For any given coffee brewed via any given method, there is an 'ideal extraction window' - a time period of brewing in which an ideal extraction will occur.
Extraction in this sense is referring to two interlinked concepts: a chemical thing, and a taste profile. Speaking at the physical level, extraction is the percentage by mass of your dry coffee that is removed into the solution that is brewed coffee at the time your brewing process finishes. At the taste level, the results of the physical definition can affect the taste of the brew you produce; an 'underextracted' brew has taken less than the ideal mass from your coffee, and will generally taste sharp, sour, and wine-dry, an 'overextracted' brew will taste harsh, bitter, and astringent, "extraction" in this sense refers to how much of either is present in your cup, or if you've hit the golden zone in the middle where neither occurs.
Your ideal extraction window is set by a combination of factors, best understood by looking at the ease by which water can 'fill' a semi-permeable physical object when we assume that we are dealing with a set total mass of many many similar objects all subject to full immersion.
If each object is smaller, it takes less time for water to fully soak into each individual one. So a smaller grind size for your coffee shortens the ideal window, as each piece you're brewing from will individually reach ideal saturation faster, and 'brew' faster as a result. Similarly, if each object is larger, it takes longer for water to fully soak into each one. By increasing your grind size, you lengthen your ideal extraction window as it will take longer to fully saturate and brew the larger pieces.
If you use more coffee, you have more total objects in the solution, resulting in more available compounds to be extracted. If they're all the same size, they should want the same time as a smaller amount of coffee ground to the same size, simply producing a more-concentrated version of what you were already making. This is the best way to get "stronger" or "more caffeinated" coffee - simply use more grounds from the start. However, having more mass in your cone or cylinder will have more total objects blocking the flow of water, so despite the fact that the ideal extraction window remains the same, in many cases using more total grounds will result in an overall slower process. Many folks will adjust their grind size as dose shifts to try and counteract this - if it's definitely going to take longer at a larger mass, we should simply grind slightly coarser so the resultant brew time increase we get is still hitting an ideal time.
If the water is hotter, it softens the cellulose structure of the grinds better. If the pieces' size is now considered fixed, when we use a hotter water it takes less time to fully saturate the same size particle than when we use a cooler water. Again, this is a way of changing the time it takes to get to the ideal brewing window. Temperature is additionally complex as there are some breakpoints involved; above 184 for instance is necessary to fully extract all the available compounds generally considered desirable in a 'complete' brew. There is some mythology around upper bounds as well, that using a too-hot water can extract compounds better left behind, but there's more and more critique of this concept and I don't like to treat it as gospel - I'd simply recommend you test with the coffees you use and see what you prefer.
Pressure also directly affects extraction. Extracting under higher pressures will result in a faster extraction if all else remains the same compared to the same extraction at lower pressure. This is why, for instance, espresso is a 30-35s extraction at 9bars pressure, while Aeropress uses a similar amount of coffee, far more water, and generally has far longer brew times and ideal windows, even if you're using something closer to an espresso grind in it. But~! For aeropress especially, this comes into play with flow rate and particle size - if you have to press harder to force a brew to flow through, you are also increasing the pressure inside the chamber and speeding up extraction. Similarly, longer times spent at higher pressures has a larger and larger effect on results, so spending a very long time yarding on a very hard press can be even worse. In most cases, if you need to press hard to hurry things up, you're already dancing with overextraction, and 'pressing harder' is not necessarily the expeditious solution it may seem - probably want a slightly coarser grind next time, instead.
The last important thing that affects extraction rate is agitation: stirring primarily, but in many cases the act of adding water at all is also providing agitation. The more agiatation you have, the faster your brew will be extracting. Similar to how volume of water allows more space between grounds and a higher concentration differential to accelerate extraction, agitation also spaces out the grounds more evenly across the total volume of water you have in play, while also accelerating each grounds' exposure to larger amounts of water it's passing through - if you add lots of water but your grounds have time to settle out, that's pretty much the same as only adding the amount of water needed for the bottom area they've settled to. Everything in the top isn't doing squat for brewing and may be causing weirdness in your cup in methods like pourover as it can potentially be exiting laterally rather than through the coffee mass and resulting in an uneven or overly-dilute-tasting brew, even if your ratios and grind size were actually correct.
