My first thought was, "I doubt a person could really play a stringed instrument for eight hours without stopping." It turns out that you definitely can.
Ultimately it will be up to you and your GM to decide what is reasonable, since the rules as written are very ambiguous in general, and don't cover this case at all.
Here is how I figure it:
- You would play for 30 minutes and pay 190 gp to generate 19 Labor. You would then spend 190 gp and up to 19 days to earn Goods, or spend 380 gp to buy Goods. Once you have the Goods and Labor, divide the total capital cost by the spending limit of the settlement (Table: Settlement Spending Limits).
- If you are playing the Lyre yourself, I would treat it as Earning Capital - i.e., 10 gp per unit of Labor.
- The Downtime rules don't cover expedited building with additional labor. This would be up to you and your DM.
Settlement Size: Large Town (25 capital limit/day)
Total if you earn the Goods: 360 gp and 21 days or less (19 days for the Goods, less with skill checks, plus 2 days to spend 38 capital).
Total if you buy the Goods: 570 gp and 2 days.
Here is my math for the Lyre:
1 man-day (unskilled) = 1 Labor (Earning Capital rules)
100 humans (unskilled) * 3 days / 30 min = 300 man-days(Labor) / 30 min =
10 Labor/min (rounded up to the nearest 30-minute increment)
10 Labor/min * 60 min/hour * 8 hours/day = 4,800 Labor/day
19 Labor / 10 Labor/min = 1.9 min for 1 Ballroom; round up to 30 min
You could generate enough Labor to build 15 Ballrooms in one 30-minute performance, but you still have to spend the gold cost for earned capital, plus the time to earn Goods (unless you buy them) and the final construction time.
Note that you can only use the Lyre to generate Labor once a week, so it would be limited to 4,800 Labor/week.
You would pay the Earned Cost for the Labor as described in the Earning Capital rules, which would be 190 gp. If you were going to buy the 19 Goods, the total cost of the building would be 570 gp (190 + 380).
Once you have the capital to construct the building, use the settlement's spending limit to determine the minimum construction time. If you are spending 38 capital in a Large Town (limit 25 capital/day spent), you would be looking at 2 days by the RAW.
It's unclear from the RAW where the 40-day build time comes into the picture. Looking through the building times for the different room types, there's no clear formula. My best guess is that it's meant to be 19 days to earn Goods, 19 days to earn Labor, and 2 days to spend the capital and finish the building (based on the capital spending limit for a Large Town). My reading of the rules is that you can just buy a room for 760 gp and have it built in 2 days, or you can spend 38 days earning the capital and build it for half the cost (total of 40 days). This would make it more or less consistent with the magic item crafting rules. Some rooms' build times deviate from this formula. The reason for this is anyone's guess.
As a DM, I would probably hand-wave the spending limit for Labor when using the Lyre. In the Large Town example, this would put you at 20 days and 360 gp, or 1 day and 570 gp, depending on whether you earn the Goods or buy them.
I don't blame you for having trouble... I reread the rules a dozen times, and had to edit this post after each reread.