definition of pH that is "the negative decimal logarithm of hydrogen ion concentration" rather than "pH = -log [H+]" bc otherwise I'd keep forgetting about the minus sign.

Check if you find deriving a formula from another formula easier than just memorizing it. Again, my personal example is I hate memorizing things so much I never really bothered to remember the equation that describes Ostwald's law of dilution - bc I knew I could easily, quickly, and painlessly derive it from the equilibrium constant for concentration + degree of dissociation (and I've done it so many times now it stuck in my brain anyway).

When all else fails, I turn to mnemotechnics. To this day I remember that Clapeyron's equation goes pV = nRT because many years ago someone on the internet shared a funny sentence whose words start with these 5 letters. The sillier the better.

If the issue is with chemistry in general:

Take it chapter by chapter. Chemistry, like most STEM subjects, is just blocks of knowledge upon blocks of knowledge. For example, if you want to learn electrolysis, you need to understand redox reactions first. Try to identify where the struggle begins and work from there.

Once you've picked a topic you want to work on, follow the reasoning in your textbook. If you get stuck, that might be a sign you're simply missing a piece of information from a previous chapter. If an example comes up, try to solve it along with the tips in the textbook.

If anything remains unclear, it's usually not the best idea to just leave it and move on. If the textbook becomes unhelpful, turn to the internet or maybe a friend. Otherwise, the next chapter may just turn out to be needlessly confusing.

Practice problems practice problems practice problems!! And not just the numerical ones. The theory-based ones where they ask you about reactions, orbitals, the properties of the elements etc. are important too.

Choose understanding over memorizing whenever possible.

Try to look at the big picture: the way certain concepts are intertwined, how one law may be a logical consequence of another law you learnt before, why some concepts are taught together, why you had to learn something else first to get to what you're studying now. Again, as an example, I think it's particularly fun to see towards the end of ochem, somewhere around the biomolecules: you need to integrate your knowledge of aromatic compounds, ketones and aldehydes, alcohols, carboxylic acids... Stack new information upon what you already know.

Study methods I'm a big fan of: spaced repetition, solving past papers (anything I can get my hands on tbh), flashcards for the things I absolutely have to memorize, exchanging questions and answers with a friend, watching related videos.

If by any chance you end up taking pchem, I have a post for that specifically.

I hope you can find something helpful here :) Good luck!

[felikatze]

OH SHIT I THINK I GET IT. so concentration is mol per liter, meaning the ratio of mol to liter is going to stay the same if mol and liter increase by the same proportion to each other, right. that's why c(H+) is constant for a specific acid/base. and our dear pH is in fact defined over c(H+), so the pH value being puffered for is just decided by the pKs of whichever acid/base you chose as your puffer.

so going further, having a ratio of 1:1 between educt and product means that the balance of them doesn't actually affect the pH. after all, the logarithm of 1:1 is 0. so in this case, pH = pKs. a solution of an equal amount of a corresponding acid-base pair is ALWAYS going to have the exact same pH value, no matter at what volume. hoooh. hourgh. I UNDERSTOOD SOMETHING!!!

[spacetimewithstuartgary]

Uncovering the role of oxygen concentration in the formation of early earth magma ocean

It is widely accepted that the early Earth largely consisted of molten magma, forming a global ocean of magma. This extreme state of Earth was likely caused by the intense heat generated from accretionary impacts, meaning the collision of smaller celestial bodies with Earth. Understanding the formation of this magma ocean is crucial for comprehending Earth’s formation. A major problem with current magma ocean formation models is the lack of consensus on the melting temperatures of deep mantle rocks. Models explaining Earth’s core formation use a specific set of experimental data to estimate mantle melting temperatures, but recent experiments have shown that these temperatures may differ by 200–250 °C from the previously accepted data.

Some studies indicate that oxygen fugacity, or the amount of oxygen available in the mantle, may strongly affect the melting temperatures of deep mantle rocks, which in turn may have influenced the formation of the magma ocean. The mantle’s oxygen fugacity is thought to have increased during accretion, core formation and subsequent mantle evolution; however, the effect of this increase on the melting temperatures of deep mantle materials remains unclear.