From the above factors to the brew time itself. Your brewing time is of course how you obtain or achieve that ideal window you're setting up with the last two steps. In any flow-through brewing (cone, AP normal), and in most steep-and-release brewing as well (clever, AP invert) your grind size will affect flow-through rates. The smaller your pieces, the tighter together they can pack, and the more time it takes water to flow through them. Think of damming up a narrow river: a single large boulder will obstruct water everywhere there's boulder, but will have space around the corners where the river still easily and rapidly flows past, dam it up with a load of sand and every individual gap is much smaller, slowing the flow of water in your river far more than the boulder did. So finer grinds need less time, but make your brew take longer, while coarser grinds need more time but make your brew actually go faster. This is why there's so much balancing act and desired precision around grind settings - you're balancing moving the brew window around changing grind size against lengthening or shortening the brew time itself as a result of the same changes.
As far as manipulating time and window to get good results, though. Tuning temperature is a very volatile way of attempting to compensate, and most folks will find tuning their process least frustrating by not changing it about lots - it can have what feels like a multiplicative effect on grind sizes and other variables. For most folks, grind is the best single variable to change around while searching for their version of the ideal brew. Depending on method, technique can help tune your ability to attain a specific brew time, without changing anything else, if you feel all your input should be good but the result is just a little off. Within cone or pour-over, the density of the brew slurry affects extraction and flow rates alike: the more water in the cone at any given time, the more space between each piece filled with water there is, and the wider the gap in concentration of extractable mass between the solution and the particulate - faster extraction; the more space between each particle filled with water, additionally, means the particles are further apart and have less of a slowing impact on flow rates - faster flow as well. With aeropress, the particle size still affects ideal brew window, but also affects how much pressure is needed to 'press through' in both methods and how much will naturally escape during brewing for AP normal. Taking steps like agitation (stirring) to prevent the brew mass from settling can again keep concentration gradients high and accelerate extraction, while also stirring or agitating directly before pressing can prevent the grounds from acting as a single mass and slowing press-through time. In either method, if the brew takes too long to finish, either in dripping through on cone or pressing on AP, you have extended your brew time and may be introducing the flavors of over-extraction to your end result.
The water you're using in your brewing (in any method) affects two things - how long it'll take to finish brewing (everything you put in the top needs to come out the bottom) as well as how diffuse or not your brew slurry will be. With more water in the brew and more space between each particle, you get faster extraction rates and faster flow rates - using a lot of water all close together in a brew process generally results in a faster brew all told, while using the same amount of water more spread out will have a slower ideal extraction rate, but also a slower overall pace.
In most cases, how much water you use in your brewing versus the amount of coffee pretty directly determines the end concentration (strength, or TDS as 'total dissolved solids'); adding water after the fact was for a very long time rather frowned on as something somehow 'bad' for the brew, but what's known as 'bypass brewing' has seemed to see a resurgance recently and most professional concensus is that it doesn't negatively affect the taste of the end up if done reasonably well. Brew a smaller batch of high-concentration, well-extracted, brew, then dilute to normal palatable levels of concentration. This got a terrible rep because when it was last heavily used, "we" didn't understand extraction as well and 99% of concentrates brewed for bypass were also horrendously overextracted - so of course the diluted result tasted horrid as well. This bypass brewing is what the manufacturer-recommended brew method for AP relies on - that diluting a well-extracted cup of concentrate will result in the same end effect as simply brewing a well-extracted larger batch 'straight', but with a far smaller footprint or hardware investment.
Bloom is ... controversial, at times. Theres a lot of folks who're pretty convinced it's either useless, ornamental, or both. There's also a lot of folks convinced it's essential and see a good bloom as a necessary indicator of a fresh coffee. I'll take somewhat a middle-point to those polar perspectives, but want to hijack a moment to talk about the gas present, why it is there, and its relationship with your brewing.
