It is never specified in canon exactly what Arithmancy is, so I had to make it up. I had fun.

(Some suggestively vulgar lyrics in this one, but no actual profanity)

~O~O~O~O~O~O~O~

~O~O~O~O~O~O~O~

The six of them disembarked from the train together and trudged their way over to the waiting row of carriages.

'Ooh!' said Luna when she saw them, and Harry knew she was reacting to the thestrals, which she could see.

'Hope they smell nicer than last year,' said Ron offhandedly.

They did not.

All of them piled into one carriage together. It was a tight fit, but none of them particularly felt like splitting up at the moment. And Harry and Ginny were prepared to take any excuse they could to sit closer together than they normally would.

They passed between a pair of dementors at the Hogwarts gates, and another wave of cold swept over them. Harry wished he'd had the presense of mind to save a bite of his chocolate.

They didn't really talk on the way up to the castle, nor on their walk up to the big oaken front doors. The site of the cavernous entrance hall lined with lit torches had a similar effect to the chocolate, though, and livened their steps a bit.

'Miss Granger!' called Professor McGonagall's voice shortly after they'd reached the Great Hall with its enchanted ceiling. 'I want to see you for a moment.' She was calling out over the sea of heads pouring into the hall.

Hermione looked curious, but fought her way over to the Transfiguation teacher. Harry and Ron, perhaps out of habit, went with her, and so did Ginny. Luna and Neville muttered 'see you later' and went to sit down.

'What is it, professor?' Hermione asked tentatively when they reached her.

'Nothing to fret over, Miss Granger. I simply need a word with you in my office. Move along then, you three.' She chivvied Harry, Ron, and Ginny along and left with Hermione, whom Harry knew was about to receive her Time Turner. He longed to secretly borrow it and study it, but even if that wouldn't be betraying both Hermione's and McGonagall's trust and potentially getting them in a lot of trouble, he wouldn't have had the slightest idea of how to go about doing that in the first place. It was, perhaps ironically, the sort of thing he would normally rely on Hermione to do.

They took their seats at the Gryffindor table, and moments later Professor Flitwick entered leading a long line of very nervous looking eleven year olds. Luna had been right; it was a much larger class than those of the previous two years.

As with the year before, Harry only half paid attention to the Sorting Hat's song, but he did take care to pay attention to the sorting itself as best he could. He was feeling more and more that it was important to at least have an idea who everybody in the school was.

'Astor, Irene,' was the first to be sorted, and she went to Hufflepuff. 'Athell, Norine,' joined her. Perhaps it was Harry's imagination, but Flitwick seemed to be moving though the list faster than McGonagall ever did.

'How are there this many of them?' Ron asked in wonder. 'Where are we going to put them all?'

'Weren't we talking about this last year?' Harry asked, vaguely remembering a conversation they'd had with Stephen and Natalie. 'How our year and Ginny's are so much smaller than usual because we were all born around the end of the war? I expect this is probably closer to normal.'

'They'd better make the tables and dormitories bigger, then,' said Ron.

'Denton, Camellia,' became the first new Gryffindor ('About time!' said Ron), and they still had yet to hit a name that Harry thought he recognized.

'Do you know any of these people?' he whispered to Ginny out of the corner of his mouth. Everyone else was too busy applauding to notice, and Hermione wasn't around, so he wasn't very concerned.

'Some of them, but not well,' she whispered back.

'Frost, Anita,' and 'Greengrass, Astoria,' both went to Slytherin, and at last there was a name Harry recognized, if only because her older sister was in his own year.

'She's engaged to Malfoy!' Ginny muttered. 'It was in the paper about a month before...' she trailed off. "Before we woke up in the past," was what she had been about to say, he was sure of it.

'Grosbeak, Phillip,' joined them at the Gryffindor table before Harry could decide if he should feel sorry for Astoria Greengrass or not.

They were more than two thirds finished before finally arriving at someone Harry definitely knew: 'Sloper, Jack', the ill-fated replacement of either Fred or George as beater on the Quidditch team during that horrible year with Umbridge, ran over to the Gryffindor crowd to raucous applause. The red-clad table had been receiving fewer students so far than the other three, so every one was cause for great celebration.