Addressing this gap, a team of researchers led by Associate Professor Takayuki Ishii from the Institute for Planetary Materials at Okayama University, Japan and Dr. Yanhao Lin from the Center for High Pressure Science and Technology Advanced Research, China, investigated the effects of oxygen fugacity on magma ocean formation during early Earth evolution. “The evolution of early Earth has been greatly influenced by oxygen fugacity, which may necessitate the reconsideration of current models. To this end, we assessed the effect of oxygen fugacity on the melting temperatures of deep mantle materials to constrain the conditions at the floor of a deep terrestrial magma ocean,” explains Prof. Ishii.

The study also involved Professor Wim van Westrenen from the Department of Earth Sciences, Faculty of Science at Vrije Universiteit Amsterdam, the Netherlands, Professor Tomoo Katsura from Bayerisches Geoinstitut, University of Bayreuth, Germany, and Dr. Ho-Kwang Mao from the Center for High Pressure Science and Technology Advanced Research, China. It was published online in the journal Nature Geoscience on July 16, 2024.

The researchers conducted melting experiments at pressures of 16–26 Gigapascals, similar to mantle depths between 470 km and 720 km, at high oxygen fugacities, on mantle pyrolite, a material composition representing Earth’s mantle. Results revealed that over this pressure range, the melting temperatures decreased with increasing oxygen fugacity and were at least 230–450 °C lower than those from experiments conducted at low oxygen fugacities. Assuming a constant temperature for the magma ocean, this implies that the magma ocean floor deepens by about 60 km for each logarithmic unit increase in mantle oxygen fugacity. This strong influence of oxygen fugacity on mantle melting suggests that current models for early Earth thermal evolution and core formation need re-evaluation.

Furthermore, these results can also explain the apparent discrepancy between the low oxygen fugacities predicted for the Earth’s deep mantle post-core formation and the high oxygen fugacities observed in magmatic rocks over 3 billion years old, formed by melting of the deep mantle.

“Beyond Earth’s formation, our findings on the dependence of melting temperatures on oxygen fugacity can also be applied to understand the formation of other rocky planets that can support human life,” remarks Dr. Lin, highlighting the potential impact of the study. He adds, “For example, these results can improve our understanding of Mars, which is a recent hot topic regarding human habitability.”

This groundbreaking study promises to improve Earth formation models, deepening our understanding of the formation of Earth and other such planets.

IMAGE: The melting temperatures of deep mantle rocks decrease with an increase in mantle oxygen concentration, suggesting the need for re-evaluation of current Earth core formation and thermal evolution models. Credit Takayuki Ishii from Okayama University

[miirshroom]

How Bloodborne astral clock investigation started:

We have the moon, a medical metaphor where the Great Ones are themed as body parts (thanks to Charred Thermos for delving into this), and a clock with 14 symbols split between 12 positions. Surely this has some structure following the 'Zodiac Man' that dictated bloodletting according to the moon in pre-Victorian times (and still shows up on websites that report info on the moon phases connected to calendar dates).

How it's going:

The Wheel of Logarius has 14 spokes which is reminiscent of the pH scale that goes from 1-14 and is a logarithmic scale based on concentration of H+ per litre of liquid. And given the medical context it would make a lot of sense that the zodiac is also split between the 4 Hippocratic humours of blood (spring), yellow bile (summer), black bile (autumn), and "phlegm" (winter). So "Formless Oedon" could naturally be "blood" which encompasses Aries/Iron, Taurus/Copper, and Gemini/Mercury. This actually makes sense in the context of Old One Coldblood being blue because that is the colour of horseshoe crab blood which has a copper metal core, but still falls under the domain of Oedon. And "quicksilver" bullets synthesized from blood have the fused property of real blood (iron based) and fantasy (hunting beasts with "quick" silver bullets). Saw something about a "flora, of the moon" which could be related to the traditional way of assigning a name to each moon instead of months. But which system? The Farmer's Almanac is a reliably documented, consistent, and accessible source but has the drawback that it is disconnected from the traditions of the native peoples. On the other hand there is the potential Osage connection with May being the 'Killer of the Flowers Moon'.

Also, the "third umbilical cord" would certainly be a body part, but I'm not so sure that's the only thing going on here. Bloodborne has a huge musical theme, and a "third chord" is a configuration of chord containing 3 tones at specific offsets from each other.