When we roast coffee, CO2 is trapped in the bean - mostly, created or released from within the bean as many of the chemical reactions fundamental to roasting coffee release CO2 as a byproduct - and that gas largely remains trapped in the bean once roasting concludes, gradually venting over time. This gas does not 'hold' flavour, and it is not desirable to keep the gas in, staling occurs mainly as a result of O2 exposure and the fact that CO2 generally slows its venting at about the time that O2s negative effect on taste begins to be perceptible is almost entirely coincidental. However, this gas does affect brewing for a relatively tritely simple physics reason: two different things cannot occupy the same space at the same time. If a ground is full of trapped O2, the water cannot soak into the same spaces to evenly extract that granule - it can only extract from regions not containing CO2. Because the CO2 is slightly pressurized, it is more difficult to displace than if the beans have fully degassed and are simply filled with neutral-pressure 'air'.
To discuss the impact on brewing, an example: in the case of cone - especially Clever - the effect of this trapped gas are very perceptible, and moreso on very fresh coffees. Using 4 day old medium roast, running a single, no intervention, fill-30s steep-drain process with no bloom results in ~%50+ of my grounds having floated on the surface of the brew and subsequently high-sided as the cone drained. Running an otherwise identical brew with a 45s bloom using 40g water to bloom 25g coffee and stirring heavily during bloom to speed up soaking-in, then the same no intervention, fill-30s steep-drain process cuts my high-siding mass down something I'd eyeball at less than 20%, with the remainder of the mass now sitting flat and even at the bottom of the cone.
A gassier ground is more volatile to your process - exact mass, volume, amount of trapped gas, hardness - all affect rate of venting, and there's too many small variances that will still play enough of an impact to be a nuisance to your brewing attempts. Similarly or more so, how that same ground will behave if immediately plunged into a brewing solution. Giving your brew process a bloom time to allow proper venting is not necessary, but certainly allows almost any brewing process, especially any using fresh coffee, to be far more consistent if all other things are allowed to remain the same.
Or for a more numerical example of the same thing, dialling in with four-day-old coffee on a Curtis G4 (a commercial on-demand automated 'pour over' brewer with a ton of process adjustment settings) using a TDS meter and extraction maths: our goal brew is one that measures a TDS of 1.2 (or 12,000 ppm) and has an extraction of 20% (of my total input dry coffee mass). Brewing straight with no bloom cycle gets me a variance of .35 TDS and ~3-7% extraction. Adding a bloom cycle using approximately (brew mass)*(1.9) and 30s hang time closes that variance down to .15 TDS and 2-4% extraction, while a 45s bloom further narrows variance to .08 TDS and 2.5% extraction. It's not perfect - longer times than that don't close the gap further, and odd shit like brew head dynamics, cone placement, and distrubtion of coffee in cone can all play roles that are exceptionally difficult to control for; but those results at least mean we're getting cups off the machine that are, if nothing else, more consistent than any of us doing the same pourover manually (at my best, I have variance of approximately .18 TDS and 5% extraction - its harder than it seems~!).
Filter rinsing, though, is a whole other bugbear. There's tons of mythos around needing to rinse paper filters to 'elimate paper taste' or wash the various processing chemicals out. They're bogus across the board. Food grade paper, no matter what colour it is, doesn't have toxic chemicals remaining - the unbleached kinds exist to pander to consumer fears that manufacturers cannot educate past but might as well make money from. The only filter I've ever encountered that imparted notable taste into coffee even when no one was primed to look for it was the Melitta #4 Unbleached Bamboo. Some folks do find that there's paper taste that they find in their cups and feel that rinsing their filters helps control for this. They're not wrong, but they're not right. At least, when I had a test audience to play with, coffee brewed to relatively normal concentrations (1:17 ratio) across a wide range of filter and even non-filter options generally was more likely to return with reports of paper taste if we asked participants about paper taste, even if no paper and/or rinsed paper were used in brewing their samples. It may still be there, in your cup - after all, you can brew an awfully paper-y tea if you run brewing water through an empty filter - but whether or not you notice it or care is much more dependent on whether or not you're looking for it than minor variations in process to try and 'correct' for it.
