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one thing that is significant for tech growth and spread is duchies, create them and hand them out, but set things up so that your vassals are balkanized and split up, hand out every spare county so that theres one count per county, make sure a duke doesnt have more than one duchie, ect, because each duke and count will have their own councilers to boost tech. vassal inheritance and intermarrage will usualy change things since vassals try to accumilate land, but gavelkind will usualy split it up unless you raise your CA high enough for vassals to implement primogenture.
dont invest your tech points in anything thats passivly gaining, you can tell what is from the cog wheels, because you will get that tech eventuly, prioritize techs based on your current situation, for military have your priority on military org to boost retinues, maybe secondary investing into techs that boost your cultural troops, for econ investing in improved keeps and castle techs will help your demense, trade stuff if your playing a republic, and for culture legalism and noble vassal opnion are also useful.
techwise the economic ones are vital for better income buildings, so when you can unlock university buildings in citys, invest in building the universitys in your city vassals in your demense because they add tech growth.
If you plan on your capital being your capital for the long term, then using the tech screen, see if any other counties in your demense have higher tech in certain areas, eg. your capital might have 2 levels of light infantry, but somewhere else might have 3, in which case, put your marshal here and evelntually your capital will catch up.
If you dont have any scheming for your spymaster to do, find the highest tech county on the map (usually Rome, Venice or Constantinople), and send him there. From here, he will leach their tech to you, as well as the % chance of a bonus.
Your chaplain can also provide a lump bonus, but this is very expensive and a very low chance of success.
Personally, I would use my steward solely for collecting taxes.
Your tech growth rate comes your ruler, along with any schools/uni's youve built - you can see on the tech screen how the rates are derived.
I'm not 100% sure, but I think the only county you need to worry about increasing tech in, is your capital, as most, if not all, bonuses are taken at your capitals level.
You can also arrange duchies in the way that their capitals border your counties from the other side, so you can take advantage from dukes growing tech there and spreading from that neighbor into your county too.
As initial stage put spymaster to spy on advanced nations and all your council members on capital, don't spend accumulated tech points. Once you caught up with advanced nations use the tech points and move councilors to your other counties to speed up tech spread from capital to there.
With liberal use of espionage and good stats/councilors you can become most technologically advanced in the world in 200 years or so even though you start way behind greeks and arabs.
What you said above seems to imply they boost tech spread TO instead of from. I'm confused...
Spymaster allows tech spread from distant land TO your capital, and putting a councilor on capital improves it(but is useless if you're not spying).
Game mechanics math:
Chances per year to gain an increase of 0.1 tech in a county are calculated like this
1)Find total positive difference in this tech from all neighbor counties, so if county has 1.3 and has 3 neighbors are 1.0, 2.0 and 1.6 this will add up to (2.0-1.3)+(1.6-1.3) = 1.0
2)Do same for demesne counties, so if county has 1.3 and your demesne has 3 more counties at 1.0, 2.0 and 1.6 this will add up to (2.0-1.3)+(1.6-1.3) = 1.0 same way
3) Base chance is 6% times values from step 1 plus step 2
3a)If county is your capital and your spymaster studies technology somewhere add 6% * his intrigue score/10* difference with county he's in
Easy to see it's huge - spymaster with intrigue of 20 in lands 2 tech levels ahead of you gives 24% increase chance on his own
4) Base chance is multiplied by buildings bonus and councilor bonus
Say your capital is 1 tech level ahead of demesne county neighboring it so base will be 12%(both demesne and neighbor parts contributed to base chance)
Putting a 16 skill councilor on that county will add 80% bonus so 12% base will become 12*1.8=21.6%
If you also have 2 universities in the cities of that county they'll add additional 60% on top so
21.6*1.6=34.56%
At least that explains why my capital tech is always so high and the neighboring ones never seem to catch up quickly enough.