The whole matter of the number of primary turns to use is an interesting subject. As mentioned elsewhere it mainly depends on the size of the capacitor and is affected by the secondary and toroid characteristics, and your first coil will most likely be happy with a static gap and a primary of 10 to 15 turns.
Later on if you decide to build a second bigger coil and decide to squeeze as much performance as possible from it, you may decide to take into account the type of spark gap being used as well. Most of the losses always occur in the spark gap, so if the type of spark gap chosen has high losses associated with it, it can sometimes be better to try and restrict the primary current by having more primary turns. This is because you will loose even more power in a gap as the current increases, simply because power lost equals ROhms x I² (the reactance of the primary x the current squared).
Now although the primary inductance may be a small value, typically 20μH to 40 μH [micro Henries] the momentary amperage flowing when the gap fires can be very high (800 amps on a bigger coil), especially when its value has been squared, so the losses can soon mount up!
With the popular RQ gap (named after Richard Quick, and consisting of multiple bits of pipe side by side) you will find that although it's a good quenching gap, you will also find this type of gap is quite lossy, so using more primary turns will keep the current (and losses) down, meaning it's best to try and aim for around 12 to 14 turns (allows for adjustment).
A better approach is to use a lower loss gap to start with, because the philosophy of using more turns to reduce current, is only a way to make up for the gap's shortfalls and is contrary to what we are trying to achieve - bright streamers. There are several gaps you could use instead (blower or sucker type of static gap, and triggered gaps), but for high power the only way is a rotary gap, be it a Synchronous type or a Asynchronous type.
With a rotary gap because it has far less losses, it means you can generally use less primary turns. Less primary turns will mean less inductive losses and therefore more current can end up in the streamers, making them appear whiter in colour (my preference), instead of purple. (The purple colour mentioned above is actually caused by the breakdown of Nitrogen in the air.)
In case you're wondering, trying to get a primary of 5 turns or less is actually tricky with typical component values, but you will find this out as you experiment more.
The reason for rotary gaps incurring less loss, is because they have far less interfaces, and also the use of a much smaller gap spacing (I use 10 thou) has some benefit. [Note: 1] So to sum up, more turns will keep the primary current down when used with RQ gaps and their high losses, but with rotary gaps, you can use less turns allowing more current, and so benefit from brighter streamers.
But remember, by allowing increased current with a rotary it will mean it will progressively incur more losses, until you reach a point where the losses outweigh any benefits. So it's all about testing through 'trial and error': just like so many other things on a tesla coil.
[Note: 1] An interface occurs where the spark leaves a solid and enters air and vice versa. So a single air gap will have two interfaces and two places for losses to occur.
As mentioned above the method of achieving a certain number of primary turns is by altering some of the coil's other component values. This would be something for a more advanced build and is not really suited for a starter coil though, I just included it out of interest if you later decided to build a bigger coil.
The same advice above generally applies for a sucker gap as well, as they also have less interfaces than a RQ type. Apart from RQ gaps needing more than 10 primary turns, there may be times when even though you are using a rotary gap, you just can't avoid having to use more than 10 turns, simply to get the tuning right. Although this is generally caused by bad planning at the outset, it's not the end of the world, and the remedy is to either proceed with using more turns, or add more capacitance to allow you to use less primary turns.