Authentic tea ceremony in/around Kyoto: where & why?

Hello there!

Me and my friends are traveling to Japan for a couple of weeks at the end of September. It will be a first visit for all of us.

We have everything booked and planned, but we still aren't sure where specifically it would be a nice place to experience an authentic tea ceremony. Our primary requests:

1. In or near Kyoto;

2. I know many of you might not remember, but we need a specific place recommended so we don't wander;

3. Beautiful and authentic (in its surroundings and performance);

4. Not TOO expensive (I think the most we would be willing to pay is around 50 Eur/$ per person, unless you have something incredible for more);

5. Ideally, we can pre-book it online somehow.

I'd appreciate it if you would be kind enough to provide links to sources (like websites) where we can read more about the places you recommend.

That's all and thank you in advance for taking your time to reply!