'There aren't many left up there,' noted Ron as 'Smalls, Saxa,' became a Ravenclaw. 'I hope we get most of them.'

Among the remaining first years, there was only one Harry thought he recognized, but only barely: Romilda Vane, the girl who had tried to dose him with a love potion in his sixth year and in so doing had inadvertently led to Ron being poisoned. He'd happily trade her to one of the other houses for anyone else still standing in the line, if he were honest.

Hufflepuff got three in a row, but then 'Swanson, Rebecca,' 'Sweet, Zachary,' and 'Swift, Scarlet' all came to Gryffindor. They'd still have the fewest of any house, but now even if they only got Romilda out of the five remaining, they'd at least have done respectably.

It turned out that they got 'Yates, Katherine,' too, which happened just as Hermione returned and joined them.

'Oh, I missed the sorting!' she moaned as she took her seat. Flitwick was gathering up the hat and stool and taking them out of the hall as McGonagall was taking her seat next to Dumbledore.

'We didn't do well,' said Ron. 'Everyone else got more than us.'

'It's not a competition, Ron!' Hermione tutted.

'You say that until every other house has a larger talent pool to pull from during Quidditch tryouts,' Ron said.

'Ah, we've won the last two years running,' said Fred, who was sitting just a little bit further down the table. 'Might be sporting to give the other houses a bit of a handicap, eh?'

'Especially with our bewilderingly talented younger sister now old enough to join the team,' added George. Ginny swelled with pride.

They didn't have time for any more conversation after that, because Dumbledore had risen to his feet. He quieted the hall and drew everyone's attention without effort, as usual, and proceeded to deliver a warning about the dementors, as though anyone needed it after that debacle on the train. He then introduced Professor Lupin, whom Harry was annoyed to see received only a lukewarm reception. People judging him based on his haggard appearance, no doubt.

'Look at Snape!' Ron hissed in his ear. Indeed Snape looked furious, and while Ron (and Harry's younger self) probably attributed this to envy over once again being beaten out of the Defense Against the Dark Arts position, Harry now understood it to be a much deeper hatred, born of Lupin's association with Snape's greatest enemies, James Potter and Sirius Black. It was also possible Snape directed some of his anger at losing Lily – first to James romantically and then (as far as he knew) to death as a direct result of of Sirius's actions – to their former friend. It was irrational, of course, but then Snape never had been the most rational of people when it came to emotions, relationships, or his own behavior.

Hagrid got a much more enthusiastic round of applause upon his announcement as the new Care For Magical Creatures professor. Harry was not surprised, of course, but he was pleased to see how many students were excited by the prospect. And this time he wasn't going to let Malfoy ruin everything.

The feast was as delicious as ever, and after making sure to stop and congratulate Hagrid on their way out, Harry, Ron, Hermione, and Ginny hurried up the staircase to bed.

'Fortuna Major,' said Ginny, who had bullied Percy into giving her the password immediately after dinner. The Fat Lady swung open and they climbed through the portrait hole. It had been a much more exhausting train ride than usual, and even Harry and Ginny didn't feel like staying up late to talk. They all said good night and were up their respective spiral staircases before the new first years even arrived. Harry made doubly sure to cast strong silencing charms on his bed curtains before going to sleep. He had zero doubt his nightmares would be potent and numerous that night.

~O~O~O~O~O~O~O~

Harry woke the next morning feeling sluggish and drained. The images dredged up by the dementor had indeed fueled his nightmares, which had woken him several times and made sure whatever sleep he did manage to get had been very fitful. When he, Ron, and Hermione entered the Great Hall in the morning, he slid onto the bench next to George Weasley, wishing not for the first time that Hogwarts allowed its students to drink coffee. At least in the morning, for Merlin's sake. He settled for a very strong cup of tea, which helped but wasn't quite the same.

'New third-year timetables,' said George, passing them along. 'What's up with you, Harry?'

'Didn't sleep well,' he said. 'That dementor on the train really got to me.'

'Ah, that explains it,' said George sympathetically. 'Foul things. Ginny said much the same when we saw her in the common room, though I don't know what horrors she has in her past to keep her up at night.'

