I am wondering about preparing spells per day. How long is a day? One session?
Also some spells last 15 minutes. How long is that during a combat encounter?
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Combat is measured in rounds. A round is 6 seconds. Note that this is not the length of a turn, but the length of a round: everyone’s turns are happening near-simultaneously, with just a slight edge to those who have higher initiative.
So there are 10 rounds to a minute, and then of course there are 60 minutes (600 rounds) to an hour, and 24 hours (1,440 minutes or 14,400 rounds) to a day.
More importantly, the general guideline is for there to be four encounters in a typical day, and each should exhaust roughly 20% of the party’s daily resources (leaving a 20% margin of error/leeway for more challenging days/allowance for recuperation for the next day). How much time takes place between these is generally a matter of plot, and of course there are lots of ways you can do things differently (more but easier fights, fewer but more dangerous fights, etc.). And, of course, this is just the guideline; you can’t do the same thing every day and still keep things interesting. An important part of DMing is knowing when and how to flout such guidelines for the sake of improving the game.
Note also that players have a fair amount of control over when and where they will go places. A wizard is probably only going to be willing to venture into the dark ‘n’ scary dungeon (a technical term) when he’s out of spell slots if things are truly dire and urgent; otherwise he’s going to be strongly pushing for the group to stop and rest. Same with a fighter who’s running low on health, or what have you.
Worth noting: arcane and divine spellcasters have different rules for defining a “day” in terms of when they get their spells back.
For divine spellcasters, they simply have a particular hour in which they are supposed to be praying – clerics of a sun god might pray at dawn, or high noon, while a druid of the twilight lady of the forest (a deity I just now made up) would likely pray at dusk. Unlike arcanists, divine spellcasters do not require sleep to refresh their spells.
On the arcane side, they can prepare/refresh spell slots at any time of day, but must get eight straight hours of sleep (there are rules for interrupted sleep and how that affects getting to refresh) between refreshes. Strictly speaking, there is nothing that prevents an arcane caster from blowing all his spells in one fight, and immediately go back to sleep and refresh again. That’s frowned upon by... almost every gaming table ever, for obvious reasons, though. Most groups expect one 8-hour rest per 24-hour day, which means the differences between arcane and divine casters don’t see a lot of play in practice. At the same time, the average Wizard, in particular, will push for about 17 hours of downtime per day: 8 hours of sleep and 1 hour of spell preparation are mandatory, and then 8 hours of time for crafting or scribing spells in his spellbook are going to be considered highly desirable (as in, he’s going to want to have that kind of time more often than not). Of course, plot may dictate that he cannot have that kind of time, but if matters aren’t urgent those are a wizard’s natural inclinations.
Anyway, despite the detail that the rules provide for these things, almost all groups I’ve played with have not paid such close attention to timing. Unless the DM specifically ambushed the group at night, or that time (as in a matter of hours) was of critical importance, these issues were hand-waved away and spellcasters simply started every day with whatever spells, which refreshed whenever the group decided to stop and make camp.
From a player’s standpoint, spells generally come in only a few durations:
“right now” – Stuff with a duration of 1 round, Instantaneous damage spells, etc. At very-low levels, also includes rounds/level spells.
“this encounter” – Anything that has a duration from about thirty seconds to a few minutes (roughly 5-100 rounds). These are usually too short to cast before combat starts, but usually last long enough to see the current encounter finished. Stuff measured in rounds/level usually falls here, and minutes/level often does too unless you’re very high level or you have reason to expect a fight soon.
“through the next encounter” – Basically, anything that you feel you can cast ahead of time and still have available through the next encounter. How long is required for that depends heavily on circumstances; if you know there are monsters all over the place, you may only need a few minutes, but if you're walking down a road you might want a couple of hours. At very low levels, hours/level spells are the main sort available for this, but by mid levels spells that have durations like “10 minutes/level” will fit the bill most of the time. In that crowded dungeon, 1 minute/level may be sufficient.
“All day” – Stuff you can cast at the beginning of the day and forget about. By mid-levels, hours/level will cover this (8 hours will cover a full day for a lot of parties, dividing the day into thirds between adventuring, downtime, and sleeping). Obviously, anything that lists 24 hours as a duration also qualifies.
“Multiple days” – Stuff you don’t need to cast every day to continue to benefit from it. There aren’t many spells with durations of days/level, but they do exist. More significantly, Permanent effects and also some things with a duration of Instantaneous (when they create something or provide a lasting benefit) go in this category.
As a DM, you will probably want to keep these ideas in mind when planning encounters – consider the spells that a party can expect to have already-running when they start a fight. Buffs that you don’t have to spend actions in combat on are automatically much better than buffs that require time during the fight. The only time I recommend paying careful attention to a spell’s duration is when a player casts a spell that is borderline between “this encounter” and “through the next encounter” – it can make a big difference whether or not that spell is still in play during the next encounter.
KRyan's answer is excellent, but just in case you're entirely new to RPGs, I should probably clarify something: The real-world hours that you spend playing the game bears little relationship with the time experienced by your character in the fictional world he or she inhabits.
As you play the game, the actions you and the other players describe your characters performing will take various amounts of in-world time. Generally, the GM will keep track of roughly how much time has passed while you adventure, and will inform you of time dependent events (such as sunset and dawn) when your characters become aware of them.
The rate at which in-game time passes compared to time in the real world is flexible. Sometimes time passes more quickly in-game than out: If nothing happens during the five day journey from one town to the next, the whole thing might be skipped over in a few sentences. Sometimes (particularly in combat) things go the other way, and an activity that takes you hours of real-world time to resolve only takes a handful of in-game seconds. (If it helps, think of it like time in a movie: An intercontinental plane flight takes hours, but will generally only ever receive a few minutes of screen time, whereas a fight scene that would reasonably only lasts a few minutes can be stretched out by gratuitous overuse of slowmo.)
As for the fifteen-minute spell duration, a 'round' of combat (the time it takes for every combatant to have their turn) is six seconds. A spell with a fifteen-minute duration will therefore last for one hundred and fifty rounds - and very, very few combats last that long.