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Your capital is the seat at which your character rules from. Depending on the character, the capital can be either a castle, city, or church/temple/mosque. Generally, Nobles, Mayors/Burghers/Doges, and Bishops/Wâlis, govern the castles, cities, and church/temple/mosque holdings respectively.
Choosing a capital is easy and at the same time difficult and here is why:
I will be going over things you should be able to find elsewhere in this wiki, but for the sake of my explanation I will be going over most if not all things associated with choosing a capital.
To change your capital click on a province thus opening the county menu screen. The county menu screen displays the following:
1. In the top left is the County Coat of Arms.
2. Next to the CCoA is a small crown in a circular window or larger crown depending on if the county is your capital or not (this area will be blank for counties you do not currently own), to make that county your capital simply click the crown icon and you're done. As explained earlier this choice is not that simple.
3. Across from the CCoA is the counties' current ruler's primary title CoA. For example: Clicking on county Powys you will see the CCoA, and the CoA of the Duchy of Gwynnedd across from it, because it is a liege county of the Duchy of Gwynedd.
4. Below the CCoA is the current county ruler's portrait as well as the diplomacy scroll next to it. Below the portrait you will see any wars that character is currently involved in as well as any truces they have.
5. To the right of the portrait you will find the county capital portrait that will be either a castle, city, or temple/church/mosque. Right clicking the portrait will bring up quick menu options: Create vassal baron (only available in baronies you own that is not already the capital), raise/dismiss barony levies, and raise/dismiss barony ships. Left clicking the holding will open the holdings menu which provides more in depth info on that particular holding as well as construction options if that holding belongs to you or one of your direct vassals.
6. Below the character and capital holding portraits, you will find a small informational bar that will display any modifiers the county currently has, whether it be negative or positive. (For example any increases to revolt chance or tech bonuses) To the right of this info bar you will find any province physical attributes, for example a river, hills, mountains, or if there is a strait crossing etc. between its neighboring counties. This is very important when selecting your capital for if a war was to come to your capital province then defending it would be easier depending on which attributes it has.
7. Below the info bar is the bread and butter of your county. Which kingdom it is De Jure a part of, which duchy it is de Jure a part of, the culture, religion, tax and income, and tech levels for military, economic, and cultural. Also you can find the buttons for raising the local liege levies/ships. To the left of those buttons are two very important factors of that county: it's supply limit and revolt risk. Be mindful of supply limits when selecting a capital, as an attacking army will face attrition damage if they have more troops than the county allows. The county revolt risk can also be crucial as the higher it gets the more likely the peasant will rise up in revolt.
8. Hint: Moving the capital to a province of Muslim/Greek origin can give you a boost in Legalism and rest of tech by 3, as these provinces generally begin with higher tech.