'Mind, our lives haven't been all that bad either and I was ready to be sick,' added Fred. 'Don't blame Dumbledore for not wanting them here.'

'True,' George agreed. 'Remember that time Dad had to go out to Azkaban? Said it was the worst place he's ever been. Came back all weak and shaking...no surprise most of the prisoners go mad in there.'

'Try and focus on something positive,' Fred suggested. 'Like flattening Slytherin in Quidditch again. First match of the season, remember?'

'How could I forget?' asked Harry, managing a grin. 'Is Oliver going to bother with tryouts or is he just going to fly the old team again? He really shouldn't keep doing that.'

'We've been on him about it,' said Fred. 'He thinks we're good enough as-is, but we reminded him about his promise to set up a reserve team, which he never did last year. And we're going to want to get someone some keeper experience to replace him next year.'

'Of course our true motivation is getting Ginny on the team, but we're not about to tell him that,' said George.

Hermione had tuned them out as soon as they started talking about Quidditch, and was now examining her timetable.

'Ooh, good, we're starting some new subjects today, she said happily.

'Hermione,' said Ron, frowning as he looked over her shoulder, 'they've messed up your schedule. Look – they've got you down for about ten subjects a day. There isn't enough time.'

'I'll manage. I've fixed it all with Professor McGonagall.'

'But look, said Ron, laughing, 'see this morning? Nine o'clock, Divination. And underneath, nine o'clock, Muggle Studies. And – ' Ron leant closer to the timetable, disbelieving, 'look – underneath that, Arithmancy, nine o'clock. I mean, I know you're good, Hermione, but no one's that good. How're you supposed to be in three classes at once?'

'Don't be silly,' she said shortly. 'Of course I won't be in three classes at once.'

'Well, then…'

'Pass the marmalade.'

Harry found this all very amusing, but he couldn't help but notice one glaring flaw in this strategy. Before, it hadn't made a difference, because he and Ron had been in all the same classes together. Now though, Ron would be in Divination with Hermione while he, Harry, was in Arithmancy with her. How on Earth was she supposed to keep that a secret? Did the teachers just not expect either of them to notice? There had to be more to it.

Maybe, Harry thought, there was some kind of jinx that interrupted or distracted people whenever they started looking into it too much. Because just as Ron was about to protest again, Hagrid approached them swinging a dead polecat, which very successfully changed the subject to his very first lesson after lunch.

'Wonder what he's getting ready?' Ron asked with a twinge of anxiety after Hagrid had moved on.

'Whatever it is, we need to make sure the lesson goes well,' said Harry. 'If something happens, it could shake his confidence and everything could go tits up fast.'

'Harry!' admonished a scandalised Hermione. Ron was looking at his watch.

'We'd better get going,' he said to Hermione. 'Divination's all the way at the top of the North Tower. It'll take us ten minutes to get there.'

That was another aspect of the subject that Harry would not miss. Though he supposed it was a good source of exercise.

He himself still had to go to the fifth floor, which wasn't a short trip, so he hastily ate the rest of his breakfast and hurried off. Peeves was floating around in circles in the entrance hall banging a wooden spoon on a helmet stolen from a suit of armor and bellowing some very rude lyrics about a young woman and her pet hippogriff. Harry forced himself not to laugh even while Filch came tearing out of one of the side corridors behind him hollering for the poltergeist to stop until he let out a loud curse of his own, having tripped over something on the floor.

Without even turning to look what had happened, Harry made a beeline for the fifth floor before Filch could somehow lay the blame on him, knowing that tune was going to be stuck in his head for a while. He was not surprised to find Hermione arriving just as he showed up.

'I thought you went with Ron to Divination?' he asked, merely to see what she would say.

'Oh!' she said, surprised and fumbling briefly with the front of her robes. (Really, Hermione, a master spy you are not, Harry laughed in his head). 'Well, I couldn't miss our first Arithmancy lesson; I've been really looking forward to it.'

He decided to take pity on her.

'Look, Hermione, if you don't want to tell me what's going on, I won't pester you. But don't treat me like I'm stupid.'

'I'm not trying to – '

'Let's go in.'

