Controlling over 1,700 Incandescent Lights ; Prepare for this massive project.

Projects General Guidance
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Feel free to skip to the "Main Question" and read backwards as needed if you don't feel like reading. It really boils down to a simple question, but I present all of this information as a fun, open challenge, welcoming any suggestions for any part of the project.

Background:
I have been away from the arduino community for a few years. I made some videos a few years back, one of them has 20k views on RFID and LCDs. I have forgotten a lot though. Anyway, I work at a roller skating rink on the weekends and their light system has not been working for years. I intended to fix that. Everything I know about this system is just from me poking around. This technology is ancient, built by Diversitronics in the early 80's. Due to it's age and condition, there is no choice but to completely replace the system. The owner of the facility has yet to commit to a budget, mostly because I haven't asked, but I would imagine it can stay under $200 using quality and professional parts.

Previous system details:
The layout consists of a main "runway" with around 50 rows, 24 bulbs in each row, as well as "stars" with 12 strips and 12 bulbs in each strip. These cover a 10,000 sq ft floor. This is all not very important for what I'm asking but it draws a picture.
There are three main control boxes, at one time controlled by a separate, main control panel. One box controls the star, the other two control the runway. For simplicity's sake I'll jump right to it, ignoring the logic control of the previous system. There are 24 "channels" or outputs in each box. 12 in each for the hot 120v and 12 for the neutral. How the lights cycle is beyond me, clearly some kind of matrix, I have never seen it operating. I intend to dig up old schematics and do some tests. Each output is controlled by a triac, recieving a 10v AC trigger signal, then directly powering the output with 120v. I don't know any specific wattages, but I believe each box has a 15a breaker.

Main Question / My plan
To start simple on the arduino end, I want to control all three boxes off the same 12 pins; meaning there will only be 12 output pins from the arduino. The code is not important right now, but I plan on making them sound active lights, like a vu meter. So, how do I high speed switch 120v AC from an arduino? Currently I am thinking of using 36 (new) triacs that can be triggered via a 5v signal, which would supply the hot 120. Or cheap solid state relays? I want to hard wire all the neutrals , but I suspect they are also currently controlled individually for a reason, perhaps how the matrix operates? Obviously relays are out of the question. The lights will be rapidly cycling on and off for long periods.

I am currently a mechanical engineering major at Clarkson University. That being said, electronics are my hobby, not my life. Any constructive comments or criticisms are very welcome. This is all the information currently available.

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I would offer that it would be best to start by fully understanding the matrix.
you may think that the world around you is real, but choose the blue pill or the red.... op's got off the track there..

first, find out why they control both it should not be that hard.
you will probably want to save as much of the old stuff as you can.

if the are working from triacs, then you should be able to replace just the logic and keep everything else.

of course, add a few tricks of your own.

I think $200 is a down payment on the lamps.
at 10 cents each, you have 170 of your budget spent.

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The lamps already exist, this is simply the control. I do plan on messing around some more, just wanted to get my ideas out there. Thanks

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Before you get too far, check the regulations.

In my area, any modifications to the electrical system of a public or commercial facility must be performed by a licensed electrician and inspected and approved by the local regulatory agencies.

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Each output is controlled by a triac, recieving a 10v AC trigger signal, then directly powering the output with 120v.

If the 10V control is isolated from the power line AC, I'd try to duplicate that control signal and keep the existing isolated TRIACs. That probably doesn't need to be AC, and maybe 5V will work. Things will be simpler if you don't need dimming.

If you use TRIACs, you need isolation. There are special opto-isolators that work with TRIACs so you are essentially building your own solid state relay. It's cheaper to build your own, but it's a lot more work than wiring-up 4 wires to a solid state relay, and there's more chance of something going wrong.

Most solid state relays will work directly from 5V, and if you build your own it can work from 5V. I don't know if you'll find a "cheap" solid state relay. In a matrix, the relay needs handle the current of an entire row or column.

I want to hard wire all the neutrals , but I suspect they are also currently controlled individually for a reason, perhaps how the matrix operates?

Yes, that's how a matrix works. The hots are the rows and the neutrals are the columns (or vice-versa). You have to control/switch both to individually address all of the lights in the matrix.

The Arduino itself doesn't need to be wired as a matrix. In fact, that's probably not the best way of doing it. With a serial shift-register (multiplexing) scheme, you can control and address an almost unlimited number of lights with 3 Arduino outputs. I've got an effect with 48 individually-addressed LEDs using serial.

Obviously relays are out of the question. The lights will be rapidly cycling on and off for long periods

Solid state is the way to go.

The owner of the facility has yet to commit to a budget, mostly because I haven't asked, but I would imagine it can stay under $200 using quality and professional parts.

Put an Arduino in a box with a power supply, and add a couple of connectors and you are up to about $100 already. You are going to need at least some additional electronics and you might want one Arduino for each of your control boxes. I don't see any way to do this for $200.

I plan on making them sound active lights, like a vu meter.

I built a 4-channel sound controlled AC lighting effect, and one of the modes is a VU meter. In the VU meter mode, the meter can be reversed, or inverted (where louder sound turns the lights off instead of on), and it has dot & bargraph modes. It's not really a "meter" because it automatically adjusts to the average volume for lots of "meter action" with loud or quiet songs, or if the volume is changed. 4-channels is not enough for a "good" meter effect, but it was easy to throw-in this effect.

I also built an 8-foot LED VU meter with 24 high-brightness LEDs spaced a couple of inches apart. This one also has the reverse/invert options and several different (non-meter) effects. There are left & right channels, so there are 48 addressable LEDs.