All that said, though, for most methods I recommend rinsing a filter anyway. A wet paper filter sticks its appropriate brewing object (AP cap, pour over cone, etc) better than the same filter dry and is at least easier to work with and in some cases even more consistent or reliable compared to the same process with a dry filter.
Now, there's a ton more variations in process and brewing that could come into play - I've tried to cover the more general theory around brewing and how some of the most common and most impactful changes to a recipe or formula can affect what you're going to get in your cup at the end. I hope I've also provided a framework by which you can at least try and assess anything I've missed; by looking at the effects of the change on how fast extraction might take, or how fast an extraction it might in turn want. The final dimension that model I'd like to present is something I rather talked past on the way to getting here: how repeatable any given change can be as far as recreating the same good or bad results again later.
If I suggest adding 'stir' step to your AP brew, say 30s after you've finished adding your full brew volume of water in inverted method: you know that with lots of water in, you've already a fairly high rate of extraction, because theres lots of solution to brew grounds, and lots of space between the grounds; you also already know that agitation accelerates brewing as well, with rate or scale of agitation relatively proportional to the resultant increase in pace. So you can guess that you already have fairly fast extraction, and similarly can guess that the 'stir' step will further accelerate extraction. If your last brew without a stir step was underextracted, this change may be what you're hoping for. If it is - your immediate concern becomes how do you make sure you can perform the same step again and get the same good result? Because scale of agitation is a factor, you want to avoid stirring at different rates - faster once or slower another will affect the scale of its effect. In adding a stir step, consider ways it could be different next time and try and control them - remember how deep you put the spoon, try and make your stirring something simple to remember and easy to repeat - "one circulation per second for five seconds" say, is much more repeatable than "stir for five seconds" not clearly defining pace.
Or using the same building-blocks approach, this understanding of extraction can make understanding this community's' fixation on grinders and expensive or high-end grinders especially a little more understandable: all of the above is most true when your grind particles are more similar in size to one another. If everything is all different sizes, there's a whole bunch of different ideal extraction windows and if you successfully hit any one of them, you must have got some of the others wrong. If there's enough 'wrong'-sized particles present, that can show up in your cups' flavour even if you hit an ideal range on the majority of your brew mass.
The final and most interconnected concept I want to introduce is also the one question that you may have found in the course of this that's not quite covered yet: why not just use tiny particles and lots of water for a very fast, very even, extraction across a narrow window. So, if smaller particles extract faster, 'why can't we' just use espresso or turkish grind in a FP or in a AP and simply have a very fast, even, brew from there? It's a very reasonable question, to be honest, and the main reason 'we' can't do that is related most fundamentally to the above concept around grind consistency. Most consumer grinders are not extremely consistent at espresso fineness - they're typically least consistent at either pole of their range (very fine and very coarse); while the shorter your extraction window is, the tighter your magins of error. A grind size that has an ideal time of 30s has a margin of error somewhere between 2 and 4 seconds; while a grind size ideal for french press has an ideal time of 4 minutes, but has almost 45s of margin of error. Longer brew times are more tolerant of particle size variance, and as a result, are generally also more tolerant of not spending several thousand dollars on a grinder. In the case of espresso fineness, without the pressurization of an espresso machine, you couldn't convince any other method to drain fast enough to not completely void your margin of error.
There's all sorts of changes and methods and practices that are ... ornamental. That's not bad. Humans are unbelievably subjective, and performing ritual with the expectation of causing better results can cause us to find those better results solely on the basis of feeling like we did something to cause them. That subjectivity is part of being human and part of what is fun about coffee; there's no reason to shun or scorn subjectively-created enjoyment as though it's any less 'real' by our experiences than things that might be more built-in for any given bean and brew.
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