They entered the classroom and took seats near the front (Harry's concession to Hermione), though not at the front (her concession to him). They took out their books and waited while the classroom filled up. There was the usual din of muttering as everyone found where they wanted to sit. This was to be a new ordeal for all of them.

For the first time in a very long time, Harry was feeling apprehensive about his immediate future. This class would be his first truly new experience since his return to Hogwarts two years ago. The first time he would be learning actually new material. He was also in an unfamiliar environment; he'd never been in this classroom before; he'd never properly met professor Vector and knew almost nothing about her, and the only other Gryffindor in the class with him was Hermione. He had initially felt that stepping outside his comfort zone would be a healthy change of pace, but was beginning to rethink that idea.

Too late to back out now, he thought. At least there were only three Slytherins in the class: Theodore Nott, and two girls named Tracey Davis and Karen Ropar. He had seen them hanging around Pansy Parkinson and her gang, but without their queen bee around, perhaps they would be tolerable. He'd never had much of a quarrel with Nott outside of his father being a Death Eater.

Most of the class was, perhaps not surprisingly, made up of Ravenclaws. Stephen and Natalie were there, along with old (future?) D.A. members Terry Boot and Anthony Goldstein, and another boy whom Harry had never heard speak a word who might have been named something Entwhistle. There were also a few Hufflepuffs, none of whom Harry knew particularly well. He was once again struck by how socially isolated he'd been during his original Hogwarts years. It wasn't even as if his year was particularly large. Hadn't he just been thinking the night before about how his year and Ginny's were two of the smallest on Hogwarts' records? Though he suspected Teddy's would come close for similar reasons, as well as perhaps the year after that.

Talking died down as Professor Vector entered the room. She was a tall, slender witch of perhaps around forty years with waves of dark brown hair reaching down to her lower back. She wore deep red robes with a tall pointed hat, and her rounded face with its small nose, small chin, and small eyes carried a stern but not unfriendly expression. A complete unknown entity to Harry – though he had heard her described by others, including Hermione, as strict but brilliant – he waited in patient silence to see where she would take the class.

'Good morning, class,' she addressed them. Her voice was soft but it carried well. She was able to project loudly with little effort; it reminded Harry of Oscar Whitenack, his first Auror instructor.

'Good morning, Professor,' replied most of the class. They, like Harry, appeared to be treading carefully around this new teacher until they knew just how accurate her reputation for strictness really was.

'I am pleased to see so many of you this year,' she said. 'Please announce yourselves appropriately.' She began to take the register. She did not pause or make any indication whatsoever of noticing anything out of the ordinary when she read Harry's name. He allowed himself a small smile; he liked her already.

When she finished the register, she said, 'Now, I don't suppose any of you could tell me – on a general level – essentially what Arithmancy is?'

Hermione's hand, as usual, shot into the air. 'Arithmancy deals with the magical properties of numbers, and with evaluating the magical efficacy of items within a set by calculating their numerical value relative to other items in the same or similar sets,' she said all in one breath. Also as usual (for the old days, anyway), Harry had no idea what she was talking about.

'That is correct, Miss Granger, though students in my class will wait to be called upon before speaking,' said Vector. It was a mild rebuke; her face remained impassive, but Hermione flushed slightly all the same.

One of the Hufflepuff girls raised her hand. 'Yes, Miss Rivers?' asked Vector.

'Excuse me, ma'am; I don't understand what that means.'

'Oh, I don't expect many of you to,' said Vector. 'Not at this stage, anyway. Don't fret; it will all make sense in time. I've prepared a small demonstration for you all; I've found many people young and old learn better with visual aids.' She conjured three small piles of straw on her desk. 'Hopefully by the end of today's lesson, each of you will have at least a general understanding of the definition which Miss Granger has so expertly repeated. Of course, this is merely a general overview. We will spend most of this year mastering the most basic elements of Arithmancy, which are numerous, if you will pardon the pun, but essential for comprehending the more advanced calculations and theorems we will be studying in later years.

'Now, then,' she continued. 'At its most simplified, Arithmancy is indeed concerned with the magical properties of numbers. Everything else we do will stem from that. It will quickly become more complicated than you can currently imagine. Most witches and wizards possess only the meanest understanding of Arithmantic principles, though in truth this is often all one will require.

