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Example: Settlement Size: Large Town (15 capital limit/day)

Total if you earn the Goods: 360 gp and 22 days or less (19 days for the Goods, less with skill checks, plus 3 days to spend 38 capital).

Total if you buy the Goods: 570 gp and 3 days.

Example: 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.

1. 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).
2. If you are playing the Lyre yourself, treat it as Earning Capital - i.e., no cost for Labor.
3. The Downtime rules don't cover expedited building with additional labor. This would be up to you and your DM.

Once you have the capital to construct the building, use the settlement's spending limit to determine the minimum construction time. If you were in a Large Town (limit 25 capital/day spent), you would be looking at 2 days by the RAW, based on the 38 capital you are spending.

1. 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).
2. If you are playing the Lyre yourself, I would treat it as Earning Capital - i.e., 10 gp per unit of Labor.
3. The Downtime rules don't cover expedited building with additional labor. This would be up to you and your DM.

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.

1. You could construct a single Ballroom in 30 minutes. If you wanted, you could construct 15 Ballrooms in one 30-minute increment.
2. If you are playing the Lyre yourself, treat it as Earning Capital - i.e., no cost for Labor.
3. The Downtime rules don't cover expedited building with additional labor. This would be up to you and your DM.

Here is my math:

This seems consistent with the Lyre's intent, which is that you could build an entire building in 30 minutes.

Note that you can only use the Lyre to generate labor once per week, so it would also be 4,800 Labor/week.

Once you have the capital to construct the building, use the settlement's spending limit to determine the minimum construction time. If you were in a Large Town (limit 15 capital/day spent), you would be looking at 3 days by the RAW, based on the 38 capital you are spending.

It's really unclear from the RAW where the 40-day build time comes into the picture. Looking through all of the building times for different room types, there's no apparent rhyme or reason to them. 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).

Using just the spending limit for the Goods, which would put you at 2 days of construction (for a Large Town), based on the 19 Goods and limit 15 Goods/day.

1. 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).
2. If you are playing the Lyre yourself, treat it as Earning Capital - i.e., no cost for Labor.
3. The Downtime rules don't cover expedited building with additional labor. This would be up to you and your DM.

Example: Settlement Size: Large Town (15 capital limit/day)

Total if you earn the Goods: 360 gp and 22 days or less (19 days for the Goods, less with skill checks, plus 3 days to spend 38 capital).

Total if you buy the Goods: 570 gp and 3 days.

Here is my math for the Lyre:

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.

Once you have the capital to construct the building, use the settlement's spending limit to determine the minimum construction time. If you were in a Large Town (limit 25 capital/day spent), you would be looking at 2 days by the RAW, based on the 38 capital you are spending.

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.