'For instance, nearly everyone is aware that seven is the most powerfully magical number,' she said. 'I wonder, however, if any of you could tell me the most magically powerful letter of the alphabet?'

Even Hermione hesitated before putting her hand up, and only two other people volunteered.

'Yes, Mr Entwhistle?'

'X,' said the usually silent Ravenclaw.

'That is correct,' said Vector. 'The letter X possesses an arithmantic value of four, the highest of any in the English alphabet. However, other writing systems can be much more potent. Runes, for instance, all possess an arithmantic value of five or higher. It is why they are so useful for wards and enchantments.'

Harry had no idea what an "arithmantic value" was, but was beginning to get his mind around what Vector was talking about. He knew some numbers were more magical than others; it only made sense that the same would be true for other things.

'Moving on, then, can anyone tell me the most magically powerful language?' asked Vector. This time, no hands went up – not even Hermione's.

'Not to worry,' Vector said. 'It isn't a fair question in any case. Researchers have yet to discover any one language that is more powerfully magical than all others, but there are some which are known to be stronger than most. Perhaps you might know of one?'

Several hands went up this time, and Harry had to laugh at Hermione's disappointment when Vector called on Stephen Cornfoot.

'Latin,' he said.

'Yes, indeed,' said Vector. 'Latin is what is described as a Tier Six Magical Language – owing to it possessing an arithmantic value of six – and is one of the few such languages currently known to exist. This is why most incantations you will learn throughout your lives are derived chiefly from Latin, even though it is no longer actively spoken anywhere. Of course in theory spells could be cast via incantations in any language, however potency will tend to diminish with lower Tiered languages. The most powerful spells and enchantments will typically need incantations in Tier Five or Six languages in order to work properly, unless cast by an extremely powerful and skilled witch or wizard. Tiers Three and Four will usually suffice for basic, everyday magic unless one is particularly untalented. English, however, is what is known as a Tier One language. Even the most powerful among us would have difficulty casting most spells using incantations derived from English.'

Here Hermione raised her hand (and waited to be called on). 'Spells can be cast in English?' she asked.

'Oh goodness, yes,' said Vector. 'Though not very well, as I say. It is time for my demonstration.'

She waved her wand at the first pile of straw and said, 'Incendio!' The straw burst into flame, startling some of the students near the front.

'Now observe what happens when I attempt to cast the same spell using an English incantation,' she said. She waved her wand at the second pile and said, 'Makefire!' A small patch near the top of the pile, directly under the tip of her wand, began to smolder, and a few sparks shot off. The fire fizzled out long before even a tenth of the pile was burned. There were several gasps of astonishment. Harry, too, was fascinated. He had never seen or heard of such a thing before.

'Clearly, the power of my spell has diminished greatly. Were I not a highly trained and experienced witch, that spell would not have had any effect whatsoever, regardless of how much power I tried to put into it. English has almost no magical properties at all, unfortunately for us. There are ways we can compensate, however. Early wizards often cast their spells in verse; the power of the metre and rhyme can amplify and enhance what little power the words do hold through arithmantic processes you will learn about several years from now. Early wizards did not always understand how or why this worked, of course, merely that it did. Nevertheless, this method can be extremely time-consuming and inefficient, not to mention the same rhyme is often unlikely to work twice unless exceptionally well-crafted.

'Wandwork is another means of compensation, and is one of the primary reasons the invention of the wand is considered so important. Watch what happens when I try my English spell again but with an additional wand twist and flourish.'

She waved her wand much more elaborately this time, then said, 'Makefire!' again. The third pile of straw caught fire – not as brightly or suddenly as the first one, but it did burn all the way down.

'This, too, is inefficient – though less so than composing spells in verse – and the additional wand movements must be specifically and precisely calculated for their value in order to correct the imbalance for each individual spell,' said Vector. 'There is often little reason to bother outside of academic curiosity when incantations in more magically potent languages are readily available.'

Karen Ropar raised her hand.

'If we don't need to be able to do that, why are you showing us?' she asked.

'I want you to understand and appreciate the importance Arithmancy plays in every aspect of magic. It permeates every stage of everything we do. When Miss Granger spoke of "sets" earlier, she did not define what that meant. A set is a group of like things that all have an arithmantic value relative to one another. The most basic set is numbers. Seven is the most powerful, but there are plenty which possess magical properties that seven does not. Thirteen, for instance, is often associated with misfortune, but can also be used in Divination. Twelve is the number most strongly associated with time and order, and seventeen with randomness and chaos. Three is a number of purification and four has strong ties to death. There are infinite numbers, and we are always learning new things about them and how they relate to one another. Most of this year will be spent studying the basic set: one to twelve. At Easter time we will begin work on the decadals, but that will continue on into next year.

'The point is that just as there are infinite numbers, so too are there infinite sets. We have already discussed letters and languages. Musical notes are a set. Plants. Elements. Animals. Colors. Everything in existence, everything you can imagine, belongs to a set. Most things can be classified by multiple sets. All have their values, all have their interactions with one another. Witches and Wizards will study deeper Arithmancy until the end of time and still not classify them all.'

Harry was immediately grateful he had not taken Arithmancy during his real school years. It was exactly the sort of thing that would have both confused and bored him senseless. At the same time, he was glad he had decided to take it now. He was mostly able to follow what Professor Vector was saying – something his thirteen year old self would most definitely have been incapable of – and it sounded fascinating. And challenging. The latter was what really had him excited. Everything – even Potions and History of Magic – had been so unbearably simple for the last two years; he was worried his brain was starting to atrophe. Now finally he had something that would demand his utmost attention and concentration once again.

Though he also thought, with a twinge of anxiety, maybe taking Ancient Runes along with it might have been biting off a little more than he could chew.

~O~O~O~O~O~O~O~

The lesson proved to be one of the most difficult Harry had ever gone through, but somehow that made him love it. His brain really had been starved for a challenge. Yet they'd spent the entire hour going over the numbers one, two, and three. He felt like he was in nursery school again.

'Transfiguration next, right?' he said to Hermione. More than just making conversation, he actually wanted to make sure he was remembering properly. His head felt like mush.

'Right,' she said. 'You go ahead; I need to use the toilet.'

'Sure,' said Harry, allowing her the pretense. 'See you there.'

He'd gone about three steps when he felt a wrenching, twisting sensation. He felt like he were being pulled through a portkey and wrung like a sponge at the same time. Everything spun around him, and the next thing he knew he was on the floor, panting heavily. There was a banging sound overhead, and a loud, booming voice making a mockery of singing.

'Oh, she bowed to him low full frontal and head on

he went down on his knees just the same

she slapped his hindquarters and held to his mane

and she rode him from sunset to dawn!'

Harry barely had time to process what was happening. Acting purely on instinct, he pulled his invisibility cloak out of his pocket and threw it over himself. He peered around, trying to get his bearings. He could see a figure walking away from him on the other side of the entrance hall, which was undoubtedly where he was, even as Filch's shouts grew louder from behind him.

Suddenly, he had the wind knocked out of him. The caretaker, unable to see him, had run headlong into him. Filch's foot plowed right into Harry's stomach, and he went careening to the floor letting out a string of oaths that it was fortunate for him Professor McGonagall was not around to hear. He hopped right back up, searching frantically around for the culprit, but saw only Peeves, who had given up on his song and was now cackling wildly at his favorite target's pratfall.

Harry's head was spinning. He'd experienced this before. Somehow, he'd ended up right where he'd been exactly an hour before. Physically, this time, as that was certainly his own past self he'd seen walking away, just as it had been his own future self he'd heard Filch trip over on his way to Arithmancy an hour ago. The only thing he knew could cause something like that was a Time Turner.

A Time Turner just like the one he knew Hermione was currently wearing around her neck.

~O~O~O~O~O~O~O~

~O~O~O~O~O~O~O~

~O~O~O~O~O~O~O~

It may seem like I churned this one out really fast, but the whole arithmancy lesson I already had written from months ago; I just had to plug it in.

I also had a lot of fun writing the verse from Peeves's song. Strictly speaking, I didn't have to. I could've just said Harry heard the same tune, but where's the fun in that?

Please leave a review and thank you in advance for doing so!